LIFE INSURANCE MATHEMATICS 2002

LIFE INSURANCE MATHEMATICS 22 Ragnar Norberg London School of Economics Absrac Since he pioneering days of Black, Meron and Scholes financial mahemaics has developed rapidly ino a flourishing area of science. Is impacs on insurance are grea by any calculaion: applicaions are virually counless and even he basic paradigms are being rehough. This alk focuses on life insurance and shows how he mahemaics of finance and of insurance doveail ino a consisen, model-based approach o measuremen and managemen of combined insurance risk and finance risk. 1 Inroducion Finance was always an essenial par of insurance. Trivially, one migh say, because any business has o aend o is money affairs. However, for a leas wo reasons insurance is no jus any business. In he firs place, is producs are no physical goods or services; hey are financial conracs wih provisions relaed o uncerain fuure evens. Therefore, pricing of insurance producs is no jus an accouning exercise involving he four basic arihmeical operaions; i is a maer of risk assessmen based on sochasic models and mehods. In he second place, insurance policies are more or less long erm conracs (in life insurance up o several ens of years) under which he cusomers pays in advance for benefis o come laer, hence he erm premium (= firs ) for he price. Therefore, he insurance indusry is a major accumulaor of capial in oday s sociey, and he insurance companies are major insiuional invesors. I follows ha he financial operaions of an insurance company may be as decisive of is revenues as is insurance operaions and ha he financial risk (or asse risk) may be as severe as, or even more severe han, he insurance risk (or liabiliy risk). Insurance risk, which is due o he random naure of he cash-flow of premiums less insurance claims, is diversifiable in he sense ha gains and losses on individual policies will average ou in a sufficienly large porfolio of independen risks: he law of large numbers is a work. Financial risk, which is due o he uncerain yields on he financial invesmens, is no diversifiable in he same simple sense. Any invesmen porfolio is affeced by booms and recessions of he marke in large, ineres rae variaions, and movemens in prices of individual socks. The composiion of an invesmen porfolio may be more or less risky (e.g. socks are more risky han governmen bonds), bu he volume of he porfolio is of course of no relevance o is riskiness; losses and gains on invesmens do no subjec hemselves o he law of large numbers. 1

On his background one may ask why insurance mahemaics radiionally ceners on measuremen and conrol of he insurance risk. The answer may parly be found in insiuional circumsances: The insurance indusry used o be heavily regulaed, solvency being he primary concern of he regulaory auhoriy. Possible adverse developmens of economic facors (inflaion, weak reurns on invesmen, low ineres raes, ec.) would be safeguarded agains by placing premiums on he safe side. The surpluses, which would ypically accumulae under his regime, were redisribued as bonuses (dividends) o he policyholders only in arrears, afer ineres and oher financial parameers had been observed. Furhermore, he insurance indusry used o be separaed from oher forms of business and proeced from compeiion wihin iself, and severe resricions were placed on is invesmen operaions. In hese circumsances financial maers appeared o be somehing he radiional acuary did no need o worry abou. Anoher reason why insurance mahemaics used o be void of financial consideraions was, of course, he absence of a well developed heory for descripion and conrol of financial risk. All his has changed. Naional and insiuional borders have been downsized or eliminaed and regulaions have been liberalized: Mergers beween insurance companies and banks are now commonplace, new insurance producs are being creaed and pu on he marke virually every day, by insurance companies and oher financial insiuions as well, and wihou prior licencing by he supervisory auhoriy. The insurance companies of oday find hemselves placed on a fiercely compeiive marke. Many new producs are direcly linked o economic indices, like uni-linked life insurance and caasrophe derivaives. By so-called securiizaion also insurance risk can be pu on he marke and hus open new possibiliies of inviing invesors from ouside o paricipae in risk ha previously had o be shared solely beween he paricipans in he insurance insurance schemes. These developmens in pracical insurance coincide wih he adven of modern financial mahemaics, which has equipped he acuaries wih a well developed heory wihin which financial risk and insurance risk can be analyzed, quanified and conrolled. A new order of he day is hus se for he acuarial profession. The purpose of his chaper is o give a glimpse ino some basic ideas and resuls in modern financial mahemaics and o indicae by examples how hey may be applied o acuarial problems involving managemen of financial risk. 2 Classical life insurance mahemaics A. The muli-sae life insurance policy. We consider an insurance policy issued a ime and erminaing a a fixed finie ime T. There is a finie se of muually exclusive saes of he policy, Z = {1,..., J Z }. By convenion, 1 is he iniial sae a ime. Le Z() denoe he sae of he policy a ime [, T ]. Taking Z o be a sochasic process wih righ-coninuous pahs and a mos a finie number of jumps, he same holds also for he associaed indicaor processes I j and couning processes N jk 2

defined, respecively, by Ij Z () = 1[Z() = j] (1 or according as he policy is in he sae j or no a ime ) and Njk Z () = {τ; Z(τ ) = j, Z(τ) = k, τ (, ]} (he number of ransiions from sae j o sae k ( j) during he ime inerval (, ]). The hisory of he policy up o and including ime is represened by he sigma-algebra F Z = σ{z(τ) ; τ [, ]}. The developmen of he policy is given by he filraion (increasing family of sigma-algebras) F Z = {F Z } [,T ]. Le B() denoe he oal amoun of conracual benefis less premiums payable during he ime inerval [, ]. We assume ha i develops in accordance wih he dynamics db() = j I Z j () db j () + j k b jk () dn Z jk(), (2.1) where each B j is a deerminisic paymen funcion specifying paymens due during sojourns in sae j (a general life annuiy), and each b jk is a deerminisic funcion specifying paymens due upon ransiions from sae j o sae k (a general life assurance). We assume ha each B j is finie-valued, righ-coninuous, and decomposes ino an absoluely coninuous par and a discree par wih a mos a finie number of jumps in [, T ]. Thus, db j () = b j ()d + B j (), where B j () = B j () B j ( ), when differen from, is a jump represening a lump sum payable a ime if he policy is hen in sae j. The funcions b j and b jk are assumed o be finie-valued and piecewise coninuous. B. The Markov chain descripion of he policy. The process Z is assumed o be a ime-coninuous Markov chain on he sae space Z. We denoe is ransiion probabiliies by p Z jk (, u) = P[Z(u) = k Z() = j], u, and he corresponding inensiies of ransiion by µ jk () = lim h p Z jk(, + h)/h, j k, implying ha hese exis for all [, n). We assume, moreover, ha he inensiies are piece-wise coninuous. The oal inensiy of ransiion from sae j is µ j = k; k j µ jk. Here and elsewhere a do in he place of a subscrip signifies summaion over ha subscrip. We shall need Kolmogorov s forward differenial equaions for s < : d p Z ij (s, ) = g; g j p Z ig (s, ) µ gj() d p Z ij (s, ) µ j () d. (2.2) We remind of he fac ha he compensaed couning processes M Z jk, j k, defined by dm Z jk () = dn Z jk () IZ j ()µ jk() d, (2.3) 3

