Unit-linked life insurance in Lévy-process financial markets

Transcription

1 CURANDO Universiä Ulm Abeilung Zahlenheorie und Wahrscheinlichkeisheorie Uni-linked life insurance in Lévy-process financial markes Modeling, Hedging and Saisics Disseraion zur Erlangung des Dokorgrades Dr. rer. na. der Fakulä für Mahemaik und Wirschafswissenschafen der Universiä Ulm UNIVERSITÄT ULM SCIENDO DOCENDO vorgeleg von Dipl.-Mah. oec. Marin Riesner, M.S. aus Schorndorf Ulm, im Juli 26

2 ii Amierender Dekan: Professor Dr. Ulrich Sadmüller 1. Guacher: Professor Dr. Ulrich Sadmüller, Universiä Ulm 2. Guacher: Professor Dr. Rüdiger Kiesel, Universiä Ulm 3. Guacher: Professor Dr. Ralf Korn, Universiä Kaiserslauern Tag der Promoion:

3 Preface Bringing ogeher he Lévy-process financial world and he heory of unilinked life insurance conracs, his hesis shall conribue o he ongoing research on he inerface beween financial and acuarial mahemaics. I concludes my docorae research which was carried ou a he Deparmen of Number Theory and Probabiliy Theory a he Universiy of Ulm in he period from Ocober 23 o July 26. My work was supervised by Professors Ulrich Sadmüller and Rüdiger Kiesel, boh Universiy of Ulm. Acknowledgemens Firs of all, I would like o express my deepes graiude o Professor Ulrich Sadmüller for being an excellen supervisor and eacher. During he las hree years we had many exremely fruiful discussions, I received valuable advice and I always go he greaes possible suppor. Moreover, I very much enjoyed being a member of he Deparmen of Number Theory and Probabiliy Theory and o conribue o is eaching duies. I also would like o express my sincere hanks o Professor Rüdiger Kiesel for his brillian supervision and numerous valuable suggesions. Especially, I am very graeful for always having been invied o he grea social evens of his Deparmen of Financial Mahemaics. Special hanks go o Professor Ralf Korn, Universiy of Kaiserslauern, for being an exernal referee of his hesis and o Professor Nick Bingham, Universiy of Sheffield, for his very helpful and encouraging commens on some of my preprins. Furhermore, I would like o hank he Head of he Deparmen of Number Theory and Probabiliy Theory a he Universiy of Ulm, Professor Helmu Maier, for offering me a pleasan research and work environmen. I am also very graeful o Professor Ulrich Rieder, Universiy of Ulm, by whom I enjoyed having my fis courses on Sochasic Analysis and Financial Mahemaics. I exend my hanks o Anja Blaer for all her suppor and o my colleagues and friends Reik Börger, Dr. Sefan Kassberger, Dr. Harmu Lanzinger, Gregor Mummenhoff, Clemens Presele, Mahias Scherer and Monika Thalmaier for many ineresing discussions.

4 iv Finally, I very much appreciae he scholarship by he federal sae Baden-Würemberg LGFG Baden-Würemberg for he enire phase of my docorae and he financial suppor graned by he Deparmen of Number Theory and Probabiliy Theory and he Deparmen of Financial Mahemaics a he Universiy of Ulm. Ulm, July 26 Marin Riesner

5 Conens Preface Acknowledgemens iii iii 1 Inroducion and summary Hisorical overview and exising lieraure Summary, conribuions and ouline of his hesis Basic conceps Preliminaries Lévy processes Definiion and basic properies Common examples Finie aciviy Lévy-processes: jump-diffusions Infinie aciviy Lévy-processes Variance gamma Lévy-process Normal inverse Gaussian Lévy-process Generalized hyperbolic disribuions and Lévy processes Financial marke Lévy-process model Föllmer-Schweizer measure Oher equivalen changes of measure Life insurance Muli-sae Markov model Typical porfolio of insured lives Analyical laws of moraliy Convenional life-insurance risk diversificaion Risk-neuraliy Combined financial and life insurance model Combined model Hedging heory of Föllmer and Sondermann Local risk-minimizaion of Schweizer

6 2 CONTENTS 3 Arbirage-free price process Inroducion A Feynman-Kac ype formula Galchouk-Kunia-Waanabe decomposiion Hedging of uni-linked pure endowmen and erm insurance Inroducion Uni-linked life insurance conracs Uni-linked pure endowmen Uni-linked erm insurance Risk-minimizing hedging sraegy Uni-linked pure endowmen Uni-linked erm insurance Why use local risk-minimizaion? Paymen sream hedging for semimaringales Inroducion The model Local risk-minimizaion Finding a locally risk-minimizing sraegy Hedging of general uni-linked life insurance conracs Inroducion General uni-linked benefi and premium paymens Locally risk-minimizing hedging sraegies Examples Single uni-linked life annuiy insurance Single uni-linked erm insurance Porfolio of n uni-linked life annuiy conracs Porfolio of n uni-linked erm insurance conracs Jump-diffusion sock model Inroducion The asymmeric double exponenial disribuion The model The asymmeric double exponenial jump-diffusion Financial marke Risk-neural measure Föllmer-Schweizer measure Kou s risk-neural measure Arbirage-free price process Valuaion of an European Call Valuaion of uni-linked pure endowmen and erm insurance wih guaranee

7 CONTENTS Spaial derivaive Saisics for jump-diffusions Inroducion Model of sock price and reurns Esimaion mehod Mahemaics behind he esimaion Fuure research A Supplemenary Maerial 165 Bibliography 169 Lis of Tables 175 Lis of Figures 177 Zusammenfassung 179

8 4 CONTENTS

9 Chaper 1 Inroducion and summary 1.1 Hisorical overview and exising lieraure Unlike radiional life insurance he benefis and possibly he premiums of an uni-linked life insurance are random and linked o he developmen of some specified reference porfolio, a muual fund or simply a sock index. Insurance companies offering hese kind of producs ofen collaborae wih muual funds and offer he policy-holder a huge variey of invesmen opporuniies including funds ha concenrae on specific counries, areas or indusry secors or even mixures of several specific funds. Such conracs are usually equipped wih a minimal capial guaranee and allow he policyholder o decide how his premiums are o be invesed. Hence, hey ransfer pars of he invesmen risk o he policyholder while, simulaneously, providing insurance, a limied exposure o he downside risk and a high level of invesmen ransparency. Moreover, he policy-holder benefis direcly from he upside poenial of his invesmen and he flexibiliy of he conracs allowing o adjus he riskiness of he invesmen depending on he remaining ime of he policy. In recen years hese specific life insurance producs, especially hose wih capial guaranees, have become very popular in mos counries wih a fully developed insurance indusry. They seem o have appeared for he firs ime around 195 in he Neherlands and around 1954 in he Unied Saes of America. Laer, in 1957, hey were inroduced o he Unied Kingdom. Uni-linked life insurance is also known as equiy-linked or equiy-based life insurance. In he Unied Saes i is called variable life insurance. These kinds of insurance producs combine financial and insurance risk and require echniques from boh financial and acuarial mahemaics for heir proper reamen. Møller 22 reviews valuaion and hedging of life and non-life insurance producs on he inerface of hose wo areas and especially describes exhausively he hisory of uni-linked life insurance producs providing an overview of relaed lieraure.

10 6 Chaper 1. Inroducion and summary The firs who analyzed uni-linked insurance producs using modern financial mahemaics were Brennan and Schwarz 1976, 1979a,b and Boyle and Schwarz 1977 who recognized ha he payoff of an uni-linked wih guaranee life insurance equals he payoff of an European call opion plus he guaraneed amoun. Furhermore, referring o a law of large numbers argumen hese auhors replaced he uncerainy of insured lives by heir expeced values and did no pay aenion o moraliy risk. For his reason hey could rea uni-linked wih guaranee life insurance conracs as coningen claims wih financial uncerainy in a complee financial marke, in which hey could apply he sandard valuaion and hedging echniques ha had been developed shorly before by Black and Scholes 1973 and Meron The idea of replacing he uncerain course of moraliy by is expeced developmen was laer also used by Delbaen 1986, Bacinello and Oru 1993a and Aase and Persson 1994 o price uni-linked life insurance conracs involving he maringale based financial echniques of Harrison and Pliska 1981 and Harrison and Kreps Moreover, Aase and Persson 1994 addiionally worked wih coninuous and no wih discree ime survival probabiliies, bu all auhors menioned above used consan ineres raes. Bacinello and Oru 1993b, Nielsen and Sandmann 1995 and Bacinello and Persson 22 added sochasic ineres raes o he exising models. Møller 1998 had he new idea no o average away moraliy reaing simulaneously he uncerain developmen of he financial marke and he insurance porfolio modeled ogeher in a produc probabiliy space as wo independen componens and shows ha he uncerainy of he insurance porfolio even makes a complee financial marke incomplee, in which riskless hedging is impossible. Moreover, for a Black-Scholes financial marke and a porfolio of eiher uni-linked pure endowmen or uni-linked erm insurance conracs, risk-minimizing hedging sraegies and he associaed hedging risk are derived. More precisely, a complee financial marke which urned incomplee by moraliy is considered and he risk-minimizing hedging heory of Föllmer and Sondermann 1986 is applied. Hedging sraegies deduced from his approach reac o he financial uncerainy as well as o he acual number of survivors in he insurance porfolio. The associaed hedging risk is a measure for he moraliy uncerainy which can no be hedged away. Since he heory of Föllmer and Sondermann 1986 requires coningen claims payable a he end of he considered ime horizon, Møller 1998 assumes ha here is only a single premium paid a issuing dae and ha he insurance benefis are deferred, while earning ineres, and cashed ou a he end of he considered ime horizon. From a pracical poin of view no admiing inermediae premium paymens is no very resricive for he pure endowmen. Bu for mos oher life insurance conracs his is a very unsaisfying assumpion, because heir benefis are usually due immediaely upon occurrence of some insurance even. Annuiy paymens are

11 1.1. Hisorical overview and exising lieraure 7 even paid on a regular basis. For his reason Møller 21a exended he risk-minimizing hedging heory for coningen claims wih fixed mauriy of Föllmer and Sondermann 1986 o paymen sreams, in paricular, insurance paymen sreams admiing inermediae premium and benefi paymens. However, his heory is only applicable in maringale financial markes implying ha he hedging risk is only inerpreable under he hedger s subjecive measure if i is already a maringale measure. Møller 21a addiionally conains he risk-minimizing hedging sraegies and he associaed hedging risk for a maringale Black-Scholes financial marke and for general uni-linked life insurance conracs, modeled in a muli-sae Markov model wih inermediae paymens. Analogously o he already menioned aricles Møller 1998, 21a, he paper Møller 21b reas risk-minimizing hedging of equiy-linked life insurance conracs in a discree ime se-up using exemplarily a Cox-Ross-Rubinsein financial marke model. More recenly, indifference pricing echniques, involving he financial variance and sandard deviaion principles, have been proposed as an alernaive way of valuaion and hedging of uni-linked life insurance conracs by Møller 21c, 23a,b. Moreover, he risk-minimizing hedging heory for paymen sreams of Møller 21a is applied in Dahl and Møller 26 for he case of radiional life insurance wih sochasic ineres raes and a moraliy inensiy of all policy-holders ha is affeced by some sochasic process. Uni-linked life insurance requires for is reamen a financial marke in which he uni is modeled. Almos all of he aricles menioned above assume a Black-Scholes kind of financial marke, in which asse prices are driven by a Brownian moion feauring a normal disribuion of asse reurns and coninuous sample pahs. Recen research, however, favors more general Lévy-process financial marke models, in which he Brownian moion is replaced by a more general Lévy process resuling in models wih more flexible asse reurn disribuions and disconinuous sample pahs. Among ohers, Chan 1999 is cerainly a seminal work inroducing general Lévy processes o financial marke modeling, bu even Meron 1976 proposed a jump-diffusion process o model asse reurns. According o Chaper 1 of Con and Tankov 24 Lévy-process models wih jumps naurally mee he following observaions of real financial markes, which are impossible or difficul o realize in diffusion based models. 1. Asse prices ofen do no follow coninuous rajecories and incorporae large, sudden movemens in prices. 2. Losses are concenraed in a few large downward moves represened by disconinuiies in he price behavior. 3. Asse reurn disribuions are heavy ailed. 4. Asse reurn disribuions may be asymmeric.

12 8 Chaper 1. Inroducion and summary 5. Financial markes are incomplee and riskless hedging is impossible. 6. Perfec hedges do no exis and opions are risky invesmens. 7. Various hedging sraegies may perform differenly. 8. Volailiy smiles in equiy and foreign exchange opion markes: The implied volailiy may be U-shaped smile as a funcion of he srike price of an opion. 9. Volailiy skews in equiy and foreign exchange opion markes: The implied volailiy may be decreasing as a funcion of he srike price of an opion. As a consequence, research in general Lévy-process financial markes has gained a major role in modern financial mahemaics over he las decade. Recenly, Jaimungal and Young 25 discussed he pricing of equiy-linked pure endowmens in a finie variaion Lévy-process financial marke using indifference pricing, more precisely, he principle of equivalen uiliy. 1.2 Summary, conribuions and ouline of his hesis The main aim of his hesis is he developmen of locally risk-minimizing hedging sraegies for uni-linked life insurance conracs whose uni is modeled in a general Lévy-process financial marke. I herefore merges he quie advanced and in recen years developed heory of Lévy-process financial markes wih he heory of uni-linked life insurances and provides moreover a quadraic hedging framework for insurance paymen sreams exposed o pure financial and pure insurance risk. Up o our knowledge, locally risk-minimizing hedging of uni-linked life insurance conracs in general Lévy-process financial markes has no been analyzed so far. In summary his hesis is par of he ongoing research on he inerface beween financial and acuarial mahemaics and generalizes he earlier work of Møller 1998, 21a in he sense ha he complee Black-Scholes financial marke is replaced by an incomplee financial marke in case of a more general geomeric Lévy-driven model. As Møller 1998, 21a we assume sochasic independence beween he financial marke and he insurance model and consider hem combined in a common produc probabiliy space. This picks up he idea o model he uncerain developmen of he sock price and he insured lives simulaneously no averaging away moraliy, ha is, he uncerain course of insured lives is no replaced by heir expeced developmen. Furhermore, modeling he insurance par of our model wih he general muli-sae Markov chain model of Hoem 1969 yields very flexible resuls which can be adaped o various

13 1.2. Summary, conribuions and ouline of his hesis 9 imaginable ypes of life insurance. In examples, however, we pu special emphasis on cerainly he mos imporan versions such as pure endowmens, erm insurances or life annuiies. We firs derive he locally risk-minimizing hedging sraegy and he associaed hedging risk for a porfolio of uni-linked pure endowmen and a porfolio of uni-linked erm insurance conracs assuming ha hey are sold agains a single premium a issuing dae and ha heir insurance benefis are deferred, while earning ineres, and cashed ou a he end of he considered ime horizon. Wih his simplifying assumpion he corresponding claims fi ino he heory of Föllmer and Sondermann 1986 and Schweizer Møller 21a provides an exension of he heory of Föllmer and Sondermann 1986 o risk-minimizing hedging of paymen sreams in maringale financial markes. Moreover, he heory of Föllmer and Sondermann 1986 has also been generalized o semimaringale financial markes by Schweizer 1991 inroducing he concep of locally risk-minimizing hedging and he Föllmer-Schweizer measure. In a separae chaper we show ha locally risk-minimizing hedging also exends o hedging of paymen sreams if he Föllmer-Schweizer measure is used. This delivers an inerpreaion of he hedging risk under an invesor s subjecive probabiliy measure and no only under some risk-neural maringale measure. The exension of locally risk-minimizing hedging o paymen sreams is new and has no been considered before. Finally we derive, in a general Lévy-process financial marke, locally risk-minimizing hedging sraegies for general uni-linked life insurance conracs admiing inermediae premiums and benefis. Since we use an incomplee financial marke from he beginning we ge wo hedging risk componens reflecing he hedging risk s origin. We call hem pure financial and pure insurance risk of hedging. Moreover, he oal hedging risk of an uni-linked life insurance claim is shown o separae exacly in hose wo erms and, in accordance wih sandard acuarial heory, he pure insurance risk is demonsraed o diversify by raising he number of insured individuals wihin he porfolio. This is however, as expeced, no he case wih he pure financial risk. The difference from Møller 1998, 21a is ha we addiionally model in compliance wih realiy he pure financial risk. Because he Black-Scholes model describes a complee financial marke, he oal hedging risk in Møller 1998, 21a corresponds o he pure insurance risk of hedging in our case. Le us give a deailed ouline of he hesis in he following. Basic conceps In Chaper 2 he essenial echnical conceps required in his hesis are reviewed and he models used in he following chapers are inroduced. Beginning wih he definiion of a Lévy process we subsequenly discuss he Lévy measure, he famous Lévy-Iô-decomposiion, being formulaed in The-

14 1 Chaper 1. Inroducion and summary orem 2.2.4, and he disribuional properies of Lévy processes such as he Lévy-Khinchin formula and infinie divisibiliy. Moreover, in Theorem we classify he pahs behavior of Lévy processes disinguishing beween finie and infinie aciviy or variaion, respecively. Subsecion conains a discussion of jump-diffusions, variance gamma, normal inverse Gaussian and generalized hyperbolic Lévy processes, since hey are commonly used for financial marke modeling. In Secion 2.3 he Lévy-process financial marke model of Chan 1999 is inroduced and he risk-neural Föllmer- Schweizer measure, which is required for locally risk-minimizing hedging, is reaed. We explain in deail he effecs of he associaed change of measure on he back-driving Lévy process possibly becoming an addiive process. Furhermore, in Lemmaa and 2.3.3, we discuss he sock price process before and afer his change of measure, respecively, and show is square-inegrabiliy in boh cases using he resul of he appendix. This resul enables us o weaken significanly an assumpion of Chan In our case i suffices now o demand only he exisence of he Lévy measure s hird momen unlike he exisence of is exponenial momens. The muli-sae Markov life insurance model of Hoem 1969, which admis o model quie general forms of life insurance, is summarized in Secion 2.4 and hisorically famous bu nowadays sill frequenly used laws of moraliy are presened in Subsecion o give examples for he absrac hazard rae of he insurance model. Following his, in Subsecion 2.4.4, we use he cenral limi heorem o review a convenional model of insurance risk diversificaion for he case of radiional life insurance and in Subsecion we explain, applying he srong law of large numbers, he risk-neuraliy of an insurance company wih respec o moraliy. Finally, Secion 2.5 is devoed o he consrucion of a common produc probabiliy space combining he Lévy-process financial marke and he insurance model under he reasonable assumpion of sochasic independence. Moreover, in he risk-neural counerpar of his space he risk-minimizing hedging heory of Föllmer and Sondermann 1986 is reviewed including a discussion of is relaion o local risk-minimizaion of Schweizer 1991 and he fundamenal Galchouk-Kunia-Waanabe decomposiion in Theorem Arbirage-free price process In Chaper 3 we discuss he arbirage-free price process of a square-inegrable coningen claim which we represen, using he Markov propery of he sock price, as a funcion of he ime, he sock value a ha ime and he due dae of he claim. The underlying financial marke is hereby he common risk-neural probabiliy space of Chaper 2, in which he process modeling he sock is an addiive process, ha is, a càdlàg, sochasically coninuous process wih independen incremens. The main emphasis of he chaper is he derivaion of he Galchouk-Kunia-Waanabe decomposiion of he

15 1.2. Summary, conribuions and ouline of his hesis 11 arbirage-free price process which is formulaed in Theorem and which has he same srucure as i was worked ou for a Lévy process. I is cerainly he mos imporan elemen of he locally risk-minimizing hedging heory and i is he basis for he derivaion of he Galchouk-Kunia-Waanabe decomposiion of he uni-linked life insurance inrinsic values in he following chapers. Roughly speaking, he decomposiion is a projecion resul in he Hilberspace of square-inegrable random variables. We prove Theorem applying direcly he orhogonaliy of he involved coninuous and purely disconinuous maringales, an approach we have no observed before. Moreover, he heorem formulaed for addiive processes seems o be new. To disinguish our conribuion from exising resuls, we provide an overview of relaed lieraure a he end of he chaper. A necessary resul of decomposing he arbirage-free price process is is Feynman-Kac ype represenaion. In our case his is a parial inegro-differenial equaion which is formulaed and proved in Theorem I follows from he maringale propery of he discouned arbirage-free price process and can be used o compue i numerically. A he beginning he applied decomposiion procedure is compleely analogous o he Black-Scholes seing, in which i already reduces wih he Feynman-Kac ype represenaion o he righ Galchouk- Kunia-Waanabe decomposiion. In our more general se-up several furher seps are necessary, which are demonsraed in Lemmaa and Hedging of uni-linked pure endowmen and erm insurance Chaper 4 is devoed o he derivaion of locally risk-minimizing hedging sraegies for a porfolio of eiher uni-linked pure endowmen or uni-linked erm insurance conracs. As menioned before, we hereby resric ourselves for he momen o he case in which here are only paymens a he beginning and a he end of he considered ime horizon. This allows o apply he locally risk-minimizing hedging heory of Schweizer 1991, which has been developed for coningen claims wih fixed mauriy in a semimaringale financial marke and which delivers an inerpreaion of he hedging risk under he subjecive probabiliy measure. The chaper serves herefore as an inermediae sep owards he general reamen of uni-linked life insurances admiing inermediae premiums and benefis. I is based on Riesner 26 and complemens Møller 1998 by replacing he complee Black-Scholes financial marke hrough a more general and incomplee Lévy-process financial marke. We sar wih a general mahemaical discussion of uni-linked life insurance benefis and derive he presen value of an enire porfolio of he wo ypes of life insurances menioned above. Afer various furher calculaions, hese reduce o he desired Galchouk-Kunia-Waanabe decomposiions being respecively formulaed in Corollaries and These calculaions include, among ohers, Iô s inegraion by pars for general semimaringales and, addiionally for he erm insurance, Fubini s he-

16 12 Chaper 1. Inroducion and summary orem for sochasic inegrals. The locally risk-minimizing hedging sraegies are subsequenly provided in Corollaries and 4.3.7, respecively. For he general Lévy-process financial marke all menioned corollaries are new resuls. We discuss and compare hem in deail o he resuls of Møller In our case he risk-minimal invesmen in he sock is a weighed sum of he Black-Scholes dela and a erm involving he jumps of he price process. Addiionally we derive he wo componens of he hedging risk, he pure financial and he pure insurance risk. Finally we jusify why local risk-minimizaion should paricularly be used for insurance claim hedging. Paymen sream hedging for semimaringales Chaper 5 parly conribues o Riesner 25 and addresses locally risk minimizing hedging of paymen sreams in general semimaringale financial markes. For maringale financial markes here is he risk-minimizing hedging heory for paymen sreams of Møller 21a. In semimaringale financial markes one usually chooses an arbirary risk-neural maringale measure, however, i remains he problem o inerpre he hedging risk under he subjecive believes. Schweizer 1991 developed herefore locally risk minimizing hedging of coningen claims wih fixed mauriy for semimaringale financial markes, which delivers an inerpreaion of he hedging risk under he subjecive probabiliy measure provided he risk-neural Föllmer- Schweizer measure is applied. We show in his chaper, using Schweizer 1991 as basis, ha risk-minimizing hedging of paymen sreams has in he same sense a local version as hedging of coningen claims wih fixed mauriy: Using he heory of Møller 21a one derives a risk-minimizing hedging sraegy under he Föllmer-Schweizer measure which is hen locally risk-minimizing wih respec o he subjecive probabiliy measure. Afer having discussed a lo of echnical machinery his main resul is summarized in Theorem I is based on he same opimaliy equaion as in he classical case which is formulaed in Theorem Furhermore, in Proposiion we relae aainable coningen claims o aainable paymen sreams in a semimaringale financial marke. The exension of locally risk-minimizing hedging o paymen sreams seems o be considered in his hesis for he firs ime. Hedging of general uni-linked life insurance conracs The opic of Chaper 6 is he derivaion of locally risk-minimizing hedging sraegies for general uni-linked life insurance conracs having inermediae premium and benefi paymens. The muli-sae Markov life insurance model of Hoem 1969 is hereby used o describe he model s insurance par. I admis sae-wise annuiy paymens as well as paymens induced by a sae ransiion. Those paymens are inerpreed as difference of benefi

17 1.2. Summary, conribuions and ouline of his hesis 13 and premium paymens. The Galchouk-Kunia-Waanabe decomposiion of he conracs inrinsic value is saed in Theorem and is proved applying he echniques already known from Chaper 4. The corresponding locally risk-minimizing hedging sraegy as well as he hedging risk are subsequenly given in Corollary For a general Lévy-process financial marke hese resuls are new. A he end of he chaper we apply his general model o uni-linked life annuiies and o uni-linked erm insurances considering firs a single conrac and subsequenly a porfolio of hese insurances. The chaper conribues o Riesner 25 and generalizes Møller 21a by replacing he complee Black-Scholes financial marke hrough a more general and incomplee Lévy-process financial marke. Jump-diffusion sock model To propose one possible concree Lévy-process financial marke, we discuss he asymmeric double exponenial jump-diffusion model of Kou 22 in Chaper 7. We jusify using his paricular model and review he asymmeric double exponenial disribuion. This is followed by a discussion of he Lévy process behind he model s jump-diffusion, which we imbed in our financial marke framework. Special emphasis is hereby pu on he jumpsize disribuion and he corresponding Lévy measure. Kou 22 uses a specific risk-neural probabiliy measure based on uiliy driven houghs and shows ha wih his change of measure one says, imposing cerain assumpions on he marke parameers like ineres rae or drif, in he class of asymmeric double exponenial jump-diffusions, whose disribuional properies are essenial for he derivaion of he model s European call opion pricing formula. Since we consider local risk-minimizaion, we are mainly ineresed in he implicaions of he Föllmer-Schweizer measure on he model and discuss hem explicily in Theorem as well as in Lemmaa and In Lemma i is hen shown ha under his measure he jumpsizes are generally no asymmeric double exponenial disribued resuling subsequenly in alernaive, he disribuional class preserving condiions on he marke parameers, which we addiionally compare o hose of Kou 22. Secion 7.5 concludes he chaper wih he formula for he arbiragefree price process of an European call, an uni-linked wih guaranee pure endowmen and an uni-linked wih guaranee erm insurance. Moreover he spaial derivaive of he arbirage-free price process is calculaed. Boh he price iself and is derivaive are essenial ingrediens of he deduced hedging sraegies. This jump-diffusion model has no been considered in connecion wih uni-linked life insurance conracs so far. Moreover, is invesigaion under he Föllmer-Schweizer change of measure has no been done before and he condiions for his change of measure no o lead ouside he class of asymmeric double exponenial jump-diffusions are new.

18 14 Chaper 1. Inroducion and summary Saisics for jump-diffusions Sudying he asymmeric double exponenial jump-diffusion of Kou 22, we realized ha saisfying parameer esimaion mehods are almos no available for jump-diffusions. Therefore we decided o address his opic in Chaper 8. Alhough we use hereby exemplarily he menioned process, our reamen is essenially also valid for oher jump-diffusions. For hose processes he probabiliy densiy is no expressible in closed form, however, heir characerisic funcion is explicily known. For ha reason we propose in Secion 8.3 o base he esimaion on he cumulan generaing funcion which we compare in he squared disance wih he complex logarihm of he empirical characerisic funcion. This leads o a sysem of wo nonlinear regression models which arise from he real and imaginary par of hese funcions and which are even linear in special cases. Addiionally we give a deermining rule for he compac suppor on which hese regression problems are considered. They may be solved using sandard leas-square rouines. The heoreical foundaion of our esimaion approach is provided in Secion 8.4. This secion conains he Glivenko-Canelli ype heorem for he empirical characerisic funcion of Csörgő and Toik 1983 which we moivae and which gives us srong consisency. Thereafer we apply he invariance principle of Csörgő 1981 o he empirical characerisic funcion resuling in a complex valued Gaussian process in he disribuional limi. Wih he help of his resul we finally sae in Theorem he asympoic finie-dimensional disribuions of he complex valued residuals of our regression problem and compue heir covariance srucure. Appendix The appendix conains a resul from Sochasic Analysis, which is based on Gronwall s inequaliy. I provides condiions under which he soluion of a cerain sochasic inegral equaion wih respec o a Lévy process is a square-inegrable process. We use i in Lemmaa and o show ha he sock process defined in 2.14 is square-inegrable under he hisorical as well as under he Föllmer-Schweizer measure. An essenial sufficien condiion is he exisence of only he hird momen of he Lévy measure which we suppose in We could herefore weaken one assumpion of Chan 1999 demanding he exisence of exponenial momens of he Lévy measure. This opens he model o a broader class of disribuions, especially o hose wih heavier ails. We designed and proved Theorem A..2 ailor-made o our siuaion, especially including a drif componen. Oher versions of he resul, wihou a drif componen, are conained wihou proofs for example in Jacod, Méléard and Proer 2 or Ma, Proer and Zhang 21.

19 Chaper 2 Basic conceps In his chaper we provide he fundamenals required in his hesis. In doing so we begin wih an inroducion o he heory of Lévy processes and review hose properies of a Lévy process which are beneficial for a Lévy-process financial marke. Moreover, we briefly discuss specific Lévy processes ha are in common use for financial marke modeling, such as jump-diffusions, variance gamma, normal inverse Gaussian or generalized hyperbolic Lévy processes. Subsequenly, we inroduce he general Lévy-process financial marke model of Chan 1999 ogeher wih changing he hisorical measure o he Föllmer-Schweizer measure. The local risk-minimizing hedging heory of Schweizer 1991, encounered in he chapers below, is based on his measure. Thereafer, we consider a muli-sae Markov model for life insurance modeling which will frequenly be used in his hesis. In our presenaion of he laer model we follow Møller 21a. Finally we describe how he financial marke and he insurance model are combined o a common model being a produc probabiliy space. This model is necessary, since we rea insurance conracs for which he insurance benefi depends on a risky asse modeled in he financial marke. For he produc space we also review boh is incompleeness and he local risk-minimizing hedging heory of Föllmer and Sondermann 1986 and Schweizer Preliminaries This lile secion a he beginning provides assumpions made hroughou he hesis and saes frequenly used and common noaions. For a deailed reamen of basic ools such as maringales, semimaringales and sochasic inegraion we refer o e.g. Jacod and Shiryaev 23 or Proer 24. Throughou his hesis we only consider insurance conracs and relaed financial markes, so i suffices o rea only he case of a finie ime horizon T [,. Definiion A filered probabiliy space Ω, F,F T, P saisfies

20 16 Chaper 2. Basic conceps he usual condiions if Ω, F, P is complee, all he null ses of F are conained in F, and F T is a righ-coninuous filraion, ha is: and F s F F are σ-algebras for s T, In paricular, we assume F = F T. F s = >s F, s T. For a filraion, F is inerpreed as he informaion known in he model a ime which increases over ime, i.e. a filraion describes how informaion is revealed in he model. All filered probabiliy spaces menioned in his hesis are assumed o saisfy he usual condiions. A sochasic process X = X T on some filered probabiliy space Ω, F,F T, P is said o be càdlàg coninu à droie, limie à gauche if is pahs are righ-coninuous admiing lef limis, i.e. lim X s = X and lim X s =: X s. s,s> s,s< For such a sochasic process we define is jump a ime T by X := X X, where we se X := X. We furher say X is adaped if X is F -measurable for each T. Occasionally we need he following measurabiliy conceps. Definiion Opional σ-algebra. The opional σ-algebra is he σ- algebra O generaed on Ω [,T] by all adaped càdlàg processes. X : Ω [,T] R is called an opional process if i is measurable wih respec o O. Definiion Predicable σ-algebra. The predicable σ-algebra is he σ-algebra P generaed on Ω [, T] by all adaped lef-coninuous processes. X : Ω [,T] R is called a predicable process if i is measurable wih respec o P. We always denoe he expecaion operaor wih respec o he canonical measure P by E[ ], whereas he expecaion wih respec o any oher measure Q is wrien as E Q [ ]. In he conex of his hesis we generally use P for he so called real-world or hisorical measure and respecively Q for he so called risk-neural measure. Definiion Le Ω, F,F T, P be some filered probabiliy space and le Q be a second probabiliy measure on i. Then P is said o be locally equivalen wih respec o Q P loc Q if P A = Q A =, A F, T. P and Q denoe he resricions of P and Q o he σ-algebra F, respecively.

21 2.2. Lévy processes 17 We remaind he reader ha a given posiive maringale Z = Z T wih Z = 1 defines a probabiliy measure Q loc P by he Radon-Nikodým derivaive d Q d P = Z. For wo random objecs Y i, i = 1,2, in mos cases Y 1 = Y 2 or ec. sands for Y 1 = Y 2 holds a.s. almos surely or Y 1 Y 2 a.s., ec.. Using he common noaions we denoe he quadraic covariaion of wo semimaringales X and Y by [X,Y ] and he predicable quadraic covariaion of wo locally square-inegrable maringales X and Y by X,Y. 2.2 Lévy processes Lévy processes are named afer he French mahemaician Paul Lévy Togeher wih Andrei N. Kolmogorov Lévy is considered as one of he founding fahers of he modern heory of sochasic processes. Taylor 1975 wries A ha ime here was no mahemaical heory of probabiliy only a collecion of small compuaional problems. Now i is a fully-fledged branch of mahemaics using echniques from all branches of modern analysis and making is own conribuion of ideas, problems, resuls and useful machinery o be applied elsewhere. If here is one person who has influenced he esablishmen and growh of probabiliy heory more han any oher, ha person mus be Paul Lévy Definiion and basic properies In his subsecion we define Lévy processes and discuss some of heir mos imporan properies. As basis we consider he filered probabiliy space Ω, G,G T, P. The nex definiion is adaped from Proer 24, Chaper I, and Con and Tankov 24, Chaper 3. Definiion An adaped càdlàg sochasic process L = L T wih values in R such ha L = is called a Lévy process if i possesses he following properies: 1. Independen incremens: 2. Saionary incremens: L L s is independen of G s, s < T. L +h L d = Lh, < + h T, ha is, he disribuion of L +h L does no depend on.