are square inegrable, muually orhogonal, zero mean F Z -maringales. Le H jk, j, k Z, j k, be predicable processes saisfying E Hjk(τ) 2 µ jk (τ)dτ <. (2.4) j k (,] Then he sochasic inegral M() = H jk () dmjk() Z j k is a zero mean square inegrable F Z -maringale. Figure 1 oulines he disabiliy model, which is ap o describe a policy on a single life, wih paymens depending on he sae of healh of he insured (e.g. a life assurance wih waiver of premium during disabiliy). a acive µ 12 i invalid µ 21 µ 13 µ 23 d dead Figure 1: Skech of a Markov chain model for disabiliies, recoveries, and deah. C. Ineres. We assume ha he invesmen porfolio of he insurance company bears ineres wih inensiy r() a ime. Thus, for a given realizaion of r( ), he value a ime of a uni payable a ime τ is e τ r if τ (an accumulaion τ or compounding facor) and e r if τ (a discouning facor). The shorhand r = r(s)ds will be in use hroughou. We assume ha r is piecewise coninuous and ha T r is finie. D. The reserve. By saue he insurer mus currenly mainain a reserve o mee fuure liabiliies in respec of he conrac. Since hese liabiliies are unknown, he reserve can only be an esimae based on he informaion currenly available. The reserve 4

a ime is defined presicely as he condiional expeced presen value of he fuure benefis less premiums, given he pas hisory of he policy: [ n ] V () = E e τ r db(τ) F. By he Markov assumpion, V () = V Z() () = j I Z j () V j (), (2.5) where he V j are he saewise reserves, [ n ] V j () = E e τ r db(τ) Z() = j n = p jg (, τ) db g (τ) + b gh (τ)µ gh (τ) dτ.(2.6) e τ r g h;h g (2.7) We need he backward (so-called Thiele s) differenial equaions dv j () = r()v j () d db j () R jk () µ jk () d, (2.8) where k; k j R jk () = b jk () + V k () V j () (2.9) is he (for eviden reasons so-called) sum a risk associaed wih a possible ransiion from sae j o sae k a ime. The differenial equaion is easily derived by differeniaing he defining expression (2.6) and using he Kolmogorov equaion (2.2). E. The principle of equivalence. The conracual paymens (or raher he premiums for given benefis) are consrained by he principle of equivalence, which lays down ha [ ] T E e τ r db(τ) =, (2.1) ha is, V 1 () = B 1 (). (2.11) The raionale of his principle, which is he basic paradigm of classical life insurance, is ha i will esablish balance on he average in a large porfolio of independen policies. 5

The reserve defined in he previous paragraph is moivaed he same way: upon providing currenly a reserve equal o he condiional expeced value of he fuure ne liabiliy on each individual reserve, he company will mee is liabiliies on he average. F. A maringale proof of Thiele s differenial equaion. We ake here he opporuniy o demonsrae a maringale echnique ha will be omnipresen hroughou he ex. Thiele s differenial equaion could be proved by elemenary calculaions as explained above, bu he following argumen gives also an insigh ino he dynamics of he oal cash-flow. Define he maringale [ n ] M() = E e τ r db(τ) F Z ] = = [ n e τ r db(τ) + e r E e τ r db(τ) + e r j Now apply Iǒ s formula o (2.12): e τ r db(τ) F Z I Z j ()V j(). (2.12) dm() = e r db() + e r ( r() d) j I Z j () V j () + e r j I Z j () dv j () + e r j k dn Z jk() (V k () V j ( )). The las erm on he righ akes care of he jumps of he Markov process: upon a jump from sae j o sae k he las erm in (2.12) changes immediaely from he discouned value of he reserve in sae j jus before he jump o he value of he reserve in sae k a he ime of he jump. Since he sae-wise reserves are deerminisic funcions wih finie variaion, hey have a mos a counable number of disconinuiies a fixed imes. The probabiliy ha he Markov process jumps a any such ime is. Therefore, we need no worry abou possible common poins of disconinuiy of he V j () and he Ij Z (). For he same reason we can also disregard he lef limi in V j ( ) in he las erm. We proceed by insering (2.1) for db() and he expression dnjk Z () = dm jk Z () Ij Z() µ jk() d obained from (2.3), and gaher dm() = e r Ij Z () db j () r() V j () d + dv j () + µ jk d R jk () j k; k j + e r R jk () dmjk Z (), (2.13) j k Since he las erm on he righ of (2.13) is he incremen of a maringale, he firs erm of he righ is he difference beween he incremens of wo maringales 6

and is hus iself he incremen of a maringale. This maringale has finie variaion and, as will be explained below, is also coninuous, and mus herefore be consan. For his o be rue for all realizaions of he indicaor funcions Ij Z, we mus have db j () r j V j () d + dv j () + µ jk d R jk () =, (2.14) k; k j which is Thiele s differenial equaion. Furhermore we obain ha dm() = e r R jk () dmjk Z (), j k which displays he dynamics of he maringale M. Finally, we explain why he (2.14) is he incremen a of a coninuous funcion. The d erms are coninuous incremens, of course. Ouside jump imes of he B j boh he B j hemselves and he V j are coninuous. A any ime where here is a jump in some B j he reserve V j jumps by he same amoun in he opposie direcion since V j ( ) = B j () + V j (). Thus, B j + V j is indeed coninuous. 3 Insurance risk A. Differenial equaions for momens of presen values. We wan o deermine higher order momens of fuure ne liabiliies. By he Markov propery, we need only he sae-wise condiional momens V (q) j () = E [( T e q τ db(τ)) r Z() = j q = 1, 2,... The funcions V (q) j are deermined by he differenial equaions d d V (q) j () = (qr() + µ j ())V (q) j () qb j ()V (q 1) j () q ( ) q µ jk () ()) p V (q p) k (), p jk k; k j valid on (, T )\D and subjec o he condiions q ( ) V (q) q j ( ) = (B j () B j ( )) p V (q p) j (), (3.1) p p= D. A rigorous proof is given in [25]. The compuaion goes as follows. Firs solve he differenial equaions in he upper inerval ( m 1, n), where he side condiions (3.1) are jus p= V (q) j (n ) = (B j (n) B j (n )) q (3.2) 7 ],

since V (q) j (n) = δ q (he Kronecker dela). Then, if m > 1, solve he differenial equaions in he inerval ( m 2, m 1 ) subjec o (3.1) wih = m 1, and proceed in his manner downwards. Leing m (q) j () denoe he q-h cenral momen corresponding o he noncenral V (q) j (), we have m (1) j () = V (1) j (), (3.3) q ) m (q) j () = p= ( 1) q p ( q p V (p) j () ( V (1) j ()) q p. (3.4) B. Numerical examples. Referring o disabiliy model, consider a male insured a age 3 for a period of 3 years. We assume ha he inensiies of ransiion a ime, when he insured is 3 + years old, are µ ad () = µ id () =.5 +.75858 1.38(3+), µ ai () =.4 +.34674 1.6(3+), µ ia () =.5, and ha he ineres rae is consan, r =.5. The cenral momens m (q) j () defined in (3.3) (3.4) have been compued for he saes a and i (sae d is unineresing) and are shown in Table 1 for (1) a erm insurance wih sum 1 (= b 2 = b 12 ); (2) an annuiy payable in acive sae wih level inensiy 1 (= b ); (3) an annuiy payable in disabled sae wih level inensiy 1 (= b 1 ); (4) for a combined policy providing a erm insurance wih sum 1 (= b 2 = b 12 ) and a disabiliy annuiy wih level inensiy.5 (= b 1 ) agains level ne premium.125 (= b ) payable in acive sae. Table 1: Cenral momens m (q) j (), q = 1, 2, 3, of he presen value of he four paymens sreams (1) (4) lised above. m (1) a () m (1) i () m (2) a () m (2) i () m (3) a () m (3) i () (1).63.63.27.27.12.12 (2) 14.748.79 4.816 6.51 39.54 61.399 (3).245 13.573 1.477 8.675 12.61 69.4 (4). 7.158.397 2.234 1.597 9.379 C. Solvency margins. Le W be he presen value of all fuure ne liabiliies in respec of an insurance porfolio. Denoe he q-h cenral momen of W by m (q). The so-called normal 8

power approximaion of he upper ε-fracile of he disribuion of W, which we denoe by w 1 ε, is based on he firs hree momens and is w 1 ε m (1) + c 1 ε m (2) + c2 1 ε 1 m (3) 6 m, (2) where c 1 ε is he upper ε-fracile of he sandard normal disribuion. Adoping he so-called break-up crierion in solvency conrol, w 1 ε can be aken as a minimum requiremen on he echnical reserve a he ime of consideraion. I decomposes ino he premium reserve, m (1), and wha can be ermed he flucuaion reserve, w 1 ε m (1). A possible measure of he riskiness of he porfolio is he raio R = ( w 1 ε m (1)) /P, where P is some suiable measure of he size of he porfolio. By way of illusraion, consider a porfolio of N independen policies, all idenical o he combined policy (4) in Table 1. Taking as P he oal premium income per year, he value of R a he ime of issue is 48.61 for N = 1, 12. for N = 1, 3.46 for N = 1, 1.6 for N = 1, and.332 for N = 1. 4 Sochasic ineres and financial risk. A. A Markov chain ineres model. The economy (or raher he par of he economy ha governs he ineres) is modeled as a homogeneous ime-coninuous Markov chain Y on a finie sae space Y = {1,..., J Y }, wih inensiies of ransiion λ ef, e, f J Y, e f. The associaed indicaor and couning processes are denoed by I Y e and N Y ef, respecively, and he filraion generaed by Y is denoed by F Y = {F Y } [,T ]. We assume ha he force of ineres akes a fixed value r e when he economy is in sae e, ha is, r() = r Y () = e I Y e ()r e. (4.1) Figure 2 oulines a simple Markov chain ineres rae model wih hree saes {1, 2, 3} and sae-wise raes of ineres r 1 =.2 (low), r 2 =.5 (medium), and r 3 =.8 (high). Direc ransiion can only be made o a neighbouring sae, and he oal inensiy of ransiion ou of any sae is.5, ha is, he ineres rae changes every wo years on he average. By symmery, he saionary (long run average) ineres rae is.5. B. The full Markov model. We assume ha he processes Y and Z are defiined on he same probabiliy space and ha hey are sochasically independen. Then (Y, Z) is a Markov chain on Y Z wih inensiies κ ej,fk () = λ ef (), e f, j = k, µ jk (), e = f, j k,, e f, j k. 9

λ 12 =.5 λ 23 =.25 r 1 =.2 r 2 =.5 r 3 =.8 λ21 =.25 λ32 =.5 Figure 2: Skech of a simple Markov chain ineres model. For he purpose of assessing he conracual liabiliy we are ineresed in aspecs of is condiional disribuion, given he available informaion a ime. We focus here on deermining he condiional momens. By he Markov assumpion, he funcions in ques are he sae-wise condiional momens [( T V (q) ej () = E e q τ db(τ)) r Y () = e, Z() = j The differenial equaions exend sraighforwardly o d d V (q) ej µ jk () k;k j Denoe by m (q) ej () = (qr e + µ j () + λ e )V (q) ej q p= ( ) q p (b jk ()) p V (q p) ek () f;f e ] () qb j()v (q 1) () ej. λ ef V (q) fj (). (4.2) (q) () he q-h cenral momen corresponding o V ej (). C. Numerical resuls for a combined insurance policy. Consider he combined life insurance and disabiliy pension policy and he Markov chain ineres model in Figure 2 modified such ha he infiniesimal marix is Λ = λ 1 1.5 1.5 1 1. (4.3) The scalar λ can be inerpreed as he expeced number of ineres changes per ime uni. Table 2 displays he firs hree cenral momens of he presen value a ime. The level premium rae (= b 1 ) is he ne premium rae in sae (2,a) (i.e. he rae ha esablishes expeced balance beween discouned premiums and benefis when he insured is acive and he ineres is a medium level.5 a ime ). The firs hree rows in he body of he able form a benchmark; λ = means no ineres flucuaion, and we herefore obain he resuls for hree cases of fixed ineres. I is seen ha he second and hird order momens of he 1