22 18 Chaper 2. Basic conceps 3. Sochasic coninuiy: ε >, lim h P L +h L ε =, ha is, given a fixed ime, he probabiliy of seeing a jump of L a is zero. The càdlàg assumpion in he definiion of a Lévy process is no necessary and can be made wihou loss of generaliy, since every Lévy process has an unique modificaion ha is càdlàg cf. e.g. Proer 24, Chaper I, Theorem 3. In Definiion II.4.1 of Jacod and Shiryaev 23 processes saisfying 1. and 2. of Definiion are called processes wih saionary independen incremens PIIS. A sraighforward choice of G could be he hisory of L up o and including ime, ha is, he compleed σ-algebra σl s, s. For he res of his hesis we always assume G o be of his form. Perhaps he mos significan propery of Lévy processes is he so called Lévy-Iô decomposiion which reveals he building blocks of Lévy processes and a he same ime a lo more srucure han he definiion does. In order o formulae his decomposiion properly he disconinuiies of a Lévy process have o be considered. Since a Lévy process has càdlàg pahs, he only disconinuiies occurring are a mos counably many jump disconinuiies cf. Riesner 23. The jump couner of a Lévy process L is for T defined as N B := 1 L s 1 L s B <s for any Borel-measurable se B R. By he càdlàg pahs of L i is clear ha N is almos surely finie whenever / B he closure of B. Proer 24, Chaper I Theorem 35, shows ha N and E[N 1 ] are σ-finie measures on R, B, where B denoes he Borel 1 -σ-algebra over R. Definiion Given a Lévy process L, he measure ν defined on R, B by νb = E[N 1 B] for any se B B is called he Lévy measure of L. νb is he expeced number, per uni ime, of jumps whose size belongs o B. Moreover for a fixed Borel-measurable se B R such ha / B, i holds ha N B T is a Poisson process 2 Proer 24, Chaper I. Tha is, i is a couning process wihou explosion which is a Lévy process cf. addiionally Lemma 2.1 in Con and Tankov 24. Therefore i is 1 Félix Edouard Jusin Émile Borel, French mahemaician named afer he French mahemaician Siméon Denis Poisson

23 2.2. Lévy processes 19 obvious ha N B has inensiy νb and ha N B νb T is a maringale. More generally, given a Lévy process L, one considers a Poisson random measure Nd,dx on [,T] R cf. e.g. Jacod and Shiryaev 23, Definiion II.1.2, Theorem II.4.8 or more elemenary Con and Tankov 24, Chaper 2.6. For any Borel-measurable se A [, T] R one ses A Nd,dx := s 1 L s 1 s, L s A, ha is, using he Dirac-measure δ a a poin a, Nd,dx := s 1 L s δ s, Lsd,dx. The measure Nd,dx and he jump couner N are conneced hrough N B = [,] B Nds,dx =: B Nds,dx for any se B B. Lemma Le f : Ω [,T] R R be a family of sochasic processes such ha f is measurable wih respec o P B. For any Borel-measurable se A [,T] R such ha E[ f,x Nd,dx] < 2.1 A i holds E[ f,xnd,dx] = E[f, x] dνdx, A A where dνdx denoes he produc of he usual Lebesgue measure and he Lévy measure. Proof. By he sandard monoone class argumen i suffices o consider fω,,x = 1 A,x 1 B s1,s 2 ] Cω,,x where A = 1, 2 ] Ã, 1 < 2 T, s 1 < s 2 T, Ã,C B and B G s1 such ha 2.1 is saisfied. According o Theorem I.2.2 of Jacod and Shiryaev 23 ses of he form B s 1,s 2 ] consiue a generaing sysem of P. Wihou loss of generaliy we assume s 1 1 and 2 s 2. By he independen incremens of L we ge ha he random variables 1 B and

24 2 Chaper 2. Basic conceps NA s 1,s 2 ] C are independen. Hence, E[ f,xnd,dx] = E[1 B 1 L s 1s, L s A s 1,s 2 ] C] s = PB E[ 1 s 2 1 L s 1 L s à C ] = PB E[N 2 à C N 1 à C] = PB 2 1 νã C = E[f, x] dνdx. The measure dνdx is called inensiy measure of Nd,dx or is compensaor. We define he compensaed measure Md,dx := Nd,dx dνdx. Le now B B and f,x be a family of sochasic processes as in he previous Lemma such ha 2.1 is saisfied. Then he observaion ha fu,xndu,dx is independen of G s s B for s < T immediaely implies ha inegrals of he form fs,xmds,dx, T, B are maringales. Wih hese observaions on he jump behavior of a Lévy process we sae he Lévy-Iô 3 -decomposiion. Theorem Lévy-Iô-decomposiion. Le L be a Lévy process and ν is Lévy measure. Then i holds x 2 νdx < and νdx <. { x 1} { x >1} Furher, here exiss a Brownian 4 moion wih variance c 2, which is a Wiener process 5 sandard Brownian moion W = W T imes a consan c,, and a consan γ = E [ L 1 L l 1] R such ha for any T L = γ + cw + L l + M, where L l = { x >1} xnds,dx and M = The processes W, L l and M are independen. { x 1} xmds,dx. 3 Kiyoshi Iô, Japanese mahemaician named afer he Scoish boanis Rober Brown named afer he American mahemaician Norber Wiener

25 2.2. Lévy processes 21 Proof. The original proof of Lévy, which has been compleed by Iô, can be found in Lévy 1934 and Iô The proof is also done in Proer 24, Chaper I.4 and in Riesner 23, where he proof of Proer 24 has been elaboraed in deail. An ouline is also conained in Con and Tankov 24. Le us briefly commen on his decomposiion following he reamen in Con and Tankov 24. Observe ha by he càdlàg pahs of L, he process L l = L s 1 L s > 1 <s is almos surely a finie sum and hence a compound Poisson process, since i is a Lévy process wih piecewise consan sample pahs cf. Con and Tankov 24, Proposiion 3.3. There is nohing special abou he hreshold of jump magniude being equal o 1. Observe ha for every < ε < 1 he inegral ε x 1 xnds,dx = <s L s 1ε L s 1 is as well a well-defined compound Poisson process. However, as ε his inegral does no necessarily converge, since here can be infiniely many small jumps. Replacing his inegral by is compensaed version M, which is a maringale, yields he desired convergence compare R min1,x2 νdx <. M could be inerpreed as an infinie superposiion of independen compensaed Poisson processes. The Lévy-Iô-decomposiion implies herefore ha every Lévy process is a combinaion of a Brownian moion wih drif and a possibly infinie sum of independen compound Poisson processes. Since every càdlàg funcion may be approximaed by a piecewise consan funcion, every Lévy process can be approximaed wih arbirary precision by a jump-diffusion process, ha is, by he sum of a Brownian moion wih drif and a compound Poisson process. The decomposiion of a Lévy process ino a maringale cw + M and a process of finie variaion γ + L l shows furher ha every Lévy process is a semimaringale cf. Definiion I.4.21 of Jacod and Shiryaev 23. In applicaions one ofen makes he addiional assumpion x νdx < 2.2 { x >1} implying E[L l 1 ] <. This allows o simplify he Lévy-Iô-decomposiion of a Lévy process L in he following way L = cw + M + a, T,

26 22 Chaper 2. Basic conceps where M = R xmds,dx and a = E[L 1 ] = γ + xνdx. { x >1} In paricular, E[L ] < for all if and only if 2.2 holds. Under his addiional assumpion he process L is a special semimaringale, since a is predicable, which implies he almos sure uniqueness of his decomposiion. cf. Corollary I.3.16 of Jacod and Shiryaev 23. This decomposiion of a Lévy process L ino a maringale cw + M and an increasing, predicable process a is called he Doob 6 -Meyer 7 -decomposiion cf. Kallenberg 22, Chaper 25 of he process L. The following Lemma saes an useful observaion abou he momens of he Lévy measure. Lemma The Lévy measure νdx of some Lévy process L saisfies x n νdx < E[ L n ] < all, { x >1} for some fixed n 1. In paricular, in his case one has ha x m νdx <, 2 m n. R Proof. The firs par follows from Theorem 25.3 in Sao Moreover, x m νdx x m νdx x n νdx and ha { x >1} { x 1} x n νdx { x >1} { x 1} x n νdx { x >1} { x 1} x 2 νdx. The maringale M is hence a square-inegrable maringale if and only if x 2 νdx <. { x >1} Observe addiionally, E[M 2 ] = E[[M,M] ] = R x2 νdx cf. Proer 24, Chaper II, Corollary 3 of Theorem 27, and Theorem 28. The nex heorem summarizes and classifies he pah behavior of a Lévy process. The riple γ,c 2,νdx is called he characerisics of L cf. Chaper II.2 of Jacod and Shiryaev 23. Theorem Pah properies. Le L be a Lévy process and γ,c 2,νdx is characerisics. Then i holds: 6 Joseph Leo Doob, American mahemaician Paul-André Meyer, French mahemaician

27 2.2. Lévy processes ν if and only if almos all pahs of L are coninuous. 2. a Finie aciviy If νr <, hen almos all pahs of L have only finiely many jumps on any compac inerval. b Infinie aciviy If νr =, hen almos all pahs of L have infiniely many jumps on any compac inerval. 3. a Finie variaion If c = and { x 1} x νdx <, hen almos all pahs of L are of finie variaion on any compac inerval. b Infinie variaion If c > or { x 1} x νdx =, hen almos all pahs of L are of infinie variaion on any compac inerval. Proof. 1. νb = implies N[,] B = almos surely for all T and for all B B. Hence by he Lévy-Iô-decomposiion almos all pahs are coninuous. If almos all pahs of L are coninuous, hen L = for all, which implies νdx =. 2. Le νr <, hen for any 1 < 2 T one has ha N[ 1, 2 ] R = 1 s 2 1 L s 1 L s R < This shows ha L has almos surely only finiely many jumps on [ 1, 2 ]. Le νr = and consider he sequence ε n given by { } ε 1 = sup r : νdx 1 { x r} { ε n = sup r : r < ε n 1, { x r} } νdx n, n 2. We define for 1 < 2 T he following sequence of independen random variables 2 Y n = Nd,dx, n 1. 1 {ε n+1 x <ε n} The variables Y n are Poisson disribued wih inensiy λ n 2 1 for all n and he oal number of jumps of L in he inerval [ 1, 2 ] is equal o n=1 Y n. Now i is no very hard o show ha a.s. PY n 1 = e λn k=1 λ k n k! = 1 1 e λn 1 1 e 2 1 all n. Hence, n=1 PY n 1 =, which implies, using he Borel-Canelli 8 - Lemma, PY n 1, i. o. = 1. 8 Francesco Paolo Canelli, Ialian mahemaician

28 24 Chaper 2. Basic conceps Therefore, n=1 Y n = almos surely. 3. Le c = and { x 1} x νdx <, hen he Lévy-Iô-decomposiion implies ha L = γ + L l + M = γ xνdx + L l + xnds,dx. { x 1} { x 1} The firs wo erms are clearly of finie variaion. I remains o consider he hird erm having oal variaion over [,T] given by TV [,T] xnds,dx = x Nds,dx. { x 1} { x 1} We have equaliy here, since he oal variaion of any càdlàg funcion is greaer or equal o he sum of is jumps. Therefore, E [ ] TV [,T] xnds,dx = T x νdx, { x 1} { x 1} which implies ha he oal variaion of L is finie almos surely. For he converse saemen we observe ha for every n 1 and every 1 < 2 T i holds, 2 TV [1, 2 ]L x Nds,dx 1 { 1 n x <1} 2 = 2 1 x νdx + x Mds,dx. { 1 n x <1} { 1 n x <1} Le now Y n := 2 1 { 1 n x <1} x Mds,dx and X n := Y n+1 Y n. Then one has ha E[X n ] = all n and Var[X n ] = E [ Xn] 2 = 2 1 x 2 νdx <. { 1 n+1 x < 1 n } n=1 n=1 Kolmogorov s hree series heorem implies hen ha n=1 X n converges o a finie limi almos surely. Therefore x νdx = TV [1, 2 ]L = a.s. { x 1} If c > hen L conains a Brownian moion componen which has almos surely pahs of infinie variaion. The echnique involving Kolmogorov s hree series heorem ha we applied in he previous proof is also used o prove ha M exiss in he Lévy- Iô-decomposiion. Le us now briefly urn o Lévy processes wih almos surely nondecreasing pahs, which play an imporan role in financial marke modeling, oo. n=1 1

29 2.2. Lévy processes 25 Definiion Subordinaor A Lévy process L is called a subordinaor if almos surely all is sample pahs are nondecreasing. Corollary A Lévy process L wih characerisic riple γ,c 2,νdx is a subordinaor if and only if c =, ν,] =, 1 xνdx < and γ 1 xνdx. Anoher immediae consequence of he Lévy-Io-decomposiion is he Lévy-Khinchin 9 formula for he characerisic funcion of a Lévy process. Theorem Lévy-Khinchin. Le L be a Lévy process and γ,c 2,νdx is characerisics. Then i holds where φ u = E [ e iul] = e ψu, u R, ψu = 1 2 c2 u 2 + iγu + e iux 1 iux 1 { x 1} νdx. R Proof. Kallenberg 22, Corollary o Under assumpion 2.2 he so-called characerisic exponen simplifies ψu = 1 2 c2 u 2 + iau + e iux 1 iux νdx. According o he Lévy-Khinchin formula he characerisics of a Lévy process uniquely deermine is characerisic funcion and hence is disribuion. Furher by he propery φ u = φ 1 u, 2.3 we see ha he disribuion of L 1 deermines he marginal disribuions of he whole Lévy process L. There is now a significan relaionship beween Lévy processes and infiniely divisible disribuions. A any fixed ime we may wrie L = n k=1 R L k n L k 1 n ha is, L represened as a sum of n i.i.d. independen and idenically disribued random variables, which holds because of he saionary and independen incremens. We cie he following definiion from Bingham and Kiesel 24. Definiion A random variable X, or is disribuion funcion F, is infiniely divisible if for each n = 1,2,... here is a disribuion funcion F n wih F as is n-fold convoluion power:, F = F n F n F n n facors 9 Aleksandr Yakovlevich Khinchin, Russian mahemaician

30 26 Chaper 2. Basic conceps ha is, X has he same disribuion as n k=1 X nk wih X ni i = 1,...,n independen wih common disribuion F n. Theorem Lévy processes and infiniely divisibiliy. Le X be any random variable. Then X is infiniely divisible if and only if X d = L 1 equal in disribuion for some Lévy process L. Proof. The if par is shown above. For he converse saemen we refer o Kallenberg 22, Theorem Thus, given an infiniely divisible disribuion i is fairly easy o consruc he corresponding Lévy process Common examples Our aim is i now o discuss some examples of Lévy processes frequenly used in financial marke modeling. Classified by heir jump aciviy cf. Theorem 2.2.6, we disinguish beween wo main caegories: finie and infinie aciviy Lévy-processes. Again we work on he filered probabiliy space Ω, G,G T, P. For a relaed discussion we refer o Chaper 4 in Con and Tankov Finie aciviy Lévy-processes: jump-diffusions The mos simple examples and a he same ime he mos well-known finie aciviy Lévy-processes are clearly he Brownian moion possessing almos sure coninuous sample pahs and suiable normally disribued incremens, and he Poisson process. This is also refleced in he Lévy-Iô-decomposiion. We recall ha in he Black 1 -Scholes 11 model he log-asse-price is modeled using a diffusion, ha is, L = β + cw, T, for β R, c > and W T a sandard Brownian moion. Anoher imporan example is he compound Poisson process which we wrie as N L = Y i, T, i=1 where he jump sizes Y i are i.i.d. wih disribuion having a densiy fx wih respec o he Lebesgue measure and where N T is a Poisson process wih inensiy λ, independen from Y i, for all i. As previously menioned compound Poisson processes are compleely characerized as Lévy processes 1 Fischer Sheffey Black, American economis Myron S. Scholes, Canadian economis 1941 and Nobel prize winner in 1997

31 2.2. Lévy processes 27 wih almos sure piecewise consan sample pahs cf. Con and Tankov 24, Proposiion 3.3. For a compound Poisson process he Lévy measure is paricularly simple. Proposiion Le L be a compound Poisson process. Then is Lévy measure is given by νdx = λfxdx. Proof. Le B B. Then, condiionally on he rajecory of he Poisson process N, he measure N[,1] B is a sum of N 1 i.i.d. Bernoulli 12 random variables aking value 1 wih probabiliy B fxdx. Hence, νb = E[N[,1] B] = E[E[N[,1] B G 1 ]] = E[N 1 fxdx] = λ fxdx. B In paricular, his shows once again ha each compound Poisson process is a finie aciviy Lévy process. Togeher wih he Brownian moion compound Poisson processes consiue so called jump-diffusions. A Lévy process of jump-diffusion ype has he following form N L = β + cw + Y i, T, i=1 where β R, c >, W T is a sandard Brownian moion and N i=1 Y i, T, is a compound Poisson process. Opposed o he Black-Scholes model using a jump-diffusion o describe log-prices in a financial marke implies ha he normal evoluion of log-prices is given by a diffusion process inerruped by jumps a random occurrence. This jumps migh represen rare evens due o unexpeced new marke informaion resuling in, for example, large drawdowns or large quoaion gains. To make he definiion of a jump-diffusion model complee one needs o specify he jump-size disribuion. The challenge here is o find he righ ail behavior o reproduce exremal evens, since he ail behavior of he process depends significanly on he ail behavior of he jump-size disribuion cf. Lemma There are essenially wo jump-size disribuions considered in he conex of financial markes. They are especially aracive since a leas for European syle opions boh lead o an almos closed opion pricing formula in he sense of being a quickly converging series. Firs, he Meron 13 model cf. Meron 1976 assumes ha he jump-sizes Y i of L follow a normal disribuion: Y i Nµ,δ 2. This implies ha he probabiliy densiy of L saisfies λ k exp x β kµ2 p x = e λ 2σ 2 +kδ 2 k! 2πσ 2 + kδ 2 k= 12 named afer he Swiss mahemaician Jakob Bernoulli Rober Carhar Meron 1944, Nobel prize winner in 1997 B

32 28 Chaper 2. Basic conceps and ha is Lévy measure is given by νdx = λ δ 2π exp x µ2 2δ 2 dx. In Chaper 7 we discuss in deail he second common jump-diffusion model known as he Kou model cf. Kou 22. Here he jump-size disribuion is assumed o follow an asymmeric double exponenial disribuion wih Lebesgue densiy fx = pη 1 e η 1x 1 [, x + qη 2 e η 2x 1, x, η 1 > 1,η 2 >, where p,q and p + q = 1. The asymmeric ail behavior seems o be an advanage compared o he Meron model, since his asymmery is observable in financial markes implied by boh over- and underreacion of marke paricipans o new marke informaion. In his case he probabiliy densiy of L is no available in closed form Infinie aciviy Lévy-processes The mos frequenly used infinie aciviy Lévy processes are generaed hrough Brownian subordinaion wih an independen increasing Lévy process. This means a Brownian moion wih a possible drif is evaluaed a a differen, new and sochasic ime scale which is given by an independen subordinaor. This ime change is inerpreed in financial erms as business ime. Those models say in he class of Lévy processes, ha is, if β+cw is a diffusion and τ T is an independen subordinaor, hen he process βτ + cw τ is again a Lévy process cf. Con and Tankov 24, Theorem 4.2. Alhough generaed hrough a Brownian moion, he Lévy-Iô-decomposiion of his kind of Lévy processes does no necessarily conain a Brownian moion leading o purely disconinuous Lévy processes. Modeling asse prices wih purely disconinuous bu infinie aciviy Lévy processes is jusified in he lieraure e.g. Carr, Geman, Madan and Yor 22 by he argumen ha he jump srucure wih an infinie number of arbirarily small jumps is already rich enough o give a nonrivial small ime behavior Variance gamma Lévy-process As a firs example we consider he variance gamma process following he exposiion in Madan 21. A variance gamma process is a Lévy process of infinie aciviy bu of finie variaion. I was firs sudied in Madan and Senea 199 for he symmeric case where β = in 2.5 below. The general case of an asymmeric variance gamma processes was hen developed in Madan and Milne 1991 and Madan, Carr and Chang Besides he volailiy, hose processes feaure only wo addiional parameers allowing

33 2.2. Lévy processes 29 o conrol skewness and kurosis of he disribuion. As subordinaor one considers a gamma process wih mean rae equal o 1, ha is, a Lévy process τ = τ T whose incremens τ +h τ over non-overlapping inervals of lengh h follow for all < +h T a gamma disribuion wih Lebesgue densiy 1 p h x = Γ x h v 1 e x v 1 h h, x, v >, 2.4 v v v where Γx denoes he gamma funcion. Wih his definiion one immediaely ges ha E[τ ] = and Var[τ ] = v. Therefore he average random ime change in unis of calender ime is whereas is variance is proporional o. The Lévy measure and he characerisic exponen of a gamma process are given by cf. Con and Tankov 24, Table 4.4 νdx = e x v vx 1, xdx and ψ τ u = 1 log 1 iuv, v respecively. Given now a gamma process, a sandard Brownian moion W T, some c > and β R, he variance gamma process is defined by L = βτ + cw τ, T, 2.5 which is a Brownian moion wih drif evaluaed a gamma ime. The characerisic exponen of he variance gamma process is given by ψu = 1 v log 1 iuβv c2 vu 2, since E [ e iul] = E [ E [ e iul G ]] = E [e τ 1 2 c2 u 2 +iβu ] = e ψτ1 2 c2 iu 2 +βu. From his represenaion i immediaely follows ha a variance gamma process is a purely disconinuous process and furher, applying l Hospial s rule 14, ha lim ψu = 1 v 2 c2 u 2 + iβu. Hence, leing he variance of he ime change end o zero resuls again in a diffusion seing. The Lévy measure of a variance gamma process cf. Madan, Carr and Chang 1998, eq. 14 has he following form νdx = 1 β v x exp c 2x 2 v + β2 c 2 c x dx. 14 Guillaume François Anoine, Marquis de l Hospial, French mahemaician

34 3 Chaper 2. Basic conceps A formula for he Lévy measure of a subordinaed Lévy process is provided for example in Theorem 4.2 of Con and Tankov 24. The form of is Lévy measure implies ha a variance gamma Lévy-process has pahs of finie variaion, since { x 1} x νdx <. Furher he asympoic behavior νdx 1 x dx, x, yields he infinie aciviy of he process. Anoher remarkable fac is ha he variance gamma process may be wrien as a difference of wo independen gamma processes implying as well he almos sure finie variaion of is sample pahs. From he momens of he variance gamma process, E[L ] = β E[L E[L ] 2 ] = β 2 v + c 2 E[L E[L ] 3 ] = 2β 3 v 2 + 3c 2 βv E[L E[L ] 4 ] = 3c 4 v + 12c 2 β 2 v 2 + 6β 4 v 3 + 3c 4 + 6c 2 β 2 v + 3β 4 v 2 2, one derives ha β is he parameer generaing skewness while kurosis is primarily influenced by v. The Lebesgue probabiliy densiy of a variance gamma process L is derived in Madan, Carr and Chang 1998 pp. 87 and 98 by making use of he mixure represenaion in 2.5. I has he form 2 p x = v v 2πcΓ e β c 2 x x 2 2v K 2 c2 v v + β2 v 1 2 c 2 2c x 2 2 v + β2, where K λ denoes he modified Bessel funcion 15 of he second kind of order λ cf. Secion 9.6 in Abramowiz and Segun Someimes he modified Bessel funcion of he second kind is also called Bessel funcion of he hird kind or MacDonald funcion. The variance gamma disribuion is a subclass of he so-called CGMY -disribuions. CGMY-disribuions are infiniely divisible and were inroduced in Carr, Geman, Madan and Yor 22 as an exension of he variance gamma disribuion o model log reurns in financial markes Normal inverse Gaussian Lévy-process A second hree parameer Lévy process is he normal inverse Gaussian process 16. I is of infinie aciviy and of infinie variaion and was inroduced o finance by Barndorff-Nielsen Here, we parially follow Con and Tankov 24, Secions 4.4.2/ The reamen of a normal inverse Gaussian process is compleely analogous o he variance Gamma process. I is defined as in 2.5 wih he excepion ha τ = τ T is an inverse 15 named afer he German mahemaician Friedrich Wilhelm Bessel named afer he German mahemaician Carl Friedrich Gauss

35 2.2. Lévy processes 31 Gaussian subordinaor, ha is, a Lévy process whose incremens τ +h τ over non-overlapping inervals of lengh h follow for all T an inverse Gaussian disribuion wih densiy p h x = h 2πv x 3 2 exp 1 2vx x h2 1, x, v >. 2.6 Noe ha E[τ ] = and Var[τ ] = v like for he Gamma subordinaor. An inverse Gaussian Lévy process has he following Lévy measure and he following characerisic exponen νdx = 1 2πv x 3/2 e x 2v 1, xdx and ψ τ u = 1 v 1 v 1 2ivu. The normal inverse Gaussian process is purely disconinuous wih characerisic exponen 1 ψu = ψ τ 2 c2 iu 2 + βu = 1 v v u 2 c 2 v 2iβuv and wih momens E[L ] = β E[L E[L ] 2 ] = β 2 v + c 2 E[L E[L ] 3 ] = 3β 3 v 2 + 3c 2 βv E[L E[L ] 4 ] = 3c 4 v + 15β 4 v c 2 β 2 v 2 + 3c 4 + 6c 2 β 2 v + 3β 4 v 2 2. Therefore skewness is again influenced by he parameer β and kurosis is again primarily conrolled by v. The Lévy measure of a normal inverse Gaussian process is νdx = β 2 + c2 v πc v p x = π e v + β c 2 x β 2 exp β x c 2 x vc v 2 β 2 + c2 v K 1 c 2 x dx and is marginal Lebesgue probabiliy densiies are given by q β K 2 + c2 v 1 c 2 x c 2 v x c 2 v. 2.7 According o Abramowiz and Segun 1964, formula 9.6.9, for λ > one has he following asympoic behavior: K λ z 1 2 λ 2 Γλ, in paricular K 1 z 1, z. 2.8 z z Therefore νdx 1 x 2 dx, x. Hence, { x 1} x νdx = and νr = which immediaely implies ha he process has pahs of infinie variaion and of infinie aciviy. We will mee again he normal inverse Gaussian disribuion in he conex of generalized hyperbolic disribuions below.

36 32 Chaper 2. Basic conceps Generalized hyperbolic disribuions and Lévy processes A he end of he Lévy process secion le us briefly have a look a generalized hyperbolic disribuions following he deailed accoun in Eberlein 21. Barndorff-Nielsen 1977 inroduced he class of generalized hyperbolic disribuions GH o model he paricle size disribuion of wind-blown sand. I is a broad class of disribuions and feaures as special case he normal inverse Gaussian disribuion and as limiing case he variance Gamma disribuion. The Lebesgue probabiliy densiy of his class is given by f GH x;λ,α,β,δ,µ = α 2 β 2 λ 2 2πα λ 1 2δ λ K λ δ δ 2 + x µ 21 α 2 β 2 K λ 1 2 α δ 2 + x µ 2 e βx µ, 2λ 1 2 where he five parameers mean and saisfy: α > he shape, β wih β < α he skewness and µ R he locaion. δ > is a scaling parameer, which corresponds o he sandard deviaion of a normal disribuion. Evenually, he parameer λ R characerizes cerain subclasses. The characerisic funcion of he GH-law has he explici form φ GH u = e iµu α 2 β 2 λ 2 K λ δ λ 2 β + iu 2 α 2 β + iu 2 K λ δ. 2.9 α 2 β 2 From his represenaion he momens of he disribuion can be derived. However, hey are no very explici and so we refer he ineresed reader o Eberlein 21. A random variable X follows a GH-disribuion if X Y = y Nµ + βy,y Y GIGλ,δ,γ, 2.1 where Nµ + βy,y denoes he normal disribuion wih mean µ + βy and variance y. GIGλ, δ, γ denoes he generalized inverse Gaussian disribuion wih parameers λ, δ and γ. Thus X follows a mean-variance mixure of normal disribuions, where he variance is sampled from a GIG-law. The Lebesgue densiy of he GIG-law is equal o γ λ 1 f GIG x;λ,δ,γ = δ 2K λ δγ xλ 1 exp 1 δ 2 2 x + γ2 x 1, x, where γ > and he remaining parameers are as inroduced above. The mixure represenaion 2.1 yields also a formula for he densiy of a generalized hyperbolic disribuion cf. Eberlein 21 f GH x;λ,α,β,δ,µ = f Nµ+βy,y xf GIG y;λ,δ, α 2 β 2 dy. 2.11

37 2.2. Lévy processes 33 Barndorff-Nielsen and Halgreen 1977 showed ha generalized inverse Gaussian laws are infiniely divisible and Kelker 1971 proved ha infinie divisibiliy is ransferred by mean-variance mixures of normal disribuions. Hence generalized hyperbolic disribuions are infiniely divisible and generae Lévy processes as described in Theorem and relaion 2.3. In general, however, only he incremens of lengh one of Lévy processes ha are consruced in he above manner follow a generalized hyperbolic disribuion. The class of GH-disribuions is universally no closed under convoluion, which is an immediae consequence of he characerisic funcion 2.9. The only excepion is he case where λ = 1 2 cf. Normal inverse Gaussian disribuions below. In Eberlein 21 i is discussed ha generalized hyperbolic Lévy-processes are purely disconinuous, of infinie variaion and of infinie aciviy. Moreover he Lévy measure is saed, bu he expression is raher complicaed as i involves inegrals of Bessel funcions of he fis and second kind. Generalized hyperbolic Lévy-processes may also be consruced via subordinaion. As subordinaor one uses a generalized inverse Gaussian process τ = τ T, where one specifies incremens of lengh one o be GIGdisribued via Theorem and furher derives he disribuion of τ via relaion 2.3. Noe ha he GIG-law is also no closed under convoluion bu has posiive mass only, so τ has almos surely only posiive incremens. The generalized hyperbolic Lévy-process is henceforh defined as L = µ + βτ + W τ, T, where W = W T is a sandard Brownian moion independen of τ compare he mixure represenaion 2.1. If he parameers of he GIGlaw are λ, δ and γ = α 2 β 2, he GH-law of he corresponding Lévy process L T has parameers λ,α,β,δ,µ. Subclass: Hyperbolic disribuions The so-called hyperbolic disribuions H, which were firs inroduced o finance by Eberlein and Keller 1995, are he subclass corresponding o λ = 1. Accordingly i is a four parameer family of disribuions wih densiy α f H x;α,β,δ,µ = 2 β 2 2αδ K 1 δ exp α δ 2 + x µ 2 + βx µ, α 2 β 2 since K1 2 z = π 2z e z cf. Wason 1944, p. 8. The logarihm of f H is a hyperbola generaed by he erm α δ 2 + x µ 2 + βx µ. For his reason his disribuion-family is called hyperbolic. Compared o a generalized hyperbolic densiy, he numerical effor o evaluae a hyperbolic

38 34 Chaper 2. Basic conceps densiy over some ime gird reduces considerably, since he Bessel funcion, as par of he normalizing facor, has o be evaluaed only once. Subclass: Normal inverse Gaussian disribuions Anoher imporan subclass of GH-disribuions are he normal inverse Gaussian N IG disribuions, which resul from he choice λ = 1 2. Hence, using K ν z = K ν z cf. Abramowiz and Segun 1964, and again K1 2 z = π 2z e z, f N IG x;α,β,δ,µ = αδ π exp δ K 1 α δ 2 + x µ 2 α 2 β 2 + βx µ. δ 2 + x µ 2 The main properies of N IG-disribuions are heir scaling propery and heir closeness under convoluion. Tha is, for independen random variables X N IGα,β,δ 1,µ 1 and Y N IGα,β,δ 2,µ 2 one has cx N IG α c, β c,cδ,cµ, c > X + Y N IGα,β,δ 1 + δ 2,µ 1 + µ 2. The scaling propery follows easily by he subsiuion rule and he densiy f N IG and he closure of convoluion follows direcly from he produc of he corresponding characerisic funcions 2.9 wih λ = 1 2. For his reason he normal inverse Gaussian Lévy-process possesses only N IGdisribued incremens. The GIG-law reduces for he parameer choice λ = 1 2 o he inverse Gaussian disribuion wih densiy f IG x;δ,γ = δ x 3 2 exp γ2 x δ 2 1 2π 2x γ, x. This explains again he name of he normal inverse Gaussian disribuion, which is a mean-variance mixure of normal disribuions, where he mixing disribuion is inverse Gaussian. Observe ha for he explici consrucion in 2.6 one has µ = and one chooses δ = h v and γ = 1 v for some v >. To derive he marginal densiies in 2.7 one applies he scaling propery and uses γ = α 2 β 2. Limiing class: Variance gamma disribuions Finally we consider he case when λ > and when we le δ. Using 2.8 one ges ha lim f GIGx;λ,δ,γ = δ γ 2 2 λ 1 Γλ xλ 1 exp γ2 2 x 1, x,

39 2.3. Financial marke 35 which is he gamma disribuion wih parameers λ and γ. The generalized hyperbolic Lévy-process reduces hence for λ > and for δ o he variance gamma Lévy-process which immediaely follows by 2.11 wih µ =. Observe ha in 2.4 one chooses λ = h v and γ = 2 v 2.3 Financial marke for some v >. Now ha we know he mos significan properies of a Lévy process and afer having seen many examples of Lévy processes we consider a Lévyprocess financial marke ha will be used in his hesis. In paricular, in he proceeding chapers our aim is o derive risk-minimizing hedging sraegies for uni-linked life insurance conracs in such a financial marke Lévy-process model The Lévy-process financial marke model ha we will work wih is inroduced in Chan On he filered probabiliy space Ω 1, G,G T, P 1 we consider a Lévy process L and assume hroughou ha he Lévy measure νdx of L saisfies x 3 νdx < { x >1} Recall 2.2 and Lemma for he consequences arising ou of condiion 2.12, especially for he Lévy-Iô-decomposiion. We suppose furher a financial marke wih only wo raded asses: a sock wih price S = S T and a bond wih price B = B T. S could refer o a single sock or o an index of asses in some financial marke. One migh hink of a muual fund or of a weighed sum of asses of a company, which could be for an insurance company he acuarial reserve of an insurance-conrac cluser. The processes S and B a any ime [,T] are assumed o be given by he sochasic differenial equaion ds = b S d + σ S dl = b + aσ S d + σ S dcw + M, 2.13 where he iniial value S is deerminisic and saisfies S >, or he differenial equaion db = r B d, B = 1, i.e. B = exp r s ds. The drif b, he volailiy σ and he risk-free ineres rae r are supposed o be coninuous and deerminisic funcions on [,T] and σ > for any [,T]. The coefficien funcions f,x = b x and g,x = σ x are obviously process-lipschiz. Therefore he soluion of 2.13 is unique cf.

40 36 Chaper 2. Basic conceps Proer 24, Chaper V, Secion 3, Theorem 6 and 7. Using he sochasic exponenial cf. Proer 24, Chaper II.8 he sock price has he following explici represenaion: S = S exp σ s dl s + bs c2 σ 2 s ds σ s L s exp σ s L s <s = S exp cσ s dw s + σ s dm s σ s M s exp σ s M s. <s aσs + b s c2 σs 2 ds In Riesner 23 his represenaion is developed in deail. In Con and Tankov 24, Chaper 8.4, he sochasic exponenial specifically for Lévy processes is reaed. In order o have S > for all, we furhermore assume ha L = M > 1/σ a.s. for all. Ŝ = B 1 S denoes he discouned sock price. Lemma Ŝ saisfies he following sochasic differenial equaion dŝ = b r Ŝ d + σ Ŝ dl = b + aσ r Ŝ d + σ Ŝ dcw + M. Furher, Ŝ is square-inegrable, i.e. E[ sup Ŝ 2 ] <. T Proof. For he firs claim we apply inegraion by pars cf. Proer 24, Chaper II, Corollary 2 of Theorem 22 o S B 1 and noe ha [S,B 1 ] = S B 1 for all, since B is coninuous and of finie variaion cf. Proer 24, Chaper II, Theorem 26 and 28. Hence, dŝ = S db 1 + B 1 ds = Ŝ r d + B 1 ds. Using 2.13 yields he claim. The square inegrabiliy follows from Theorem A..2. One idenifies he Lévy process wih cw + M and one ses f,x = aσ + b r x and g,x = σ x. Noe ha κ = cw + M,cW + M = c 2 + R x2 νdx. The previous lemma saes he unique decomposiion of he special semimaringale Ŝ, which we can rewrie in inegral form as Ŝ = S + σ s Ŝ s dcw s + M s + aσ s + b s r s Ŝs ds, where σ sŝs dcw s + M s is a P 1 -square-inegrable local maringale, in fac i is a maringale Proer 24, Chaper I, Theorem 51, and aσ s+ b s r s Ŝs ds is a coninuous adaped, and hence predicable, process.

41 2.3. Financial marke 37 Some auhors e.g. Eberlein 21 model he sock price using he ordinary insead of he sochasic exponenial of a Lévy process and moivae his in considering disribuional properies of logarihmic sock reurns or jusify i by he Black-Scholes seing, where he sock price is a geomeric Brownian moion, ha is, he ordinary exponenial of a Lévy process. I can be seen ha S = e L does no in general saisfy he sochasic differenial equaion However, Goll and Kallsen 2 Lemma 5.8 show ha he ordinary exponenial of a Lévy process is he sochasic exponenial of anoher cerain Lévy process and provide a formula ha ransforms he corresponding Lévy processes. Hence he wo models are no compleely differen. For he derivaion of hedging sraegies we are more ineresed in he infiniesimal ime behavior of he asse price as opposed o he explici srucure of he process iself. For his reason we work wih he sochasic differenial equaion 2.13 and he so implied sochasic exponenial Föllmer-Schweizer measure Risk-neural valuaion requires a locally P 1 -equivalen measure such ha Ŝ is a maringale under his measure. We call such a measure equivalen maringale measure. In Chan 1999 i is shown ha here are arbirary many equivalen maringale measures in a general Lévy-process financial marke. Such markes are well-known o be free of arbirage, bu riskless hedging is no possible, ha is, hose markes are by far no complee. For he heory of arbirage-free and complee markes we refer o he classical paper of Harrison and Pliska 1981 or o he fundamenal paper of Delbaen and Schachermayer A review of his heory is also conained in Bingham and Kiesel 24 or in Con and Tankov 24. In his hesis he incompleeness of general Lévy markes becomes apparen o some exen. For he incomplee Lévy marke one such equivalen maringale measure is he so-called Föllmer-Schweizer measure needed for he hedging heory applied in he chapers below and denoed by Q 1. In Chan 1999 his change of measure is discussed in deail: In analogy o he equivalen change of measure in he Black-Scholes model, he densiy process of Q 1 wih respec o P 1 is here given by Z = 1 + cg s Z s dw s + G s xz s Mds,dx, T, 2.15 where, wih v = R x2 νdx, he Girsanov 17 parameer is R G = r b aσ σ c 2 + v The marke price of risk in he Black-Scholes world corresponds o G. The explici represenaion of Z yields ha if G L > 1 for all [,T] 17 named afer he Russian mahemaician Igor Vladimirovich Girsanov 1967

42 38 Chaper 2. Basic conceps hen Q 1 is a rue measure and no merely a signed measure. This holds if σ 1 G < 1 for all, since he jumps of L are assumed o be bounded from below by 1/σ for all. If L has also unbounded jumps from above, i suffices o have addiionally G > for all [,T] cf. Riesner 23. The Lévy process L under Q 1 is given by cf. Chan 1999 L Q = cw Q + M Q + A Q, where he predicable par A Q has he following represenaion A Q = a + c 2 G s ds + R G s x 2 νdxds = r s b s σ s ds. Furher, he process cw Q = cw c 2 G s ds is a sandard Brownian moion under Q 1 wih variance c 2, and M Q Q 1 -maringale given by M Q = M R G s x 2 νdxds = R xm Q ds,dx. is a M Q d,dx denoes he compensaed jump measure of L under Q 1, namely and M Q d,dx = N Q d,dx ν Q dxd, ν Q dx = 1 + G xνdx. N Q d,dx is he jump measure of he process L Q ; i is defined like he measure Nd,dx in Secion I is imporan o noe ha he se of possible pahs of he process is invarian under an equivalen change of measure. However, he likelihood of seeing one specific pahs is changed and of course he saisical properies of he pahs, such as he inensiy of jumps or he independence of incremens, are no preserved. I is henceforh no surprising ha one does no necessarily ge again a Lévy process afer an equivalen change of measure. Jacod and Shiryaev 23, Theorem II.4.15, show ha a semimaringale has independen incremens if and only if is characerisics admi a version ha is deerminisic. This is he case wih L under Q 1, since G is deerminisic. However, in Corollary II.4.19 hey also show ha incremens of a semimaringale are saionary if and only if hey are linear in ime. This is no necessarily rue in our case and only holds if G does no depend on. For compleeness we consider he following definiion and refer he reader for a furher discussion o Con and Tankov 24, Chaper 14.