Table 2: Cenral momens m (q) ej () of orders q = 1, 2, 3 of he presen value of fuure benefis less premiums for he combined policy in ineres sae e and policy sae j a ime, for some differen values of he rae of ineres changes, λ. Second column gives he ne premium π of a policy saring from ineres sae 2 (medium) and policy sae 1 (acive). e, j : 1, 1 1, 2 2, 1 2, 2 3, 1 3, 2 λ π q 1.72 1.26. 7.158.3 5.318.125 2 1.156 6.24.397 2.234.159.938 3 6.845 36.93 1.597 9.379.465 2.947 1.3 9.166. 7.254.18 5.843.5.127 2.838 5.622.438 3.151.236 1.793 3 4.652 21.814 1.895 7.923.86 3.11 1.1 7.61. 7.212.1 6.854.5.126 2.467 3.14.416 2.767.37 2.467 3 2.7 1.546 1.74 8.88 1.462 7.357 1. 7.27. 7.165. 7.123 5.125 2.44 2.33.399 2.32.394 2.274 3 1.643 9.491 1.613 9.322 1.585 9.155 1. 7.158. 7.158. 7.158.125 2.397 2.234.397 2.234.397 2.234 3 1.597 9.379 1.597 9.379 1.597 9.379 11

presen value are srongly dependen on he (fixed) force of ineres and, in fac, heir absolue values decrease when he force of ineres increases (as could be expeced since increasing ineres means decreasing discoun facors and, hence, decreasing presen values of fuure amouns). I is seen ha, as λ increases, he differences across he hree pairs of columns ge smaller and in he end hey vanish compleely. The obvious inerpreaion is ha he iniial ineres level is of lile imporance if he ineres changes rapidly. The overall impression from he wo cenral columns corresponding o medium ineres is ha, as λ increases from, he variance of he presen value will firs increase o a maximum and hen decrease again and sabilize. This observaion suppors he following piece of inuiion: he inroducion of moderae ineres flucuaion adds uncerainy o he final resul of he conrac, bu if he ineres changes sufficienly rapidly, i will behave like fixed ineres a he mean level. Presumably, he values of he ne premium in he second column reflec he same effec. 5 Geing rid of financial risk A. Non-diversifiable risk. The principle of equivalence ress on he implici assumpion ha he experience basis, ha is he ransiion inensiies, ineres, and adminisraion coss hroughou he conrac period, are known a he ime of incepion of he conrac. In realiy, however, he experience basis may undergo significan and unforeseeable changes wihin he ime horizon of he conrac, hus exposing he insurer o a risk ha is non-diversifiable. In he presen paper, which focuses on he inerplay beween insurance and finance, we will concenrae on he ineres rae and assume ha all oher elemens of he experience basis are known and fixed hroughou he erm of he conrac. The risk semming from he uncerain developmen of he ineres rae can, under cerain ideal marke condiions, be eliminaed by leing he conracual paymens depend on he reurns on he company s invesmens. Producs of his ype, known as uni-linked insurances, have been gaining increasing marke shares ever since hey emerged some few decades ago, and oday hey are also heoreically well undersood, see Aase and Persson (1994), Møller (1998), and references herein. Unlike he uni-linked concep, a sandard life insurance policy specifies conracual paymens in nominal amouns, binding o boh paries hroughou he enire erm of he conrac. Thus, an adverse developmen of he ineres raes can no be counered by raising premiums or reducing benefis and also no by cancelling he conrac (he righ of wihdrawal remains one-sidedly wih he insured). The only way he insurer can preven he non-diversifiable financial risk is o charge premiums o he safe side. In pracice his is done by calculaing premiums on a conservaive so-called echnical basis or firs order basis, which represens a provisional wors-case scenario for he fuure developmen 12

of he experience basis. In our simplified se-up, wih ineres as he only uncerain elemen in he experience basis, his means ha premiums and reserves are calculaed under he assumpion ha he ineres rae is r, ypically lower han expeced. Denoe he corresponding firs order sae-wise reserve by Vj. Premiums are se in accordance wih he principle of equivalence, [ n ] E e τ r db(τ) =, (5.1) or, equivalenly, V 1 () = B 1 (). (5.2) B. Definiion of he surplus. Wih premiums based on pruden firs order assumpions, he porfolio will creae a sysemaic echnical surplus if everyhing goes well. We define he surplus a ime as S() = = e r e τ r d( B)(τ) VZ() () (5.3) e τ δ db(τ) VZ() (), (5.4) which is pas ne income (premiums less benefis), compounded wih he facual second order ineres, minus expeced discouned fuure liabiliies valuaed on he conservaive firs order basis. This definiion complies wih pracical accounancy regulaions in insurance since S() is precisely he difference beween he curren cash balance and he firs order reserve ha by saue has o be provided o mee fuure liabiliies. Noice ha S() =, a consequence of (5.1), and S(T ) = T e T τ δ db(τ), as i ough o be. Differeniaing (5.4) gives ds() = e r r() d e τ r db(τ) db() dv Z() () = r() d S() + r() d V Z() () db() dv Z() (). Upon subsiuing db() from (2.1) and VZ() () from (2.5), using he general Iô formula o wrie dv Z() () = j I Z j () dvj () + {Vk () V j ()} dn jk Z () j k (here are almos surely no common jumps of he deerminisic sae-wise reserves and he couning processes), and picking dvj () from (2.8), we find where ds() = δ() d S() + dc() + dm(), (5.5) dc() = j I Z j () c j() d, 13

wih c j () = (r() r ) V j () (5.6) and dm() = j k R jk() dm Z jk(), wih he Mjk Z defined in (2.3). The process M is a zero mean H-maringale in he condiional model, given G n, ha is, E[M() H s G n ] = M(s) for s, and M() =. Then i is also a zero mean F-maringale in he full model since E[M() F s ] = E [ E[M() H s G n ] F s ] = E[M(s) F s ] = M(s). The erm dm() in (5.5) is he purely accidenal par of he surplus incremen. The wo firs erms on he righ of (5.5) are he sysemaic pars, which make he surplus drif o somehing wih expeced value differen from. The firs erm is he earned ineres on he surplus iself, and wha remains is quie naurally he policy-holder s conribuion o he echnical surplus. To pu i anoher way, le us swich he firs erm on he righ of (5.5) over o he lef and muliply he equaion wih e δ o form a complee differenial on he lef hand side. Inegraing from o and using he fac ha S() = C() = M() =, we arrive a e δ S() = e τ δ dc(τ) + e τ δ dm(τ), (5.7) showing ha he discouned surplus a ime is he discouned oal conribuions plus a maringale represening noise. We can rewrie (5.7) as S() = e τ δ dc(τ) + e τ δ dm(τ), (5.8) displaying he surplus a ime as he compounded oal of conribuions and accidenal maringale incremens. (Beware ha he las erm on he righ of (5.8) is no a maringale alhough is expeced value is for all.) C. Redisribuion of surplus Bonus. The echnical surplus belongs o he insured and has o paid back as bonus. There are many possible schemes. The simples is cash bonus: The rae a which bonus will be paid a some fixed fuure ime u, provided he insured is hen alive, is W = (r(u) r )V (u). Adoping he Markov chain ineres model, we can make model-based predicion of his quaniy. A ime < u, given r() = r e, W is prediced by is condiional expeced value W e () = E[W r() = r e ]. 14