43 2.3. Financial marke 39 Definiion An adaped càdlàg sochasic process X = X T wih values in R such ha X = is called an addiive process if i possesses independen incremens and if i is sochasically coninuous. The process M Q is a Q 1 -maringale, since is incremens are independen and is expecaion is zero. I is moreover Q 1 -square-inegrable, since E Q 1[M Q 2 ] = R x2 νs Q dxds <. This follows immediaely from he coninuiy of G and condiion 2.12 implying R x3 νdx <. Lemma Under Q 1 Ŝ saisfies he following sochasic differenial equaion dŝ = σ Ŝ dcw Q + M Q, where W Q and M Q are as defined above. Furher Ŝ is a Q 1-square-inegrable maringale, ha is, E Q 1[sup [,T] Ŝ 2 ] <. Proof. From Lemma we know he sochasic differenial equaion for Ŝ under P 1. Hence under Q 1 Ŝ saisfies dŝ = b r Ŝ d + σ Ŝ dl Q = b r Ŝ d + σ Ŝ dcw Q = σ S dcw Q + M Q. + M Q + σ r b Ŝ d Ŝ is a Q 1 -square-inegrable maringale by Theorem A..2. Compare he proof of Lemma Oher equivalen changes of measure Chan 1999 provides a general formula for he densiy process Z = Z T of any o P 1 locally equivalen measure Q. I is for T described as he sochasic differenial equaion dz = Z G dcw + Z H,x 1Md,dx, Z = 1, 2.17 R where G is a predicable process and H,x is a predicable process for fixed x and a Borel-measurable funcion for fixed. G and H,x are called he Girsanov parameers and have o saisfy cerain condiions such ha Z is a maringale. If hey saisfy addiionally c 2 σ G + aσ + b r + σ xh,x 1νdx = 2.18 R for T, hen Ŝ is a maringale under Q. Using Girsanov s heorem for semimaringales Jacod and Shiryaev 23, Theorem III.3.24 Chan 1999 develops a special Girsanov heorem for Lévy processes elling how σ

44 4 Chaper 2. Basic conceps o ransform a given Lévy process under a given change of measure. The Girsanov parameers and Girsanov s heorems are inroduced and discussed in Chaper III of Jacod and Shiryaev 23. The Föllmer-Schweizer measure is an example of his general heory wih H,x = 1+G x and G defined as above. Anoher imporan equivalen maringale measure for a Lévy process driven asse is he so-called Esscher measure, which we do no need here. The ineresed reader migh consider he exposiion in Chan 1999 or he more pracically oriened reamen in Con and Tankov Life insurance Besides he financial marke we require a life insurance model ha evolves independenly of he financial marke. We denoe he life insurance model by he filered probabiliy space Ω 2, H,H T, P 2 and use i o describe he sochasic life saes of an insured person, which are relevan for he considered insurance conrac. Furher, we rea he sochasic evoluion of a whole porfolio of idenical insurance conracs and review some conceps of classical acuarial mahemaics. For a basic inroducion o acuarial mahemaics, such as he fundamenal acuarial principle of equivalence, we refer o Bowers, Gerber, Hickman, Jones and Nesbi 1997 or o Gerber and Cox The precise mahemaical descripion of specific uni-linked life insurances, such as uni-linked pure endowmens or uni-linked erm insurances, follows in he subsequen chapers Muli-sae Markov model One possible model o describe Ω 2, H,H T, P 2 is he classical mulisae Markov 18 model of Hoem 1969; see also Møller 21a and Norberg One considers he se J = {,1,...,J} of possible saes of a policy where usually one assumes o be is iniial sae. J = {acive, disabled, dead} describes exemplarily hree possible saes of an insured individual. A càdlàg Markov process Z = Z T wih values in J and iniial disribuion 1,,..., is now used o indicae he sae of he policy a ime. One furher defines H o be he compleed σ-algebra σz s, s, i.e. H T is he naural compleed filraion of Z. In order o coun he number of ransiions from sae j o sae k in he ime inerval,] a mulivariae couning process N jk j k is defined by N jk = #{s s,],z s = j,z s = k} j,k J, j k, T. Moreover he processes I j = 1 {Z=j}, j J, are inroduced indicaing wheher he policy is in sae j a ime or no. The Markov chain Z is 18 Andrei Andreyevich Markov, Russian mahemaician

45 2.4. Life insurance 41 furher assumed o posses ransiion raes λ jk given by λ jk = I j µjk, T, where he inensiies of ransiion µ jk are supposed o exis and o be deerminisic, coninuous funcions. The ransiion raes λ jk compensae he couning processes N jk resuling in square-inegrable, muually orhogonal and zero-mean maringales M jk = N jk λ jk u du, T The orhogonaliy follows from he fac ha he N jk do no have any simulaneous jumps cf. Kallenberg 22, Lemma The predicable quadraic variaion process M jk,m jk is hence given by M jk,m jk = λ jk u du = I j uµ jk u du, T, 2.2 and for predicable processes F jk, j,k J, j k, T, such ha inegrals of he form E[ Fs jk 2 1 j s µjk s ds] <, Fs jk dms jk, T, are again square-inegrable, zero-mean maringales Proer 24, Chaper IV, Lemma before Theorem 28. Given he inensiies of ransiion he ransiion probabiliies p jk,u = P 2 Z u = k Z = j of Z are deermined by Kolmogorov s backward differenial equaions d d p jk,u = µ jl p jk,u p lk,u, u T, 2.21 l:l j subjec o he condiions p jk, = 1 {j=k}. Conrarily, given he ransiion probabiliies p jk,u = P 2 Z u = k Z = j of Z one has for j k ha µ jk p jk, + h = lim. h h

46 42 Chaper 2. Basic conceps Typical porfolio of insured lives Anoher way o describe he probabiliy space Ω 2, H,H T, P 2 is o consider a porfolio of independen and idenically disribued lives as i is used in Møller Applying he usual acuarial noaion, we sar wih a group of l x individuals seleced from a cohor of equal age x. The remaining lifeimes of hose individuals are assumed o be a sequence of i.i.d. non-negaive random variables T 1,...,T lx defined on Ω 2, H, P 2. The disribuion of T 1 is hereby assumed o exhibi a coninuous densiy f wih respec o he Lebesgue measure. Is hazard rae funcion µ x+,, is hen given by µ x+ = R f fs ds, where he denominaor is sricly posiive, and zero oherwise. By he following argumen involving he mean value heorem presupposing he necessary echnical condiions he hazard rae funcion may be inerpreed as he probabiliy ha an individual dies in a small period of ime. Le,h >, h small, hen P 2 T 1 + h P 2 T 1 P 2 T 1 > The observaion µ x+ = d d log he survival funcion given by p x = P 2 T 1 > = exp f fsds h = µ x+h. fsds, for fsds >, leads o µ x+τ dτ 2.22 and he number of deahs in he group is recorded by he couning process N I = N I T, where l x N I = 1T i, T. i=1 H in his model is assumed o be he compleed σ-algebra σ{1t i u; u, i = 1,...,l x }. The process N I is compensaed by he sochasic inensiy process cf. e.g. Jacod and Shiryaev 23, Proposiion II.3.32 λ l x N I µ x+, T, which is he produc of he number of individuals surviving unil ime and he hazard rae, leading o he compensaed process M I = M I T, given by M I = NI λ u du, T. I is a square-inegrable, zero-mean maringale wih predicable quadraic variaion process M I,M I = λ u du, T.

47 2.4. Life insurance 43 This porfolio model is, however, jus a special applicaion of he mulisae Markov model discussed in he previous subsecion, where he sae space is modeled as J = {,1,...,l x } and sae j corresponds o exacly j policy holders having died. The ransiion raes of he Markov process Z are hen λ jk = I j 1 {k=j+1}n jµ x+ for j,k =,...,l x 1. Noe ha he inensiies of ransiion are given by µ jk = n jµ x+ k = j + 1, n jµ x+ k = j, else. Moreover, he ransiion probabiliies are k < j, p jk,u = u exp n jµ x+τ dτ k = j, and for k > j Kolmogorov s backward differenial equaions deermine p jk by d d p jk,u = n jµ x+ p jk,u p j+1,k,u Analyical laws of moraliy Previously we considered he hazard rae funcion of he remaining lifeimes of individuals in an insurance porfolio. To make his more precise, we discuss now some analyical choices of hazard rae funcions, also called laws of moraliy cf. Bowers, Gerber, Hickman, Jones and Nesbi If an analyical law is applied o generae moraliy ables, hey are commonly used and heir simple analyical form has he advanage o allow for furher laborious mahemaical compuaions. However, i migh be quesionable if one simple law describes accuraely he moraliy for a whole group of individuals spanning a whole range of differen ages. Furher, wih he increasing speed and sorage capaciy of compuers i is nowadays possible o apply much more advanced laws involving more parameers o generae moraliy ables. One of he firs laws of moraliy µ x = ω x 1, x < ω, where ω is he heoreical maximal age for individuals, was proposed in 1729 by De Moivre 19. Anoher law was inroduced by Gomperz 2 in 1825: µ x = βγ x, β >, γ > 1, x. 19 Abraham de Moivre, French mahemaician Benjamin Gomperz, English mahemaician

48 44 Chaper 2. Basic conceps A furher famous choice is he Gomperz-Makeham hazard rae funcion. I complemens he Gomperz law, which capures he hazard of aging, by a consan inerpreed as capuring he acciden hazard. I is given by µ x = α + βγ x, β >, α β, γ > 1, x. Finally, in 1939, Weibull 21 proposed o apply µ x = kx n, k >, n >, x Convenional life-insurance risk diversificaion The purpose of his subsecion is o demonsrae how life insurance companies convenionally diversify moraliy risk wihin an insurance porfolio. Therefore we consider he classical life insurance wih deerminisic premium and benefi paymens. A similar consideraion is conained in Bowers, Gerber, Hickman, Jones and Nesbi We assume he porfolio model of Subsecion and furher ha some insurance company issues a ime = o each of he l x individuals an idenical conrac same insurance period T, same insured sum K agains a single premium P. Exemplarily, we hink of a erm insurance paying he insured sum o he heirs, in case he insured person dies before T. The presen value of he insurer s oal liabiliy is hen described by H = l x i=1 H i, where H i = 1T i Te rt i K for some riskless rae of ineres r, and wih H 1,...,H lx forming a sequence of i.i.d. random variables. We suppose furher ha he insurance company chooses he premium P such ha P 2 l x P H α, for some risk-level α being close o 1 and presuppose H 1 L 2 P 2. Le µ = E[H 1 ] and σ 2 = Var[H 1 ]. For l x sufficienly large he cenral limi heorem implies hen ha H lx µ P 2 σ l xp l x µ l x σ lx P l x µ Φ l x σ l x where Φ denoes he cumulaive sandard normal disribuion funcion. Hence P is approximaely chosen such ha P = µ + z α σ lx, 21 Erns Hjalmar Waloddi Weibull, Swedish engineer and mahemaician

49 2.5. Combined financial and life insurance model 45 where z α is he α-quanile of he sandard normal disribuion. This shows σ ha he risk premium charged, i.e. z α lx, becomes smaller if he number of insured persons in he porfolio is increased. To accoun for ha risk premium life insurance companies usually include a safey loading in heir moraliy ables Risk-neuraliy We follow Aase and Persson 1994 and Møller 1998, 21a in assuming risk-neuraliy of an insurance company owards moraliy. This is a common assumpion jusified boh by he reasoning in he previous secion and by a law of large numbers argumen saing ha he moraliy probabiliies in a large porfolio of sochasically independen insured individuals develop as expeced. Tha is, using he porfolio model from Secion of i.i.d. remaining lifeimes T 1,...,T lx, he relaive number of survivors unil ime [,T] 1 l x l x i=1 1T i > p x a.s. for l x. Roughly speaking, he acual number of survivors l x i=1 1T i > equals he expeced number l x p x, if he porfolio size is sufficienly large. This forms also he basis for he acuarial fundamenal principle of equivalence saying ha he expeced presen values of benefi and premium paymens are equal. 2.5 Combined financial and life insurance model Combined model A uni-linked or equiy-linked life insurance is a cerain kind of insurance linking he insurance benefi o he marke value of some specified reference porfolio which can consis of socks, bonds or shares of a muual fund. To model such conracs properly, we need o combine a financial marke model for a reference porfolio and an insurance model for he sochasic of he insured lives. The foundaion of our subsequen model is herefore a filered probabiliy space Ω, F,F T, P modeled as a produc space of wo sochasically independen probabiliy spaces. The firs, Ω 1, G,G T, P 1, describes a Lévy process financial marke as inroduced in Secion 2.3 and he second, Ω 2, H,H T, P 2, describes an insurance model as discussed in Secion 2.4. To be more precise: Ω = Ω 1 Ω 2, F = G H, T, PA = P 1 P 2 A, A F = G T H T.

50 46 Chaper 2. Basic conceps For risk-neural valuaion we also need o decide which equivalen maringale measure o use in he combined model. In conras o our seing Møller 1998 considers a sandard Black-Scholes financial marke, which is complee, and chooses is unique equivalen maringale measure. Because of he uncerainy of he insured lives in he second probabiliy space his combined model becomes also incomplee. In fac Møller 1998 shows ha in he combined model here are, even wih a complee financial marke, arbirary many equivalen maringale measures o choose from. In our case he financial marke is already incomplee and since we assume risk neuraliy of he insurer owards moraliy, we obain an equivalen maringale measure in he combined model building he produc measure of he Föllmer Schweizer measure Q 1 and he moraliy law P 2. We denoe ha risk-neural produc measure by Q Hedging heory of Föllmer and Sondermann We already know now ha he above inroduced combined model, represened by Ω, F,F T, Q, is incomplee. Therefore we briefly review he concep of risk -minimizing hedging of Föllmer and Sondermann 1986 who exended he hedging heory for complee markes o he case of incomplee markes. For a deailed reamen of his heory in he insurance conex we refer o Møller In he beginning we summarize he mos imporan facs abou hedging in financial markes in a single definiion. Le L 2 QŜ be he space of square-inegrable predicable processes ξ T saisfying E Q [ ξs 2 d Ŝ,Ŝ s ] <. Wih his assumpion we have ha for ξ L 2 QŜ inegrals of he form ξ s dŝs are square inegrable maringales cf. Proer 24, Chaper IV, Lemma before Theorem 28. Definiion A random variable H L 2 Q is a coningen claim wih mauriy T if i is F T -measurable and if i describes a payoff a ime T. 2. A pair of sochasic processes ϕ = ξ,η is called a rading or porfolio sraegy if a ξ = ξ T L 2 QŜ, b η = η T is adaped, c he discouned porfolio process ˆV ϕ, given by ˆV ϕ = ξ Ŝ +η for T, saisfies ˆV ϕ L 2 Q and is càdlàg for all T.

51 2.5. Combined financial and life insurance model For a coningen claim H a porfolio sraegy is called H-admissible if H = ˆV T ϕ Q-a.s.. 4. The cumulaive cos process Cϕ associaed wih a sraegy ϕ is C ϕ = ˆV ϕ ξ s dŝs, T. 5. The inrinsic risk process Rϕ associaed wih a sraegy ϕ is R ϕ := E Q [ C T ϕ C ϕ 2 F ], T. 6. The inrinsic value process V T of a coningen claim H is V = E Q [H F ], T. The firs componen of a porfolio sraegy, ξ, is usually inerpreed as he number of socks and he second componen, η, as he number of bonds held a ime [,T] wihin he porfolio. The undiscouned porfolio value is naurally given by V = ˆV B = ξ S + η B. In he following we concenrae on H-admissible sraegies ϕ for a coningen claim H. In his case a hedger is able o generae H, bu maybe only a some coss. The cumulaive cos process Cϕ represens he value of he porfolio less accumulaed gains originaed hrough rading in he sock S. If ha cos process is no Q-a.s. consan, a hedger has o inves addiional capial afer having made he iniial invesmen V ϕ = C ϕ. Is risk is hereby measured by he squared value of fuure coss semming from rading. I is given by he inrinsic risk process. Conrarily, if he cos process is Q- a.s. consan Harrison and Pliska 1981 called he corresponding sraegy self-financing. They also proved ha in a complee marke every coningen claim H is aainable, ha is, H = V ϕ + ξ s dŝs Q-a.s. for a self-financing rading sraegy ϕ. Clearly in his case we have no risk involved and R ϕ =, Q-a.s. Exending his heory Föllmer and Sondermann 1986 call a rading sraegy mean-self-financing, if he cos process is a maringale. We have already seen ha we face an incomplee marke, where no every coningen claim is self-financed aainable. So for hedging purposes he quesion arises of how o inves in he sock and he bond so ha he involved risk is minimal in some sense. Föllmer and Sondermann 1986 define a rading sraegy o be risk-minimizing as follows cf. also Schweizer 1991.

52 48 Chaper 2. Basic conceps Definiion Le ϕ = ξ,η be a rading sraegy and [,T. An admissible coninuaion of ϕ from on is a rading sraegy ϕ = ξ, η saisfying and ξ s = ξ s for s, η s = η s for s <, ˆV T ϕ = ˆV T ϕ Q-a.s. The rading sraegy ϕ is called risk-minimizing if for any [,T and for any admissible coninuaion ϕ of ϕ from on we have R ϕ R ϕ Q-a.s. Föllmer and Sondermann 1986 furhermore show ha a risk-minimizing sraegy is found wih he Galchouk-Kunia-Waanabe decomposiion of he inrinsic value process of a coningen claim H. We adap he following heorem from Con and Tankov 24. Theorem Galchouk-Kunia-Waanabe decomposiion. Le Ŝ T be a square-inegrable maringale wih respec o Q. For any random variable H L 2 Q which is measurable wih respec o F T here exiss a sochasic process ξ H L 2 QŜ and a square inegrable F T -random variable KT H such ha H = E Q [H] + ξ H dŝ + KT H. Furher he zero-mean square-inegrable maringale K H = E Q [K T F ], T, is orhogonal o all sochasic inegrals wih respec o Ŝ. Tha is, K H ξ s dŝs, T, is again a maringale for any ξ L 2 QŜ or equivalenly if K H,Ŝ = for all T. Proof. For he original resul we refer o Kunia and Waanabe 1967 and Galchouck A compac and recen reamen is Ansel and Sricker Using his decomposiion V is now expressed as V = E Q [H] + ξ H s dŝs + K H, T, 2.23 where K H = K H T is a zero-mean square-inegrable maringale, orhogonal o he process Ŝ and ξh L 2 QŜ. Using he orhogonaliy Föllmer and Sondermann 1986 prove he following heorem. Theorem Föllmer and Sondermann Assume Ŝ is a square-inegrable maringale. For every coningen claim H exiss an unique H-admissible

53 2.5. Combined financial and life insurance model 49 risk-minimizing rading sraegy ϕ H. I is mean self-financing and is componens are given by ξ,η = ξ H,V ξ H Ŝ, T, where ξ H is he inegrand in Furher, he associaed risk process is given by R ϕ = E Q [ K H T K H 2 F ], T, where K H is aken from Proof. Föllmer and Sondermann Local risk-minimizaion of Schweizer Föllmer and Sondermann 1986 considered merely he case where he hisorical, subjecive measure P is already a maringale measure for he discouned sock price. Opposed o he risk-neural word, only in his case i makes sense o alk abou hedging-risk. So in he previous secion we are a firs glance a lile bi unprecise, if we do no presuppose Ŝ o be a maringale under P. However, we work under he Föllmer-Schweizer measure Q which has he following jusificaion. Schweizer 1991 shows ha he approach of Föllmer and Sondermann 1986 fails for he case, where Ŝ is a semimaringale and no a maringale under P. I is hen no possible o obain a risk-minimizing hedging sraegy reducing he involved risk uniformly over ime. For a counerexample we refer he reader o Schweizer 21. The reason is ha he Galchouk-Kunia-Waanabe projecion echnique fails for semimaringales because one is unable o conrol heir finie-variaion drif par. Is influence on R P ϕ = E [ C T ϕ C ϕ 2 F ], T, is no eviden. Insead Schweizer 1991 proposes o consider a class of smaller variaions of rading sraegies and finds he so-called locally riskminimizing sraegies under P. The idea is o change he subjecive probabiliy measure o he Föllmer-Schweizer measure which ransforms he semimaringale Ŝ ino a maringale. Under his measure one finds hen a riskminimizing sraegy according o Föllmer and Sondermann 1986 which is hen in urn locally risk-minimizing under P. The perhaps mos significan feaure of he Föllmer-Schweizer measure is ha i ransforms only he semimaringale Ŝ ino a maringale while preserving orhogonaliy relaions. Tha is, orhogonal maringales under P are also orhogonal maringales under Q and vice versa. For compleeness we remark ha he exension of he Galchouk-Kunia-Waanabe decomposiion o he semimaringale case is referred o as he Föllmer-Schweizer decomposiion. The orhogonaliy holds hen wih respec o he maringale par of he semimaringale. In

54 5 Chaper 2. Basic conceps oher words, if one derives he Galchouk-Kunia-Waanabe decomposiion under some equivalen maringale measure and wries i down under he original measure P, i is called Föllmer-Schweizer decomposiion. For furher deails abou local risk-minimizaion we refer o Schweizer 1991 or for is exension owards hedging of general paymen sreams o Chaper 5 of his hesis.

55 Chaper 3 Arbirage-free price process 3.1 Inroducion In Subsecion we observed he imporance of he Galchouk-Kunia- Waanabe decomposiion cf. Theorem for locally risk-minimizing hedging of coningen claims. Therefore in his chaper our focus is he derivaion of ha significan decomposiion in our financial marke keeping uni-linked life insurance in view. Since normally an insurance benefi has o be cashed ou a he ime a cerain even occurs, we ofen deal in he insurance conex wih ime-dependen paymen sreams, a siuaion being more general compared o simple coningen claim hedging. For his reason we are considering ime-dependen payoff funcions and call heir value arbirage-free price process. We presuppose for he whole chaper he riskneural combined model Ω, F,F T, Q as discussed in Chaper 2 wih all he noaions and assumpions made. We remark, however, ha we could perform he same analysis only for he financial marke, as inroduced in Secion A Feynman-Kac ype formula We begin wih he arbirage-free price process of cerain payoff funcions in he financial marke. As previously menioned we consider he Lévy process sock model S cf and a ime-dependen funcion g : [,T] [, R such ha g,s is G -measurable, T, and such ha [ sup E Q B 1 u gu,s u ] 2 <. 3.1 u [,T] This echnical assumpion corresponds o he square-inegrabiliy of simple coningen claims cf. Definiion Noe ha g,s is only influenced by he uncerainy of he financial marke and is inerpreed as a payoff specified a ime depending on he asse S. A coningen claim H from

56 52 Chaper 3. Arbirage-free price process Definiion is a special case of a ime-dependen payoff funcion, where one ses g,s = for [,T and gt,s T = H. Following risk-neural valuaion we obain he nex definiion. Definiion The arbirage-free price process a ime T of a payoff gu,s u a ime u T is denoed by F,S,u and is given by F,S,u = { E Q [B Bu 1gu,S u G ], < u T, B Bu 1 gu,s u, u T. Naurally we are only ineresed in he case < u T. Similar o he Black-Scholes differenial equaion, which is fulfilled by he price process in a Black-Scholes financial marke, here is an useful characerizaion of he price process in he more general Lévy-process seing in erms of an inegro-differenial equaion. To formulae his inegro-differenial equaion properly we need a few observaions firs. Lemma The arbirage-free price process saisfies F,S,u = E Q [ B B 1 u gu,s u S ], < u T. Proof. This follows immediaely from he Markov propery of Lévy processes cf. e.g. Proer 24, Chaper I, Theorem 32, which carries over by he independen incremens o he possible addiive process L Q cf. Subsecion and Con and Tankov 24, Chaper 14. In he following we herefore consider for x > and for < u T he funcion F,x,u = E Q [,x B Bu 1 gu,s u ] := E Q [ B Bu 1 gu,s u S = x ], and assume ha F,,u C 1,2 [,T] [, for each u [,T]. Tha is, i is once coninuously differeniable wih respec o he firs and wice coninuously differeniable wih respec o he second variable. For noaional convenience le us denoe by F,, = F,, and by F,x, = xf,x,. The nex Lemma saes a simple fac, bu is ofen used in he sequel, especially in he proof of he heorem below. Lemma Given a sock price as inroduced in Secion 2.3. Then he coninuous par of is quadraic variaion is given by [S,S] c = σ 2 ss 2 sc 2 ds.

57 3.2. A Feynman-Kac ype formula 53 Proof. The claim is no very hard o proof and follows by a simple calculaion using he well-known rules for he quadraic variaion cf. Proer 24, Chaper II, Secion 6. [S,S] c = [ σ s S s cdws Q + dms Q + r s S s ds, = = σ s S s cdw Q s + dm Q s + r ss s ds ] c σ 2 ss 2 s c 2 d[w Q,W Q ] s + 2 σ s S 2 s cr s d[w Q, ] c s + σ s S 2 s r s d[m Q, ] c s + σ 2 ss 2 sc 2 ds. σ 2 ss 2 s cd[w Q,M Q ] c s σ 2 ss 2 s d[m Q,M Q ] c s r 2 ss 2 s d[, ] c s All erms, excep of he las one, vanish because W Q and M Q are orhogonal. Theorem F,x,u saisfies for x and < u T he following inegrodifferenial equaion: F,x,u + A F,x,u r F,x,u =, Fu,x,u = gu,x, 3.2 where A F,x,u = 1 2 c2 σ 2 x 2 F,x,u + r xf,x,u { + F,x + σ xy,u F,x,u σ xyf,x,u } ν Q dy. R 2. Le g C[,T] [, be a coninuous and bounded funcion on is range R. Then for u [,T] any bounded soluion F,,u C 1,2 [,T] [, of he parabolic Cauchy problem 3.2 admis he represenaion F,x,u = E Q [,x B Bu 1 gu,s u ], u T. Proof. Firs par: We fix u [,T]. The iniial value follows immediaely. Furher he process B 1 F,S,u is a G, Q-maringale, hence i is a special semimaringale and any decomposiion ino a local maringale and a

58 54 Chaper 3. Arbirage-free price process predicable process of finie variaion has o saisfy ha he predicable par is idenically zero. To derive such a represenaion we apply Iô s formula o B 1 F,S,u. To begin wih, he produc rule yields db 1 F,S,u = F,S,udB 1 + B 1 df,s,u, since B is coninuous and of finie variaion. Using now db 1 = r B 1 d and applying Iô s formula cf. e.g. Proer 24, Chaper II, Theorem 33 o he funcion F,,u, we ge B 1 F,S,u = F,S,u + Bs <s r s Bs 1 Fs,S s,uds Fs,S s,uds + Bs 1 F s,s s,uds s Bs 1 F s,s s,ud[s,s] c s { Fs,Ss,u Fs,S s,u F } s,s s,u S s. B 1 s 3.3 Recall ha N Q is he measure ha couns he jumps of L Q. Using S = σ S L Q we observe = <s R Bs 1 { Fs,Ss,u Fs,S s,u F } s,s s,u S s Bs 1 { Fs,Ss + σ s S s y,u Fs,S s,u F s,s s,uσ s S s y } N Q ds,dy. Wriing S as he produc of Ŝ and B he produc rule yields 3.4 ds = dŝb = Ŝ db + B dŝ = S r d + B dŝ. 3.5 We proceed now and plug 3.4, 3.5 and he resul of Lemma ino equaion 3.3, obaining + + db 1 R R F,S,u = r B 1 [ F,S,ud + B 1 F,S,ud + F,S,uS r d + F,S,uB dŝ F,S,uc 2 σ 2 S 2 d { F,S + σ S y,u F,S,u F,S,uσ S y } M Q d,dy { F,S + σ S y,u F,S,u F,S,uσ S y } ν Q dyd ].

59 3.2. A Feynman-Kac ype formula 55 Hence, B 1 F,S,u can be wrien as he sum of he local Q-maringale F,S,u + F s,s s,udŝs { + Fs,Ss + σ s S s y,u Fs,S s,u R B 1 s F s,s s,uσ s S s y } M Q ds,dy and he predicable, finie-variaion process Bs 1 Fs,S s,u + A s Fs,S s,u r s Fs,S s,u ds, which has o be idenically zero for any T. The coninuiy of F and he non-negaiviy of he sock price S immediaely imply F,x,u + A F,x,u r F,x,u =, u, x. To prove he second par of he heorem, we fix again u [,T] and assume ha F,,u is a bounded soluion of 3.2. Consider he bounded funcion Differeniaion yields and Hence, vτ,s τ := B B 1 τ Fτ,S τ,u τ u. vτ,s τ = B B 1 Fτ,S τ,u + B B 1 r τ Fτ,S τ,u τ A τ vτ,s τ = B Bτ 1 A τ Fτ,S τ,u. vτ,s τ + A τ vτ,s τ = B B 1 τ. Using his and Iô s formula, we ge Fτ,S τ,u + A τ Fτ,S τ,u r τ Fτ,S τ,u τ dvτ,s τ = vτ,s τ dτ + v τ,s τ ds τ v τ,s τ d[s,s] c τ + vτ,sτ + σ τ S τ y vτ,s τ R v τ,s τ σ τ S τ y N Q dτ,dy = vτ,s τ + A τ vτ,s τ dτ + v τ,s } {{ } τ σ τ S τ cdw Q τ + dmτ Q = + vτ,sτ + σ τ S τ y vτ,s τ R v τ,s τ σ τ S τ y M Q dτ,dy.

60 56 Chaper 3. Arbirage-free price process Hence, vτ,s τ is a local Q-maringale. Since i is furhermore bounded, i is a rue maringale. Thus we obain Therefore, = E Q,x [vτ,s τ v,s ] = E Q,x [vu,s u v,s ] = E Q,x [vu,s u] F,x,u. F,x,u = E Q [,x vu,su ] = E Q,x [B Bu 1 Fu,S u,u]. The claim follows, since Fu,S u,u = gu,s u. Theorems of his ype are called Feynman 1 -Kac 2 formulas and especially he firs par of he heorem is one of he main ools o derive he Galchouk- Kunia-Waanabe decomposiion of he arbirage-free price process. For his reason we worked i ou ailor-made o our siuaion. The second par of he heorem complemens our consideraion. The operaor A in 3.2 is he infiniesimal generaor of he ransiion operaor ha corresponds o he addiive process in he background cf. Con and Tankov 24, Chaper 14. Of course, here are more general versions of he heorem boh in erms of he operaor A and wih respec o he condiions imposed on he arbiragefree price process or on he payoff funcion. For example Kallenberg 22 Chaper 24, Theorem 24.1 presens he heorem more general wih respec o he process in he background in considering merely a generaor of a Feller-semigroup. Feynman-Kac formulas are addiionally useful in numerical compuaions of he price process F,S,u. This fac is paricularly recognized in Con and Tankov 24 Chaper 12, where he parial inegro-differenial equaion is analyzed in deail. I is invesigaed wha condiions he price process or he payoff funcion have o saisfy such ha a soluion exiss. For his reason he parial inegro-differenial equaion is formulaed separaely for differen kind of payoff funcions such as European or American ype opions. Especially hey deermine condiions on an European-syle payoff under which F C 1,2 [,T] [,. Karazas and Shreve 24 rea he case of a Brownian moion, where he measure ν and hence he inegral erm in 3.2 is no presen. A he end we have again a brief glance a A F,x,u = 1 2 c2 σ 2 x2 F,x,u + r xf,x,u { + F,x + σ xy,u F,x,u σ xyf,x,u } ν Q dy. R 1 Richard Phillips Feynman, American physicis and Nobel prize winner Mark Kac, Polish mahemaician

61 3.3. Galchouk-Kunia-Waanabe decomposiion 57 If he erms under he inegral are separaed, he inegral does in general no converge. Le us, however, consider he special case where he parameers of he financial marke are consan, ha is, r r, σ σ and b b. Then he measure ν Q dy = 1 + Gyνdy is independen of cf Furhermore assume ha { y 1} y νdy < and ha he process is of finie aciviy, ha is νr < cf. Lemma 2.2.5, Theorem and Then he parial inegro-differenial equaion of Theorem simplifies o F,x,u c2 σ 2 x 2 F,x,u + r σ r + ν Q R F,x,u + R R y ν Q dy xf,x,u F,x + σxy,uν Q dy =, which should considerably simplify he analysis for a numerical soluion. 3.3 Galchouk-Kunia-Waanabe decomposiion Le us now derive he Galchouk-Kunia-Waanabe decomposiion cf. Theorem of he discouned price process B 1 F,S,u. In he following chapers his urns ou o be vial for he Galchouk-Kunia-Waanabe decomposiion of he inrinsic value of uni-linked life insurance conracs. A reference for he heory of orhogonal local maringales and is relaion o he predicable quadraic covariaion is, for example, Secion I.4, Theorem I.4.4 and Theorem II.1.33 in Jacod and Shiryaev 23. In he following we always assume ha here exiss a non-random consan c 1 such ha F,x,u c1 <,x,u Q-a.s. 3.6 Definiion For < u T we define he discouned jump of he price process F,S,u induced by a jump of he underlying sock process S by J,x,u = B 1 {F,S + σ S x,u F,S,u}. Recall ha S = σ S L Q. Hence, for LQ = x he induced jump of he price process is J,x,u. The nex lemma saes a firs decomposiion of he discouned price process. Lemma For < u T he process B 1 F,S,u admis he decomposiion where B 1 F,S,u = F,S,u + K u = R F s,s s,udŝs + K u, Js,y,u F s,s s,uσ s Ŝ s y M Q ds,dy.

62 58 Chaper 3. Arbirage-free price process Proof. This represenaion was developed in he firs par of he proof of Theorem Opposed o a sandard Black-Scholes seing one easily verifies ha his decomposiion is no he Galchouk-Kunia-Waanabe decomposiion, since he process Ŝ is no orhogonal o he process K. Recall he differenial of he discouned sock price Ŝ, which was derived in Lemma 2.3.3: Hence, for T, dŝ = cσ Ŝ dw Q + σ Ŝ R ym Q d,dy. 3.7 Ŝ, Ku = cσ s Ŝ s dws Q, Ku + σ s Ŝ s ym Q ds,dy, Ku, where he firs erm on he righ hand side is equal o zero, since he inegral wih respec o he Brownian moion is coninuous and he inegral wih respec o M Q d,dy is purely disconinuous. I remains he second erm being equal o R R σ s Ŝ s y Js,y,u F s,s s,uσ s Ŝ s y νs Q dyds, which is generally no zero. A more careful consideraion of he decomposiion above reveals ha some erms cancel ou. This is explained in he nex lemma. Lemma For < u T he process B 1 F,S,u admis he decomposiion where B 1 F,S,u = F,S,u + K u = R F s,s s,ucσ s Ŝ s dw Q s + K u, Js,y,uM Q ds,dy. This decomposiion is orhogonal in he sense ha W Q is orhogonal o he process K. Proof. We plug 3.7 ino he decomposiion saed in Lemma This yields he claimed decomposiion. The orhogonaliy is immediae, since he Brownian moion is coninuous and he process K is purely disconinuous i.e. quadraic pure-jump.

63 3.3. Galchouk-Kunia-Waanabe decomposiion 59 Of course, his is again no he desired decomposiion. We do no only wan o inves in he coninuous par of he sock process, alhough i is orhogonal. Bu in any case his decomposiion shows a nice represenaion propery of he price process in erms of he Brownian moion W Q and he compensaed jump measure M Q d,dy, which are he componens of he process L Q in he background. The previous wo lemmas lead o he nex heorem, which saes he Galchouk-Kunia-Waanabe decomposiion of B 1 F,S,u. Theorem Le v Q = R x2 ν Q dx and κ = c 2 + v Q, [,T]. For < u T he Galchouk-Kunia-Waanabe decomposiion of F,S,u is given by B 1 B 1 F,S,u = F,S,u + ξ s udŝs + K u, where and ξ u = c2 κ F,S,u + K u = ζ 1 s 1 σ Ŝ κ udws Q + R R xj,x,uν Q dx, ζ s 2 y,um Q ds,dy is orhogonal o Ŝ. Furher he processes ζ1 u and ζ 2 y,u are given by ζ 1 u = cσ Ŝ F,S,u ξ u, ζ 2 y,u = J,y,u σ Ŝ ξ uy. Proof. The proof consiss of wo seps. Firs we fix u [,T] and derive ha B 1 F,S,u has he claimed decomposiion. Second we show ha for he claimed choice of ξ u his decomposiion is he desired Galchouk- Kunia-Waanabe decomposiion. We build on he decomposiion ha we derived in Lemma and inser some predicable process ξ u resuling in B 1 F,S,u = F,S,u + + ξ s udŝs F s,s s,u ξ s u dŝs + K u.

64 6 Chaper 3. Arbirage-free price process The differenial 3.7 of he discouned sock price Ŝ yields furher B 1 F,S,u = F,S,u ξ s udŝs cσ s Ŝ s F s,s s,u ξ s u dws Q σ s Ŝ s y F s,s s,u ξ s u M Q ds,dy R + K u. Since K u = R Js,y,u F s,s s,uσ s Ŝ s y M Q ds,dy we deduce B 1 F,S,u = F,S,u R ξ s udŝs cσ s Ŝ s F s,s s,u ξ s u dws Q Js,y,u σ s Ŝ s ξ s uy M Q ds,dy. Now we have o choose ξ u such ha his decomposiion is orhogonal. Consider for < u T he predicable quadraic covariaion Ŝ,Ku = Ŝ, ζ s 1 udws Q + ζ s 2 y,um Q ds,dy. 3.8 We know ha inegrals wih respec o W Q and M Q d,dy are orhogonal. Hence wih 3.7 expression 3.8 is equal o cσ s Ŝ s ζ s 1 ud W Q,W Q s + σ s Ŝ s yζ s 2 y,uνs Q dyds. Insering he expressions for ζ s 1 u and ζ s 2 y,u yields = + + R c 2 σ sŝ2 2 s F s,s s,u ξ s u ds σ s Ŝ s y Js,y,u yσ s Ŝ s ξ s u νs Q dyds c 2 σ sŝ2 2 s F s,s s,uds κ s σ sŝ2 2 s ξ s uds σ s Ŝ s yjs,y,uνs Q dyds. R Finally, for ξ s u = c2 κ s F 1 s,s s,u + R xjs,x,uνq s dx he expression is equal o zero for any < u T, which yields he desired R R σ s Ŝ s κ s

65 3.3. Galchouk-Kunia-Waanabe decomposiion 61 orhogonal decomposiion. I remains o show ha he inegrals wih respec o Ŝ, W Q and M Q d,dy are really square-inegrable. Since i is coninuous in and 2.12 holds we have ha sup x 2 ν Q dx x 2 νdx + sup G x 3 νdx <. [,T] R R [,T] R The mean-value heorem yields J,x,u = F,x,uσ Ŝ x for some inermediae value x = x S,σ,x, and hence ξ u = c2 F,S,u + 1 F,x,ux 2 ν Q dx. 3.9 κ κ We conclude by 3.6 ha c1 sup ξ u sup c 2 + [,T] [,T] κ R R x 2 ν Q dx < u Q-a.s. 3.1 This non-random boundedness of ξ implies ha i is an elemen of L 2 QŜ cf. he beginning of Subsecion for is definiion and so ξ sudŝs is a square-inegrable maringale. Likewise we infer herefore ha here exis non-random consans c 2,c 3 > such ha sup 1 ζ u c2 sup Ŝ u Q-a.s [,T] [,T] sup ζ 2 y,u c 3 y sup Ŝ y,u Q-a.s [,T] [,T] The Q-square inegrabiliy of Ŝ cf. Lemma ogeher wih 3.11 implies now ha ζ1 s udws Q is a square-inegrable maringale cf. Proer 24, Chaper IV, Lemma before Theorem 28. Furher wih 3.12 we have ha E Q T 2] [ ζ s 2 y,um ds,dy Q R = E Q [ ζ s 2 y,u 2 ]ν Q s dyds R Tc 2 3 EQ [ sup Ŝ 2 ] sup y 2 ν Q s dy <, [,T] s [,T] R R ζ2 so s y,um Q ds,dy is also a square-inegrable maringale. This finally yields ha Ku is a square-inegrable maringale. A he end of he chaper we give, o he bes of our knowledge, a shor overview of exising lieraure dealing wih he Galchouk-Kunia- Waanabe decomposiion in he Lévy-process conex. Jacod, Méléard and

66 62 Chaper 3. Arbirage-free price process Proer 2 invesigae he so-called Clark-Ocone-Haussmann formula in a general Markov-process seing. Imposing cerain condiions his formula reduces o he Galchouk-Kunia-Waanabe decomposiion. Since our seing is simpler, we are able o sae a raher elemenary and independen proof. In addiion, our heorem is much more explici. For cerain funcionals of Lévy processes he Clark-Ocone-Haussmann formula is also addressed in he papers of Lokka 24 and Benh, Di Nunno, Løkka, Øksendal and Proske 23 in erms of Malliavin Calculus. In paricular, Malliavin derivaives are used o prove and o express he decomposiion, which is by far less explici as our expression. Con and Tankov 24 derive in Secion 1.4 a o our formulaion similar explici represenaion of he inegrand wih respec o he discouned sock. However, hey neiher explicily sae he Galchouk-Kunia-Waanabe decomposiion nor he process K u. Furhermore, hey only rea European syle opions and formulae he heorem only for he case where he sock price has consan coefficiens and is driven by a Lévy process. We consider a much more general sock process, which migh be under Q merely driven by an addiive process and which has nonconsan coefficiens. Moreover, we prove he heorem differenly from Con and Tankov 24. To be more precise, hey derive he inegrand wih respec o he discouned sock by minimizing he variance of a cerain hedging error applying sandard calculus echniques whereas we direcly make use of he predicable quadraic covariaion and he srucure of he underlying Hilberspace.