I is easy o show ha he funcions W e () are he soluion o he differenial equaions d d W e() = λ ef (W e () W f ()), subjec o he condiions l;f e W e (u) = (r e r )V u, e = 1,..., J Y. By erminal bonus he surpluses are accumulaed and paid back as a lump sum a he erm of he conrac T, provided he insured is hen alive. The amoun paid is where W = T e T τ = W () W () = e T W () = r (r(τ) r )V (τ) dτ e τ r (r(τ) r )V (τ) dτ + W (), r, T e T τ r (r(τ) r )V (τ) dτ. The random variables W () and W (), which are unknown a ime, are prediced by Wriing W e () = E[W () r() = r e ], W e () = E[W () r() = r e ]. W () = e r() d W ( + d), W () = W () (r() r ) V () d + W ( + d), we easily show by a backward argumen ha he funcions W e() and W e () are he soluion o he differenial equaions d d W e () = r e W e () + λ ef (W e () W f ()), f;f e d d W e () = W e()(r e r )V () + subjec o he condiions e = 1,..., J Y. W e (T ) = 1, W e (T ) =, f;f e λ ef (W e () W f ()), 15

6 Reinroducing financial risk, and eliminaing i again A. Guaraneed ineres. Recall he basic rules of he wih profi insurance conrac: On he one hand, any surplus is o be redisribued o he insured. On he oher hand, benefis and premiums se ou in he conrac canno be alered o he insured s disadvanage. This means ha negaive surplus, should i occur, canno resul in negaive bonus. Thus, he wih profi policy comes wih an ineres rae guaranee o he effec ha bonus is o be paid as if facual ineres were no less han firs order ineres, roughly speaking. For insance, cash bonus is o be paid a rae (r() r ) + V ( ) per survivor a ime, hence he insurer has o cover (r r()) + V (). (6.1) Similarly, erminal bonus (ypical for e.g. a pure endowmen benefi) is o be paid as a lump sum ( ) T e T τ r (r(τ) r )V (τ) dτ per survivor a ime n, hence he insurer has o cover ( ) T e T τ r (r r(τ))v (τ) dτ + +. (6.2) (We wrie a + = max(a, ) = a.) An ineres guaranee of his kind represens a liabiliy on he par of insurer. I canno be offered for free, of course, bu has o be compensaed by a premium. This can cerainly be done wihou violaing he rules of game for he paricipaing policy, which lay down ha premiums and benefis be se ou in he conrac a ime. Thus, for simpliciy, suppose a single premium is o be colleced a ime for he guaranee. The quesion is, how much should i be? Being brough up wih he principle of equivalence, we migh hink ha he expeced discouned value of he liabiliy is an agreeable candidae for he premium. However, he raionale of he principle of equivalence, which was o make premiums and benefis balance on he average in an infiniely large porfolio, does no apply o financial risk. Ineres rae variaions canno be eliminaed by increasing he size of he porfolio; all policy-holders are faring ogeher in one and he same boa on heir once-in-a-lifeime voyage hrough he roubled waers of heir chaper of economic hisory. This risk canno be averaged ou in he same way as he risk associaed wih he lenghs of he individual lives. 16

None he less, in lack of anyhing beer, le us find he expeced discouned value of he ineres guaranee, and jus anicipae here ha his acually would be he correc premium in an exended model specifying a so-called complee financial marke. Those who are familiar wih basic arbirage heory know wha his means. Those who are no should a his sage jus imagine ha, in addiion o he bank accoun wih he ineres rae r(), here are some oher invesmen opporuniies, and ha any fuure financial claim can be duplicaed perfecly by invesing a cerain amoun a ime and hereafer jus selling and buying available asses wihou any furher infusion of capial. The iniial amoun required o perform his duplicaing invesmen sraegy is, quie naurally, he price of he claim. I urns ou ha his price is precisely he expeced discouned value of he claim, only under a differen probabiliy measure han he one we have specified in our physical model. Wih hese reassuring phrases, le us proceed o find he expeced discouned value of he ineres guaranee. B. Pricing guaraneed ineres. Consider cash bonus wih guranee given by (6.1): Given ha r() = r e (say), he price of he oal claims under he guaranee, averaged over an infiniely large porfolio, is [ ] T E e τ r (r r(τ)) + V (τ) τ p x dτ r() = r e. (6.3) A naural saring poin for creaing some useful differenial equaions by he backward consrucion is he price of fuure claims under he guaranee in sae e a ime, [ ] T W e () = E e τ r (r r(τ)) + V (τ) τ p x dτ r() = r e, (6.4) e = 1,..., J Y, T. The price in (6.3) is precisely W e (). Condiioning on wha happens in he ime inerval (, + d] and neglecing erms of order o(d) ha will disappear in he end anyway, we find W e () = (1 λ ( e d) (r r e ) + V () p x d + e red W e ( + d) ) + λ ef d W f (). From here we easily arrive a he differenial equaions d d W e() = (r r e ) + V () p x +r e W e () f; f e which are o be solved subjec o he condiions f; f e λ ef d (W f () W e ()), (6.5) W e (T ) =. (6.6) 17

Nex, consider erminal bonus a ime n given by (6.2): Given r() = r e, he price of he claim under he guaranee, averaged over an infiniely large porfolio, is [ ( ) ] T E e n r e T τ r (r r(τ))v (τ) dτ T p x r() = r e [ ( ) T = E e τ r (r r(τ))v (τ) dτ + + ] r() = r e T p x. (6.7) The price of he claim a ime should be he condiional expeced discouned value of he claim, given wha we know a he ime: [ ( ) ] T E e T r e T τ r (r r(τ))v (τ) dτ T p x r(τ); τ where = E [ ( U() + n e U() = + τ r (r r(τ))v (τ) dτ ) + e τ r (r r(τ))v (τ) dτ. ] r(τ); τ T p x. (6.8) The quaniy in (6.8) is more involved han he one in (6.4) since i depends effecively on he pas hisory of ineres rae hrough U(). We can, herefore, no hope o end up wih he same simple ype of problem as above and in all oher siuaions encounered so far, where we essenially had o deermine he condiional expeced value of some funcion depending only on he fuure course of he ineres rae. Which was easy since, by he Markov propery, we could look a sae-wise condiional expeced values W e (), e = 1,..., J Y, say. These are deerminisic funcions of he ime only and can be deermined by solving ordinary differenial equaions. Le us proceed and see wha happens. Due o he Markov propery (condiional independence beween pas and fuure, given he presen) he expression in (6.8) is a funcion of, r() and U(). Dropping he unineresing facor T p x, consider is value for given U() = u and r() = r e, [ ( ) ] W e (, u) = E u + T e Use he backward consrucion: W e (, u) = (1 λ e d)e [ ( τ r (r r(τ))v (τ) dτ u + (r r e )V () d + e re d T 18 e +d + r() = r e τ +d r (r r(τ))v (τ) dτ. ) + ] r( + d) = r e