67 Chaper 4 Hedging of uni-linked pure endowmen and erm insurance 4.1 Inroducion The following is based on Riesner 26. We begin his chaper discussing uni-linked insurance in general and also how o model uni-linked pure endowmen or erm insurance conracs. Thereafer we derive hedging sraegies for a porfolio of eiher one of hose conracs in an incomplee Lévyprocess financial marke. Presupposing he financial marke inroduced in Secion 2.3 and assuming he insurance porfolio discussed in Subsecion we work on he risk-neural combined financial and insurance model Ω, F,F T, Q of Secion 2.5. In paricular, we make use of he heory developed in Chaper 3. We sudy he model of Møller 1998 in deail and demonsrae how his resuls are affeced when he complee Black-Scholes marke is replaced by an incomplee financial marke in case of a more general geomeric Lévydriven model as inroduced by Chan In Møller 1998 he uncerainy of boh he insured lives and he sock price is modeled simulaneously as a produc probabiliy space and, unlike he acuarial principle of equivalence, moraliy is no averaged away. Consequenly he uncerain developmen of insured lives is no replaced by he expeced developmen, and he insurance claims depend on boh sources of uncerainy. However, in a complee financial marke here is no pure financial hedging risk and so wih our replacemen hrough an incomplee financial marke, an insurance company issuing uni-linked life insurance conracs now addiionally faces pure financial risk. The inrinsic risk of he company arises hen no longer only from he insurance porfolio bu also from he financial marke. We derive his risk and he hedging sraegies wih he local risk-minimizaion

68 64 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance heory of Föllmer and Sondermann 1986 and Schweizer 1991 cf. Subsecions and This requires coningen claims payable a he end of he considered ime horizon only. Thus for he momen we deal wih he simple siuaion, in which insurance conracs do no generae inermediae paymen imes. For conracs having insurance benefis before ime T, for example if he insured person dies, we assume ha he benefis are deferred earning ineres wih he risk-free ineres rae and are cashed ou a he end of he considered ime horizon. We also suppose he premiums o be paid as a single premium a he beginning. An exension, allowing inermediae paymen imes, is reaed in Møller 21a, however, only for a financial marke in which he sock is a maringale under he hisorical measure. A generalizaion admiing he sock also o be a semimaringale under he hisorical measure is developed in Chapers 5 and 6 of his hesis. For he pure endowmen we discuss in deail our resuls and show ha he financial risk is, as expeced, no diversifiable by raising he number of insured persons wihin he porfolio. In accordance wih sandard acuarial heory cf. Subsecion his is however he case wih risk originaed by moraliy. Addiionally we derive he conrac s inrinsic value process which is a naural candidae for he reserve ha an insurance company should hold back and which provides simulaneously an idea of a fair premium. For he resuls of he uni-linked erm insurance he same reasoning applies. In considering he pure endowmen and he erm insurance separaely we also rea he case of he popular endowmen insurance being simply he sum of hem. 4.2 Uni-linked life insurance conracs Le us sar wih a general inroducion in he heory of uni-linked life insurance conracs. As briefly menioned earlier, uni-linked or equiylinked life insurance conracs link he insurance benefi and possibly he premiums o he marke value of some specified reference porfolio mos likely consising of socks, bonds or shares of a muual fund represened in our model by he sock S. In Aase and Persson 1994 uni-linkage is discussed in deail. Therefore we only summarize common ypes of unilinked payoffs no aking ino accoun for he momen he due dae of he insurance benefi. We denoe he payoff by gs for some suiable funcion g : [, R. Wih a pure uni-linked insurance conrac he amoun o be paid is given by he value of S a expiraion, ha is, gs = s. In his case all financial risk ransfers o he policy-holder. Opposed o his, an unilinked wih guaranee conrac is equipped wih a guaranee ha assures he policy-holder a minimum amoun if he reference porfolio falls below a cerain level a due dae. For a prefixed guaranee k > he payoff

69 4.2. Uni-linked life insurance conracs 65 has hen he form gs = maxs,k. For some insurance producs i is also imaginable having a guaranee ha is no fixed. For example i could be he case ha he guaranee depends on ime and is k = k exp r u du for some ineres rae r and some prefixed amoun k >. Tha is, he amoun of guaranee increases by earning ineres over he insurance period. Of course here are many oher possible choices for he payoff, whereas from a pracical poin of view one is resriced by being able o evaluae he associaed price process F,S,u cf. Chaper 3. For insance, he funcion gs = maxs,k is he sum of an European call opion s k + and a consan k for which price process evaluaion echniques exis. In his chaper we consider he insurance porfolio model ha we inroduced in Subsecion and assume ha o each of he l x individuals an uni-linked insurance conrac is issued a ime agains a single premium a issuing dae. Those uni-linked insurance conracs are supposed o be eiher of pure endowmen or of erm insurance ype and hence specify benefi paymens being linked o he developmen of he financial marke and coningen on he remaining lifeime of he policyholder Uni-linked pure endowmen In case he issued uni-linked conracs are uni-linked pure endowmen conracs we suppose he insurance benefi o be a coningen claim gs T for some suiable funcion g such ha gs T L 2 Q. Recall ha wih a pure endowmen he sum insured is o be paid a he end of he erm T if he insured person is sill alive. Therefore he presen value of each insured person s insurance benefi is given by H i = 1T i > TB 1 T gs T, which is F T -measurable and a he same ime he individual obligaion of he insurance company a ime. Therefore he enire porfolio generaes he discouned claim H = B 1 T gs T l x i=1 1T i > T = B 1 T gs Tl x N I T, 4.1 where l x NT I represens he number of insured persons surviving he insurance period Uni-linked erm insurance Now le us assume ha he issued conracs are uni-linked erm insurance conracs. A erm insurance is in some sense he opposie of a pure endowmen. Is benefi is due immediaely upon deah before ime T. Hence, paymens can occur a any ime during [,T] and he conrac funcion g,s is allowed o be ime-dependen. Such claims do no fi ino our framework

70 66 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance of hedging coningen claims cf. Subsecion 2.5.2, since hey are generally due before ime T. As in Møller 1998 we herefore simply assume ha all paymens are deferred o he erm of he conrac and are accumulaed wih he risk-free rae of ineres r. Hence if T i < T he insurance company has o pay B T B 1 T i gt i,s Ti a ime T, and his is a coningen claim fiing in our framework, provided sup u [,T] E Q [ Bu 1 gu,s u 2] <. The presen value of he insurer s liabiliies generaed by l x erm insurances wih deferred and accumulaed paymens is hen given by H = B 1 T = l x i=1 l x i=1 which is of course equal o H = B T B 1 T i gt i,s Ti 1T i T B 1 u gu,s ud1t i u B 1 u gu,s udn I u. P-a.s., 4.3 Risk-minimizing hedging sraegy The hedging sraegies ha we derive in his secion are risk-minimizing wih respec o he risk-neural measure Q. Whereas for a hedger, i is more imporan o know he risk in erms of his subjecive beliefs represened by he hisorical, subjecive measure P. Since we work under he Föllmer-Schweizer measure he risk-minimizing sraegies under Q are locally risk-minimizing wih respec o P. Compare he relaed discussion in Subsecion Uni-linked pure endowmen Consider he claim developed in 4.1: H = B 1 T gs Tl x N I T. The inrinsic-value process V T of H is given by V = E Q [H F ] = E Q [l x N I T H ]B 1 E Q [B B 1 T gs T G ], applying he independence of he financial marke and he insurance porfolio. The firs facor of V is jus he condiional expeced number of insured persons surviving unil ime T given he number of insured persons a ime. This is a binomial experimen wih survival probabiliy as success probabiliy. Hence E Q [l x N I T F ] = l x N I T p x+.

71 4.3. Risk-minimizing hedging sraegy 67 The second facor is exacly he arbirage-free price process for u = T cf. Definiion Hence V = l x N I T p x+ B 1 F,S. 4.2 Since he conrac funcion gs T is ime-independen, in wha follows we omi he hird argumen u of F,S,u since i is consan and equal o T. V may be inerpreed as he marke value associaed wih he enire porfolio of pure endowmens and furher, in accordance wih he acuarial principle of equivalence, he iniial value V = l xtp x F,S is a naural candidae for he single premium of he enire porfolio cf. Møller However, boh values are considered under he risk-neural pricing rule Q. Hence, he process V could be seen as he prospecive reserve of he insurance porfolio under he measure Q. To find he risk-minimizing hedging sraegy we need o develop he Galchouk-Kunia-Waanabe decomposiion of V cf. Subsecion Applying Iô s inegraion by pars cf. Proer 24, Chaper II, Theorem 22, Corollary 2 o 4.2 yields V = + B 1 s Fs,S s d l x N I s T s p x+s l x N I s T s p x+s d B 1 s Fs,S s + [ B 1 F,S,l x N I T p x+ ]. For he quadraic covariaion one has: [ B 1 F,S,l x N I T p x+ ] = V + [ B 1 F,S,l x N I c T p x+ ] + Bs 1 Fs,S s l x Ns I T s p x+s <s Obviously, l x N I T p x+ is a process of finie variaion, hence i is a quadraic pure-jump process and he coninuous par of he quadraic covariaion is idenically. Furher for fixed s T Q B 1 s Fs,S s l x Ns I T s p x+s > =, 4.3 since a leas Bs 1 Fs,S s is coninuous in probabiliy and he processes i.e. S and N I are independen by assumpion. So he sum over he jump erms vanishes Q-a.s., because we face a mos counably many jump disconinuiies in considering only càdlàg processes. The produc rule for Lebesgue-Sieljes inegrals yields d l x N I T p x+ = lx N I d T p x+ + T p x+ dl x N I = l x N I T p x+ µ x+ d T p x+ dn I = T p x+ dm I.

72 68 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance Recall he expression for p x in Puing hings ogeher, we ge V = V + + l x N I s T s p x+s d B 1 s Fs,S s B 1 s Fs,S s T s p x+s dm I s, Q -a.s. 4.4 Alhough our sock process is no coninuous, hence Iô s formula is quie differen from he case of a coninuous sock process, he inegral wih respec o M I is exacly he same as obained by Møller 1998 and we can use Theorem o ge: Corollary For he pure endowmen in 4.1 he process V has he following Galchouk-Kunia-Waanabe decomposiion: V = V + where K H = l x N I s T s p x+s ξ s dŝs + K H, T, + + l x Ns I T sp x+s ζ s 1 dws Q R l x Ns I T sp x+s ζ 2 ym Q ds,dy B 1 s Fs,S s T s p x+s dm I s is orhogonal o Ŝ and ξ, ζ 1 and ζ 2 are aken from Theorem Proof. The decomposiion of V follows from 4.4 and Theorem Noe ha i is indeed he desired Galchouk-Kunia-Waanabe decomposiion. The inegrals wih respec o Ŝ, W Q and M Q d,dy are sill squareinegrable, since l x N I T p x+ is Q-a.s. bounded by he non-random consan l x. Furher he orhogonaliy beween Ŝ and he inegrals wih respec o W Q and M Q is kep. For boh claims compare he proof of Theorem 3.3.4, especially he relaions 3.1, 3.11 and The inegral B 1 s Fs,S s T s p x+s dms I is orhogonal o Ŝ by he independence of Ŝ and M I, i.e. he process cw Q + M Q MI is clearly a Q-maringale. I is a square-inegrable maringale, since by Doob s inequaliy cf.proer 24, Chaper I, Theorem 2 i holds ha E Q [ sup [,T] 4 sup E Q [ [,T] B 1 s Fs,S s T s p x+s dm I s 2] B 1 s Fs,S s T s p x+s 2 d M I,M I s ], s

73 4.3. Risk-minimizing hedging sraegy 69 and furher we know ha d M I,M I s = λ s ds. Since λ s, Bs 1 and T s p x+s can be bounded by a non-random consan, i remains o consider he expecaion E Q [ F 2 s,s s ds] for which we have ha sup E Q [ [,T] F 2 s,s s ds] cons. E Q [g 2 S T ] <. This follows by Jensen s inequaliy and he square-inegrabiliy was presupposed cf Using Theorem i is now easy o derive a risk-minimizing sraegy for he claim of an uni linked pure endowmen insurance. Corollary The unique admissible risk-minimizing sraegy ϕ under Q for he pure endowmen in 4.1 is for T given by ξ = l x N I T p x+ ξ, η = l x N I T p x+ B 1 F,S ξ Ŝ. For he associaed inrinsic risk process Rϕ one has, for s T and ρ s = E Q [ζ s 1 2 F ] + E Q [ ζ s 2 y 2 F ]νs Q dy, R ϕ = l x N I T p x+ + l x N I T p x+ R T sp x+s 1 s p x+ + l x N I T p x+ ρ sds T sp x+s µ x+s Bs 2 E Q [F 2 s,s s F ]ds. Proof. Only he second par needs jusificaion. According o Theorem he risk process is given by R ϕ = E Q [ KT H K H 2 ] F. Using he orhogonaliy of he inegrals wih respec o W Q, M Q ds,dy and M I we ge R ϕ = E Q [ + E Q [ + E Q [ T sp x+s l x Ns I ζ s 1 Independence and Tonelli s heorem yields R ϕ = R 2 d W Q,W Q s F ] T sp x+s l x N I s ζ2 s y 2 ν Q s dyds F ] T sp x+s Bs 1 Fs,S s 2 d M I,M I s F ]. T sp 2 x+s E Q [ l x N I s 2 F ] ρ sds + E Q [ T sp 2 x+s B 2 s F 2 s,s s dλ s F ].

74 7 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance Furher, dλ s = l x N I s µ x+s ds and given he pas F we already observed ha l x N I s follows a binomial disribuion for s [,T]. So and E Q [ l x N I s F ] = lx N I s p x+. E Q [ l x N I s 2 F ] = lx N I s p x+ 1 s p x+ + l x N I 2 s p 2 x+. A simple compuaion shows T s p x+ss p x+ = T p x+. Hence R ϕ = l x N I T p x+ + l x N I T p x+ T sp x+s 1 s p x+ + l x N I T p x+ ρ sds T sp x+s µ x+s Bs 2 E Q [F 2 s,s s F ]ds. A his poin we discuss our resuls in comparison o he resuls of Møller 1998, which are obviously a special case of ours. Møller 1998 obained l x N I T p x+ F,S as he risk-minimal invesmen in he sock. We ge a similar resul: The opimal invesmen in he sock ξ depends also on he expeced number of insured persons surviving from o T. However, if we allow addiionally he sock process o have jumps, we inves no only he Black-Scholes -hedge F,S bu ξ = c2 κ F,S + = c2 κ F,S + 1 κ 1 xj,xν Q σ Ŝ κ dx R F,x x 2 ν Q dx, for some inermediae value x resuling from he mean-value heorem. The difference from a classical Black-Scholes seing is apparen. If our Lévy process was merely a Brownian moion he measure ν Q dx would be equal o and κ c 2 : we would have ξ = F,S as expeced. In he presence of jumps he locally risk-minimizing invesmen in he sock is nearly a weighed sum of F,S and of he jump J,x. Noe ha c 2 is he variance of he Brownian par cw Q and ha κ s ds = c 2 + R x2 νs Q dxds is he variance of L under Q. Furher he insurance company is able o reduce is oal porfolio risk o he inrinsic risk, deermined by he process K H. Møller 1998 inerpres K H as he insurer s loss, in our case: dk H = l x N I T p x+ ζ 1 dw Q + l x N I T p x+ ζ 2 ym Q d,dy R + B 1 F,S T p x+ dm I. R

75 4.3. Risk-minimizing hedging sraegy 71 The inegral driven by dm I = dn I λ d can be inerpreed as he driver of pure insurance risk and is exacly he cos process obained by Møller A deah will produce an immediae gain for he insurer due o he downward adjusmen of he expeced number of survivors, whereas no deah will cause a small loss. We inerpre he remaining erms of K H as he driver for financial risk, i.e. he inegrals wih respec o W Q and M Q. Those do no appear in he complee financial marke of Møller For he incomplee financial marke his shows he influence on he insurer s risk of an invesmen in he possible disconinuous sock. I is of ineres ha boh he disconinuous i.e. M Q and he coninuous i.e. W Q par of he sock price influence K H. Addiionally, ζ 1 and ζ 2 depend on F as well as on J, x. Hence, he more general Lévy-process seing yields an increase of risk also influenced by he coninuous par of he sock process, as one migh perhaps no expec. However, if we considered an incomplee financial marke wih a coninuous price process, here would be also an addiional erm for financial risk. Noe ha he financial risk driver is adjused according o he expeced number of insured persons surviving from o T in he sense ha a deah resuls in a downward adjusmen of he loss. Le us consider he iniial inrinsic risk R more deailed. R ϕ = l xt p x + l xt p x T sp x+s 1 s p x + l xt p x ρ sds T sp x+s µ x+s Bs 2 E Q [F 2 s,s s ]ds. Møller 1998 suggess aking he iniial inrinsic risk R as a measure associaed wih he non-hedgeable par of he claims and he examines he raio R /l x. This is in analogy o he classical acuarial heory, where he relaive porfolio risk decreases as he size l x of he porfolio increases cf. Subsecion By classical heory we mean he case of deerminisic premiums and deerminisic insurance benefis. The difference of he classical heory from our seing is ha he uni-linked conracs are all linked o he same asse and hence are no independen. Considering consequenially he same raio in our case, we see for he insurance risk par of R ha one obains, in accordance wih Møller 1998, 1 T l xt p x T sp x+s µ x+s Bs 2 E Q [F 2 s,s s ]ds as l x. l 2 x The financial risk par of R, l xt p x T s p x+s 1 s p x + l xt p x ρ sds, when divided by l 2 x apparenly does no end o if l x is increased. This only holds rue for he firs par of i. However, he second par of he raio

76 72 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance is independen of l x. This means here is some risk originaing from rading in he marke ha can no be eliminaed by an increase of he number of insured individuals. In Møller 1998 his effec is no presen. Of course, he same resuls are obained a any ime [,T] if he raio R /l x N I is considered insead. Finally le us consider examples for he same choice of a conrac funcion as in Møller 1998, o see how his resuls are affeced. Example We ake gs = s. Thus we consider a pure-uni linked insurance promising he insured person, if hen sill alive, he value of he sock a mauriy dae. No surprisingly, his yields F,S = E Q [B B 1 T S T F ] = S. Hence, he inrinsic value process of he conrac is V = l x N I T p x+ Ŝ and in paricular a fair premium could be V = l xtp x S. A simple calculaion of he variables of Theorem yields ξ 1, ζ 1 ζ 2. Observe ha ζ 1 and ζ 2 influence he financial risk par of he company s inrinsic risk. In case of a pure-uni linked conrac he insurance company does no bear he invesmen risk, hence i is clear ha hey have o be. I is also quie obvious ha he bes hedge for his kind of conrac is o inves he whole capial in he sock. This is refleced by ξ. Corollary gives he risk-minimizing hedging sraegy ξ,η = l x N I T p x+, N I T p x+ Ŝ and he inrinsic risk R ϕ = l x N I T p x+ T sp x+s µ x+s E Q [Ŝ2 s F ]ds. A any ime he insurance company holds exacly he expeced number of socks o be cashed ou a ime T and each deah adjuss he reserve V by he amoun N I T p x+ Ŝ. Example We ake gs = k, k >. This choice of g represens he case of deerminisic benefis. Then F,S = B B 1 T k and V = l x N I T p x+ B 1 T k. Furher Hence, ξ ζ 1 ζ 2. ξ,η =,l x N I T p x+ kb 1 T. The inrinsic risk can be compued explicily, as R ϕ = l x N I T p x+ 1 T p x+ k 2 B 2 T.

77 4.3. Risk-minimizing hedging sraegy 73 The previous wo examples do no involve financial risk for he insurer. Therefore he obained hedging sraegies and he inrinsic risks are exacly he same as in Møller Example The perhaps mos ineresing case is he choice gs = maxs,k for some k > represening an uni-linked wih guaranee conrac. In case of a sandard Black-Scholes financial marke cf. Møller 1998 we have an explici expression for he price process F,S, since gs = k + s k +. However his is no he case in a more general Lévyprocess seing. For a concree choice of a Lévy process one is forced o apply, for example, Mone-Carlo simulaion echniques or numerical mehods o solve he parial inegro-differenial equaion in Theorem In Chaper 7 we rea he compuaion of F,S for an asymmeric double exponenial jump-diffusion process Uni-linked erm insurance To complee his chaper we include he erm insurance, alhough we have o make he addiional assumpion ha in case a deah occurs he insurance benefi is deferred, while earning ineres, and cashed ou a he end of he considered ime horizon only. Of course, from a pracical poin of view his is no saisfacory and as menioned in he inroducion we will come back o his in Chaper 6 allowing insurance conracs o have inermediae paymens. In ha chaper he erm insurance will only be briefly menioned as a special case and we will refer o he curren secion insead. Moreover, he decomposiion echnique ha is applied here is a special case of he general proof of Theorem in Chaper 6. Recall ha he presen value of he insurer s liabiliies for a porfolio of uni-linked erm insurances is given by H = B 1 u gu,s u dn I u, 4.5 where sup E[Bu 1 gu,s u 2 ] <. u [,T] Again we need o compue he Galchouk-Kunia-Waanabe decomposiion of he inrinsic value process V = E Q [H F ]. For he firs seps we follow he ideas of Møller 1998 and use he measurabiliy o see ha V = Bu 1 gu,s udnu I + EQ [ Bu 1 gu,s udnu I F ].

78 74 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance Wih he compensaor λ of N I we ge V = = + Bu 1 gu,s u dnu I + E Q [ Bu 1 gu,s u l x Nuµ I x+u du F ] B 1 u gu,s u dn I u B 1 F,S,ul x N I u p x+ µ x+u du, where F,S,u is defined as in Definiion We compare now B 1 F,S,ul x N I u p x+ 4.6 wih equaion 4.2, which is V of he pure endowmen insurance, and noe hey are almos equal. We only have o subsiue u for T. Hence, we plug decomposiion 4.4 wih u insead of T ino 4.6 yielding V = + B 1 u gu,s udn I u + Using Theorem we ge where V = + + F,S,ul xu p x l x N I s u sp x+s d B 1 s Fs,S s,u Bs 1 Fs,S s,u u s p x+s dms I µ x+u du. B 1 u gu,s udn I u + α u s dmi s + βs u dŝs + } δs u ym Q ds,dy du, R { F,S,ul xu p x µ x+u γ u s dw Q s α u := B 1 F,S,u u p x+ µ x+u β u := l x N I u p x+ µ x+u ξ u γ u := l x N I u p x+ µ x+u ζ 1 u δ u y := l x N I u p x+ µ x+u ζ 2 y,u.

79 4.3. Risk-minimizing hedging sraegy 75 Furher we change limis of inegraion o ge V = B 1 u gu,s udn I u + { F,S,ul xu p x µ x+u 1 {s u} α u s dmi s + 1 {s u} βs u dŝs + } 1 {s u} δs u ym Q ds,dy du R u { F,S,ul xu p x µ x+u + β u s dŝs + u γ u s dw Q s + u u α u s dmi s R 1 {s u} γ u s dw Q s } δs u ym Q ds,dy du. The las inegral wih respec o du includes in curely brackes again he expression Bu 1 Fu,S u,ul x Nu I p x+u wrien in erms of he decomposiion 4.4 for he special case = u. Noe ha p x+u = 1. Hence, V = B 1 u gu,s u dn I u B 1 F,S,ul xu p x µ x+u du + 1 {s u} β u s dŝs du + R u Fu,S u,ul x N I uµ x+u du 1 {s u} δ u s ym Q ds,dydu. 1 {s u} α u s dmi s du 1 {s u} γ u s dw Q s du We simplify his furher noing ha using Fu,S u,u = gu,s u. Thus, F,S,ul xu p x µ x+u du = V and V = V B 1 s gs,s s dm I s + 1 {s u} β u s dŝs du + R 1 {s u} δ u s ym Q ds,dydu. 1 {s u} α u s dmi s du 1 {s u} γ u s dw Q s du The funcion ω,,u α u ω is now O B[,T]-measurable and by 3.1 we have 1 {s u} α u s dmi s du < Q-a.s.

80 76 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance Since M I is of finie variaion we apply he sandard Fubini heorem and obain ha = B 1 s gs,s s dm I s + B 1 s gs,s s + s α u s du 1 {s u} α u s dmi s du dm I s Q-a.s. The funcion ω,,u β u ω is P B[,T]-measurable and uniformly bounded by a non-random consan cf. 3.1 in he proof of Theorem Fubini s heorem for sochasic inegrals cf. Proer 24, Chaper IV, Theorem 64 yields hen: 1 {s u} β u s dŝs du = s βs u du dŝs. Also ω,,u γ u ω is P B[,T]-measurable and furher here is a non-random consan c 2 such ha sup γ u c 2 sup Ŝ, [,T] [,T] u Q-a.s., since 3.11 holds and l x N I u p x+ µ x+u is non-randomly bounded. Hence for all s T we have ha s 1 γs u 2 2 du c 2 sup Ŝ. [,T] The square-inegrabiliy of Ŝ implies now ha he inegral of s γu s du wih respec o W Q exiss cf. Proer 24, Chaper IV, Lemma before Theorem 28. Therefore we are allowed o apply he second version of he Fubini Theorem for sochasic inegrals cf. Proer 24, Chaper IV, Theorem 65 yielding 1 {s u} γs u dws Q T du = γs u du dws Q. Analogously, he funcion ω,,u,y δ u yω is O B[,T] BR- measurable, and by 3.12 we have ha sup δ u y c 3 y sup Ŝ, y,u Q-a.s., [,T] [,T] for a non-random consan c 3. This implies for all s T ha 1 δs u y2 2 du c 3 y sup Ŝ. s [,T] s

81 4.3. Risk-minimizing hedging sraegy 77 Hence, E Q [ R 2 δs M ds,dy u ydu Q ] < s by he square-inegrabiliy of Ŝ cf. proof of Theorem So we may apply again Fubini s heorem for sochasic inegrals: 1 {s u} δs u ym Q ds,dydu = R R s δs u ydu M Q ds,dy Q-a.s. V admis herefore Q-a.s. for T he following decomposiion. V = V R s βs u du dŝs + δs M u ydu Q ds,dy s B 1 s gs,s s + s s α u s du dms I. γs u du dws Q Corollary For he erm insurance in 4.5 he process V has he following Galchouk-Kunia-Waanabe decomposiion: where ξ = K H = V = V + l x Ns I ξ s dŝs + K H, T, + u p x+ µ x+u ξ udu, and l x N I s ζ1 s ϑ s dm I s is orhogonal o Ŝ wih ζ 1 = ζ 2 y = dws Q + R u p x+ µ x+u ζ 1 u du, u p x+ µ x+u ζ 2 y,udu, ϑ = B 1 g,s l x Ns I ζ2 ym Q ds,dy B 1 F,S,u u p x+ µ x+u du ξu, ζ 1 u and ζ 2 y,u are aken from Theorem s

82 78 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance Proof. To prove ha he derived decomposiion is indeed he Galchouk- Kunia-Waanabe decomposiion one follows he same lines as in he proof of Corollary The boundedness and orhogonaliy resuls carry over o he erm insurance wihou difficulies. Corollary The unique admissible risk-minimizing sraegy ϕ under Q for he erm insurance in 4.5 is for T given by ξ = l x N I u p x+ µ x+u ξ udu, η = ξ Ŝ. B 1 u gu,s u dn I u + l x N I B 1 F,S,u u p x+ µ x+u du For he associaed inrinsic risk process Rϕ one has, for s T and ρ s = E Q [ζ s 1 2 F ] + E Q [ ζ s 2 y 2 F ]νs Q dy, R ha R ϕ = l x N I s p x+ 1 s p x+ + l x N I s p x+ ρ sds + l x N I E Q [ϑ 2 s F ] s p x+ µ x+s ds. Proof. As for he pure endowmen his is jus an applicaion of Theorem The risk process is derived analogously. We only briefly commen on hese resuls, since he same reasoning as for he pure endowmen applies. The sraegy ϕ in his case is chosen such ha he porfolio value ˆV ϕ = V = Bu 1 gu,s u dnu I + E Q [ Bu 1 gu,s u dnu I F ], ha is, a any ime i is equal o he sum of paymens for deahs already occurred and expeced fuure paymens of possible fuure deahs. Compared o he pure endowmen, ξ is addiionally inegraed over all possible imes a deah could occur from ime on. Oher han ha i posses he same properies as he risk-minimal sock invesmen of he pure endowmen. The insurer reduces again his oal risk o he inrinsic risk driven by he insurer s loss, dk H = l x N I ζ1 dw Q + R l x N I ζ2 ym Q d,dy + ϑ dm I,

83 4.4. Why use local risk-minimizaion? 79 where he inegrals wih respec o W Q and M Q represen again he financial risk driver. Addiionally, as for ξ and in conras o he pure endowmen insurance, ζ 1 u and ζ 2 y,u are inegraed over all possible imes a deah could occur. Moreover, each ime a deah occurs he invesmen in he sock and hence he financial risk is reduced according o he number of insured persons. The inegral wih respec o M I is he pure insurance risk driver and is equal o he cos process obained by Møller Considering he process ϑ we consiue ha a deah resuls in he insurer seing aside he paymen B 1 g,s o be cashed ou a T and in reducing simulaneously his reservers by he amoun B 1 F,S,u u p x+ µ x+u du. Of course, he examinaion of he raio R /l x N I, [,T], yields he same resul as for he pure endowmen insurance: he insurance risk can be reduced by raising l x, whereas he financial risk is independen of l x. 4.4 Why use local risk-minimizaion? One migh ask why we use locally risk-minimizing hedging in he conex of insurance claims insead of some oher hedging heory. Insurance claims may arise arbirarily in he inerval [,T] herefore i is quie reasonable o apply local risk-minimizaion for insurance claim hedging, since i guaranees he opimal porfolio value V ϕ being exacly equal o he inrinsic value V a any ime T. This is he bes approximae we have on he insurance claim a ime corresponding simulaneously o he prospecive reserve of he insurance claim under Q. For oher hedging approaches his equaliy is usually no rue in he open inerval,t and is only required a ime T, when some coningen claim is due. Therefore he solvency of he insurer is enhanced wih local risk-minimizaion compared o oher hedging approaches. For example consider a erm insurance, where a deah and hence a claim can occur a any random ime during he inerval [,T] and no only a T. Moreover, in case of adverse developmen wihin he insurance porfolio, a difference of he opimal porfolio value from he inrinsic value could be relevan o he solvency as well see also Møller Le us briefly describe mean-variance hedging as one possible alernaive belonging, as well as local risk-minimizaion, o he heory of quadraic hedging approaches. Here an iniial invesmen and a hedging sraegy is chosen such ha i is self-financing on he inerval [,T and such ha he variance beween he porfolio value a ime T and a coningen claim iniial inrinsic risk is minimal under he hisorical measure, which is no an easy ask. In paricular, finding he so-called variance opimal maringale measure needs a lo of echnical machinery and seems o be quie advanced for models wih jumps cf. Con and Tankov 24, Chaper 1.4. For he special case having he original measure already as maringale mea-

84 8 Chaper 4. Hedging of uni-linked pure endowmen and erm insurance sure for Ŝ, he ξ-componen is idenical for he mean-variance and he local risk-minimizing hedging sraegy. However, he η-componens differ by he self-financing propery of mean-variance hedging sraegies. The major disadvanage for our purposes is ha during he insurance period for a meanvariance hedging sraegy he porfolio value V will in general no equal he condiional expeced presen value of he claim V. I jus minimizes he iniial inrinsic risk.

85 Chaper 5 Paymen sream hedging for semimaringales 5.1 Inroducion In he previous chaper we did no have a locally risk-minimizing hedging heory for insurance paymen sreams forcing us o defer paymens o he end of he considered ime horizon. Therefore in his chaper we jusify ha he risk-minimizing hedging heory for paymen sreams of Møller 21a can be exended o locally risk-minimizing hedging of paymen sreams. The saring poin for all hose consideraions is he fundamenal hedging heory of Föllmer and Sondermann 1986 for incomplee financial markes. Considering he sock price being already a maringale under he hisorical measure, Föllmer and Sondermann 1986 developed a risk-minimizing hedging heory for square-inegrable coningen claims wih fixed mauriy. Laer, generalizing his heory o semimaringale financial markes, Schweizer 1991 inroduced he noion of local risk-minimizaion. Sill considering he maringale financial marke, Møller 21a enlarged he heory of Föllmer and Sondermann 1986 o risk-minimizing hedging of general paymen sreams, where he hedger s liabiliies are given by a square-inegrable paymen process and no, as before, by a coningen claim wih fixed mauriy. Such paymen processes occur, for insance, wih uni-linked life insurances admiing boh premium and benefi paymens o occur a any ime in he considered ime inerval. Møller 21a explicily poses he open quesion of locally risk-minimizing hedging of paymen sreams. This includes, in paricular, he choice of he equivalen risk-neural maringale measure such ha he hedging risk is inerpreable under he hisorical measure. In his chaper we are now able o answer his quesion in reaing he remaining semimaringale case. The chaper is self-conained and o a large exend independen from he previous chapers of his hesis. Moreover, i is he basis for one par of Riesner 25.

86 82 Chaper 5. Paymen sream hedging for semimaringales 5.2 The model Aligning our seup o Schweizer 1991, we invesigae local risk-minimizaion o be applicable o general paymen sreams. Throughou he chaper we work on some filered probabiliy space Ω, F,F T, P saisfying he usual condiions of righ-coninuiy and compleeness. Especially F = {,Ω} and F T = F. T R is fixed and denoes he finie ime horizon. Furher we consider a financial marke wih ime horizon T consising of one risky asse sock wih discouned price process X and a riskless invesmen alernaive bond assuming is value o be consan and equal o one. X = X T is defined o be a semimaringale wih decomposiion X = X + M + A, such ha he following assumpions cf. Schweizer 1991 hold: X1 M = M T is a square-inegrable maringale wih M = and A = A T is a predicable process of finie variaion A wih A =. X2 For P almos all ω, he measure induced by M ω has he whole inerval [,T] as is suppor. X3 A is coninuous. X4 A is absoluely coninuous wih respec o M wih a densiy α saisfying E M [ α log + α ] <. X5 X is coninuous a T P-a.s. Assumpion X1 implies he exisence of he predicable variaion process M of M wih respec o P. PM denoes he measure P M on he produc space Ω := Ω [,T] wih he σ-algebra P of predicable ses. Definiion A rading sraegy ϕ = ξ,η is a process consising of he predicable process ξ = ξ T such ha E[ ξu 2 d M 2] u + ξ u d A u <, 5.1 i.e. ξ L 2 P M and ξ u d A u L 2 P, and consising of he adaped process η = η T such ha he porfolio value V ϕ := ξ X + η is càdlàg and saisfies V ϕ L 2 P, T. As usual, a ime, ξ denoes he amoun of shares held in he porfolio and η describes he invesmen in he bond. The predicabiliy assumpion on ξ reflecs he fac ha we have o make he invesmen decision before acually knowing he value of he sock a some specific ime.

87 5.2. The model 83 Definiion A paymen sream or process is an adaped, càdlàg squareinegrable process Y = Y T and he cos process Cϕ is defined by C ϕ := V ϕ ξ u dx u + Y, T, where ξ saisfies he condiions of Definiion The paymen process Y may be inerpreed as a hedger s discouned liabiliies owards a buyer of he conrac o be hedged, ha is, i represens conracual paymens. More specifically, from he hedger s poin of view, Y Y s, s < T, is aken o be he oal discouned ougoings less income during he inerval s,]. If Y is negaive, he hedger receives money from he buyer and if i is posiive he buyer receives conracual paymens from he hedger. The inroducion of such a process o describe more general paymen sreams insead of coningen claims i.e. H L 2, F T -adaped and describing a paymen a ime T is due o Møller 21a. Noe ha he paymen process Y = 1 { T } H, T, is a coningen claim for an appropriae choice of H. Furhermore, he cos process Cϕ, being clearly càdlàg and square-inegrable, represens he oal coss of he hedger ha are generaed by conracual paymens and addiional invesmens in he porfolio being no covered by rading gains. Especially he iniial coss C ϕ = V ϕ + Y reflec he sum of he iniial invesmen and he firs conracual paymen. This immediaely delivers an inerpreaion of he porfolio value V ϕ: I is he value of he porfolio held a ime afer he paymens Y have been made. Since we are primarily concerned o hedge he paymen process Y, i is apparen o concenrae on so called -admissible sraegies ϕ, i.e. V T ϕ = P-a.s. In oher words, V T ϕ is he value of he porfolio afer seling all liabiliies. Such a sraegy always exiss. Jus choose ξ η resuling in C ϕ = Y, T. Probably his is no he bes way of dealing wih he considered problem of hedging he paymen process Y and so, as Møller 21a, we addiionally consider he following risk process: R ϕ = E[C T ϕ C ϕ 2 F ], T, given by is càdlàg version. Møller 21a calls a sraegy Y -self-financing if C ϕ C ϕ P-a.s. This is in analogy o he original framework of Föllmer and Sondermann 1986 o which Y -self-financing is only equivalen, if Y Y P-a.s. Y -self-financing means ha all flucuaions of he value process are eiher rading gains or paymens described by Y. I is possible ha for a Y -self-financing sraegy ϕ: PV T ϕ < V ϕ > >, so Møller 21a calls a paymen process Y aainable, if here exiss a sraegy ϕ which is Y -self-financing and -admissible. To link he new and he classical heory we consider he following proposiion, which is a combinaion of a

88 84 Chaper 5. Paymen sream hedging for semimaringales proposiion in Schweizer 1991 and a proposiion in Møller 21a. As in Møller 21a we denoe eniies wih a bar if hey are relaed o he classical heory. Proposiion Le Y be a paymen process. The following saemens are equivalen: a H = Y T is classically aainable, i.e. i exiss a classically selffinancing sraegy ϕ = ξ, η such ha C ϕ = V ϕ ξ u dx u C ϕ P-a.s. and V T ϕ = H. b There exiss a -admissible rading sraegy ϕ wih R ϕ =, T, P-a.s. c Y is aainable. Proof. c b: There exiss a Y -self-financing sraegy ϕ wih V T ϕ =. Especially, his implies C ϕ C ϕ P-a.s. b a: Since R ϕ =, we have C ϕ = C ϕ, T, P-a.s. Given ϕ define ϕ = ξ, η wih ξ := ξ and η := η + Y, T. Then on he one hand we ge V T ϕ = ξ T X T + η T = V T ϕ + Y T = Y T and on he oher hand for T C ϕ = V ϕ = V ϕ ξ u dx u ξ u dx u + Y = C ϕ C ϕ P-a.s. a c: Define ϕ = ξ,η by ξ := ξ and η := η Y, T. Then V T ϕ = ξ T X T + η T = V T ϕ Y T =, and also C ϕ = V ϕ = V ϕ ξ u dx u + Y ξ u dx u C ϕ P-a.s. In general no every paymen process is aainable. Møller 21a calls hus, in analogy o Föllmer and Sondermann 1986, a rading sraegy Y - mean-self-financing, if is cos process C ϕ is a maringale. For hedging general paymen processes in a semimaringale marke, we consider now he same opimizaion problem as in he classical case. The following definiion is aken from Schweizer 1991.