+ λ ef d W f (, u) = f; f e (1 λ e d)e re d W e ( + d, e re d u + (r r e )V d) + λ ef d W f (, u). f; f e Inser here e ±re d = 1 ± r e d + o(d), W e ( + d, e re d u + (r r e )V () d) = W e (, u) + W e(, u) d + u W e(, u)(u r e + (r r e )V ()) d + o(d), and proceed in he usual manner o arrive a he parial differenial equaions W e(, u)+(ur e +(r r e )V ()) u W e(, u) r e W e (, u)+ These are o be solved subjec o he condiions W e (T, u) = u +, f; f j λ ef (W f (, u) W e (, u)) =. e = 1,..., J Y. Since he funcions we are ineresed in involved boh and U(), we are lead o sae-wise funcions in wo argumens and, herefore, quie naurally end up wih parial differenial equaions for hose. 7 A Markov chain financial marke A. Moivaion. The heory of diffusion processes, wih is wealh of powerful heorems and model variaions, is an indispensable oolki in modern financial mahemaics. The seminal papers of Black and Scholes [6] and Meron [21] were crafed wih Brownian moion, and so were mos of he almos counless papers on arbirage pricing heory and is bifurcaions ha followed over he pas quarer of a cenury. A main course of curren research, iniiaed by he maringale approach o arbirage pricing ([15] and [16]), aims a generalizaion and unificaion. Today he core of he maer is well undersood in a general semimaringale seing, see e.g. [9]. Anoher course of research invesigaes special models, in paricular various Levy moion alernaives o he Brownian driving process, see e.g. [1] and [27]. Pure jump processes have been widely used in finance, ranging from plain Poisson processes inroduced in [22] o quie general marked poin processes, see e.g. [4]. And, as a pedagogical exercise, he marke driven by a binomial process has been inensively sudied since i was launched in [8]. We will here presen a model where he financial marke is driven by a coninuous ime homogeneous Markov chain. The idea was launched in [26] and 19

reappeared in [11], he conex being limied o modelling of he spo rae of ineres. This will allow us o synhesize insurance and finance wihin he mahemaical model framework already familiar o us. Some addiional noaion and resuls are presened in Appendix H. B. The Markov chain marke We are going o exend he ineres model in 4. Thus le {Y } be a coninuous ime Markov chain wih finie sae space Y = {1,..., J Y }. Recall ha he associaed indicaor and couning processes are denoed by Ie Y and Nef Y. The F Y = {F Y }, is aken o saisfy he usual condiions of righ-coninuiy (F = u> F u ) and compleeness (F conains all subses of P-nullses), and F is assumed o be he rivial (, Ω). This means, essenially, ha Y is righconinuous (hence he same goes for he Ie Y and he Nef Y ) and ha Y deerminisic. We assume ha Y is ime homogeneous so ha he ransiion probabiliies p Y ef (s, ) = P[Y = f Y s = e] depend only on he lengh of he ransiion period, s. Henceforh we herefore wrie p Y ef (s, ) = py ef ( s). This implies ha he ransiion inensiies p Y ef λ ef = lim (), (7.1) e f, exis and are consan. To avoid repeiious reminders of he ype e, f Y, we reserve he indices e and f for saes in Y hroughou. We will frequenly refer o Y e = {f; λ ef > }, he se of saes ha are direcly accessible from sae e, and denoe he number of such saes by J Y e = Y e. Pu λ ee = λ e = f;f Y e λ ef (minus he oal inensiy of ransiion ou of sae e). We assume ha all saes inercommunicae so ha p Y ef () > for all e, f (and > ). This implies ha > for all e (no absorbing saes). The marix of ransiion probabiliies, J Y e and he infiniesimal marix, P Y () = (p ef ()), Λ = (λ ef ), are relaed by (7.1), which in marix form reads Λ = lim 1 (PY () I), and by he backward and forward Kolmogorov differenial equaions, d d PY () = P Y ()Λ = ΛP Y (). (7.2) 2

Under he side condiion P Y () = I, (7.2) inegraes o In he represenaion (H.2), P Y () = exp(λ). (7.3) P Y () = Φ D e=1,...,j Y (e ρe ) Φ 1 = n e ρe φ j ψ j, (7.4) he firs (say) eigenvalue is ρ 1 =, and corresponding eigenvecors are φ 1 = 1 and ψ 1 = (p 1,..., p n ) = lim (p e1 (),..., p ej Y ()), he saionary disribuion of Y. The remaining eigenvalues, ρ 2,..., ρ n, are all sricly negaive so ha, by (7.4), he ransiion probabiliies converge exponenially o he saionary disribuion as increases. We now urn o he subjec maer of our sudy and, referring o inroducory exs like [5] and [32], ake basic noions and resuls from arbirage pricing heory as prerequisies. C. The coninuous ime Markov chain marke. We consider a financial marke driven by he Markov chain described above. Thus, Y represens he sae of he economy a ime, F Y represens he informaion available abou he economic hisory by ime, and F Y represens he flow of such informaion over ime. In he marke here are m + 1 basic asses, which can be raded freely and fricionlessly (shor sales are allowed, and here are no ransacion coss). A special role is played by asse No., which is a locally risk-free bank accoun wih sae-dependen ineres rae e=1 r() = r Y = e I e ()r e, where he sae-wise ineres raes r e, e = 1,..., J Y, are consans. Thus, is price process is ( ) ( ) S () = exp r(s) ds = exp r e I e (s) ds, wih dynamics e ds () = S () r() d = S () e r e I e () d. (Seing S () = 1 a jus a maer of convenion.) The remaining m asses, henceforh referred o as socks, are risky, wih price processes of he form S i () = exp α ie I e (s) ds + β ief N ef (), (7.5) e e f Y e 21