89 5.2. The model 85 Definiion Le ϕ = ξ, η be a rading sraegy and [, T]. a An admissible coninuaion of ϕ from on is a rading sraegy ϕ = ξ, η saisfying ξ s = ξ s for s, η s = η s for s <, and V T ϕ = V T ϕ P-a.s. b An admissible variaion of ϕ from on is a rading sraegy = δ,ε such ha ϕ + is an admissible coninuaion of ϕ from on. c ϕ is called risk-minimizing, if for any [,T] and for any admissible coninuaion ϕ of ϕ from on one has R ϕ R ϕ P-a.s. or if, equivalenly, R ϕ + R ϕ P-a.s. for every admissible variaion of ϕ from on. I is now possible o resae he following Lemma, which is for he classical case formulaed in Schweizer Lemma Le Y be a paymen process, ϕ = ξ,η be a -admissible rading sraegy and [,T]. Then here exiss a rading sraegy ˆϕ saisfying a V T ˆϕ = b C ˆϕ = E[C T ˆϕ F ], P-a.s., for T. c R ˆϕ R ϕ, P-a.s., for T. If we choose :=, hen ˆϕ is Y -mean-self-financing. If ϕ is risk-minimizing, hen C ϕ is a maringale. Proof. Le ˆξ := ξ and { η, <, ˆη := E[Y T ξ u dx u F ] + ξ u dx u Y ξ X, T, { η, <, = E[Y T Y ξ u dx u F ] ξ X, T. We choose he càdlàg version. This implies ha { V ϕ, <, V ˆϕ = E[Y T Y ξ u dx u F ], T,

90 86 Chaper 5. Paymen sream hedging for semimaringales is càdlàg and square-inegrable. ˆϕ is herefore a rading sraegy. Immediaely we have ha V T ˆϕ = and C T ϕ = Y T ξ u dx u = Y T ˆξ u dx u = C T ˆϕ. Furher, E[C T ˆϕ F ] = E[C T ˆϕ C ˆϕ F ] + C ˆϕ = C ˆϕ + E[Y T ˆξ u dx u V ˆϕ + ˆξ u dx u Y F ] = C ˆϕ + E[Y T Y = C ˆϕ, T. Using he esablished resuls, one deduces R ˆϕ = E[C T ˆϕ C ˆϕ 2 F ] = E[C T ϕ C ϕ + C ϕ C ˆϕ 2 F ] ˆξ u dx u F ] V ˆϕ = R ϕ + C ϕ C ˆϕ 2 + 2C ˆϕ C ϕc ϕ C ˆϕ = R ϕ C ϕ C ˆϕ 2 R ϕ, T P-a.s. To prove ha Cϕ is a maringale if ϕ is risk-minimizing, we noe ha ˆϕ is an admissible coninuaion of ϕ and ha by he above for T R ϕ = R ˆϕ + C ϕ E[C T ϕ F ] 2 R ˆϕ R ϕ. Therefore, C ϕ = E[C T ϕ F ], T. If X is already a square-inegrable P-maringale, he quesion of finding a risk-minimizing rading sraegy for general paymen processes is compleely solved in Møller 21a. In fac i is enough o have a locally square-inegrable local maringale X. The cenral idea is o uilize he wellknown Galchouk-Kunia-Waanabe decomposiion cf. Theorem For compleeness we review he resul of Møller 21a and include is proof. We assume for he momen ha X is a square-inegrable maringale. For a square-inegrable process Y one ges for T V := E[Y T F ] = V + ξ Y u dx u + K Y, P-a.s., 5.2 where ξ Y L 2 P X and K Y = K Y T is a square-inegrable maringale orhogonal o X wih K Y =, P-a.s. As usual we call wo maringales orhogonal if heir produc is a maringale, or equivalenly for squareinegrable maringales if heir predicable quadraic covariaion is idenically zero.

91 5.2. The model 87 Proposiion Møller. Assume ha X is a square-inegrable maringale. For every paymen process Y here exiss an unique -admissible risk-minimizing rading sraegy ϕ = ξ, η for Y given by ξ,η = ξ Y,V Y ξ Y X, T. The associaed risk-process is given by R ϕ = E[ K Y T K Y 2 F ]. Proof. Firs one proves exisence. Using 5.2 we ge Y T = VT = V + = V + ξ Y u dx u + K Y T ξ Y u dx u + K Y T KY. Le now ϕ be a -admissible rading sraegy. So, since ϕ is -admissible, C T ϕ C ϕ = V T ϕ V ϕ ξ u dx u + Y T ξ u dx u + Y = V Y V ϕ + KT Y KY + ξ Y u ξ u dxu. 5.3 Using he orhogonaliy of X and K and he measurabiliy of he firs erm in 5.3 we ge R ϕ = E[ KT Y 2 KY F ] + V Y V ϕ 2 + E[ ξ Y u ξ 2 u d X u F ]. Choosing now ξ = ξ Y and η = V Y ξ X, T, yields ha he las wo erms vanish and ha ϕ = ξ,η is a risk-minimizing sraegy, since he firs erm does no depend on he sraegy. For he uniqueness noe ha if ˆϕ is risk-minimizing and -admissible, one mus have ˆξ = ξ Y, since R is minimized. Furher by Lemma ˆϕ is Y -mean-self-financing, hence C ˆϕ is a maringale and so by 5.3 V ˆϕ = V Y, i.e. ˆη = V Y ˆξ X, T. The difference from he classical heory is ha a risk-minimizing sraegy ϕ is chosen such ha a ime, T, he value of he porfolio is equal o he condiional expeced value of he hedger s balance, i.e. V ϕ = E[Y T Y F ].

92 88 Chaper 5. Paymen sream hedging for semimaringales For coningen claim hedging in he semimaringale case we already menioned in Chaper 2 ha hings are differen. According o Schweizer 1991 i is impossible o find a risk-minimizing hedging sraegy ha minimizes he process R ϕ uniformly over ime and over all admissible coninuaions of a rading sraegy. For a deailed counerexample we refer o Schweizer 21. The echnical reason is he finie-variaion par of he semimaringale X and is influence on he process Rϕ. The Kunia-Waanabe projecion echnique fails in his case. Since coningen claim hedging is jus a special case Y = 1 { T } H of our more general seing, he projecion echnique mus of course fail here, oo. To solve his problem Schweizer 1991 inroduced locally risk-minimizing hedging sraegies and showed ha hey are he righ ones o consider in he classical semimaringale case. 5.3 Local risk-minimizaion We invesigae now if local risk-minimizaion can be exended from coningen claim hedging o hedging of general paymen processes. By inuiion his migh be quie reasonable, however, in echnical erms i is somewha sophisicaed. In paricular, he influence of he paymen process Y on he concep is no so obvious. For his reason we decided o go ino a lile bi more deail and work hrough he heory of Schweizer 1991 while considering he expanded se-up. The following definiions are aken from Schweizer Definiion A rading sraegy = δ, ε is called a small perurbaion if δ and δ u d A u are bounded and furher δ T = ε T =. For a subinerval s, ] of [, T] and a small perurbaion, he small perurbaion is given by s,] := δ s,],ε [s, δ s,] ω,u := δ u ω 1 s,] u, ε [s, ω,u := ε u ω 1 [s, u. Noe ha δ s,] is predicable and ε [s, is adaped and obviously, for any small perurbaion and for any -admissible sraegy ϕ i holds: and ϕ+ are -admissible. The boundedness condiion for a small perurbaion can be inerpreed as no allowing unlimied sysemaic rading gains. Recall herefore ha A is he finie variaion predicable par of X. The idea of Schweizer 1991 is o consider local variaions of a rading sraegy. To perform his we need a pariion of he inerval [,T]. Le his pariion be given by τ = i i N where = < 1 < < N = T, wih mesh size τ := max 1 i N i i 1. A sequence of pariions τ n n N is called increasing, if τ n τ n+1 for all n and -convergen if lim τ n =. n

93 5.3. Local risk-minimizaion 89 Definiion Given a rading sraegy ϕ, a small perurbaion and a pariion τ of [,T], consider he following quoien: r τ [ϕ, ]ω, := R i ϕ + i, i+1 ] R i ϕ E[ M M ω 1 i, F i τ i+1 i ] i+1 ]. i A sraegy ϕ is called locally risk-minimizing, if lim inf n rτn [ϕ, ], P M -a.e., for every small perurbaion and every increasing -convergen sequence τ n of pariions of [,T]. The assumpion X2 avoids a division by in he above definiion, i.e. he measure induced by M is no consan over some subinerval of [,T]. We generalize now a Lemma of Schweizer Lemma Le Y be a paymen process presupposing X o saisfy assumpions X1 and X2. Furhermore le ϕ be a -admissible rading sraegy. If ϕ is locally risk-minimizing, hen i is Y -mean-self-financing. Proof. As in he proof of Lemma we define for T, ˆξ := ξ and ˆη := E[Y T Y ξ u dx u F ] ξ X. Le now := ˆϕ ϕ, hen is a small perurbaion = δ,ε = ˆξ ξ, ˆη η =, ˆη η wih ε T =. Le τ n be he n-h dyadic pariion of [,T] and denoe by d := d + 2 n T T he successor in τ n of d τ n. Then V d ϕ + d,d ] = ξ d X d + η d + ε d 1 [d,d d = V d ϕ + ǫ d = V d ˆϕ, and V T ϕ + d,d ] = V T ˆϕ =. Furher, C T ϕ + d,d ] C d ϕ + d,d ] = Y T ξ u dx u V d ϕ + d,d ] = C T ˆϕ C d ˆϕ, d τ n, for all n N. d ξ u dx u + Y d Using hese ideniies, one considers he influence of on he risk process: R d ϕ + d,d ] R d ϕ = E[ C T ϕ + d,d ] C d ϕ + d,d ] 2 Fd ] R d ϕ = E[C T ˆϕ C d ˆϕ 2 F d ] R d ϕ = R d ˆϕ R d ϕ.

94 9 Chaper 5. Paymen sream hedging for semimaringales In he proof of Lemma we saw ha R d ˆϕ = R d ϕ C d ϕ E[C T ϕ F d ] 2. Hence, r τn [ϕ, ] = d τ n C d ϕ E[C T ϕ F d ] 2 E[ M d M d F d] 1 d,d ]. Since r τ does no depend on he specific srucure of Cϕ, he proof is now exacly he same as in he classical case: Assume here exiss a dyadic raional d and a se B of posiive probabiliy such ha C d ϕω E[C T ϕ F d ]ω ω B. The righ-coninuiy of C ϕ and E[C T ϕ F ] implies now ha ω B here exiss γω > and βω such ha C d ϕ E[C T ϕ F d ] ω γω >, for every dyadic raional d [d,d + βω] yielding immediaely lim inf n rτn [ϕ, ]ω, < for all ω B and for any d,d + βω. This conradics lim inf n rτn [ϕ, ]ω, P M -a.e., since M is sricly increasing P-a.s. So we mus have C d ϕ = E[C T ϕ F d ] dyadic raionals d, P-a.s. The claim follows hen by righ-coninuiy. In he classical case i suffices o vary only he ξ componen o find a locally risk-minimizing sraegy. The same is rue for he more general seing. Lemma Le Y be a paymen process and ϕ be a sraegy which is -admissible and Y -mean-self-financing. Then ϕ is uniquely deermined by is ξ componen. Proof. Cϕ is a maringale wih erminal value C T ϕ = Y T P-a.s. Hence η, T, is deermined by he following ideniy: ξ u dx u, E[C T ϕ F ] = C ϕ = ξ X + η ξ u dx u + Y, P-a.s.

95 5.3. Local risk-minimizaion 91 The previous Lemma ells us in addiion ha choosing η = E[C T ϕ F ] ξ X + ξ u dx u Y, < T, and η T = yields a Y -mean-self-financing sraegy. Considering now a paymen process Y and a -admissible, Y -mean-self-financing sraegy ϕ, which is deermined by is ξ-componen, we wrie Cξ := Cϕ and Rξ := Rϕ. Leing moreover = δ,ε be a small perurbaion and τ a pariion of [,T] we compare for i τ he sraegy ϕ+ i, i+1 ], which is -admissible bu no necessarily Y -mean-self-financing, wih he -admissible Y -meanself-financing rading sraegy associaed o ξ + δ i, i+1 ]. Lemma Under hese assumpions, one has R i ϕ + i, i+1 ] = R i ξ + δ i, i+1 ] + ε i + E[ and hence, r τ [ϕ, ] = r τ [ξ,δ] + i τ i+1 i δ u da u F i ] 2, ε i + E[ i+1 i δ u da u F i ] 2 E[ M i+1 M 1 i, i F i ] i+1 ], 5.4 wih r τ [ξ,δ] := i τ R i ξ + δ i, i+1 ] R i ξ E[ M i+1 M i F i ] 1 i, i+1 ]. Proof. δ T = ε T =, since is a small perurbaion, and so C T ϕ + i, i+1 ] = = C T ϕ For i τ we have ha ξ u dx u i+1 i+1 i δ u dx u + Y T i δ u dx u = C T ξ + δ i, i+1 ]. i C i ϕ + i, i+1 ] = V i ϕ + i, i+1 ] ξ u dx u + Y i i = ξ i X i + η i ξ u dx u + Y i + ε i = C i ϕ + ε i and since ϕ is Y -mean-self-financing, C i ξ + δ i, i+1 ] = E[C T ξ + δ i, i+1 ] F i ] = C i ϕ E[ = C i ϕ E[ i+1 i δ u dx u F i ] i+1 i δ u da u F i ]

96 92 Chaper 5. Paymen sream hedging for semimaringales 5.5 and 5.6 imply i+1 C i ϕ + i, i+1 ] = C i ξ + i, i+1 ] + ε i + E[ δ u da u F i ], i and hence C T ϕ + i, i+1 ] C i ϕ + i, i+1 ] = C T ξ + δ i, i+1 ] C i ξ + δ i, i+1 ] ε i + E[ The maringale propery of Cξ + δ i, i+1 ] yields hen i+1 i δ u da u F i ]. R i ϕ + i, i+1 ] = R i ξ + δ i, i+1 ] + i+1 ε i + E[ δ u da u F i ] 2. i We now adap he following Lemma of Schweizer Lemma Assume X saisfies X1-X5. Le Y be a paymen process and ϕ = ξ, η a -admissible rading sraegy. Then he following saemens are equivalen: i ϕ is locally risk-minimizing. ii ϕ is Y -mean-self-financing, and lim inf n rτn [ξ,δ] P M -a.e. for every δ saisfying Definiion and for every -convergen sequence τ n of pariions of [, T]. iii ϕ is Y -mean-self-financing, and he maringale Cϕ is orhogonal o M. Proof. If i or ii holds, hen C ϕ is a square-inegrable maringale. This follows, since ϕ is Y -mean-self-financing in boh cases. Furhermore he second erm on he righ hand side of 5.4 he sum is no influenced by he paymen process Y. Hence he equivalence of i and ii can be derived from Lemma 2.2 in Schweizer 1991 and Proposiion 3.1 in Schweizer 199. The equivalence of i and iii follows from Theorem 3.2 in Schweizer 199. Remark a The orhogonaliy resuls of Schweizer 1991 and Schweizer 199 suppose he cos process C ϕ o be a square-inegrable maringale only. For his reason, adding he square-inegrable paymen process Y does no disrup he classical heory. b Noe ha ii i immediaely follows from 5.4 and Lemma

97 5.3. Local risk-minimizaion 93 c For i ii i suffices o show ha i τ n ε i + E[ i+1 i δ u da u F i ] 2 E[ M M 1 i, F i+1 i ] i+1 ] i converges o as lim n τ n = for any increasing -convergen sequence of pariions τ n. For a special case his is done in he proof of Lemma 2.2 in Schweizer 1991 and in general in he proof of Proposiion 3.1 in Schweizer 199. One now proceeds o characerize a locally risk-minimizing sraegy by he Kunia-Waanabe decomposiion of C T ϕ wih respec o P and M. This yields an opimaliy equaion for he ξ componen and he problem of finding a locally risk-minimizing sraegy is hence reduced o solve his sochasic opimaliy equaion. Theorem Assume ha X saisfies X1-X5. Le Y be a paymen process and le ϕ = ξ,η be a -admissible rading sraegy. Then ϕ is locally risk-minimizing if and only if ϕ is Y -mean-self-financing and ξ saisfies he opimaliy equaion: µ Y ;P ξ µ ξ,a;p = P M -a.e. 5.7 Proof. We consider he Galchouk-Kunia-Waanabe decomposiion cf. Theorem of V := E[Y T F ], which is for T given by V = E[Y T F ] = V + µ Y ;P u dm u + K Y ;P, P-a.s., 5.8 as well as he Galchouk-Kunia-Waanabe decomposiion ξ u da u = E[ ξ u da u ] + µ u ξ,a;p dm u + K ξ,a;p T P-a.s. Noe ha ξ u da u is square-inegrable, since ϕ is a rading sraegy. The wo decomposiions ogeher yield C T ϕ = Y T = V + = V E[ ξ u dx u ξ u da u ] } {{ } =C ϕ + K Y ;P T µ Y u ;P dm u + K Y ;P T + K ξ,a;p T, P-a.s. Lemma yields hen he resul. µ Y ;P u ξ u dm u + A u ξ u µ ξ,a;p u dmu

98 94 Chaper 5. Paymen sream hedging for semimaringales 5.4 Finding a locally risk-minimizing sraegy In his secion i is shown ha an unique soluion o he opimaliy equaion exiss. Firs one adds some more srucure on he considered probabiliy space cf. Schweizer Definiion Two sochasic processes M and N are said o form a P-basis of L 2 P if boh M and N are square-inegrable maringales under P and M M and N N are P-orhogonal as maringales. Furher, if every H L 2 P has an unique represenaion H = E[H] + µ H;P u dm u + ν H;P u dn u P-a.s. for wo predicable processes µ H;P L 2 P M and ν H;P L 2 P N. The following assumpions are added o he model. P1 There exiss a process N = N T such ha M and N form a P-basis of L 2 P. P2 There exiss a probabiliy measure Q equivalen o P such ha X and N form a Q-basis of L 2 Q. Because of P 2 he considered financial marke mus be free of arbirage. Schweizer 1991 provides moreover a formula for he Radon-Nikodým densiy of he measure Q, called he Föllmer-Schweizer measure, and shows ha i is essenially unique. This formula provides an insigh ino he performed Girsanov change of measure removing he drif A from X. I gives furhermore an idea of how o consruc he measure Q. Consider now a paymen process Y = Y T which is also squareinegrable wih respec o Q. Under assumpion P 2 we apply Proposiion o he hedging problem and find an unique hedging sraegy ϕ Y ;Q = ξ Y ;Q,η Y ;Q which is risk-minimizing wih respec o Q and saisfies V T ϕ Y ;Q =. ξ Y ;Q is given by he Kunia-Waanabe decomposiion and Y T = E Q [Y T ] + ξ Y ;Q u dx u + ν Y ;Q u dn u Q-a.s. 5.9 η Y ;Q = E Q [Y T F ] Y ξ Y ;Q X, Q-a.s., T. The idea is now o use ϕ Y ;Q as candidae for a locally risk-minimizing sraegy under P. Theorem Assume ha X saisfies X1-X5 and ha P1 and P 2 hold. Le Y be a paymen process and square-inegrable wih respec o Q. Suppose addiionally ha ν Y ;Q L 2 P N and ha ξ Y ;Q saisfies 5.1. Then ϕ Y ;Q is locally risk-minimizing wih respec o P.

99 5.4. Finding a locally risk-minimizing sraegy 95 Proof. According o Theorem firs one has o show ha ξ Y ;Q is a soluion o he opimaliy equaion 5.7. Since ξ Y ;Q saisfies he condiions of a rading sraegy and because of assumpion P1, we may wrie ξu Y ;Q da u = E[ + ξu Y ;Q da u ] + ν ξy ;Q,A;P u dn u. Using his resul, and E[Y T ] = E Q [Y T ] + E[ ξu Y ;Q da u ], µ ξy ;Q,A;P u dm u which holds because of P2, one rewries 5.9 as follows: Y T = E Q [Y T ] + = E[Y T ] + + ν Y ;Q u ξu Y ;Q dm u + ξu Y ;Q da u + ξ Y ;Q u + µ ξy ;Q,A;P u dmu + ν ξy ;Q,A;P u dnu. νu Y ;Q dn u We compare his represenaion of Y T wih 5.8 and refer o he uniqueness of he Galchouk-Kunia-Waanabe decomposiion o conclude ha he opimaliy equaion 5.7 is saisfied: µ Y ;P = ξ Y ;Q + µ ξy ;Q,A;P P M -a.e. I remains now o show ha ϕ Y ;Q is Y -mean-self-financing wih respec o P. 5.9 yields C T ϕ Y ;Q = V T ϕ Y ;Q ξu Y ;Q dx } {{ } u + Y T = E Q [Y T ] + νu Y,Q dn u. = Furher, C ϕ Y ;Q = V ϕ Y ;Q = ξ Y ;Q X + η Y ;Q ξu Y ;Q dx u + Y ξ Y ;Q u dx u + Y. Recall ha η Y ;Q = E Q [Y T F ] Y ξ Y ;Q X. Using P2 we ge E Q [Y T F ] = E Q [Y T ] + ξ Y ;Q u dx u + ν Y ;Q u dn u.

100 96 Chaper 5. Paymen sream hedging for semimaringales Hence, C ϕ Y ;Q = E Q [Y T ] + ν Y ;Q u dn u, T, The claim follows, since N is a maringale under P and ν Y ;Q L 2 P N. Summarizing his chaper we remark ha he local risk-minimizaion heory for he classical case ransfers o he case of general paymen processes and is applied in he same manner. Firs, one removes he drif of he semimaringale X performing an equivalen change of measure o he Föllmer-Schweizer measure Q and second, one applies he heory of Møller 21a o find a risk-minimizing hedging sraegy under Q. This sraegy is hen locally risk-minimizing wih respec o he original measure P.

101 Chaper 6 Hedging of general uni-linked life insurance conracs 6.1 Inroducion The following chaper conribued o one par of Riesner 25 and generalizes he consideraions of Chaper 4 wih respec o wo aspecs. Firs, in Chaper 4 only insurance conracs ha do no generae inermediae paymens were reaed. We resriced ourselves o he case in which only a single premium paymen a he ime of issue occurs and in which benefi paymens are always deferred o he end of he considered ime horizon. Benefis only cashed ou in his way are no resricion for he pure endowmen bu cerainly for he erm insurance for which i is simply no realiy. The reason for his consrain is he local risk-minimizaion heory of Schweizer 1991 which can only be applied for coningen claims wih fixed mauriy. Using Chaper 5, where we jusified locally risk-minimizing hedging o work also for general paymen sreams, we allow here he insurance conracs o have boh inermediae premium and benefi paymens. The second aspec of generalizaion is he insurance model which is now based on he muli-sae Markov model of Hoem 1969 which we discussed in Subsecion and which allows o model quie general forms of life insurance. To be more specific, we consider life insurance conracs driven by a Markov jump process and having uni-linked benefis and possibly premiums based on our Lévy-process sock model of Secion 2.3. Those benefis are usually due immediaely upon occurrence of some insurance even a random imes. In life insurance such an even could be for insance a disabiliy or he deah of he policy-holder. Similarly, premiums are paid according o a predefined premium scheme for a fixed ime or as long as he policy-holder is alive. For hose general conracs we derive locally risk-minimizing hedging

102 98 Chaper 6. Hedging of general uni-linked life insurance conracs sraegies and compue addiionally he involved hedging risk, which has now a local inerpreaion under he hisorical measure. This is an imporan figure, since, as in realiy, we face an incomplee financial marke in which riskless hedging is no possible. The model and he resuls presened here complemen Møller 21a who derives risk-minimizing hedging sraegies for uni-linked life insurance paymen sreams in he same muli-sae Markov model bu for a Black-Scholes maringale financial marke. Moreover, in analogy o Chaper 4 we show for he generalized model ha he involved hedging risk can be separaed in wo componens: financial risk and insurance risk. In Møller 21a he pure financial risk is no presen, since for a complee financial marke, urned incomplee by he moraliy law, only he insurance risk appears. We conclude he chaper wih examples of uni-linked life annuiy and erm insurance conracs boh in a single insurance and in a porfolio model consideraion. In paricular, we apply he general heory in deail and reduce i o he specific insurance models. Le us briefly recall he necessary echnicaliies. As in Chaper 4 he Lévy-process financial marke, Ω 1, G,G T, P 1, of Secion 2.3 including he change of measure is supposed and as menioned for he insurance par of our model, Ω 2, H,H T, P 2, he muli-sae Markov model discussed in Subsecion is assumed. Using hose wo models we consider he risk-neural combined financial and insurance model Ω, F,F T, Q of Subsecion recalling ha Ω, F,F T, P is supposed o be he produc of he wo independen probabiliy spaces Ω 1, G,G T, P 1 and Ω 2, H,H T, P 2. We also remind he reader ha he measure Q denoes he produc measure of he Föllmer-Schweizer measure Q 1 and he measure P 2 which is assumed o be risk-neural. Moreover, all he heory developed in Chapers 3 and 5 is a prerequisie for his chaper. 6.2 General uni-linked benefi and premium paymens The muli-sae Markov insurance model admis quie general forms of benefi and premium paymens cf. Møller 21a. Firs, i migh be he case ha a ransiion from sae j o sae k a ime immediaely induces a paymen g jk = g jk,s and second, i is possible ha depending on he policy sojourning in sae j he insurance company coninuously pays he rae g j = gj,s a ime. Paymens of he firs ype usually occur wih general life insurances whereas sae-wise life annuiies ypically generae paymens of he second ype. Being in sae j, he policy shall addiionally admi lump-sum annuiy paymens g j = gj,s a fixed deerminisic imes Γ = {τ 1,...,τ n } for some n 1. The amoun payable a ime [,T], depending on he policy being in sae j, is herefore equal o G j = Gj,S = g j + gj 1 { Γ} and up o ime he oal sae-wise annuiy

103 6.2. General uni-linked benefi and premium paymens 99 paymen is equal o Gj u,s u du = gj u,s u du + τ Γ,τ gj τ,s τ. Here and for he res of his chaper we rea 1 {u Γ} du as discree couning measure on he se Γ wih jump high one. This enables us o discuss he coninuous and he discree case simulaneously. Moreover, we assume benefi and premium paymens represened as difference in he funcions g jk, g j and g j, where negaive values are premium paymens. For echnical reasons,s g jk,s,,s g j,s and,s g j,s 1 { Γ} are supposed o be measurable for j,k J, and we need ha sup [ E Q B 1 u gu,s u ] 2 <, for all g jk,g j, g j 1 { Γ}. 6.1 u [,T] Tha condiion is se in accordance wih 3.1 and guaranees ha each welldefined sochasic inegral of Bu 1gu,S u wih respec o a square-inegrable maringale is a square-inegrable maringale. Following now Definiion we consider he arbirage-free price process of hose insurance claims given for u T by Obviously we have F jk,s,u := E Q [B B 1 u g jk u,s u G ], F j,s,u := E Q [B B 1 u G j u,s u G ]. F j,s,u = E Q [B B 1 u g j u G ] + E Q [B B 1 u g j u 1 {u Γ} G ]. Recall ha F,S,u, u T, is he price a ime of an insurance claim due a ime u. The superscrip eiher sands for he sae j or for a ransiion jk from sae j o sae k. In order o perform he following calculaions properly we again suppose ha he funcions F,,u C 1,2 [,T] [, for each u [,T], ha is, once coninuously differeniable in he firs and wice coninuously differeniable in he second variable. For noaional convenience we denoe by F,x,u = d dx F,x,u. Lemma and he remarks hereafer apply similarly. In analogy o 3.6 we furher assume F,x,u o be uniformly bounded, ha is, here exiss a non-random consan c 1 > such ha F,x,u c1 <,x,u and for all F j,f jk Q-a.s. 6.2 Having defined he price processes we inroduce now he paymen process Y T cf. Definiion and Møller 21a which is described by general uni-linked insurance conracs and for which we will derive a hedging sraegy. I is given by Y = Y + j J Bu I 1 u j Gj u du + gu jk dnjk u, T. k:k j

104 1 Chaper 6. Hedging of general uni-linked life insurance conracs Here i is imporan ha S and N jk jump simulaneously only wih zero probabiliy. Hence he inegrals B 1 u gu jk dnu jk are Q-a.s. well-defined. Considering he process Y he noion uni-linked becomes finally apparen. The process describes paymens depending on he curren sae of he policy and on he marke value of he sock S. 6.3 Locally risk-minimizing hedging sraegies To find now he locally risk-minimizing hedging sraegy under P for he paymen process Y, we have o derive a risk-minimizing sraegy for i under he measure Q cf. Chaper 5. As we already know, he inrinsic value process V = E Q [Y T F ], T, of Y under Q and is Galchouk- Kunia-Waanabe decomposiion play a crucial role. We proceed as in Møller 21a: V = Y + E Q [Y T Y F ] = Y + E Q [ I ug j j u du + j J = Y + E Q [ j J = Y + B 1 j J B 1 u B 1 u k:k j I j u Gj u du + k:k j gu jk dnu jk F ] p Z,j,u F j,s,u + In he hird row we used ha Bu 1 gu jk dnu jk gu jk Ij u µjk u du F ] k:k j µ jk u F jk,s,u du. I j uµ jk u du are maringales cf Furhermore he independence of he financial marke and he insurance porfolio allows o facorize he expecaions and o calculae E Q [I j u F ] = E P 2 [I j u H ] = p Z,j,u, u T. In order o abbreviae he expression of V he following auxiliary processes V i, T and i J, are inroduced: V i,s = j J [ p ij,u F j,s,u + k:k j µ jk u F jk,s,u du ]. As previously menioned, in Møller 21a he original measure P is already a maringale measure for he sock admiing o evaluae he processes V i direcly under P. They reflec hen he curren marke value of fuure benefis less premiums a ime condiional on he policy being in sae i a ime and on he value of he sock being S. In his case he processes V i do acually represen he classical sae-wise prospecive reserve which is he expeced value of discouned benefis less premiums under he physical

105 6.3. Locally risk-minimizing hedging sraegies 11 measure P. In our more general seing we have o inerpre his under he risk-neural measure Q calling V i he sae-wise Q-prospecive reserve. In conclusion we express he processes V i as V i,s = E Q [B Y T Y Z = i,s = s] and finally wrie V = Y + IB i 1 V i,s. 6.3 i J Theorem The Galchouk-Kunia-Waanabe decomposiion of V under Q is given by where and K = V = V + ξ i = j J + i J i,k:i k I i τ ζ1i τ is orhogonal o Ŝ wih ζ 1i = j J ζ 2i y = j J ϑ ik = B 1 i J I i τ ξi τ p ij,u ξ j u + ϑ ik τ dmik τ dwτ Q + dŝτ + K, T, R k:k j i J p ij,u ζ 1j u + p ij,u ζ 2j µ jk u ξjk u du Iτ i ζ2i τ y M Q dτ,dy µ jk u ζ 1jk k:k j y,u + g ik + V k,s V i,s µ jk u ζ2jk k:k j. ξ u, ζ1 u and ζ 2 y,u are aken from Theorem u du, y,u du, Proof. The firs sep is o decompose he process B 1 V i,s, i J. As in Møller 21a we le Y i,u = j J p ij,ub F 1 j,s,u + µ jk u F jk,s,u k:k j

106 12 Chaper 6. Hedging of general uni-linked life insurance conracs for all i J and u T and apply Iô s inegraion-by-pars formula cf. Proer 24, Chaper II, Theorem 22, Corollary 2 yielding dy i,u = B F 1 j,s,u + µ jk u F jk,s,u dp ij,u j J k:k j + [ p ij,u db 1 F j,s,u + ] µ jk u db 1 F jk,s,u. j J k:k j The quadraic covariaion erm is idenically equal o, since p ij,u is of finie variaion and coninuous for each u. This follows immediaely from 2.21 saing ha i is differeniable in he firs variable wih a bounded derivaive on [,T]. Subsiuion of 2.21 for dp ij,u yields dy i,u = µ ik pij,u p kj,u j J k:k i F j,s,u + B 1 k:k i k:k j µ jk u F jk,s,u d + p ij,u [ ] j J = µ ik Y i,u Y k,u d + p ij,u [ ]. j J Now we apply Theorem o he differenials db 1 F,S,u, which yields dy i,u = k:k i µ ik Y i,u Y k,u d where + j J + = α i,u α i,u β i,u k:k j [ p ij,u ξ j udŝ + ζ 1j udw Q µ jk u ξ jk d + β i,u := k:k i := j J µ ik + R udŝ + ζ 1jk + R ζ 2j y,um Q d,dy ζ 2jk dŝ + γ i,u dw Q + Y i,u Y k,u, p ij,u ξ j u + udw Q ] y,um Q d,dy k:k j R δ i,u ym Q d,dy, µ jk u ξjk u,

107 6.3. Locally risk-minimizing hedging sraegies 13 This implies γ i,u p ij,u := j J δ i,u y := p ij,u j J Y i,u = Y i,u + + Noe now ha B 1 V i,s = B 1 V i,s = + + R Y i,u ζ 1j ζ 2j α i,u τ dτ + u + µ jk u ζ1jk k:k j y,u + δ i,u τ ym Q dτ,dy. du + R k:k j β i,u τ dŝτ + u, µ jk u ζ2jk y, u. γτ i,u dwτ Q Y i,u du. Hence cf. Møller 21a, 1 {τ u} α i,u τ dτdu 1 {τ u} βτ i,u dŝτ du + 1 {τ u} δτ i,u ym Q dτ,dydu 1 {τ u} γτ i,u dwτ Q du Yu i,u du. Le us evaluae each of he six summands on he righ hand sigh separaely realizing firs ha Y i,u du = V i,s. Furhermore, as in Møller 21a, i holds ha = = Yu i,u du j J B 1 u p ij u,u } {{ } =1 {i=j} B G i u + k:k i 1 u F j u,s u,u + µ ik u gik u du, k:k j µ jk u F jk u,s u,u du and 1 {τ u} α i,u τ dτ du = = = τ α i,u τ dudτ µ ik τ k:k i τ µ ik τ Bτ 1 k:k i Y i,u τ Yτ k,u dudτ V i τ,s τ V k τ,s τ dτ.

108 14 Chaper 6. Hedging of general uni-linked life insurance conracs The applicaion of Fubini s heorem is allowed here Q-a.s., since he funcion ω,,u α i,u ω, i J, is O B[,T]-measurable, and since by 6.1 we have uniformly for all,u ha F,S,u < Q-a.s. This implies ogeher wih he coninuiy of µ ha 1 {τ u} α i,u dτ du < τ Q-a.s. Addiionally, he funcion ω,,u β i,u ω, i J, is P B[,T]- measurable and uniformly bounded by a non-random consan cf. 3.1 in he proof of Theorem Fubini s heorem for sochasic inegrals cf. Proer 24, Chaper IV, Theorem 64 yields hen: 1 {τ u} β i,u τ dŝτ du = τ βτ i,u du dŝτ = ξ i τ dŝτ. Also ω,,u γ i,u ω, i J, is P B[,T]-measurable and furher here is a non-random consan c 2 such ha sup γ i,u c 2 sup Ŝ, u Q-a.s., [,T] [,T] since 3.11 holds and he remaining erms are non-randomly bounded. Hence for all τ T we have ha τ 1 γτ i,u 2 2 du c 2 sup Ŝ. [,T] The square-inegrabiliy of Ŝ implies now ha he inegral of s γi,u s du wih respec o W Q exiss cf. Proer 24, Chaper IV, Lemma before Theorem 28. Therefore we may apply he second version of Fubini s Theorem for sochasic inegrals cf. Proer 24, Chaper IV, Theorem 65 yielding 1 {τ u} γτ i,u dwτ Q du = γτ i,u du dwτ Q = ζ τ 1i dwτ Q. Analogously, he funcion ω,,u,y δ i,u y, i J, is O B[,T] BR-measurable. So, in similariy o he consideraions above and o he calculaions before Corollary in he discussion of he erm insurance we are allowed o apply again Fubini s heorem for sochasic inegrals: 1 {τ u} δτ i,u ym Q dτ,dydu = R R τ δτ i,u y du M Q dτ,dy = τ R ζ τ 2i ym Q dτ,dy Q-a.s.

109 6.3. Locally risk-minimizing hedging sraegies 15 The compuaion so far yields he desired decomposiion of B 1 V i,s. B 1 V i,s = V i,s + + R ξ i τ dŝτ + ζ τ 2i ym Q dτ,dy B 1 τ G i τ + k:k i ζ τ 1i dwτ Q µ ik τ ϑik τ dτ, Q-a.s. Using 6.3 and inegraion by pars we now obain he decomposiion of V : V = Y + dy τ + d Iτ i B 1 τ V i τ,s τ i J I τg i i τ dτ + = Y + + i J B 1 τ i J + [ I i,b 1 V i,s ] k:k i I i τ d B 1 τ V i τ,s τ +. gτ ik dnτ ik Bτ 1 V i τ,s τ diτ i Since I i is of finie variaion, i is a quadraic pure-jump semimaringale cf. Proer 24, Chaper II, Theorem 26. So he coninuous par of he quadraic covariaion above is idenically, and we ge [ I i,b 1 V i,s ] = Ii B 1 V i,s + <τ I i τ B 1 τ V i τ,s τ. This erm simplifies considerably because of he iniial sae of he Markov chain Z implying I i = 1 {i=} and because of he sum of jumps being <τ I i τ B 1 τ V i τ,s τ = Q-a.s., T. This holds, since a leas V i,s is coninuous in probabiliy and he processes I i and S are independen by assumpion. Moreover, we only rea càdlàg processes and hence face only counably many jump disconinuiies which yields he claim by he σ-addiiviy of Q cf. he consideraion in 4.3. Addiionally observe ha di i = k:k i dn ki dn ik.

110 16 Chaper 6. Hedging of general uni-linked life insurance conracs Hence, Q-a.s., R V = V + Iτ i ξi τ dŝτ + i J + Iτ ζ i τ 2i ym Q dτ,dy + + i,k:i k Bτ 1 i J Iτ i i J = V + Iτ i ζ1i τ dwτ Q Bτ 1 V k τ,s τ V i τ,s τ dnτ ik i J I i τg i τ dτ + B 1 τ G i τ + I i τ ξi τ k:k i k:k i µ ik τ ϑik τ dŝτ + K. gτ ik dnτ ik dτ In he las sep we used ha dn ik I i µ ik d = dm ik. By he same lines as in he proof of Corollary one shows now ha his is indeed he desired Galchouk-Kunia-Waanabe decomposiion. Observe herefore ha i J Ii ξi is Q-a.s. bounded by a non-random consan cf. proof of Theorem and 3.1 and so he inegral wih respec o Ŝ is a squareinegrable maringale. Also by he boundedness of i J Ii ζ 1i and of i J Ii ζ2i y in he sense of 3.11 and 3.12, respecively he inegrals wih respec o W Q and M Q, are square-inegrable maringales. The square-inegrabiliy of he ϑ ik, which follows from 6.1, implies ha he inegrals wih respec o he M ik are also square-inegrable maringales cf. addiionally 2.2. The orhogonaliy of K and Ŝ is a direc consequence on he one hand of he orhogonal decomposiion of B 1 F,S,u and on he oher hand of he independence of S and M ik. Corollary The unique -admissible locally risk minimizing sraegy ϕ under P for he paymen process Y of an uni-linked life insurance conrac is given by ξ,η = I ξ i, i i J i J IB i 1 V i,s I ξ i i Ŝ, T. i J

111 6.3. Locally risk-minimizing hedging sraegies 17 The inrinsic risk process under Q is for T given by R ϕ = I i E Q [ ζ τ 1j 2 F ] + E Q [ ζ τ 2j y 2 F ]ντ Q dy i,j J + i J p ij,τdτ I i j,k:j k E Q [ ϑ jk 2 τ F ]p ij,τµ jk τ dτ, where ξ i, ζ 1j, ζ 2j and ϑ jk are aken from Theorem Proof. The hedging sraegy is an immediae consequence of Proposiion and Theorem Furher R ϕ = E Q [K T K 2 F ] wih K as defined in Theorem To simplify his expression we use he pairwise orhogonaliy of he inegrals wih respec o W Q, M Q, and M ik. Noe ha in our case W τ Q is a coninuous, R M Q dτ,dy is a purely disconinuous maringale and ha he M ik are independen of W Q and M Q by assumpion. For T one ges E Q [ = j J = i,j J I i j J I j τ ζ1j τ R dw Q τ 2 F ] E Q [Iτ j F ] E Q [ ζ τ 1j 2 F ]dτ, E Q [ ζ τ 1j 2 F ]p ij,τdτ. Similarly, E Q T [ = i,j J I i R j J R I j τ ζ2j τ ym Q dτ,dy 2 F ] E Q [ ζ τ 2j y 2 F ]p ij,τντ Q dydτ. Finally, 2.2 and he muual orhogonaliy of he M ik iself yields E Q [ = E Q [ = i J j,k:j k I i j,k:j k j,k:j k 2 ϑ jk τ dmjk τ F ] ϑ jk τ 2λ jk τ dτ F ] E Q [ ϑ jk 2 τ F ]p ij,τµ jk τ dτ.