i = 1,..., m, where he α ie and β ief are consans and, for each i, a leas one of he β ief is non-null. Thus, in addiion o yielding sae-dependen reurns of he same form as he bank accoun, sock No. i makes a price jump of relaive size γ ief = exp (β ief ) 1 upon any ransiion of he economy from sae j o sae k. By he general Iô s formula, is dynamics is given by ds i () = S i ( ) α ie I e () d + γ ief dn ef (). (7.6) e e f Y e Taking he bank accoun as numeraire, we inroduce he discouned sock prices S i () = S i ()/S (), i =,..., m. (The discouned price of he bank accoun is B 1, which is cerainly a maringale under any measure). The discouned sock prices are S i () = exp (α ie r e ) I e (s) ds + β ief N ef (), (7.7) e e f Y e wih dynamics d S i () = S i () e (α ie r e )I e () d + e γ ief dn ef (), (7.8) f Y e i = 1,..., m. We sress ha he heory we are going o develop does no aim a explaining how he prices of he basic asses emerge from supply and demand, business cycles, invesmen climae, or whaever; hey are exogenously given basic eniies. (And God said le here be ligh, and here was ligh, and he said le here also be hese prices.) The purpose of he heory is o derive principles for consisen pricing of financial conracs, derivaives, or claims in a given marke. D. Porfolios. A dynamic porfolio or invesmen sraegy is an m + 1-dimensional sochasic process θ () = (η(), ξ ()), where η() represens he number of unis of he bank accoun held a ime, and he i-h enry in ξ() = (ξ 1 (),..., ξ m ()) represens he number of unis of sock No. i held a ime. As i will urn ou, he bank accoun and he socks will appear o play differen pars in he show, he laer being he more visible. I is, herefore, convenien o cosume 22

he wo ypes of asses and heir corresponding porfolio enries accordingly. To save noaion, however, i is useful also o work wih double noaion θ() = (θ (),..., θ m ()), wih θ () = η(), θ i () = ξ i (), i = 1,..., m, and work wih and S = (S 1 (),..., S m ()). S = (S (),..., S m ()). The porfolio θ is adaped o F Y (he invesor canno see ino he fuure), and he shares of socks, ξ, mus also be F Y -predicable (he invesor canno, e.g. upon a sudden crash of he sock marke, escape losses by selling socks a prices quoed jus before and hurry he money over o he locally risk-free bank accoun.) The value of he porfolio a ime is V θ () = η()s () + m ξ i ()S i () = η()s () + ξ ()S() = θ ()S () i=1 Henceforh we will mainly work wih discouned prices and values and, in accordance wih (7.7), equip heir symbols wih a ilde. The discouned value of he porfolio a ime is Ṽ θ () = η() + ξ () S() = θ () S (). (7.9) The sraegy θ is self-financing (SF) if dv θ () = θ () ds () or, equivalenly, dṽ θ () = θ () d S () = m ξ i () d S i (). (7.1) We explain he las sep: Pu D() = S () 1, a coninuous process. The dynamics of he discouned prices S () = D()S () is hen d S () = dd()s ()+ D() ds (). Thus, for Ṽ θ () = D()V θ (), we have dṽ θ () = dd() V θ () + D() dv θ () = dd() θ () S () + D()θ () ds () i=1 = θ () (dd()s () + D() ds ()) = θ () d S (), hence he propery of being self-financing is preserved under discouning. The SF propery says ha, afer he iniial invesmen of V θ, no furher invesmen inflow or dividend ouflow is allowed. In inegral form: Ṽ θ () = Ṽ θ + θ(s) d S(s) = Ṽ θ + ξ(s) d S(s). (7.11) Obviously, a consan porfolio θ is SF; is discouned value process is Ṽ θ () = θ S (), hence (7.1) is saisfied. More generally, for a coninuous porfolio θ we would have dṽ(θ) = dθ () S () + θ () d S (), and he self-financing 23

condiion would be equivalen o he a budge consrain dθ () S () =, which says ha any purchase of asses mus be financed by a sale of some oher asses. We urge o say ha we shall ypically be dealing wih porfolios ha are no coninuous and, in fac, no even righ-coninuous so ha dθ() is meaningless (inegrals wih respec o he process θ are no well defined). E. Absence of arbirage. An SF porfolio θ is called an arbirage if, for some >, V θ < and V θ () a.s. P, or, equivalenly, Ṽ θ < and Ṽ θ () a.s. P. A basic requiremen on a well-funcioning marke is he absence of arbirage. The assumpion of no arbirage, which appears very modes, has surprisingly far-reaching consequences as we shall see. A maringale measure is any probabiliy measure P ha is equivalen o P and such ha he discouned asse prices S () are maringales (F, P). The fundamenal heorem of arbirage pricing says: If here exiss a maringale measure, hen here is no arbirage. This resul follows from easy calculaions saring from (7.11): Forming expecaion Ẽ under P and using he maringale propery of S under P, we find E[Ṽ θ ()] = Ṽ θ + E[ ξ(s) d S(s)] = Ṽ θ (he sochasic inegral is a maringale). I is seen ha arbirage is impossible. We reurn now o our special Markov chain driven marke. Le Λ = ( λ ef ) be an infiniesimal marix ha is equivalen o Λ in he sense ha λ ef = if and only if λ ef =. By Girsanov s heorem, here exiss a measure P, equivalen o P, under which Y is a Markov chain wih infiniesimal marix Λ. Consequenly, he processes M Y ef, j = 1,..., J Y, f Y e, defined by d M Y ef () = dn ef () I e () λ ef d, (7.12) and M ef () =, are zero mean, muually orhogonal maringales w.r.. (F Y, P). Rewrie (7.8) as d S i () = S i α ie r e + γ ief λef I e () d + γ ief d M ef Y () (7.13), e f Y e e f Y e i = 1,..., m. The discouned sock prices are maringales w.r.. (F Y, P) if and only if he drif erms on he righ vanish, ha is, α ie r e + f Y e γ ief λef =, (7.14) 24