112 18 Chaper 6. Hedging of general uni-linked life insurance conracs By Proposiion we know ha he porfolio value of he locally risk-minimizing sraegy saisfies V ϕ = E Q [Y T Y F ] = i J I i B 1 V i,s. If we worked under he physical measure P, V ϕ would exacly correspond o he prospecive reserve of a classical life insurance. Thus we call here V ϕ he Q-prospecive reserve indicaing he evaluaion under he measure Q. In he following we commen addiionally on he sraegy and he hedging risk. For a relaed discussion we also refer o Subsecion The locally risk-minimal invesmen in he sock a ime depending on he sae i of he policy is where ξ i = j J ξ u = c2 κ F,S,u + p ij,u ξ i u + 1 σ Ŝ κ k:k j R µ jk u ξjk u du, xj,x,uν Q dx. The difference from a classical Black-Scholes seing like in Møller 21a is apparen. If our Lévy process was merely a Brownian moion he measure ν Q dx would be equal o and κ c 2 : we would only inves F,S,u, as expeced. In he presence of he underlying having jumps he locally riskminimizing invesmen in he sock is nearly a weighed sum of F,S,u and he jump J,x,u = B 1 {F,S + σ S x,u F,S,u}. Noe ha c 2 is he variance of he Brownian par cw Q and ha κ s ds = c 2 + R x2 νs Q dxds is he variance of L under Q. For he jump par he weigh is no so obvious bu, as seen before, he mean-value heorem yields J,x,u = F,x,uσ Ŝ x for some suiable inermediae value x making he weighing more clear. Formula 6.4 shows ha he insurer s inrinsic risk R ϕ of he locally risk-minimizing sraegy has wo componens dependen on he policy being in sae i: a financial risk driven by rading in he marke only presen in an incomplee financial marke E Q [ ζ τ 1j 2 F ] + E Q [ ζ τ 2j y 2 F ]ντ p Q dy ij,τdτ. j J b insurance risk driven by moraliy j,k:j k R E Q [ ϑ jk 2 τ F ]p ij,τµ jk τ dτ.

113 6.4. Examples 19 Noe ha he pure financial risk appears only in an incomplee financial marke and ha Møller 21a, considering merely a complee financial marke, neglecs his significan risk figure. The insurance risk is driven by he sum-a-risk, ϑ jk = B 1 g jk + V k,s V j,s, and resuls from he uncerainy of he insured lives. I is equal o he oal inrinsic risk in Møller 21a. Each ime a ransiion of he policy from sae j o sae k akes place, he paymen g jk has o be cashed ou conribuing immediaely o he insurer s loss. A he same ime he sae-wise Q-prospecive reserve V j,s has o be adjused o he new sae k and is hen given by V k,s. If V k,s V j,s is negaive, i is used o cover he paymen g jk. 6.4 Examples Wihou specifying he ype of insurance conrac, here are wo simple choices of conrac funcions a which we briefly have a look a. If he conracual paymens g jk and G j are deerminisic wih respec o he financial marke, nohing is invesed in he sock. Theorem immediaely implies here ha ξ and ha no financial risk is presen, i.e. ζ 1 ζ 2. In his case he insurance risk becomes he risk of a classical life insurance. For a relaed resul compare Example If he conrac funcions are given by g jk = G j = S, for all j,k and for all, he insured akes over he whole financial risk and consequenly i is no presen for he insurer. From Theorem one immediaely derives ha ξ 1 and ζ 1 ζ 2. This is again similarly discussed in Example As an applicaion of he advanced and general heory presened so far, we will model now he uni-linked life annuiy and he uni-linked erm insurance boh as a single conrac and as a porfolio of n idenical conracs. This illusraes he heory and helps o undersand he foregoing discussion. For a sandard Black-Scholes marke he erm insurance examples are also reaed in Møller 21a. Since here are some differences from he incomplee general Lévy-process financial marke, we reconsider hem for compleeness of his hesis Single uni-linked life annuiy insurance We discuss a life annuiy insurance issued o a single person agains a single premium P a ime. This conrac immediaely sars o pay he insured person an annuiy condiional on he person being alive. Furhermore we assume ha his paymen is of uni-linked wih guaranee ype. To be more precise, condiional on being alive he insured receives a ime a guaraneed deerminisic annuiy γe δ > increasing coninuously by compounding ineres wih some rae δ >. However, if he value of some

114 11 Chaper 6. Hedging of general uni-linked life insurance conracs reference porfolio S exceeds his minimal paymen he insured ges he amoun S. For his conrac he sae space of he muli-sae Markov model is J = {,1}, where represens he sae policy holder alive and 1 he sae policy holder dead. Le he policy holder be of age x a ime wih remaining lifeime T x afer ime. The policy holder dying a ime = 1T x and moreover he inensiy µ 1 of he only possible ransiion from sae o sae 1 is he hazard-rae funcion µ of T x. Noe ha naurally here are no ransiions form sae 1 o sae. The ransiion probabiliies can be deermined by Kolmogorov s backward equaions 2.21: T x implies N 1 Hence, d d p,u = µ 1 p,u. p,u = exp u µ τ dτ. The probabiliy p,u is he survival probabiliy unil ime u given he policy holder is alive a ime, in acuarial noaion his is wrien as u p x+. In conras p 1,u = u q x+ = 1 p,u is herefore he probabiliy ha he person a age x + dies before ime u. The conrac funcions are all equal o zero excep of G = G,S = g,s P 1 =, g,s = maxs,γe δ. where This resuls in only one price process F,S,u, which has o be evaluaed depending on he paricular Lévy-process model of he financial marke cf. Chaper 7. In case he Lévy process is a Brownian moion he Black-Scholes formula can be applied. The locally risk-minimal invesmen in he sock is now given by cf. Corollary ξ = I ξ + I 1 ξ 1 = I ξ = 1T x ξ since ξ 1, ha is, here are no more paymens afer he policyholder s deah. By Theorem we have now, since all oher erms are zero, ξ = 1 = j= where cf. Theorem p j,u ξ j u + ξ up,udu, ξ u = c2 κ F,S,u + 1 σ Ŝ κ k:k j R µ jk u ξjk u du xj,x,uν Q dx. 6.4

115 6.4. Examples 111 Similarly, η = 1T x > B 1 V,S + 1T x B 1 V 1,S ξ Ŝ = 1T x > B 1 V,S ξ Ŝ, since V 1,S = and where V,S = = 1 [ j= p j,u F j,s,u + F,S,up,udu. k:k j µ jk u F jk,s,u du] V,S is he expeced oal amoun o be cashed ou o he policy-holder in he inerval [,T], if he policy-holder is sill alive. The inrinsic hedging risk cf. Corollary is given by R ϕ = i,j {,1} + I i p ij,τdτ 1 i= I i = 1T x > j,k:j k p,τdτ + 1T x > E Q [ ζ τ 1j 2 F ] + R E Q [ ϑ jk 2 τ F ]p ij,τµ jk τ dτ, E Q [ ζ τ 1 2 F ] + E Q [ ϑ 1 2 τ F ]p,τµ 1 τ dτ, R E Q [ ζ τ 2j y 2 F ]ντ Q dy E Q [ ζ 2 τ y 2 F ]ν Q τ dy since ζ 11 ζ 21 y µ 1 p 1. In analogy o 6.4 one obains ζ 1 = ζ 1 up,udu and ζ 2 y = where ζ 1 u = cσ Ŝ F,S,u ξ u, ζ 2 y,u = J,y,u σ Ŝ ξ uy. I remains o consider he sum a risk ϑ 1 = B 1 g 1 + V 1,S V,S = B 1 V,S. ζ 2 y,up,udu,

116 112 Chaper 6. Hedging of general uni-linked life insurance conracs This shows he amoun by which he insurance company can reduce is reserve if he insured person dies a ime. Recall Definiion of he iniial hedging coss. Moreover we have ha C ϕ = V ϕ + Y = V,S P. I is reasonable o demand C ϕ =, hence an idea of a minimal premium is P = F,S,up,udu. We noe ha he hedging sraegy is generally no self-financing and ha here are addiional coss of hedging during he inerval,t] Single uni-linked erm insurance In his example we assume he same se-up as we did for he single life annuiy excep ha now we consider a single uni-linked life erm insurance wih guaranee. The conrac is again issued a ime agains he single premium P and specifies he insurance benefi payable immediaely upon deah of he policy holder if occurred before ime T. Again we have wo conrac funcions differen from zero: g 1,S = maxs,γe δ and G = g 1 {=} = P 1 {=}. Thus we only have one price process F 1,S,u and he locally risk-minimizing sraegy cf. Corollary for he sock is given by where ξ = 1 = ξ = I ξ + I1 ξ1 = I ξ = 1T x ξ, j= p j,u ξ j u + ξ 1 up,uµ u du, k:k j µ jk u ξjk u du wih ξ 1 u = c2 κ F 1,S,u + 1 σ Ŝ κ R xj 1,x,uν Q dx. As for he single life annuiy he locally risk-minimal invesmen in he bond is η = 1T x > B 1 V,S ξ Ŝ,

117 6.4. Examples 113 where V,S = = 1 [ j= p j,u F j,s,u + F 1,S,up,uµ u du. k:k j µ jk u F jk,s,u du] Noe ha p,uµ u is he probabiliy ha he person survives from o u and dies immediaely a u. The inrinsic risk is cf. Corollary R ϕ = I i E Q [ ζ τ 1j 2 F ] + E Q [ ζ τ 2j y 2 F ]ντ Q dy where wih i,j {,1} + p ij,τdτ 1 i= I i = 1T x > j,k:j k p,τdτ + 1T x > E Q [ ϑ jk 2 τ F ]p ij,τµ jk τ dτ, E Q [ ζ τ 1 2 F ] + E Q [ ϑ 1 2 τ F ]p,τµ 1 τ dτ, ζ 1 = ζ 2 y = R R ζ 11 up,uµ u du, ζ 21 y,up,uµ u du, ζ 11 u = cσ Ŝ F 1,S,u ξ 1 u, ζ 21 y,u = J 1,y,u σ Ŝ ξ 1 uy. In his case he sum a risk is given by ϑ 1 = B 1 g 1 V,S, E Q [ ζ τ 2 y 2 F ]ντ Q dy which clearly shows he origin of he insurance risk. If he person dies a ime he company s effor is he paymen g 1 less reserve V,S. Furher, he iniial coss are C ϕ = V ϕ P = An idea of a minimal premium could herefore be P = F 1,S,up,uµ u du P. F 1,S,up,uµ u du.

118 114 Chaper 6. Hedging of general uni-linked life insurance conracs Porfolio of n uni-linked life annuiy conracs Saring form he single uni-linked life annuiy wih guaranee cf. Subsecion 6.4.1, we consider now a porfolio of n idenical such conracs issued o n policy holders wih i.i.d. remaining lifeimes and common hazard-rae funcion µ. In Subsecion 2.4.2, o which we refer, we already discussed he embedding of a porfolio of independen and idenically disribued lives in he muli-sae Markov model. Recall ha in his case he sae space is J = {,1,...,n}, where sae j corresponds o exacly j policy holders having died, and ha he ransiion probabiliies saisfy p jk,u = for k < j. Since each policy holder receives he same annuiy, he non-zero sae-dependen paymen funcions are given by G j,s = g j,s np 1 =, g j,s = n jmaxs,γe δ. where This immediaely resuls in he price process F j,s,u = n jf,s,u, where F,S,u denoes he price of he paymen: maxs,γe δ. The locally risk-minimizing hedging sraegy for he sock invesmen cf. Corollary is hen ξ = = = n i= n I i ξi = ξz j=z n j=z p Z j,u ξ j u + p Z j,uξ j udu = k:k j n j=z µ jk u ξ jk u du p Z j,un jξ udu, where ξ u = c2 κ F,S,u + 1 σ Ŝ κ R xj,x,uν Q dx. One observes now ha n Z u given Z u > follows a binomial disribuion wih parameers n Z,p,u, where p,u = u p x+ = exp u µ τ dτ. Hence, n j=z p Z,j,un j = E Q [n Z u Z ] = n Z p,u, and ξ = n Z ξ up,udu.

119 6.4. Examples 115 Tha is, he invesmen is adjused according o he number of persons in he porfolio a ime. For he bond we ge where η = i J V Z,S = = IB i 1 V i,s ξ Ŝ = B 1 V Z,S ξ Ŝ, n [ p Zj,u F j,s,u + j=z k:k j n j=z = n Z p Zj,un jf,s,udu F,S,up,udu, µ jk u F jk,s,u du] 6.5 which is he single uni-linked life annuiy s Q-prospecive reserve imes he number of insured persons a ime. The compuaion of he hedging risk cf. Corollary yields ha R ϕ = n j=z + since, we recall, E Q [ ζ τ 1j 2 F ] + R E Q [ ζ τ 2j y 2 F ]ντ Q dy p Zj,τdτ n E Q [ ϑ j,j+1 2 τ F ]p Zj,τµ j,j+1 τ dτ, j=z µ jk = n jµ k = j + 1, n jµ k = j, else. Le us consider n 1j 2 j=z ζ τ more deailed cf. Theorem n ζ 1j 2 n n τ = j=z l= n n = j=z = j=z n Z j= l=j n j l=z τ τ n = n j 2 T j=z τ τ p jl τ,u ζ τ 1l u + µ lk u ζ1lk τ k:k l 2 p jl τ,un lζ τ 1 u du 2 p Z+j,l+jτ,un j lζ τ 1 u du pτ,uζ 1 τ udu 2, u 2 du

120 116 Chaper 6. Hedging of general uni-linked life insurance conracs where ζ τ 1 u = cσ Ŝ τ F τ,s τ,u ξ τ u. Analogously we derive n j=z ζ 2j 2 n τ = n j 2 T 2, pτ,uζ τ y,udu 2 j=z τ where ζ τ 2 y,u = Jτ,y,u σ τ Ŝ τ ξ τ uy. Le now hen ρ τ = E Q [ R ϕ = τ + E Q T [ R τ n j=z + n j=z pτ,uζ 1 τ udu 2 F ] pτ,uζ 2 τ y,udu 2 F ]ν Q τ dy, ρ τn j 2 p Zj,τdτ E Q [ ϑ j,j+1 2 τ F ]n jp Zj,τµ τ dτ. This simplifies furher, using and n j=z n j 2 p Zj,τ = E Q [n Z τ 2 Z ] ϑ j,j+1 τ = n Z p,τ1 p,τ + n Z 2 p 2,τ, = Bτ 1 g j,j+1 τ + V j+1 τ,s τ V j τ,s τ, for which we have wih 6.5 and j =,...,n Z ha = B 1 n Z j 1 n Z j ϑ Z+j,Z+j+1 τ τ τ = B 1 τ Fτ,S τ,upτ,udu τ Fτ,S τ,upτ,udu. This shows exacly he amoun by which he insurance company can reduce is reserve if a person in he porfolio dies. Hence, R ϕ = n Z + n Z p,τµ τ dτ. p,τ 1 p,τ + n Z p,τ ρ τdτ E Q [ B 1 τ τ Fτ,S τ,upτ,udu 2 F ]

121 6.4. Examples 117 For he premium we obain again ha C ϕ = V ϕ + Y = V,S np = n F,S,up,udu np. Noe ha all he porfolio resuls reduce o he single life annuiy of Subsecion if n = Porfolio of n uni-linked erm insurance conracs To conclude he chaper, le us finally consider he same insurance porfolio as in Subsecion 6.4.3, however, for n idenical uni-linked erm insurance conracs being already discussed for he single policy case in Subsecion Since he compuaions here are very similar o he compuaions in he previous examples, we only briefly summarize he resuls. If an insured individual dies, we observe a ransiion from sae j o sae j +1 and he company has o cash ou g jk,s = 1 {k=j+1} maxs,γe δ, which resuls in he price process F jk,s,u = 1 {k=j+1} F,S,u. The premium is due a issuing dae, hus G = g 1 {=} = np 1 {=}. The locally risk-minimal sock invesmen is cf. Corollary wih ξ = = n i= n j=z I i ξi = ξz = = n Z n j=z p Z j,u p Z j,un jµ u ξ udu ξ up,uµ u du ξ u = c2 κ F,S,u + Analogously o 6.5 one derives η = i J I i B 1 1 σ Ŝ κ R k:k j µ jk u ξjk u du xj,x,uν Q dx. V i,s ξ Ŝ = B 1 V Z,S ξ Ŝ wih V Z,S = n Z F,S,up,uµ u du.

122 118 Chaper 6. Hedging of general uni-linked life insurance conracs Also he hedging risk is R ϕ = n j=z E Q [ ζ τ 1j 2 F ] + R E Q [ ζ τ 2j y 2 F ]ντ Q dy p Zj,τdτ n + E Q [ ϑ j,j+1 2 τ F ]p Zj,τµ j,j+1 τ dτ, j=z where n j=z ζ 1j 2 n τ = n j 2 T 2, ζ τ 1 upτ,uµ u du j=z τ and n j=z ζ 2j 2 n τ = n j 2 T 2. ζ τ 2 y,upτ,uµ u du j=z τ Again we have here, ζ τ 1 u = cσ Ŝ τ F τ,s τ,u ξ τ u and ζ τ 2 y,u = Jτ,y,u σ τ Ŝ τ ξ τ uy. Furhermore we observe ha ϑ j,j+1 τ = B 1 τ since for j =,...,n Z i holds ha maxs τ,γe δτ + V j+1 τ,s τ V j τ,s τ = ϑ τ, V Z+j+1 τ,s τ V Z+j τ,s τ = Fτ,S τ,upτ,uµ u du. τ This has he following inerpreaion. If an insured person dies a ime τ, he insurance company reduces is Q-prospecive reserve by he amoun τ Fτ,S τ,upτ,uµ u du o cover he paymen maxs τ,γe δτ. Le again which yields ρ τ = E Q T [ τ + E Q T [ R τ R ϕ = n Z + n Z ζ 1 τ upτ,uµ u du 2 F ] ζ 2 τ y,upτ,uµ u du 2 F ]ν Q τ dy, p,τ 1 p,τ + n Z p,τ ρ τdτ E Q [ϑ 2 τ F ]p,τµ τ dτ.

123 6.4. Examples 119 An idea of a minimal premium may again derived from he iniial hedging coss: C ϕ = V,S np = n F,S,up,uµ u du np. In Subsecion we already considered a porfolio of uni-linked erm insurance conracs, however, wih he consrain ha he insurance benefis are deferred and cashed ou a he end T of he insurance period. In Chaper 4 we had o make his unrealisic addiional assumpion, since we were no able o rea inermediae paymen imes. Compared o our resul here, allowing inermediae paymen imes, we recall ha in Corollary we obained almos he same resul. Only he bond invesmen, η, differs by he amoun B τgτ,s τ d n 1 j= Nj,j+1 τ which represens in Chaper 4 he paymens already se aside o be cashed ou a ime T.

124 12 Chaper 6. Hedging of general uni-linked life insurance conracs

125 Chaper 7 Jump-diffusion sock model 7.1 Inroducion In he previous chapers we derived hedging sraegies for uni-linked life insurance conracs modeling he uni in a Lévy-process financial marke. However, so far we have no specified he Lévy process behind he financial marke and so, given a cerain payoff gu,s u for a conrac, he risk-neural price process F,S,u of Definiion and is derivaive F,S,u were an absrac ingredien for he hedging sraegy and he involved hedging risk. Of course, depending on he uni, one has o decide which Lévy process fis bes. In case he Lévy process of he financial marke is a Brownian moion one has he Black-Scholes opion pricing formula o express explicily he price process of uni-linked wih guaranee life insurance conracs. For almos all oher Lévy processes, besides possibly very few jump-diffusions, or more general payoff srucures one usually has no explici formula. In paricular for infinie aciviy Lévy-processes cf. Chaper 2 one has o rely a an early sage on numerical mehods o compue he price process. A useful ool for numerical evaluaions in hose cases is he Feynman-Kac parial inegro-differenial equaion of Theorem In Chaper 12 of Con and Tankov 24 various mehods o solve his equaion numerically are discussed. For finie aciviy Lévy-processes hings are a lile bi beer. The wo mos common jump-diffusion models, he Meron and he Kou model, feaure an almos closed formula for he price process of common opions; i is a quickly converging series. We decided o consider he jump-diffusion model for opion pricing proposed in Kou 22. The model assumes ha he logarihm of he asse price follows a jump-diffusion, where he jump sizes are asymmeric double exponenial disribued. The jumps may be inerpreed as rare evens due o unexpeced new marke informaion. According o Kou 22 his model feaures several advanages compared o a Black-Scholes financial marke. Firs, from empirical invesigaions i is widely known ha asse reurn dis-

126 122 Chaper 7. Jump-diffusion sock model ribuions incorporae asymmeric lepokuric feaures. This means hey may be skewed o he lef and may have a higher peak and wo asymmeric heavier ails han hose of a normal disribuion. This follows by he fac ha markes end o have boh overreacion and underreacion o various good or bad news. Furher, in an opion pricing framework one usually observes a volailiy smile effec i.e. he implied volailiy curve is a convex funcion of he srike price. Boh he lepokuric feaures and he volailiy smile effec can be explained by he asymmeric double exponenial jumpdiffusion model. Second, under a specific change of measure, i is possible o derive in his model closed-form soluions o a number of opion pricing problems such as European call and pu opions as well as pah dependen opions like American or barrier opions cf. Kou 22, Kou and Wang 23, and Kou and Wang 24. The decisive poin, however, is he change of he hisorical probabiliy measure o an equivalen risk-neural probabiliy under which he price process of an opion is calculaed. In Kou 22 he risk-neural measure is chosen considering a raional expecaions equilibrium framework in which an invesor wih a power uiliy funcion ries o maximize his uiliy. Under some resricive assumpions on he model parameers, like ineres rae and drif, he logarihm of he asse price process says in he class of asymmeric double exponenial jump-diffusion processes. The derivaion of he opion pricing formula relies heavily on he disribuional properies of his class cf. Secion 7.2 and i remains unclear wha happens if anoher risk-neural probabiliy is chosen perhaps leading ouside he class of asymmeric double exponenial jump-diffusions. Moreover, i is an open quesion o find a risk-minimizing hedging sraegy under an arbirary chosen risk-neural measure. This is a significan issue, since riskless hedging is no possible. We know ha a Lévy-process financial marke has in general an arbirary number of equivalen risk-neural measures and we are mainly ineresed in he derivaion of locally risk-minimizing hedging sraegies using he Föllmer-Schweizer measure as risk-neural probabiliy. For his reason we will invesigae he behavior of an asymmeric double exponenial jump-diffusion process under an equivalen change of measure o he Föllmer-Schweizer measure and show ha in his case he process says only in his disribuion class, if he corresponding Girsanov parameer is equal o one or if i is equal o zero represening he case where no change of measure is necessary. I seems ha only in hose wo special cases one has an analogous closed-form soluion for he price process. The reason is he effec of he Girsanov parameer i.e. he marke price of risk on he jumpsize disribuion. Since his effec does no depend on a specific disribuion, our reamen is exemplarily for similar cases. Before discussing he change of measure, we review some fundamenal and needed resuls of Kou 22. We imbed he model in our Lévy-process framework of Subsecion and in he financial marke of Secion 2.3.

127 7.2. The asymmeric double exponenial disribuion 123 Finally, we derive exemplarily he price process for an uni-linked pure endowmen and erm insurance wih guaranee. Moreover, we compue he spaial derivaive of he price process. 7.2 The asymmeric double exponenial disribuion Le us begin discussing he asymmeric double exponenial disribuion and some of is main properies. In he following we work on some probabiliy space Ω, G, P. Mos of wha is presened here can be raced back o Kou 22. Definiion A random variable Y is said o follow an asymmeric double exponenial disribuion if i has a densiy wih respec o he Lebesgue measure of he form f Y y = pη 1 e η 1y 1 [, y + qη 2 e η 2y 1, y, η 1 > 1, η 2 >, where p,q and p + q = 1. If Y is an asymmeric double exponenial disribued random variable, hen one has { Y = d ξ +, wih probabiliy p ξ, wih probabiliy q, where ξ + and ξ are exponenial random variables wih means 1 η 1 and 1 η 2, respecively. Furhermore i is no very hard o compue he following momens of Y : E[Y ] = p η 1 q η 2, Var[Y ] = 2p η q η 2 2 p q 2, η 1 η 2 E[e Y ] = p η 1 η q η 2 η The condiion η 1 > 1 in Definiion implies ha E[e Y ] <. As described in he inroducion we will consider jump-diffusions whose jumps follow an asymmeric double exponenial disribuion. I will urn ou o be useful for i o know he disribuion of he sum of asymmeric double exponenial random variables on he one hand and on he oher hand he disribuion of he sum of a normal and a gamma disribued random variable, i.e. he sum of a normal random variable and a sum of exponenial random variables. Lemma Le ξ + and ξ be independen exponenial disribued random variables wih parameers η 1 and η 2, respecively. Then

128 124 Chaper 7. Jump-diffusion sock model a Pξ + > ξ = η 2 η 1 +η 2 and Pξ + < ξ = η 1 η 1 +η 2. b Pξ + ξ x ξ + > ξ = Pξ + x for x. c Pξ + ξ x ξ + < ξ = P ξ x for x. Proof. By symmery i suffices o consider for he firs claim Pξ + > ξ = η 1 e η1u η 2 e η2v du,v = η 2. η 1 + η 2 {u v } For he second par we look for x a Pξ < ξ + x + ξ = and use par a o obain { v<u v+x} = η 2 η 1 + η 2 1 e η 1 x, η 1 e η 1u η 2 e η 2v du,v Pξ + ξ x ξ + > ξ = Pξ < ξ + x + ξ Pξ + > ξ = Pξ + x. Le now x. Then Pξ + ξ x,ξ + < ξ = Pξ x + ξ +,ξ > ξ + and using again par a yields = Pξ ξ + x = η 1 e η1u 1 [, uη 2 e η2v du,v {v u x} = η 1 η 1 + η 2 e η 2x, Pξ + ξ x ξ + < ξ = Pξ x + ξ +,ξ > ξ + Pξ + < ξ = P ξ x. Applying he previous lemma one shows ha he sum of i.i.d. asymmeric double exponenial random variables is equal in disribuion o a randomly mixed gamma random variable. Proposiion Le Y i, i = 1,...,n, n 1, be i.i.d. asymmeric double exponenial random variables. Then k n ξ i +, wih probabiliy P n,k, k = 1,...,n, d Y i = i=1 k i=1 ξi, wih probabiliy Q n,k, k = 1,...,n, i=1

129 7.2. The asymmeric double exponenial disribuion 125 where ξ + i and ξ i are i.i.d. exponenial random variables wih raes η 1 and η 2, respecively. For 1 k n 1 he probabiliies are given by and n 1 n k 1 n P n,k = i k i i=k n 1 n k 1 n Q n,k = i k i i=k Moreover, P n,n = p n and Q n,n = q n. η1 η 1 + η 2 η1 η 1 + η 2 i k η2 η 1 + η 2 n i η2 η 1 + η 2 n i p i q n i, i k p n i q i. Proof. The claim follows by a random walk argumen using Lemma 7.2.2, cf. Kou 22. Proposiion Le X be a normal disribued random variable wih mean and variance σ 2. Le furher ξ i, i = 1,...,n, n 1, be i.i.d. exponenial random variables wih rae η >. Then for he random variables X + n i=1 ξ i and X n i=1 ξ i holds: a The densiy funcions are given by f P σηn X+ n i=1 ξ = i σ /2 2π eση2 e η Hh n 1, σ + ση f P σηn X n i=1 ξ = i σ /2 2π eση2 e η Hh n 1 σ + ση, where he funcion Hh n x for n is defined as follows: Hh n x := and Hh 1 x := e x2 /2. b The ail probabiliies are given by P X + P X i=1 x Hh n 1 ydy = 1 n! x x n e 2 /2 d, n ξ i x = σηn σ /2 2π eση2 I n 1 x; η, 1, σ, ση n ξ i x = σηn σ /2 2π eση2 I n 1 x;η, 1, σ, ση i=1 where I n c;α,β,δ := c e αx Hh n βx δdx, n.

130 126 Chaper 7. Jump-diffusion sock model Proof. We include he proof of Kou 22, since i is no very long. Recall n i=1 ξ i follows a gamma disribuion wih parameers n and η, hence is densiy is Therefore, f P ηn n i=1 ξ x = i n 1! xn 1 e ηx 1, x. f X+ P n i=1 ξ i = = e η η n f P n i=1 ξ i xf Xxdx = e η η n e ση2 /2 e xη x n 1 1 n 1! σ 2π e x n 1 n 1! Subsiuing now y = x σ 2 η/σ yields x 2 2σ 2 dx 1 σ x σ 2 η 2 2π e 2σ 2 dx. /σ y ση n 1 f P X+ n i=1 ξ = i e η e ση2/2 σ n 1 η n σ ση n 1! = σηn σ /2 2π eση2 e η Hh n 1, σ + ση 1 2π e y2 /2 dy where he las sep follows from he subsiuion y = u. The oher densiy is derived analogously noing ha Hence, P n i=1 f P X n i=1 ξ = i eη η n e ση2 /2 x ξ i x = f P n i=1 ξ udu. i x n 1 n 1! Subsiuing y = x + σ 2 η/σ yields he claimed resul. 1 σ x+σ 2 η 2 2π e 2σ 2 dx. For a deailed reamen of he funcions Hh n and I n we refer o Kou 22 who also refers o Abramowiz and Segun Those funcions can be evaluaed using he cumulaive sandard normal disribuion funcion Φ and he following resuls. Obviously, Hh x = 2πΦ x, and furhermore for all n 1 one has ha n Hh n x = Hh n 2 x xhh n 1 x.

131 7.2. The asymmeric double exponenial disribuion 127 Depending on he parameers, he funcion I n can be compued hrough he following relaions. For n i holds ha I n c;α,β,δ = eαc α β + α for α, β > and ha n i= I n c;α,β,δ = eαc α β α β n i Hh i βc δ α n+1 2π β e αδ β + α2 2β 2 Φ n i= β n i Hh i βc δ α n+1 2π β e αδ β + α2 2β 2 Φ for α <, β <. Finally for n one uses ha βc + δ + α β βc δ α β 1 β Hh n+1βc δ for α =, β >, I n c;α,β,δ = for α, β, Hh n δe αc for α <, β =. The nex proposiion is vial for he evaluaion of European call opions o be considered below. In paricular he derivaion of he pricing formula depends on hese resuls. Proposiion Le X be a normal random variable wih mean and variance σ 2 and le ξ i, i = 1,...,n, n N, be i.i.d. exponenial random variables wih rae η. Then, for any real numbers α, β, K R, K/α > and K h = log β, α one ges for n 1 he following expecaions. For η > 1, E[ αe β+x+p n i=1 ξ i K + ] = αe β σηn σ 2π eση2 /2 I n 1 h;1 η, 1 σ, ση and for η >,,, K σηn σ 2π eση2 /2 I n 1 h; η, 1 σ, ση E[ αe β+x P n i=1 ξ i K + ] = αe β σηn σ 2π eση2 /2 I n 1 h;1 + η, 1 σ, ση K σηn σ 2π eση2 /2 I n 1 h;η, 1 σ, ση.

132 128 Chaper 7. Jump-diffusion sock model Proof. Observe ha E[ αe β+x+p n i=1 ξ i K + ] = αe β e f P X+ n X i=1 ξ d K P + i h = αe β σηn σ 2π eση2 /2 K P X + h n ξ i h i=1 n i=1 ξ i h e e η Hh n 1 σ + ση d and ha his expecaion only exiss if η > 1. Moreover for all η > we have ha E[ αe β+x P n i=1 ξ i K + ] = αe β e f P X n X i=1 ξ d K P i h = αe β σηn σ 2π eση2 /2 K P X h n ξ i h. i=1 The claim follows hen from Proposiion The model n i=1 e e η Hh n 1 σ + ση d ξ i h In his secion we discuss he model of Kou 22 in he conex of Subsecion Basic properies of Lévy processes and Secion 2.3 Financial marke. We work on a filered probabiliy space Ω, G,G T, P for some finie number T >. As always in his hesis he probabiliy space is supposed o saisfy he usual condiions and we assume ha G = {,Ω}, G T = G The asymmeric double exponenial jump-diffusion We sar wih a Lévy process, L = L T, which is defined by N L = cw + V i 1, T, 7.2 i=1 where W T is a sandard Brownian moion, c > is deerminisic, N T is a Poisson process wih rae λ and V i is a sequence of i.i.d. nonnegaive random variables such ha Y = logv follows an asymmeric

133 7.3. The model 129 double exponenial disribuion. Moreover, W, N and Y i are supposed o be sochasically independen. As before, we le he filraion G T be he naural, compleed filraion of L T. Lemma The Lévy measure of he process L has he form: where νdx = λf V 1 xdx, f V 1 x = pη x 1+η 1 1 [, x + qη x 1 η 2 1 1, x, wih η 1 > 1, η 2 >, p,q and p + q = 1. f V 1 x denoes he Lebesgue densiy of he jump size disribuion of he process L. Proof. The firs claim follows from Proposiion Observe furher ha he disribuion of V 1 is given by F V 1 x = PV 1 x = F Y logx + 1, x > 1. Le herefore x > 1. Then for he Lebesgue densiy f Y x of Y one has ha which yields he claim. 1 f V 1 x = f Y logx x = pη x e η 1 log1+x 1 [, log1 + x Theorem The decomposiion where + qη x eη 2 log1+x 1, log1 + x, L = cw + M + a, T, N M = V i 1 E[V 1 1]λ, i=1 and a = E[V 1 1]λ is he Lévy-Iô-decomposiion of L. Furhermore, M = M T is a pure-jump maringale and if η 1 > 2, hen M is a squareinegrable maringale. A = A T = a T is predicable and of finie variaion implying L o be a special semi-maringale.

134 13 Chaper 7. Jump-diffusion sock model Proof. The decomposiion follows from Theorem and condiion 2.2. Tha condiion, being x νdx <, { x >1} is saisfied here, since he form of he Lévy measure cf. Lemma implies ha 1+x 1+η 1 is he only relevan par o consider. For his par i holds now ha x1 + x 1+η 1 behaves asympoically as x 1+ε for some ε > and x, since we generally have ha η 1 > 1. Moreover, N 1 a = E[L 1 ] = E[ V i 1] = i=1 n E[ V i 1] PN 1 = n. Using he i.i.d. assumpion of he V i and he fac ha N 1 follows a Poisson disribuion we ge ha a = n=1 n= n E[V 1 1] λn n! e λ = E[V 1 1] i=1 n= λ n+1 n! e λ = E[V 1 1]λ. The funcion A = a is obviously of finie variaion and predicable, since i is monoonically increasing and coninuous. Finally, if η 1 > 2 one has ha x 2 x 2 νdx = λpη x 1+η dx <, 1 { x >1} since x 2 1+x 1+η 1 also behaves asympoically as x 1+ε for some ε > and x. Hence in his case M is square-inegrable cf. Lemma We remark ha he predicable par A also could have been compued using he Lévy measure, ha is, a = E[L 1 ] = xνdx = λ xf V 1 xdx. R This shows again he necessiy demanding η 1 > 1, since E[V ] = E[e Y ] = p η 1 η q η 2 η Financial marke Le us now consider he financial marke of Secion 2.3 wih he Lévy process L of 7.2. Using he sochasic differenial equaion 2.13 we describe he sock S = S T via N ds = bs d + cs dw + S d V i 1, 7.3 R i=1

135 7.3. The model 131 where S > is deerminisic and b, modeling he drif of he sock-price movemen, is supposed o be a deerminisic real consan. Also recall ha for he risk-free ineres rae r he riskless invesmen in his financial marke is given by B = e r, T. Lemma The process S has he following explici form: S = S exp cw + b c2 N + Y i, T. 2 By convenion we se i=1 V i = 1. Proof. The explici formula for he sock price is generally formulaed in 2.14 and reduces in our case o S = S exp i=1 L + b c2 1 + L s exp L s. 2 <s Ns Having now L s = i=1 V i 1 = V Ns 1 N s, we ge ha S = S exp cw + N k=1 = S exp cw + b c2 N V i. 2 V k 1 + b c2 N N V i exp V j 1 2 i=1 The claim follows, since Y i = logv i. To ge an impression of he sock price behavior Figure 7.1 exemplarily displays four sample pahs for a fixed se of parameers. From he previous lemma we see ha he process S is he exponenial funcion of he asymmeric double exponenial jump-diffusion process θ+cw + N i=1 Y i, where θ = b c2 2. However, i is defined in 7.3 as he sochasic exponenial of he Lévy process L = b + L, i.e. S = E L. The consrucion of S illusraes herefore Lemma 5.8 in Goll and Kallsen 2 which shows ha E L = e X, where X = L c2 2 + <s i=1 log1 + Ls L s N = θ + cw + Y i = log E L. i=1 j=1

136 132 Chaper 7. Jump-diffusion sock model S S S S Figure 7.1: Sock price sample pahs. Parameers: S = 1, c =.2, b =.5, λ = 5, η 1 = 17, η 2 = 15, p =.5. Moreover, Goll and Kallsen 2 sae a formula for he Lévy measure of X which we denoe by ν X. According o his formula one has for any 1 < a < b ha ν X a,b = 1 a,b log1 + xνdx = 1 log1+a,log1+b uλf V 1 e u 1e u du = 1 log1+a,log1+b uλf Y udu Of course, in our case we already know he Lévy measure of he jumpdiffusion process X. I is given by ν X dx = λf Y xdx cf. Proposiion As always for he financial marke we require addiionally he condiion x 3 νdx < { x >1} o be saisfied. I is mainly formulaed in 2.12 o guaranee he squareinegrabiliy of he maringale M boh under he hisorical and under he

137 7.4. Risk-neural measure 133 Föllmer-Schweizer measure. The following lemma provides a sufficien condiion. Lemma If η 1 > 3, hen condiion 2.12 is saisfied. Proof. We use he same argumen as before. For x > 1 i holds ha νdx = λpη x 1+η 1 dx. If η 1 > 3, hen x x 1+η 1 behaves asympoically as x 1+ε for ε > and for x which yields he claim. 7.4 Risk-neural measure Here we work again on Ω, G,G T, P of he previous secion Föllmer-Schweizer measure As before in his hesis, Q denoes he Föllmer-Schweizer measure and Z T he Radon-Nikodým derivaive of Q wih respec o P. From 2.15 we know ha he densiy process Z = E[Z T F ], T, saisfies dz = GZ cdw + dm, where he Girsanov parameer G cf has he form r b + a G = c 2 wih v = x 2 νdx. + v Lemma The explici represenaion of Z is given by Z = exp cgw c2 G 2 2 Z is posiive for < G 1. N e λg E[V 1 1] R 1 + GV i 1, T. Proof. The formula for he sochasic exponenial cf. Proer 24, Chaper II.8 ogeher wih Theorem and M s = V Ns 1 N s yields Z = exp GcW + M c2 G 2 2 k=1 i=1 <s 1 + G M s exp G M s = exp cgw c2 G 2 N 2 + G V k 1 λg E[V 1 1] N 1 + GV i 1exp G N V j 1, i=1 from which he claim follows. Furhermore, since he random variables V i are concenraed on [,, we have ha Z is sricly posiive if < G 1. j=1

138 134 Chaper 7. Jump-diffusion sock model Noe ha if G =, hen Z 1 and no change of measure is performed. Moreover, i is quie easy o check ha he process Z is indeed a densiy. Using he independence, for every T one has ha E[Z ] = E[exp cgw c2 G 2 2 N ]e λg E[V1 1] E[ 1 + GVi 1 ] i=1 k = e λg E[V 1 1] λ λk E[ 1 + GV i 1]e k! k= i=1 = e λ1+g E[V 1 1] 1 + G E[V 1 1] k λ k k! = 1. k= The general heory of Subsecion yields now he following heorem, which describes he Lévy process L under he measure Q. Theorem Under he measure Q, L denoed by L Q is again a Lévy process and has he following Lévy-Iô-decomposiion L Q = cw Q + M Q + A Q, where cw Q = cw c 2 G is a Brownian moion under Q. For he maringale M Q one has ha N M Q = V i 1 xνdx Gx 2 νdx i=1 R R N = V i 1 xν Q dx. i=1 R and he Lévy measure ν Q dx = 1 + Gxνdx. M Q is a square-inegrable, quadraic pure-jump maringale. Furhermore, A Q = r b is predicable and of finie variaion. Alhough he previous heorem follows from he general heory we consider he nex lemma which complemens he heorem s asserion and explains furher his specific change of measure. Lemma Under he measure Q we have ha a cw Q = cw c 2 G, T, is a Brownian moion wih variance c 2. b N i=1 V i 1, T, is a compound Poisson process wih rae E Q [N ] = 1 + G E[V 1 1]λ =: λ Q,

139 7.4. Risk-neural measure 135 and N M Q = V i 1 E Q [V 1 1]λ Q. i=1 c he Lévy measure of L is given by where ν Q dx = λ Q f Q V 1 xdx, f Q V 1 x = λ λ Q1 + Gxf V 1x. Proof. a This is a sandard argumen and is repeaed for compleeness only. Girsanov s heorem for local maringales cf. Jacod and Shiryaev 23, Theorem III.3.11 saes ha cw Q 1 := cw d cw,z Z s, T, s is a local Q-maringale. Recalling he definiion of Z we compue under P cw,z = cw,g Z s dcw s + M s = c2 G Z s ds. We used ha cw,cw = c2 and cw,m =, since M is quadraic pure-jump and W is coninuous. The associaiviy of sochasic inegrals yields hen cw Q = cw c 2 G. Furher, W Q is Q-a.s. coninuous and E Q [cw Q 2 ] = E Q [cw c 2 G 2 ] = c 2 E[W cg 2 Z ] = c 2 1 2π R = c 2 = cw Q,cW Q. x cg 2 e x cg2 2 Lévy s heorem cf. Proer 24, Chaper II, Theorem 38 yields hen ha W Q is a sandard Q-Brownian moion. b Le T. Firs we compue he expecaion of N under Q. E Q [N ] = E[N Z ] = e λ1+g E[V 1 1] = 1 + G E[V 1 1]λ =: λ Q. dx k 1 + G E[V 1 1] k λ k k= k!

140 136 Chaper 7. Jump-diffusion sock model Le now H := N i=1 V i 1 and g j := 1 + GV j 1, hen N E Q [e iuh ] = E[e iuh Z ] = E[e iuh e λ λq g j ] N = e λ λq E[ = e λq k= = e λq exp = exp R j=1 e iuv j 1 g j ] j=1 k E[e iuv1 1 λ k g 1 ] R k! e iux λ1 + Gxf V 1 xdx e iux 1 ν Q dx. This shows he characerisic funcion of H is ha of a compound Poisson process. c The Lévy measure of L under Q is given by cf. Proposiion ν Q dx = λ Q f Q V 1 xdx. Togeher wih ν Q dx = 1 + Gxνdx = λ1 + Gxf V 1 xdx we have f Q V 1 x = λ λ Q1 + Gxf V 1x. Before proceeding wih he nex lemma, le us remark ha R νq dx = λ Q, which is he expeced resul, and moreover ha = E Q [V 1 1]λ Q = p η 1 1 R q η G xν Q dx 2p η 1 1η q η 2 + 1η λ. The nex lemma shows ha under he measure Q he sock price is he exponenial funcion of a jump-diffusion process like under P. Lemma The explici represenaion of S under Q is given by S = S exp cw Q N + i=1 Y i + r c2 2 EQ [V 1 1]λ Q, T. Proof. We apply he sochasic differenial equaion 7.3 wih L Q. In anal-

141 7.4. Risk-neural measure 137 ogy o he proof of Lemma we have, using Theorem 7.4.2, ha S = S exp L Q + b c2 1 + L Q 2 s exp L Q s <s = S exp cw Q + M Q + r c2 2 <s = S exp cw Q + r c2 2 EQ [V 1 1]λ Q 1 + M Q s exp M Q s N V i. Recall ha he Föllmer-Schweizer measure is chosen such ha he discouned sock price Ŝ := B 1 S is a maringale under his measure. In our case his is relaively simple o check. One jus uses he independen and saionary incremens and immediaely ges ha i=1 E Q [ Ŝ Ŝ s F s ] = 1, s T. In he inroducion we already menioned ha for he opion pricing formula of Kou 22 i is fundamenal o say in he class of asymmeric double exponenial jump diffusions afer he change of measure. According o Lemma he sock price under Q is he exponenial funcion of a jump-diffusion process. I is now he quesion, if his is again an asymmeric double exponenial jump-diffusion. This can be answered if one knows he jump-size disribuion, ha is, he disribuion of Y i under Q. Lemma The Q-disribuion of he random variables Y i is described by he following Lebesgue densiy: f Q Y y = p λ λ Q 1 Gη 1 e η1y + G η 1 η 1 1 η 1 1e η 1 1y 1 [, y + q λ λ Q 1 Gη 2 e η 2y + G η 2 η η 2 + 1e η 2+1y 1, y. Proof. The cumulaive disribuion funcion of Y under Q is given by Hence, F Q Y y = QY y = QV 1 ey 1 = F Q V 1 ey 1. f Q Y y = f Q V 1 ey 1e y = λ λ Q1 + Gey 1e y f V 1 e y 1 = λ λ Q1 + Gey 1e y pη 1 e 1+η1y 1 [, y + qη 2 e η2 1y 1, y. Simplifying his expression yields he claimed densiy.

142 138 Chaper 7. Jump-diffusion sock model Recall ha we require G 1. Unforunaely for G,1 he densiy of Y under Q is no convenien for furher explici compuaions. Boh he upward and he downward jump follow a cerain linear combinaion of he densiies of wo differen exponenial disribuions. This shows ha he Föllmer-Schweizer change of measure does generally no resul anymore in an asymmeric double exponenial jump-diffusion model. In hese cases we can herefore no expec o obain a similar closed-form opion pricing formula as Kou 22. This has several reasons. Firs, for an asymmeric double exponenial random variable, he sum of upward and he sum of downward jumps follow a gamma disribuion, respecively. This is he opion pricing formula s basis relying on he disribuion of he sum of a normal random variable and a gamma disribued random variable cf. Proposiion Under Q, however, we neiher generally know he disribuion of he sum of upward nor he sum of downward jumps. Second, a furher key feaure is he memoryless propery of he exponenial disribuion cf. Lemma which is used o spli he sum of asymmeric double exponenial random variables wih a cerain probabiliy and in disribuion ino a sum of upward and a sum of downward jumps. Such a memoryless propery does in our case neiher hold for he upward nor for he downward jump disribuion. I seems ha o compue expecaions under Q for G,1, involving he sock S, one is forced o apply Mone-Carlo simulaion echniques using Lemma Having an absoluely coninuous disribuion wih respec o he Lebesgue measure Lemma ogeher wih he inversion mehod can be applied o simulae Y. There remain wo special cases for which one says in he class of asymmeric double exponenial jump-diffusions. If G =, hen one does no perform a change of measure or, in oher words, he hisorical measure is already he risk-neural measure. If G = 1, hen we ge he following Corollary. Corollary Le G = 1. Then Y follows under Q an asymmeric double exponenial disribuion wih densiy f Q Y y = pq η 1 1e η 1 1y 1 [, y + q Q η 2 + 1e η 2+1y 1, y, where p Q := p λ λ Q η 1 η 1 1 and q Q := q λ λ Q η 2 η Le us discuss his a lile bi furher assuming from now on G = 1. Observing λ Q = λ E[V ], one immediaely ges ha p Q + q Q = λ E[V ] = 1. λq Moreover, one has ha cf. 7.1 E Q [V ] = p Qη 1 1 η qqη η 2 + 2,

143 7.4. Risk-neural measure 139 and ha, under Q, { Y = d ξ +, wih probabiliy p Q ξ, wih probabiliy q Q, where ξ + and ξ are exponenial random variables wih parameers η 1 1 and η 2 + 1, respecively. For pracical purposes he assumpion G = 1 is very resricive. I implies consrains on he parameers of he financial marke such as ineres rae r, drif b, volailiy c 2 + v and jump sizes influenced by a. In paricular, his assumes a marke price of risk equal o 1. From a heoreical poin of view, his special case offers an ineresing insigh in his change of measure. The subjecive probabiliies of an upward or a downward jump p and q change o he risk-neural probabiliies p Q and q Q. Noe here ha λ/λ Q, η 1 /η 1 1 and η 2 /η are he proporions of he subjecive parameers λ, η 1, η 2 o he risk-neural parameers λ Q, η 1 1, η 2 + 1, respecively Kou s risk-neural measure Finally we consider he change of measure in Kou 22 and compare he resuling jump-size disribuion wih he jump-size disribuion derived under he Föllmer-Schweizer measure. We already menioned ha in Kou 22 he choice of he risk-neural measure is based on a raional expecaions equilibrium framework, where he invesor uses he power uiliy funcion Uc, = e θ cγ γ for some < γ < 1. According o Kou 22, his resuls in he following densiy: Z Kou = exp cγ 1W 1 2 c2 γ 1 2 λ E[V γ 1 ] 1 N i=1 V γ 1 i. In equaion 2.17 of Subsecion we briefly considered he general sochasic differenial equaion describing he densiy of any equivalen maringale measure in a Lévy-process financial marke. If he Girsanov parameers are ime-independen, hen is explici expression is given by Z Kou = exp cgw 1 N 2 c2 G 2 Hx 1 νdx R i=1 HV i 1 for some G R and some Borel-measurable funcion H : R [,, provided ha H = 1. I is an equivalen maringale measure only if he Girsanov parameers saisfy he necessary maringale condiion 2.18 for he sock: c 2 G + b r + xhxνdx =. 7.4 R

144 14 Chaper 7. Jump-diffusion sock model We recall ha he choice Hx = 1 + Gx wih G = r b+a belongs o he c 2 +v Föllmer-Schweizer measure. Comparing now he general form of he densiy wih he densiy of Kou 22 yields ha G Kou = γ 1 and H Kou x = 1 + x γ 1 for a γ such ha 7.4 is saisfied. Noe ha λe[v γ 1 ] 1 arises from he following equaion H Kou x 1 νdx = 1 + x γ 1 1 λf V 1 xdx. R In analogy o Lemma we derive ha Hence, f Q,Kou Y y = f Q,Kou V 1 ey 1e y R f Q,Kou V 1 x = λ λ QHKou xf V 1 x. = p λ λ Q η 1 η γ η γe η 1+1 γy 1 [, y + q λ λ Q η 2 η γ η γe η 2 1+γy 1, y. 7.5 This resaes he resul of Kou 22 ha for any γ one says in he class of asymmeric double exponenial jump-diffusions. However, recall ha < γ < 1 has o be chosen such ha equaion 7.4 is saisfied, which is no an easy ask. In Kou 22 his condiion is formulaed differenly and i is presupposed o be saisfied by γ. However, finding or compuing γ remains an open quesion. One migh argue ha he specificaion of γ is a problem in he same way as i is for our choice G = 1 for he Föllmer-Schweizer measure. In any case, if he parameers η 1 and η 2 are no oo small he densiy 7.5 is similar o he densiy in Corollary Arbirage-free price process We work now on he probabiliy space Ω, G,G T, Q, where Q is he Föllmer-Schweizer measure wih G = 1. Wih his assumpion he sock price S is he exponenial funcion of an asymmeric double exponenial jump-diffusion Valuaion of an European Call Le us consider an European call opion which is due a ime u, u T, wih payoff S u K + for some deerminisic srike price K >. For his opion we will formulae he risk-neural price formula of Kou 22 in he nex heorem. We derive i in deail, since he formula is quie advanced and in our case he jump-size disribuion is slighly differen from he one in Kou 22. Noe ha he involved series are quickly convergen.

145 7.5. Arbirage-free price process 141 Theorem A ime < u T, he arbirage-free price process of S u K + is given by F,S,u = e ru E Q [S u K + G ] F,S,u = π,u S e λq µu Φd +,u Ke ru Φd,u n + π n,u P Q n,k η1 1c u k n=1 S e λ Q µ+ c2 2 + π n,u n=1 S e λ Q µ+ c2 2 k=1 u a1,u Ke ru b 1,u n k=1 Q Q n,k η2 + 1c u k u a2,u Ke ru b 2,u, where Φx = 1 x 2π e y2 /2 dy is he cumulaive probabiliy funcion of a sandard normal disribuion, log S K + r ± c2 2 λq µ u d ±,u = c, µ = E Q [V ] 1, u and Furhermore, wih λ Q u n π n,u = QN u = n = e λq u. n! 1 a 1,u = f 1,uI k 1 h,u;2 η 1, c u,1 η 1c u, 1 b 1,u = f 1,uI k 1 h,u;1 η 1, c u,1 η 1c u, 1 a 2,u = f 2,uI k 1 h,u;2 + η 2, c u, 1 + η 2c u, b 2,u = f 2,uI k 1 h,u;1 + η 2, f 1,u = ec1 η 1 2 u /2 c, 2πu f 2,u = ec1+η 2 2 u /2 c, 2πu 1 c u, 1 + η 2c u

146 142 Chaper 7. Jump-diffusion sock model and K h,u = log + λ Q µ r + c2 u. S 2 The probabiliy weighs are for 1 k n 1 as follows: n 1 P Q n,k = n k 1 n η1 1 i k η2 + 1 n i p Qi q Qn i i k i η 1 + η 2 η 1 + η 2 and i=k n 1 Q Q n,k = n k 1 n η1 1 n i η2 + 1 i k p Qn i q Qi. i k i η 1 + η 2 η 1 + η 2 i=k For k = n one has ha P Q n,n = p Qn and Q Q n,n = q Qn. Proof. Le < u T. Using he saionary and independen incremens we ge ha F,S,u = e ru E Q S u + [ S K G ] S + = e r u E Q [ S e λq µu e X u N u K G ], where X u n = r c2 n u + cw Q u 2 + Y i. 7.6 Applying Proposiion and he Poisson disribuion of N u wih rae λ Q u we obain ha F,S,u = e ru n= i=1 + π n,u E Q [ S e λq µu e Xu n K G ] + = π,ue ru E Q [ S e λq µu e Xu K G ] n + + e ru π n,u P Q n,k EQ [ S e λq µu e X+ u k K G ] where n=1 + k=1 n + Q Q n,k EQ [ S e λq µu e X u k K G ], k=1 X u ± k = r c2 k u + cw Q u 2 ± ξ i ±. i=1

147 7.5. Arbirage-free price process 143 ξ + and ξ are exponenial random variables wih rae η 1 1 and η 2 + 1, respecively. We evaluae now each of he hree expecaions above separaely. Leing α = S e λq µu and β = r c2 2 u we apply Proposiion which yields + E Q [ S e λq µu e X+ u k K G ] = η 1 1c u k e ru S e λ Q µ+ c2 u 2 a1,u Kb 1,u and + E Q [ S e λq µu e X u k K G ] = 1 + η 2 c u k e ru S e λ Q µ+ c2 u 2 a2,u Kb 2,u, where K K h,u = log β = log + λ Q µ r + c2 u. α S 2 Finally, he Black-Scholes formula implies + e ru E Q [ S e λq µu e Xu K G ] = S e λq µu Φd +,u Ke ru Φd,u. This complees he proof. Corollary Le < u T and le wih F BS,S,u = S Φd BS +,u Ke ru Φd BS,u, d BS ±,u = log S K + r ± c2 2 u c u he Black-Scholes opion pricing formula. Then one has ha a F,S,u = F BS,S,u for λ =. b F,S,u F BS,S,u for η 1 and η 2. Proof. The firs par is obvious. Since he process X in 7.6 converges in disribuion o a Brownian moion wih drif if boh η 1 and η 2 end o infiniy, he second claim follows. For deails we refer o Kou 22.

148 144 Chaper 7. Jump-diffusion sock model Valuaion of uni-linked pure endowmen and erm insurance wih guaranee Le us sar wih an uni-linked pure endowmen wih guaranee, for which he insurance company pays o he insured, if alive a ime T, he amoun gs T = maxs T,K for some deerminisic capial guaranee K >. As in Chaper 4 we se u = T in his case and suppress he explici dependence on he due dae u of he insurance claim. Corollary A ime T, he arbirage-free price process of an uni-linked pure-endowmen wih guaranee K > is given by F pe,s = F,S + Ke rt, where F,S is aken from Theorem Proof. The claim follows immediaely from Theorem once noiced ha maxs T,K = S T K + + K. In case of an uni-linked erm insurance wih guaranee he conrac funcion is ime dependen and addiionally we assume ha he capial guaranee K > earns ineres wih some risk-free ineres rae δ >. Hence, he conrac funcion is gu,s u = maxs u,ke δu for u T. Corollary A ime < u T, he arbirage-free price process of an uni-linked erm-insurance wih guaranee Ke δu is compuable via he formula: F i,s,u = F,S,u + Ke δ ru+r, where F,S,u is aken from Theorem wih he following adapaions log S Ke + r ± c2 d ±,u = δu 2 λq µ u c, u Ke δu h,u = log + λ Q µ r + c2 u. S 2 Proof. Analogously o Corollary he claim follows from Theorem Spaial derivaive In he previous chapers we saw ha in a Lévy-process financial marke he derivaive of he arbirage-free price process wih respec o is space variable is required. This derivaive occurs wih he Galchouk-Kunia-Waanabe decomposiion cf. Theorem of he arbirage-free price process and corresponds o he well-known dela in a Black-Scholes financial marke. We recall from Chaper 3 ha for x > and < u T we have ha F,x,u = x e ru E Q [gu,s u S = x].

149 7.5. Arbirage-free price process 145 Corollary For < u T i holds F,S,u = π,ue λq µu Φd +,u n + π n,u P Q n,k η1 1c u k n=1 k=1 {e λ Q µ+ c2 u 2 a1,u + e λ Q µ+ c2 u +2 η 2 1 h,u K e ru +1 η 1h,u S h,u f 1,uHh k 1 c u + η 1 1c } u n + π n,u η2 + 1c u k Q Q n,k n=1 k=1 {e λ Q µ+ c2 u 2 a2,u + e λ Q µ+ c2 u +2+η 2 2 h,u K e ru +1+η 2h,u S f 2,uHh k 1 h,u c u η 2c u Proof. We consider he formula from Theorem and compue firs x π,u xe λq µu Φd +,u Ke ru Φd,u x=s = π,u e λq µu Φd +,u + xe λq µu ϕd +,u Ke ru ϕd,u x d +,u, x=s where ϕy = 1 2π e 1 2 y2 denoes he densiy of he sandard normal disribuion. Moreover, he derivaives of d +,u and d,u coincide, since d,u = d +,u c u. I holds now ha and hence ha d +,uc u = log S K + r + c2 2 λq µ } u, S e λq µu 1 2π e 1 2 d2 +,u Ke ru 1 2π e 1 2d +,u c u 2 = 1 e 1 2 d2,u d + S e λq µu Ke ru e +,uc u c2 2 u 2π = 1 2π e 1 2 d2 +,u S e λq µu Ke ru S K eru e λq µu =.

150 146 Chaper 7. Jump-diffusion sock model This yields he firs erm of he derivaive and we proceed in compuing he derivaive of he second summand of F,S,u. xe λ Q µ+ c2 2 x = e λ Q µ+ c2 2 u a1,u Ke ru b 1,u u a1,u + xe λ Q µ+ c2 u 2 Ke ru x b 1,u. x=s For x = S one obains ha x a 1,u = f 1,u x I k 1 x h,u. h,u;2 η 1, x=s x a 1,u x=s 1 c u,1 η 1c u Since y I ny;α,β,δ = y e αx Hh n βx δdx = e αy Hh n βy δ y and x h,u x=s = 1. S we ge for x = S ha x a 1,u = f 1,u h,u e 2 η1h,u Hh k 1 c u + η 1 1c u. S Analogously for x = S we have ha x b 1,u = f 1,u h,u e 1 η1h,u Hh k 1 c u + η 1 1c u. S I remains o consider he derivaive of he las summand of F,S,u. xe λ Q µ+ c2 2 x = e λ Q µ+ c2 2 u a2,u Ke ru b 2,u u a2,u + xe λ Q µ+ c2 u 2 Ke ru x b 2,u. x=s x=s x a 2,u x=s Performing for x = S similar compuaions as for a 1,x and b 1,x we ge ha x a 2,u = f 2,u h,u e 2+η2h,u Hh k 1 S c u η 2c u x b 2,u = f 2,u h,u e 1+η2h,u Hh k 1 c u η 2c u. S This finally yields he resul.

151 Chaper 8 Saisics for jump-diffusions 8.1 Inroducion In his chaper we consider parameer esimaion for he asymmeric double exponenial jump-diffusion sock model reaed in he previous chaper. However, he presened heory, including he esimaion mehod, is o a large exen independen of a concree jump-size disribuion and hence applicable o various ypes of jump-diffusions, for which i seems ha saisfying parameer esimaion mehods are almos no available. The paricular difficuly of saisical inference for jump-diffusions having only finiely many observaions is o discriminae he diffusion and he jump par of he process. I ofen remains unobserved, wheher a discree realizaion sems from a large movemen of he diffusion or from a jump. Aï-Sahalia 24 shows ha maximum likelihood esimaion is heoreically able o perfecly disenangle Brownian noise from jumps provided one samples frequenly enough or, in oher words, a leas in he limi of infiniely frequen sampling. Usually jump-diffusions do no exhibi a ransiion probabiliy densiy in closed form. Insead heir densiy is only expressible as a series of improper convoluion inegrals coming from he summaion of he relaed random variables. This funcion may be non-concave and may have several local maxima in a several dimensions spanning parameer space. Thus, i is far from rivial o apply numerical mehods for maximum likelihood esimaion in his case. Anoher disadvanage is ha he ransiion densiy may be unbounded, which is discussed in Honoré 1998 in connecion wih pifalls of maximum-likelihood esimaion for he Meron jumpdiffusion model. We propose o rely he esimaion on he characerisic and he cumulan generaing funcion which are boh explicily known for jump-diffusions and which carry he same amoun of informaion like he probabiliy densiy. Because of he saionary and independen incremens, he logarihmic sock reurns consiue an i.i.d. sample of he jump-diffusion. Comparing

152 148 Chaper 8. Saisics for jump-diffusions his sample s empirical wih he heoreical cumulan generaing funcion in heir squared disance leads o a sysem of nonlinear regression problems, whose componens arise from he real and imaginary par of hese funcions. The regression models can be solved using sandard numerical rouines and in case he expeced jump sizes are known, hey even reduce o linear regression problems. We demonsrae our esimaion mehod wih examples of wo special cases, whereas we pospone he more involved numerical reamen of he general case o fuure research. To jusify our approach, we discuss srong consisency of he empirical characerisic funcion and show ha in our case an invariance principle is applicable leading o a complex-valued Gaussian process in he disribuional limi. Using he laer we derive he asympoic finie-dimensional disribuions of he empirical characerisic funcion and formulae, moreover, a heorem for he asympoic finie-dimensional disribuions of he complex residuals of our nonlinear regression problem including heir covariance srucure. Belomesny and Reiß 26a,b also base he calibraion of exponenial jump-diffusion models on he cumulan generaing funcion and esimae he Lévy measure of he process in a nonparameric way applying Fourier inversion. However, hey use opion daa o draw inference on he underlying risk-neural price process of he sock, ha is, hey do no calibrae he model under he hisorical measure. Hence, heir approach can neiher be applied o asse models for which opion daa is no available nor o siuaions in which a paricular risk-neural measure such as he Föllmer- Schweizer measure for hedging purposes is chosen. 8.2 Model of sock price and reurns Suppose we are given a filered probabiliy space Ω, G,G T, P for some finie number T > which suppors he following sochasic process: S = S exp cw + b c2 N + Y j, T, for some S >. Like in Chaper 7, W T is a sandard Brownian moion, c > is deerminisic, N T is a Poisson process wih rae λ and Y i is a sequence of independen and asymmeric double exponenial disribued random variables cf. Secion 7.2 and Lemma Recall Definiion in which he probabiliy densiy of Y 1 is given by f Y y = pη 1 e η 1y 1 [, y + qη 2 e η 2y 1, y, η 1 >, η 2 >, 8.2 where p,q and p + q = 1. Only for echnical reasons we addiionally required η 1 > 1 or η 1 > 3 in Chaper 7. We assume now ha we observe S j=1

153 8.2. Model of sock price and reurns 149 a n > poins in ime, say < 1 <... < n T, and se S k := S k, k n, for noaional convenience. Given hese discree observaions heir relaive reurn is usually defined by R k = S k S k 1 S k 1 = S k S k 1 1, k = 1,...,n. Because of he exponenial form of he asse price i is more convenien o apply he logarihmic reurn, commonly known as log-reurn, which is of he form: X k = log1 + R k = log S k, k = 1,...,n. S k 1 Of course, considering X k insead of R k has o be jusified. A simple jusificaion is he following. The order of R k is ypically less han 1 2 and in his case a simple Taylor series expansion of log1 + R k immediaely yields X k R k R2 k 2 + R k 3 + R4 k , which shows ha his difference is numerically negligible. Moreover, he independen and saionary incremens of he Brownian moion and of he compound Poisson process in 8.1 imply on he one hand ha he X k, k = 1,...,n, are independen and on he oher hand ha, wihou loss of generaliy, X k d = L1 := cw 1 + b c2 2 N 1 + Y j, k = 1,...,n. 8.3 If k k we rescale he relevan parameers by k k 1 =. The random variable L 1 is a sum of a normal disribued random variable and a Poisson random sum having a cerain jump-size disribuion. I follows immediaely from he convoluion of he corresponding densiies ha he probabiliy densiy of L 1 is only expressible as a series involving erms of improper inegraion, which are no expressible in closed form cf. Proposiion This makes i almos impossible o apply maximum-likelihood esimaion, since he local behavior of he densiy in he six dimensional parameer space j=1 Θ = R,, 1,, [,1] of he parameer θ = b,c,λ,η 1,η 2,p is no known. We do know, however, he characerisic funcion of L 1, which is described by he Lévy-Khinchin formula discussed in Theorem and which sores he same amoun of informaion like he densiy. We have ha φu := φ 1 u = E [ e iul 1 ] = e ψu, u R,

154 15 Chaper 8. Saisics for jump-diffusions where ψu = 1 2 c2 u 2 + iau + e iux 1 iux νdx R is he characerisic exponen, also called he cumulan generaing funcion. Noe ha condiion 2.2 is saisfied here. Lemma The cumulan generaing funcion of L 1 has he following explici form: ψu, θ = u 2 c λ p η q u2 η2 2 + u2 + iu b c2 2 + λ pη1 η1 2 + qη 2 u2 η u2 Proof. We firs observe ha a = E[L 1 ] = b c2 + λ E[Y 1 ] 2 cf. he discussion afer Theorem Moreover, from Proposiion we know ha νdx = λf Y xdx yielding e iux 1λf Y xdx = λpη 1 e iu η1x dx + λqη 2 e iu+η2x dx λ R pη1 = λ η 1 iu + qη 2 η 2 + iu 1, and Hence, R iuxλf Y xdx = iuλ E[Y 1 ]. ψu = 1 2 c2 u 2 + iu b c2 pη1 + λ 2 η 1 iu + qη 2 η 2 + iu 1. We use now p + q = 1 o obain ha pη 1 η 1 iu + qη 2 η 2 + iu 1 = ipu η 1 iu Furher for he complex numbers i holds ha and Finally he claim follows. ipu η 1 iu = u2 p η i pη 1 u u2 η1 2 + u2, iqu η 2 + iu = u2 q η2 2 + i qη 2 u u2 η2 2 + u2. iqu η 2 + iu.

155 8.3. Esimaion mehod 151 For noaional convenience we le ψ 1 u, θ := u 2 c λ be he real and p η u2 q η2 2 + u2 ψ 2 u, θ := u b c2 2 + λ pη1 η1 2 + qη 2 u2 η2 2 + u2 be he imaginary par of ψu, θ. 8.3 Esimaion mehod Our aim is o esimae θ Θ for he random variable L 1 of he previous secion. As already menioned, maximum likelihood esimaion is hard o apply, so we concenrae on he characerisic funcion insead. For a given random sample X 1,...,X n of i.i.d. observaions of L 1 we firs define he empirical characerisic funcion. Definiion The empirical characerisic funcion ˆφ n u of i.i.d. random variables X 1,...,X n is defined by where ˆφ n u = ˆF n x = 1 n e iux d ˆF n x, 8.4 n 1X k x, x R, k=1 is he empirical disribuion funcion of X 1,...,X n. Equaion 8.4 immediaely implies ha ˆφ n u = 1 n n e iux k = 1 n k=1 n cosux k + i 1 n k=1 n sinux k. To ge an esimae for θ, our idea is o compare he empirical wih he heoreical characerisic funcion. Since he cumulan generaing funcion ψ already carries all significan informaion, we consider his funcion insead of he characerisic funcion which leads o a sysem of nonlinear regression models. Le C := C \ {x + iy : x,],y = } be he complex plain excluding he negaive real line and le Log : C C be he complex logarihm. Recall ha for any z C wih represenaion z = z e i Arg z one has ha Log z = log z + iarg z, where Arg z π,π. Using his lile review of he complex logarihm we define he empirical cumulan generaing funcion. k=1

156 152 Chaper 8. Saisics for jump-diffusions Definiion The empirical cumulan generaing funcion ˆψ n u of i.i.d. random variables X 1,...,X n is defined by ˆψ n u = Log ˆφ n u = log ˆφn u + iarg ˆφn u. Furhermore, we le ˆψ n,1 u = log ˆφn u and ˆψn,2 u = Arg ˆφ n u. Le us express ˆψ n,1 u and ˆψ n,2 u explicily: ˆψ n,1 u = log 1 n 2 1 n 2 n 2 cosux k + n 2 sinux k k=1 k=1 k=1 = log n + 1 n 2 n 2 2 log cosux k + sinux k and n arcan k=1 sinux k n k=1 cosux k ˆψ n,2 u = n k=1 arcan sinux k n k=1 cosux k, if + π, if k=1 n k=1 cosux k > n k=1 cosux k <. Following our idea we now compare ψu, θ wih ˆψ n u in heir squared disance, ha is, ψu, θ ˆψ n u 2 = ψ 1 u, θ ˆψ 2 2, n,1 u + ψ 2 u, θ ˆψ n,2 u which should be as small as possible for an esimaed ˆ θ, for each u R. Figure 8.1 shows plos of he real par ˆψ n,1 u lef column and he imaginary par ˆψ n,2 u righ column of he empirical cumulan generaing funcion for an i.i.d. sample of L 1 random variables. The dashed lines represen he corresponding real and imaginary pars, ψ 1 u, θ and ψ 2 u, θ, of he rue cumulan generaing funcion, respecively. The firs row of Figure 8.1 clearly indicaes ha for large absolue values of u he empirical funcion sars o blur. For ha reason we sugges o reduce is suppor o some compac inerval [ U,U] on which hose funcions are considered and on which he parameer esimaion is based. For he suppor reducion we propose o choose U such ha he slope of ˆψ n,1 u does no change is sign on [ U,U]. The second row of Figure 8.1 picures he resul for he reduced suppor. Now we discreize his suppor by an equidisan pariion U u 1 <... < u m U leading generally o a sysem of wo nonlinear regression models of he form: ˆψ n,l u j = ψ l u j, θ + ε n,l u j, j = 1,...,m, l = 1,2, 8.5

157 8.3. Esimaion mehod 153 psi_ psi_ u u psi_ psi_ ur ur Figure 8.1: Empirical vs. heoreical cumulan generaing funcion. Lef column: Real par. Righ column: Imaginary par. Dashed line: Theoreical funcions. Parameers: Sample size n = 1, c 2 =.2, b =.8, λ = 3, η 1 = η 2 = 1, p =.4. for some residuals ε n,l u, l = 1,2. Le us consider hese regression models a lile bi more deailed saring wih wo special cases. Special case 1 Assuming ha he asymmeric double exponenial disribuion underlying L 1 saisfies η 1 = η 2 = ρ, where ρ > is known, leads in fac o a sysem of wo linear regression models, which are: ˆψ n,1 u j = c 2 u 2 j 2 + λ u 2 j ρ 2 + u 2 j + ε n,1 u j, j = 1,...,m and ˆψ n,2 u j = b c2 u j ρ u j + λ2p 1 2 ρ 2 + u 2 j + ε n,2 u j, j = 1,...,m. The firs linear regression yields esimaes for c 2 and λ denoed by ĉ 2 and ˆλ. The second linear model produces subsequenly esimaes for e 1 := b c2 2

158 154 Chaper 8. Saisics for jump-diffusions and e 2 := λ2p 1 denoed by ê k, k = 1,2 which imply he following esimaes for b and p: ˆb = ê1 + ĉ2 2 and ˆp = 1 2 ê2 ˆλ + 1. These regression problems are easily solved using a sandard saisics package such as S-Plus. Le us illusrae his case wih an example, which demonsraes ha his esimaion approach really separaes c 2 and λ, which is he basic difficuly of esimaing jump-diffusions having only discree observaions and no furher indicaion disinguishing beween he coninuiies and he disconinuiies of he pahs. Example We assume η 1 = η 2 = 1. The descripive saisics in Tables 8.1 and 8.2 resul from 5 independen esimaions each based on an i.i.d. sample of L 1 wih he following parameers: b =.8, c 2 =.2, λ = 2 and p =.4. We disinguish beween wo sample sizes being n = 1 and n = 5. Moreover he u j were chosen such ha u j u j 1 = 1 1. n = 1 ˆb ĉ 2 ˆλ ˆp Mean Median Sd. dev s. Qu rd. Qu Minimum Maximum Table 8.1: Descripive saisics of 5 independen parameer esimaes in he special case η 1 = η 2 = 1. Sample size: n = 1. n = 5 ˆb ĉ 2 ˆλ ˆp Mean Median Sd. dev s. Qu rd. Qu Minimum Maximum Table 8.2: Descripive saisics of 5 independen parameer esimaes in he special case η 1 = η 2 = 1. Sample size: n = 5.

159 8.3. Esimaion mehod 155 Special case 2 Generalizing he previous case we suppose now η 1 = η 2 and ge a nonlinear regression problem ha is parially linear: and ˆψ n,1 u j = c 2 u 2 j u 2 j + λ 2 η ε n,1 u j, j = 1,...,m u2 j ˆψ n,2 u j = b c2 η1 u j u j + λ2p 1 2 η ε n,2 u j, j = 1,...,m. u2 j In his case we apply nonlinear regression o he firs model and esimae c 2, λ and η 1. In S-Plus his is done using he procedure nls wih he opion plinear for parial linear, which accouns for he lineariy in c 2 and λ. Once η 1 is esimaed, we consider ˆψ n,2 u j = b c2 ˆη1 u j u j + λ2p 1 2 ˆη ε n,2 u j, j = 1,...,m. u2 j To his model simple linear regression is applied like in special case 1. I urned ou ha he following equivalen ransformed sysem of regression models sabilizes he numerical esimaion procedure, provided u j, j = 1,...,m. ˆψ n,1 u j u 2 j = c λ 2 η ε n,1 u j, j = 1,...,m u2 j and ˆψ n,2 u j u j = b c2 + λ2p 1 2 η 1 η u2 j + ε n,2 u j, j = 1,...,m. We consider again an example and poin ou ha he parameer esimaion does no ake much compuaional ime, in fac i only akes a few seconds, since i relies on implemened sandard soluions for leas-square problems. Example We assume η 1 = η 2. The descripive saisics in Tables 8.3 and 8.4 resul from 5 independen esimaions each based on an i.i.d. sample of L 1 wih he following parameers: b =.1, c 2 =.2, λ = 2, η 1 = 2 and p =.4. We disinguish beween wo sample sizes being n = 1 and n = 5. Moreover he u j were chosen such ha u j u j 1 = 1 1 and he iniial value for he nonlinear leas-square procedure deermining η1 2 was se arbirarily o 8 for boh sample sizes.

160 156 Chaper 8. Saisics for jump-diffusions n = 1 ˆb ĉ 2 ˆλ η1 ˆp Mean Median Sd. dev s. Qu rd. Qu Minimum Maximum Table 8.3: Descripive saisics of 5 independen parameer esimaes in he special case η 1 = η 2. Sample size: n = 1. n = 5 ˆb ĉ 2 ˆλ η1 ˆp Mean Median Sd. dev s. Qu rd. Qu Minimum Maximum Table 8.4: Descripive saisics of 5 independen parameer esimaes in he special case η 1 = η 2. Sample size: n = 5. General case In general we ge he following sysem of nonlinear regression models. and ˆψ n,1 u j u 2 j ˆψ n,2 u j u j = = c e 1 2 η e 2 u2 j η ε n,1 u j, u2 j b c2 + λp 2 η 1 η u2 j λ1 p η 2 η u2 j + ε n,2 u j, where j = 1,...,m, e 1 = λp and e 2 = λ1 p. Nonlinear regression of he firs model yields ĉ 2, ˆη 1, ˆη 2, ˆλ = ê 1 + ê 2 and ˆp = ê1 ê 1 + ê 2. These esimaes can hen be used o consider he second regression model as a linear regression problem. Noe ha if η 1 and η 2 are known, boh regressions are linear.

161 8.4. Mahemaics behind he esimaion Mahemaics behind he esimaion In his secion our aim is o derive, a leas asympoically, some saemens abou he disribuion and he covariance of he residuals in 8.5. To achieve his we firs invesigae he asympoic behavior of he empirical characerisic funcion for an increasing sample size. For a sequence of i.i.d. random variables X 1,X 2,X 3,... wih common disribuion funcion Fx we denoe by ˆF n x he empirical disribuion funcion of he firs n variables. φu, ˆφn u, ψu and ˆψ n u are analogously he rue and he empirical characerisic or cumulan generaing funcions, respecively. For noaional convenience we suppress he explici dependence on he parameer vecor θ in his subsecion. Le us recall he Glivenko-Canelli heorem which saes ha he empirical disribuion funcion converges uniformly almos surely o he rue disribuion funcion, ha is, lim ˆF n x Fx = a.s. sup n x R This resul is no self-eviden for he empirical characerisic funcion, since one generally has an improper Riemann-Sieljes inegral. However, he properies of a disribuion funcion imply ha for a given ε > here exiss a M > such ha F M < ε/6 and 1 FM < ε/6. Furhermore, he Glivenko-Canelli Theorem yields ha for sufficienly large random n one has ˆF n M < ε/6, 1 ˆF n M < ε/6 and ˆF n F < ε/6 a.s., where f := sup x R fx. Hence for any compac se K R and sufficienly large n i holds ha sup ˆφ n u φu = sup e iux d ˆFn x Fx u K u K M d ˆF n x + M dfx ˆF n F + sup u K ε + sup u 2M ˆF n F u K M u M M d ˆF n x + M ˆF n x Fx a.s. dx dfx for any compac se K R. Choosing K = [ T n,t n ], he law of ieraed logarihm for he empirical process implies T n = on/log log n 1/2 cf. Csörgő More general one has he following heorem: Theorem Csörgő and Toik. Le X 1,X 2,X 3,... be a sequence of i.i.d. random variables. If log T n lim n n =,

162 158 Chaper 8. Saisics for jump-diffusions hen he empirical characerisic funcion saisfies: lim sup ˆφ n u φu = n u T n a.s. Proof. Csörgő and Toik The heorem s rae T n = expon is bes possible for almos sure convergence. Noe ha Figure 8.1 parly reflecs his behavior. Since he complex logarihm is coninuous, he almos sure convergence of Theorem carries forward o he cumulan generaing funcion and gives us srong consisency. In he nex lemma we invesigae he momens of he empirical characerisic funcion. Lemma For given i.i.d. random variables X 1,...,X n he empirical characerisic funcion has he following properies. 1. E[ˆφ n u] = φu, u R. 2. Cov [ˆφn u, ˆφ n v ] = 1 n φu v φuφ v, u,v R. As usual, z denoes he conjugae, x iy, for a given complex number z = x + iy, x,y R. Proof. Le u R. By he i.i.d. assumpion one has ha E[ˆφ n u] = 1 n n E[e iux k ] = 1 n j=1 n E[e iux 1 ] = φu. j=1 Now le u,v R. The covariance is hen given by Cov [ˆφn u, ˆφ n v ] = E [ˆφn uˆφ n v ] φuφv [ 1 n = E n 2 e iu vx k + 1 n n 2 k=1 Since φv = φ v, he claim follows. j,k=1 j k ] e iux k e ivx j φuφv = 1 n φu v φuφv φuφv n = 1 n φu v φuφv. Noe ha he previous Lemma implies ha Var [ˆφn u ] = 1 n 1 φu 2, u R.

163 8.4. Mahemaics behind he esimaion 159 For he empirical characerisic funcion i is even possible o formulae an invariance principle o derive a limiing process for he convergence in disribuion. Therefore we need he following process. Definiion A sochasic process B = B, 1, is called a Brownian bridge if B = W W1 for a sandard Wiener process W = W, 1. Definiion yields ha a Brownian bridge B is a Gaussian process wih E[B] = and E[BBs] = min,s s for all s, 1. Donsker s invariance principle cf. Chaper 4 in Csörgő and Révész 1981 saes now for β n x = n ˆFn x Fx and a Brownian bridge B ha β n x d BFx, n. This has been ransferred o he characerisic funcion by Csörgő However, he ransformaion is again no as easy as i migh seem a firs glance. I uses he following sequence of sochasic processes, Y n u = n ˆφn u φu = e iux dβ n x, u R, and is formulaed in a very general heorem by Csörgő 1981 which we reduce o our siuaion. Theorem Csörgő. Le X 1,X 2,X 3,... be a sequence of i.i.d. random variables and le < T 1 < T 2 <. If E[ X α ] < for arbirary large α, hen here exiss for each n a Brownian bridge B n such ha sup Y nu e iux db n Fx = On 1/2 log n a.s. T 1 u T 2 In paricular, here exiss a Brownian bridge B such ha for all u R Y n u d Y u = e iux dbfx, n. Y u is a complex valued Gaussian process saisfying E[Y u] = and d E[Y uy v] = φu v φuφ v for all u,v R. Here, denoes convergence in disribuion. Proof. The proof is given in Csörgő For compleeness le us noe ha Y u = iu = 1 e iux BFxdx = e iuf 1 x dwx W1 1 1 e iuf 1 x dbx e iuf 1 x dx,

164 16 Chaper 8. Saisics for jump-diffusions where F 1 y = sup{x Fx y}, y 1. Since e iux BFx is almos everywhere coninuous, Y u exiss as an improper Riemann-Sieljes inegral whenever F x and 1 Fx decrease fas enough for x, which is guaraneed by he exisence of all momens. Moreover, Y u exiss also as a sochasic inegral, since E[ 1 eiuf 1 x dwx 2 ] 1 a.s. for each u R. Hence, for each u R, he sochasic process eiuf 1 x dwx, 1, is a maringale and E[Y u] =. For u,v R he covariance is compued using he following observaion: Cov[Y u,y v] = E[Y uy v] = [Y u,y v] 1 = 1 e iu vf 1 x d[w,w] x φuφ v[w,w] 1 = φu v φuφ v. This implies he covariance srucure for he real and imaginary par of he process. We denoe by Re z = z+z 2 he real par and by Imz = z z 2i he imaginary par of a complex number z. These relaions yield for all u,v R: Cov[Re Y u,re Y v] = 1 2 Reφu + v + Reφu v Re φure φv, Cov[ReY u,im Y v] = 1 2 Im φu v Im φu + v + ReφuIm φv, Cov[Im Y u,im Y v] = 1 2 Reφu + v Reφu v + Im φuim φv. The nex lemma demonsraes ha Theorem is applicable in our case. Lemma The random variable L 1 in 8.3 saisfies he condiion E[ L 1 α ] < for arbirary large α. Proof. According o Lemma i suffices o have { x >1} x α νdx <, which is cerainly saisfied for νdx = λf Y xdx, where f Y x is he densiy of he asymmeric double exponenial disribuion in 8.2. An immediae consequence of Theorem is he following asympoic saemen of he finie-dimensional disribuions of he empirical characerisic funcion. Corollary Le X 1,X 2,X 3,... be a sequence of i.i.d. random variables and le u 1,...,u m R. Then he empirical characerisic funcion saisfies: n ˆφn u 1 φu 1 Y u 1. n ˆφn u m φu m d., n, Y u m where Y u is he limi process from Theorem wih he covariance srucure described above.

165 8.4. Mahemaics behind he esimaion 161 Le us come back o our sysem of nonlinear regression models in 8.5. Using he heory developed so far, we are now able o derive an asympoic saemen abou he finie-dimensional disribuions of he residuals ε n,l u, l = 1,2. Theorem Le X 1,X 2,X 3,... be a sequence of i.i.d. random variables and le u 1,...,u m R such ha φu j C for all j = 1,...,m. Then i holds: n εn,1 u 1 + iε n,2 u 1 where. n εn,1 u m + iε n,2 u m d Ỹ u = Y u φu Ỹ u 1. Ỹ u m, n, for he limi process Y u from Theorem Moreover, Ỹ u has he following covariance srucure: φu v E[Ỹ uỹ v] = 1, u,v R. φuφ v Proof. There is a R > such ha he open balls B R φu j = {z C : z φu j < R} C for all j = 1,...,m. Furhermore, according o Theorem we find a n such ha for τ 1 he sraigh lines z n,j τ = φu j + τ ˆφn u j φu j B R φu j a.s. for all n n and j = 1,...,m. We have herefore for each j = 1,...,m almos surely ha ε n,1 u j + iε n,2 u j = ˆψn,1 u j ψ 1 u j + i ˆψn,2 u j ψ 2 u j = Log ˆφ n u j Log φu j = ˆφnu j φu j 1 z dz = 1 = ˆφn u j φu j 1 z n,j τ z n,j τ dτ Again by Theorem i exiss a ñ n such ha Hence, 1 z n,j τ dτ, n n. z n,j τ φu j ˆφ n u j φu j φu j, a.s. n ñ. 2 1 z n,j τ 2 φu j, a.s. n ñ,

166 162 Chaper 8. Saisics for jump-diffusions so by dominaed convergence we ge ha 1 1 z n,j τ dτ 1, a.s. n. φu j Slusky s Lemma and Corollary yield hen he claim. The previous heorem implies in paricular ha he residuals of he regression problem are no independen. Le us finally consider again he special case 1 of Secion 8.3. In his case we assume for a given ρ ha η 1 = η 2 = ρ for he disribuion of he random variable L 1, which is defined in 8.3. This implies a linear regression wih he following marix represenaion: ˆψ n = A θ + ε n, where ˆψ n u 1 ˆψ n =. ˆψ n u m, A = u u 2 1 ρ 2 +u 2 1. u 2 m 2 u 2 m ρ 2 +u 2 m iu 1. iu m i u 1ρ ρ 2 +u 2 1. i umρ ρ 2 +u 2 m, θ = c 2 λ e 1 e 2 and ε n,1 u 1 + iε n,2 u 1 ε n =.. ε n,1 u m + iε n,2 u m Recall ha u 1,...,u m R and ha m 1 is fixed. The leas-square esimae ˆ θn for θ R 4 is given by ˆ θ n = A H A 1 Re A H ˆψn, where A H denoes he ranspose of he conjugae marix A of A. The covariance of nˆ θn is herefore asympoically deermined by lim n Cov [ nˆ θn ] = A H A 1 lim n n Cov[ReAH ε n ] A H A 1, where A Cov[ReA H H ε n + A ε n ] = E[ H ε n A H ε n + A H ε n H] 2 2 = 1 2 Re A H E[ ε n ε H n ]A Re A H H E[ ε n ε n ]A.

167 8.5. Fuure research 163 Hence, because of Theorem 8.4.7, lim n n Cov[ReAH ε n ] = C for some marix C R 4 4, which shows ha 8.5 Fuure research lim Cov [ ] nˆ θn = A H A 1 C A H A 1. n There are sill some open quesions o be answered in connecion wih he suggesed esimaion mehod. We pospone hem o fuure research raising here, however, he mos imporan issues from our poin of view. 1. Esimaion of special case 1 applying he complex linear regression ha we discussed a he end of Secion 8.4. The resuls should hen be compared o he resuls in Secion 8.3, where we esimaed he parameers using separaely he wo linear models corresponding o he real and imaginary par. 2. Numerical reamen of he general case of Secion 8.3, where he problem is o separae η 1 and η 2. A nonlinear leas-square rouine has o be applied and he quesion of how o choose he iniial values for he rouine has o be answered. An ad-hoc applicaion of he S-Plus rouine nls ofen erminaes irregularly because of singulariies or oo many ieraion seps. 3. Improvemen of he esimaions in special case 2, where he sandard deviaion of some esimaors is sill no saisfying. 4. Describing in analogy o he linear case he esimaor s covariance srucure in he nonlinear case. 5. Opimizaion of he suppor for he funcion ψ 1 u, θ of Secion 8.3. Considering his funcion suggess o discriminae large values of u o esimae c 2 and possibly λ from small values of u o esimae η 1 and η Finding an opimal se of supporing poins for he regression models. This includes heir number as well as heir locaion. Noe ha he residuals a differen supporing poins are dependen. 7. Applicaion of he esimaion mehod o real financial daa o ge parameer esimaes for an asymmeric double exponenial jump-diffusion sock model.

168 164 Chaper 8. Saisics for jump-diffusions

169 Appendix A Supplemenary Maerial Lemma A..1 Gronwall s inequaliy 1. Le [a,b] be a closed inerval in R and f : [a,b] R be non-negaive and locally bounded. If here exis C,h such ha for all [a,b] f C + h a fsds, hen we have ha f C e h a for all [a,b]. Proof. This follows from Theorem in Applebaum 24. Theorem A..2. Suppose ha x R and ha he process X = X T is given by he following sochasic inegral equaion X = x + fs,x s ds + gs,x s dl s, T, where, on some filered probabiliy space Ω, F,F T, P, L = L T is a square-inegrable maringale and a Lévy process wih L,L = κ for some κ >. Furher le f,g be coninuous funcions on [,T] R ha are coninuously differeniable in he spaial variable x. Addiionally assume ha here exiss a consan K > such ha sup T f, + g, K x f,x + x g,x K. 1 named afer he Swedish mahemaician Thomas Hakon Grönwall

170 166 Chaper A. Supplemenary Maerial Then for any T here exiss a consan C = CK,T,κ >, such ha E[ sup Xs 2 ] C 1 + x 2. s Especially, if f, x = for all, x hen X is a maringale. Proof. By he mean-value-heorem we have ha f,x K1+ x for all,x. The same holds for g,x. Furher he inegral gs,x s dl s is a local maringale, since X s is a lef coninuous process and hence locally bounded Proer 24, Chaper IV, Theorem 29 and remark before Theorem 15. Tha is, here exiss a reducing sequence of sopping imes τ n n=1 wih lim n τ n = such ha he sopped process τ n gs,x s dl s is a maringale for each n 1. We observe now ha E[ sup Xs] 2 4x 2 s 2] + E[ sup 4 fu,x u du s s s 2]. + E[ sup 2 gu,x u dl u s By he Cauchy-Schwarz-inequaliy and he above esimae for f we have ha Nex, s E[ sup 4 s s E[ sup 2 s 2] fu,x u du 4T E[ f 2 u,x u du] gu,x u dl u 2] = E[ lim 8TK 2 T + 8TK 2 T + sup 2 n s τ n E [ Xu 2 ] du E[ sup Xs. 2 ]du s u s 2]. gu,x u dl u Using now Faou s lemma as well as Doob s maximal quadraic inequaliy we ge ha E[ sup 2 s s 2] τn 2] gu,x u dl u 8 lim E[ gu,x u dl u n τn = 8 lim E[ g 2 u,x u d L,L n u ] = 8κ E[g 2 u,x u ]du.

171 Chaper A. Supplemenary Maerial 167 Noe ha he inerchange of inegraion in boh cases is jusified by Tonelli s heorem. Moreover, using he above esimae for g yields 8κ Alogeher we obain ha where E[g 2 u,x u ]du 16κK 2 T + 16κK 2 T + E[ sup Xs 2 ] 4x2 + 8TK 2 T + 2κ + ht,k,κ s Lemma A..1 yields now ht,k,κ = 8KT + 2κ. E [ Xu 2 ] du E[ sup Xs. 2 ]du s u E[ sup Xs 2 ]du, s u E[ sup Xs] 2 4x 2 + 8TK 2 T + 2κ e ht,k,κt C1 + x 2. s If f,x = for all,x hen X is a local maringale and by Theorem I.51 in Proer 24 a maringale.

172 168 Chaper A. Supplemenary Maerial

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179 Lis of Tables 8.1 Descripive saisics of 5 independen parameer esimaes in he special case η 1 = η 2 = 1. Sample size: n = Descripive saisics of 5 independen parameer esimaes in he special case η 1 = η 2 = 1. Sample size: n = Descripive saisics of 5 independen parameer esimaes in he special case η 1 = η 2. Sample size: n = Descripive saisics of 5 independen parameer esimaes in he special case η 1 = η 2. Sample size: n =

180 176 LIST OF TABLES

181 Lis of Figures 7.1 Sock price sample pahs. Parameers: S = 1, c =.2, b =.5, λ = 5, η 1 = 17, η 2 = 15, p = Empirical vs. heoreical cumulan generaing funcion. Lef column: Real par. Righ column: Imaginary par. Dashed line: Theoreical funcions. Parameers: Sample size n = 1, c 2 =.2, b =.8, λ = 3, η 1 = η 2 = 1, p =

182 178 LIST OF FIGURES

183 Zusammenfassung Im Mielpunk dieser Arbei seh die Enwicklung von lokal risikominimalen Hedgingsraegien für fondsgebundene Lebensversicherungsverräge, wobei der Fonds in einem allgemeinen Lévy-Prozess Finanzmarkmodell modellier wird. Dadurch soll die in den lezen Jahren enwickele und ziemlich wei vorangeschriene Theorie der Lévy-Prozess Finanzmärke mi der Theorie für fondsgebundene Lebensversicherungen verbunden und außerdem ein Rahmen bereigesell werden, der quadraisches Hedgen von Zahlungssrömen ermöglich, die einem Invesiionsrisiko und einem Versicherungsrisiko ausgesez sind. Unserem Wissen zur Folge, wurde lokal risikominimales Hedgen von fondsgebundenen Lebensversicherungsverrägen in allgemeinen Lévy-Prozess Finanzmärken bisher noch nich analysier. Diese Arbei liefer daher einen Beirag zur laufenden Forschung an der Schniselle zwischen Finanz- und Versicherungsmahemaik und verallgemeiner die früheren Arbeien von Møller 1998, 21a, indem der vollsändige Black- Scholes Finanzmark durch einen unvollsändigen Finanzmark, am Beispiel eines allgemeineren geomerischen Lévy-Prozess Modells, ersez wird. Wie Møller 1998, 21a nehmen wir sochasische Unabhängigkei zwischen dem Finanzmark und dem Versicherungsmodell an und berachen beide zusammengefass in einem gemeinsamen Produkwahrscheinlichkeisraum. Dies greif die Idee auf, die Unsicherhei des Finanzmarkes und die der versicheren Leben gleichzeiig zu modellieren und nich die Serblichkei durch ihren erwareen Verlauf zu ersezen. Zudem liefer die Verwendung des Markovkeenmodells von Hoem 1969 zur Beschreibung der Lebensversicherungen sehr flexible Resulae, die an die verschiedensen vorsellbaren Lebensversicherungsaren angepass werden können. In Beispielen lieg jedoch unser Schwerpunk auf den sicherlich wichigsen Varianen wie die Erlebensfallversicherung, die Todesfallversicherung und die Leibrenenversicherung. Wir beginnen mi der Herleiung von lokal risikominimalen Hedgingsraegien und dem dami verbundenen Hedgingrisiko für ein Porfolio von fondsgebundenen Erlebensfallversicherungen und für ein Porfolio von fondsgebundenen Todesfallversicherungen, wobei wir zunächs annehmen, dass diese gegen eine Einmalprämie zum Aussellungsdaum verkauf und dass ihre Versicherungsleisungen bis zum Ende des beracheen Zeihorizons

184 18 ZUSAMMENFASSUNG verzinslich aufgeschoben und ers dann ausbezahl werden. Mi dieser vereinfachenden Annahme passen die ensprechenden Zahlungsansprüche in die lokal risikominimale Hedgingheorie von Schweizer Die risikominimale Hedgingheorie von Föllmer und Sondermann 1986 für Zahlungsansprüche mi fesem Auszahlungszeipunk wurde von Møller 21a für allgemeine Zahlungssröme erweier. Beide Theorien gelen jedoch in Maringalfinanzmärken, in denen das Hedgingrisiko uner dem jeweils beracheen Maringalmaß inerpreier werden kann. Schweizer 1991 verallgemeinere die Theorie von Föllmer und Sondermann 1986 für Semimaringalfinanzmärke, indem er lokal risikominimales Hedgen und das risikoneurale Föllmer-Schweizer Maß eingeführ ha. Eine uner diesem Maß risikominimale Hedgingsraegie is lokal risikominimal bezüglich dem ursprünglichen Maß und kann somi nich nur uner einem möglicherweise beliebigen, risikoneuralen Maringalmaß hinsichlich ihres Risikos inerpreier werden. Aus diesem Grund unersuchen wir, ob sich lokal risikominimales Hedgen auch auf allgemeine Zahlungssröme anwenden läss. Diese Erweierung auf Zahlungssröme is neu und wurde zuvor noch nich berache. Schließlich leien wir in einem allgemeinen Lévy-Prozess Finanzmark lokal risikominimale Hedgingsraegien für allgemeine fondsgebundene Lebensversicherungsverräge her, welche innerhalb ihrer Laufzei Versicherungsleisungen und Prämienzahlungen zulassen. Da wir von Anfang an einen unvollsändigen Finanzmark benuzen, erhalen wir zwei Besandeile des Hedgingrisikos, welche dessen Ursprung widerspiegeln. Wir bezeichnen sie als das reine Finanz- und das reine Versicherungsrisiko. Das gesame Hedgingrisiko eines fondsgebundenen Lebensversicherungsanspruchs eil sich genau in diese beiden Größen auf, wobei das reine Versicherungsrisiko durch eine Erhöhung der Anzahl der versicheren Personen innerhalb des Porfolios diversifizierbar is. Dies gil jedoch nich für das reine Finanzrisiko, was man ohnehin in der Realiä nich erwaren würde. Der Unerschied zu Møller 1998, 21a is, dass wir in Übereinsimmung zur Realiä zusäzlich das reine Finanzrisiko modellieren. Da das Black-Scholes Modell einen vollsändigen Finanzmark beschreib, simm das gesame Hedgingrisiko von Møller 1998, 21a mi dem reinen Versicherungsrisiko in unserem Fall überein. Im Folgenden geben wir eine deailliere Inhalsübersich der Disseraion. Grundlegende Konzepe In Kapiel 2 werden die in dieser Arbei benöigen wesenlichen echnischen Grundlagen besprochen und es werden die in den folgenden Kapieln häufig verwendeen Modelle eingeführ. Nach der Definiion eines Lévy- Prozesses diskuieren wir das Lévy-Maß, die berühme Lévy-Iô-Zerlegung, welche in Theorem formulier wird, und die Vereilungseigenschafen von Lévy Prozessen, zu denen die Lévy-Khinchin Formel und die un-

185 ZUSAMMENFASSUNG 181 endliche Teilbarkei gehören. Zusäzlich klassifizieren wir in Theorem das Pfadverhalen von Lévy-Prozessen, indem wir jeweils zwischen endlicher und unendlicher Akiviä oder Variaion unerscheiden. Der Unerabschni beinhale eine Diskussion von üblicherweise zur Modellierung von Finanzmärken verwendeen Lévy-Prozessen. Im Anschluss daran führen wir im Abschni 2.3 das Lévy-Prozess Finanzmarkmodell von Chan 1999 ein und behandeln das für lokal risikominimales Hedgen benöige Föllmer-Schweizer Maß. Wir erklären ausführlich die Auswirkungen des dami zusammenhängenden Maßwechsels auf den Lévy-Prozess des Modells, welcher zu einem addiiven Prozess wird. Außerdem diskuieren wir jeweils in den Lemmaa und den Akienpreisprozess vor und nach diesem Maßwechsel und zeigen in beiden Fällen seine quadraische Inegrierbarkei, indem wir das Resula aus dem Anhang dieser Arbei verwenden. Dieses Resula ermöglich es uns, die von Chan 1999 vorausgeseze Exisenz exponenieller Momene des Lévy-Maßes wesenlich abzuschwächen, indem wir nur noch die Exisenz des drien Momens benöigen. Im darauffolgenden Abschni 2.4 fassen wir das auf einer zeiseigen Markovkee basierende Lebensversicherungsmodell von Hoem 1969 zusammen, welches die Modellierung von ziemlich allgemeinen Lebensversicherungsverrägen zuläss. Im Unerabschni präsenieren wir dann hisorische und zugleich noch heue häufig benuze Serblichkeisgeseze, um Beispiele für die Hasardrae des Versicherungsmodells zu geben. Anschließend verwenden wir den zenralen Grenzwersaz, um im Unerabschni ein wei verbreiees Diversifikaionsmodell für das Versicherungsrisiko am Beispiel der gewöhnlichen Lebensversicherung zu besprechen. Weierhin zeig eine Anwendung des sarken Gesezes der großen Zahlen im Unerabschni die Risikoneuraliä eines Versicherungsunernehmens in Bezug auf die Serblichkei. Schließlich is Abschni 2.5 der Konsrukion eines gemeinsamen Produkwahrscheinlichkeisraums gewidme, welcher das Lévy-Prozess Finanzmarkmodell mi dem Versicherungsmodell uner der begründeen Annahme von sochasischer Unabhängigkei zusammenfass. Zudem wiederholden wir im risikoneuralen Gegensück dieses Raumes die risikominimale Hedgingheorie von Föllmer und Sondermann 1986, einschließlich der für sie grundlegenden Galchouk-Kunia-Waanabe Zerlegung in Theorem und einer Diskussion ihres Zusammenhangs zum lokal risikominimalen Hedgen von Schweizer Arbirage-freier Preisprozess In Kapiel 3 diskuieren wir den arbiragefreien Preisprozess eines quadraisch inegrierbaren Zahlungsanspruchs, welchen wir mi Hilfe der Markoveigenschaf des Akienprozesses als eine Funkion der Zei, des Akienwers zu dieser Zei und dem Fälligkeisdaum des Anspruchs darsellen. Dabei lieg der gemeinsame risikoneurale Finanzmark aus Kapiel 2 zu Grunde.

186 182 ZUSAMMENFASSUNG In diesem wird die Akie durch einen addiiven Prozess, das heiß, einem càdlàg, sochasisch seigen Prozess mi unabhängigen Zuwächsen, modellier. Der Schwerpunk dieses Kapiels is die in Theorem hergeleiee Galchouk-Kunia-Waanabe Zerlegung des arbiragefreien Preisprozesses. Sie ha dieselbe Srukur, wie wenn sie für einen Lévy-Prozess ausgearbeie worden wäre und is sicherlich der wichigse Teil der lokal risikominimalen Hedgingheorie. Darüber hinaus is sie die Grundlage der Galchouk-Kunia- Waanabe Zerlegung des inneren Wers der fondsgebundenen Lebensversicherungen in den folgenden Kapieln. Grob umschrieben is diese Zerlegung ein Projekionsresula im Hilberraum der quadraisch inegrierbaren Zufallsvariablen. Wir beweisen Theorem 3.3.4, indem wir die Orhogonaliä der beeiligen seigen und rein unseigen Maringale direk verwenden. Diese Herangehensweise haben wir zuvor noch nich beobache. Darüber hinaus schein die Formulierung des Theorems für addiive Prozesse neu zu sein. Um unseren Beirag von bereis exisierenden Resulaen abzugrenzen, sellen wir am Ende des Kapiels einen Überblick über dami zusammenhängende Lieraur zur Verfügung. In Theorem berachen wir die Feynman-Kac parielle Differenzial- und Inegralgleichung, welche vom arbiragefreien Preisprozess nowendigerweise erfüll und zum Beweis der Galchouk-Kunia-Waanabe Zerlegung benöig wird. Diese Gleichung kann zur numerischen Berechnung des arbiragefreien Preisprozesses verwende werden und folg aus der Maringaleigenschaf des diskonieren arbiragefreien Preisprozesses. Die angewendee Zerlegungsmehode is am Anfang sehr ähnlich zur der für Black-Scholes Finanzmärke bekannen Herangehensweise. Allerdings erhäl man in diesem Fall durch die ensprechende Feynman-Kac Darsellung bereis die richige Galchouk-Kunia-Waanabe Zerlegung. In unserem Fall sind dagegen weiere Schrie erforderlich, welche in den Lemmaa und genauer ausgeführ werden. Hedging von fondsgebundener Erlebens- und Todesfallversicherung Kapiel 4 basier auf Riesner 26 und is der Enwicklung von lokal risikominimalen Hedgingsraegien für ein Porfolio von enweder fondsgebundenen Erlebens- oder Todesfallversicherungsverrägen gewidme. Wie bereis erwähn, schränken wir uns dabei zunächs auf den Fall von Zahlungen nur zu Beginn und zum Ende des beracheen Zeiinervalls ein. Diese Einschränkung ermöglich es uns, die für Zahlungsansprüche mi fesem Auszahlungszeipunk enwickele lokal risikominimale Hedgingheorie von Schweizer 1991 anzuwenden. Das Kapiel dien als Zwischenschri bevor wir Prämienzahlungen und Versicherungsleisungen auch innerhalb der Versicherungsperiode berachen und verallgemeiner darüber hinaus Møller 1998, indem der vollsändige Black-Scholes Finanzmark durch einen allgemeineren und unvollsändigen Lévy-Prozess Finanzmark ersez wird. Wir beginnen das Kapiel mi einer allgemeinen mahemaischen Diskussion

187 ZUSAMMENFASSUNG 183 fondsgebundener Lebensversicherungsverräge und besimmen die Barwere des gesamen Porfolios der oben erwähnen Versicherungsaren. Verschiedene weiere Umformungen führen uns dann jeweils in den Korollaren und zur gewünschen Galchouk-Kunia-Waanabe Zerlegung der zu diesen Barweren gehörenden inneren Were der beracheen Versicherungsporfolios. Diese Umformungen beinhalen uner anderem die Anwendung von Iôs parieller Inegraionsformel für allgemeine Semimaringale und, zusäzlich bei der Todesfallversicherung, von Fubinis Theorem für sochasische Inegrale. Die lokal risikominimalen Hedgingsraegien werden anschließend jeweils in den Korollaren und wiedergegeben. Alle erwähnen Korollare sind für einen allgemeinen Lévy-Prozess Finanzmark neue Resulae, welche wir ausführlich diskuieren und mi denen von Møller 1998 vergleichen. In unserem allgemeineren Fall is der risikominimale Invesmenaneil der risikobehafeen Anlage eine gewichee Summe des Black-Scholes Dela and einem weieren, aus den Sprüngen des Preisprozesses resulierenden Term. Anschließend berachen wir das Hedgingrisiko und leien seine zwei Besandeile, das reine Finanz- und das reine Versicherungsrisiko, her. Am Ende des Kapiels begründen wir, warum lokale Risikominimierung speziell für Versicherungszahlungsansprüche verwende werden solle. Hedging von allgemeinen Zahlungssrömen für Semimaringale In Kapiel 5 wird lokal risikominimales Hedgen von Zahlungssrömen in einem allgemeinen Semimaringalfinanzmark unersuch. Für Maringalfinanzmärke exisier die risikominimale Hedgingheorie für Zahlungssröme von Møller 21a. Diese liefer eine Inerpreaion des Hedgingrisikos uner dem jeweils beracheen Maringalmaß. Is das subjekive Maß jedoch kein Maringalmaß, so sell sich die Frage wie man das Hedgingrisiko auch uner diesem Maß und nich nur uner einem, möglicherweise beliebig gewählen, risikoneuralen Maringalmaß inerpreieren kann. Schweizer 1991 ha die lokal risikominimale Hedgingheorie für Zahlungsansprüche mi fesem Auszahlungszeipunk für Semimaringalfinanzmärke enwickel. Diese ermöglich eine lokale Inerpreaion des Hedgingrisikos uner dem subjekiven Wahrscheinlichkeismaß, vorausgesez das Föllmer-Schweizer Maß wird als risikoneurales Maß verwende. Wir benüzen nun Schweizer 1991, um in diesem Kapiel zu zeigen, dass das risikominimale Hedgen von Zahlungssrömen im selben Sinne eine lokale Version besiz wie das Hedgen von Zahlungsansprüchen mi fesem Auszahlungszeipunk. Man benuz die Theorie von Møller 21a und leie uner dem Föllmer-Schweizer Maß eine risikominimale Hedgingsraegie her. Diese is dann lokal risikominimal in Bezug auf das subjekive Wahrscheinlichkeismaß. Nach einigen echnischen Deails, fassen wir dieses Haupresula in Theorem zusammen. Es basier lezendlich auf derselben Opimaliäsgleichung wie im klassischen Fall, welche wir in Theorem formulieren. In Proposiion verglei-

188 184 ZUSAMMENFASSUNG chen wir außerdem den Begriff von erreichbaren Zahlungssrömen mi dem klassischen Begriff von erreichbaren Zahlungsansprüchen. Die Erweierung von lokal risikominimalem Hedgen auf Zahlungssröme is ein neues Resula. Ein Teil von Riesner 25 is diesem Kapiel ennommen. Hedging von allgemeinen fondsgebunden Lebensversicherungsverrägen Das Thema von Kapiel 6 is die Herleiung von lokal risikominimalen Hedgingsraegien für allgemeine fondsgebundene Lebensversicherungsverräge, welche innerhalb ihrer Laufzei sowohl Prämien- als auch Leisungszahlungen zulassen. Zur Modellierung der Lebensversicherungsverräge verwenden wir das zeiseige Markovkeenmodell von Hoem Es ermöglich sowohl zusandsabhängige als auch durch einen Zusandsübergang ausgelöse Zahlungen, welche wir als Differenz von Prämien- und Versicherungsleisungen inerpreieren. Das Kapiel räg zu Riesner 25 bei und verallgemeiner Møller 21a, indem der vollsändige Black-Scholes Finanzmark durch einen allgemeineren und unvollsändigen Lévy-Prozess Finanzmark ersez wird. Die Galchouk-Kunia-Waanabe Zerlegung des inneren Wers dieser allgemeinen Versicherungen is als Haupresula in Theorem wiedergegeben und wird mi aus Kapiel 4 bereis bekannen Techniken bewiesen. Die daraus anschließend abgeleiee lokal risikominimale Hedgingsraegie sowie das dami zusammenhängende Hedgingrisiko sind in Korollar enhalen. Für einen allgemeinen Lévy-Prozess Finanzmark sind diese Resulae neu. Am Ende des Kapiels wenden wir das allgemeine Modell auf die fondsgebundene Leibrenen- und die fondsgebundene Todesfallversicherung an, wobei wir zunächs jeweils einen einzelnen Verrag und im Anschluss daran ein Porfolio dieser Verräge berachen. Modellierung der Akie durch eine Sprungdiffusion Um einen möglichen, konkreen Lévy-Prozess Finanzmark vorzuschlagen, diskuieren wir in Kapiel 7 das asymmerisch doppel exponeniale Sprungdiffusionsmodell von Kou 22. Am Anfang des Kapiels begründen wir warum wir dieses Modell für geeigne halen und besprechen die asymmerisch doppele Exponenialvereilung. Danach diskuieren wir den Lévy- Prozess, der der Sprungdiffusion zu Grunde lieg und binden das Modell in unseren Finanzmark ein. Dabei legen wir besonderen Wer auf die Sprunghöhenvereilung und das dami zusammenhängende Lévy-Maß. Als risikoneurales Wahrscheinlichkeismaß verwende Kou 22 ein spezielles, auf Nuzenüberlegungen basierendes Maß und zeig, dass man mi dem ensprechenden Maßwechsel, uner gewissen Annahmen an die Markparameer wie Zins- und Drifrae, innerhalb der Klasse der asymmerisch doppel exponenialen Sprungdiffusionen bleib. Die Vereilungseigenschafen dieser Klasse sind für das Aufsellen der Preisformel des Modells für europäische

189 ZUSAMMENFASSUNG 185 Callopionen wesenlich. Da wir lokal risikominimales Hedgen berachen, sind wir an den Auswirkungen des Föllmer-Schweizer Maßes auf das Modell ineressier. Diese diskuieren wir ausführlich in Theorem sowie in den Lemmaa und Im Anschluss daran zeigen wir in Lemma 7.4.5, dass im Allgemeinen die Sprunghöhen uner diesem Maß keiner asymmerisch doppelen Exponenialvereilung mehr folgen und leien daraus ensprechende Bedingungen an die Markparameer ab, uner denen man auch nach diesem Maßwechsel innerhalb der gewünschen Vereilungsklasse bleib. Diese Bedingungen vergleichen wir anschließend mi denen von Kou 22. Abschni 7.5 beende das Kapiel mi den Formeln für den arbiragefreien Preisprozess einer europäischen Callopion, einer fondsgebundenen Erlebensfallversicherung und einer fondsgebundenen Todesfallversicherung, wobei beide eine Mindeszahlung garanieren. Außerdem berechnen wir die räumliche Ableiung des ensprechenden arbiragefreien Preisprozesses. Sowohl der Preisprozess als auch seine räumliche Ableiung sind Besandeile der in dieser Arbei aufgesellen Hedgingsraegien. Das Sprungdiffusionsmodell von Kou 22 wurde bisher noch nich im Zusammenhang mi fondsgebundenen Lebensversicherungen und dem Föllmer-Schweizer Maßwechsel berache. Außerdem sind die Bedingungen neu, uner denen dieser Maßwechsel nich aus der Klasse der asymmerisch doppel exponenialen Sprungdiffusionen herausführ. Saisik für Sprungdiffusionen Während dem Sudium der asymmerisch doppel exponenialen Sprungdiffusion von Kou 22 fiel uns auf, dass es keine zufriedensellenden Parameerschäzmehoden für Sprungdiffusionen gib. Aus diesem Grund behandeln wir dieses Thema in Kapiel 8. Obwohl wir den genannen Prozess dabei beispielhaf verwenden, is unsere Berachung im wesenlichen auch für andere Sprungdiffusionen gülig. Da die Wahrscheinlichkeisdiche dieser Prozesse keine geschlossene Form besiz, ihre charakerisische Funkion aber explizi bekann is, basieren wir in Abschni 8.3 die Schäzung auf die kumulanenerzeugende Funkion. Diese vergleichen wir mi dem komplexen Logarihmus der empirischen charakerisischen Funkion im Hinblick auf ihren quadraischen Absand und erhalen somi ein Sysem aus zwei nichlinearen Regressionsmodellen, welche aus dem Real- und Imaginäreil dieser Funkionen hervorgehen. Zusäzlich geben wir eine Regel zur Besimmung des kompaken Trägers an, auf dem diese Regressionsprobleme berache werden. Diese können dann mi Hilfe von Sandardverfahren zur Lösung von Kleinsquadraproblemen gelös werden und sind sogar in einigen Spezialfällen linear. Die heoreischen Grundlagen unserer Schäzmehode sellen wir in Abschni 8.4 berei. Dieser Abschni enhäl das Glivenko-Canelli Theorem für die charakerisische Funkion von Csörgő und Toik 1983, welches wir moivieren und aus welchem wir sarke Konsisenz erhalen. Da-

190 186 ZUSAMMENFASSUNG nach wenden wir das Invarianzprinzip von Csörgő 1981 auf die empirische charakerisische Funkion an und bekommen einen komplexwerigen Gaußschen Prozess als Grenzwer in Vereilung. Dieses Resula führ uns dann asympoisch in Theorem zu den endlichdimensionalen Vereilungen der komplexwerigen Residuen unseres Regressionsproblems einschließlich ihrer Kovarianzsrukur. Anhang Der Anhang enhäl ein Resula aus der sochasischen Analysis, welches auf der Ungleichung von Gronwall basier. Es liefer Bedingungen, uner denen die Lösung einer besimmen sochasischen Inegralgleichung bezüglich eines Lévy-Prozesses ein quadraisch inegrierbarer Prozess is. Wir benuzen es in den Lemmaa und 2.3.3, um zu zeigen, dass der in 2.14 definiere Akienprozess sowohl uner dem hisorischen als auch uner dem Föllmer-Schweizer Maß quadraisch inegrierbar is. Mi diesem Ergebnis genüg uns die Exisenz des drien Momens des Lévy-Maßes, welche wir in 2.12 voraussezen. Wir konnen deswegen die von Chan 1999 gefordere Exisenz exponenieller Momene abschwächen und dadurch das Modell für eine größere Klasse von Vereilungen, insbesondere für diejenigen mi dickeren Tails, öffnen. Wir haben Theorem A..2 speziell an unsere Erfordernisse angepass und bewiesen. Andere Versionen des Resulas sind zum Beispiel in Jacod, Méléard and Proer 2 oder Ma, Proer and Zhang 21 enhalen, allerdings ohne explizie Beweisführung und ohne eine Drifkomponene, welche wir hinzugefüg haben.