j = 1,..., J Y, i = 1,..., m. From general heory i is known ha he exisence of such an equivalen maringale measure P implies absence of arbirage. The relaion (7.14) can be cas in marix form as j = 1,..., J Y, where 1 is m 1 and α e = (α ie ) i=1,...,m, r e 1 α e = Γ e λe, (7.15) ) Γ e = (γ ief ) f Ye i=1,...,m, λe = ( λef. f Y e The exisence of an equivalen maringale measure is equivalen o he exisence of a soluion λ e o (7.15) wih all enries sricly posiive. Thus, he marke is arbirage-free if (and we can show only if) for each j, r e 1 α e is in he inerior of he convex cone of he columns of Γ e. Assume henceforh ha he marke is arbirage-free so ha (7.13) reduces o d S i () = S i () γ ief d M ef Y (), (7.16) e f Y e where he M ef defined by (7.12) are maringales w.r.. (F Y, P) for some measure P ha is equivalen o P. Insering (7.16) ino (7.1), we find ha θ is SF if and only if dṽ θ () = e m f Y e i=1 ξ i () S i ( )γ ief d M ef Y (), (7.17) implying ha Ṽ θ is a maringale w.r.. (F Y, P) and, in paricular, Ṽ θ () = Ẽ[Ṽ θ () F ]. (7.18) Here Ẽ denoes expecaion under P. (Noe ha he ilde, which in he firs place was inroduced o disinguish discouned values from he nominal ones, is also aached o he equivalen maringale measure and cerain relaed eniies. This usage is moivaed by he fac ha he maringale measure arises from he discouned basic price processes, roughly speaking.) F. Aainabiliy. A T -claim is a conracual paymen due a ime T. Formally, i is an F Y T - measurable random variable H wih finie expeced value. The claim is aainable if i can be perfecly duplicaed by some SF porfolio θ, ha is, Ṽ θ T = H. (7.19) If an aainable claim should be raded in he marke, hen is price mus a any ime be equal o he value of he duplicaing porfolio in order o avoid 25

arbirage. Thus, denoing he price process by π() and, recalling (7.18) and (7.19), we have π() = Ṽ θ () = Ẽ[ H F ], (7.2) or [ ] π() = Ẽ e T r H F. (7.21) By (7.2) and (7.17), he dynamics of he discouned price process of an aainable claim is d π() = e m f Y e i=1 G. Compleeness. Any T -claim H as defined above can be represened as ξ i () S i ( )γ ief d M ef Y (). (7.22) T H = Ẽ[ H] + ζ ef ()d M ef Y (), (7.23) e f Y e where he ζ ef () are F Y -predicable and inegrable processes. Conversely, any random variable of he form (7.23) is, of course, a T -claim. By virue of (7.19), and (7.17), aainabiliy of H means ha T H = Ṽ θ + = Ṽ θ + T dṽ θ () ξ i () S i ( )γ ief d M ef Y (). (7.24) e f Y e Comparing (7.23) and (7.24), we see ha H is aainable iff here exis predicable processes ξ 1 (),..., ξ m () such ha m ξ i () S i ( )γ ief = ζ ef (), i=1 for all j and f Y e. This means ha he J Y e -vecor i ζ e () = (ζ ef ()) f Ye is in R(Γ e ). The marke is complee if every T -claim is aainable, ha is, if every n j - vecor is in R(Γ e ). This is he case if and only if rank(γ e ) = J Y e, which can be fulfilled for each e only if m max e J Y e. 26

8 Arbirage-pricing of derivaives in a complee marke A. Differenial equaions for he arbirage-free price. Assume ha he marke is arbirage-free and complee so ha prices of T -claims are uniquely given by (7.2) or (7.21). Le us for he ime being consider a T -claim of he form H = h(y (T ), S l (T )). (8.1) Examples are a European call opion on sock No. l defined by H = (S l (T ) K) +, a caple defined by H = (r(t ) g) + = (r YT g) +, and a zero coupon T -bond defined by H = 1. For any claim of he form (8.1) he relevan sae variables involved in he condiional expecaion (7.21) are, Y (), S l (), hence π() is of he form π() = J Y e=1 I e ()f e (, S l ()), (8.2) where he f e (, s) = [e Ẽ T r H ] Y () = e, S l () = s (8.3) are he sae-wise price funcions. The discouned price (7.2) is a maringale w.r.. (F Y, P). Assume ha he funcions f e (, s) are coninuously difereniable. Using Iô on π() = e Y J r e=1 I e ()f e (, S l ()), (8.4) we find d π() = e r e +e r e I e () ( r e f e (, S l ()) + f e(, S l ()) + s ) f e(, S l ())S l ()α lj d f Y e (f f (, S l ( )(1 + γ lef )) f e (, S l ( ))) dn ef () = e r I e () ( r e f e (, S l ()) + f e(, S l ()) + s f e(, S l ())S l ()α le e + ) {f f (, S l (1 + γ lef )) f e (, S l ( ))} λ ef d f Y e +e r (f f (, S l ( )(1 + γ lef )) f e (, S l ( ))) d M ef Y (). (8.5) e f Y e 27

By he maringale propery, he drif erm mus vanish, and we arrive a he non-sochasic parial differenial equaions r e f e (, s) + f e(, s) + s f e(, s)sα le + f Y e (f f (, s(1 + γ lef )) f e (, s)) λ ef =, (8.6) e = 1,..., J Y, which are o be solved subjec o he side condiions e = 1,..., J Y. In marix form, wih f e (T, s) = h(e, s), (8.7) R = D j=1,...,j Y (r e ), A l = D j=1,...,j Y (α le ), and oher symbols (hopefully) self-explaining, he differenial equaions and he side condiions are Rf(, s) + f(, s) + sa l s f(, s) + Λf(, s(1 + γ)) =, (8.8) f(t, s) = h(s). (8.9) B. Idenifying he sraegy. Once we have deermined he soluion f e (, s), e = 1,..., J Y, he price process is known and given by (8.2). The duplicaing SF sraegy can be obained as follows. Seing he drif erm o in (8.5), we find he dynamics of he discouned price; d π() = e r (f f (, S l ( )(1 + γ lef )) f e (, S l ( ))) d M ef Y ().(8.1) e f Y e Idenifying he coefficiens in (8.1) wih hose in (7.22), we obain, for each sae j, he equaions m ξ i ()S i ( )γ ief = f f (, S l ( )(1 + γ lef )) f e (, S l ( )), (8.11) i=1 f Y e. The soluion ξ e () = (ξ i,e ()) i=1,...,m (say) cerainly exiss since rank(γ e ) m, and i is unique iff rank(γ e ) = m. Furhermore, i is a funcion of and S( ) and is hus predicable. This simplisic argumen works on he open inervals beween he jumps of he process Y, where d M ef Y () = I e() λ ef d. For he dynamics (8.1) and (7.22) o be he same also a jump imes, he coefficiens mus clearly be lef-coninuous. We conclude ha ξ() = J Y e=1 I e ( )ξ(), 28

which is predicable. Finally, η is deermined upon combining (7.9), (7.2), and (8.4): η() = e Y J r e=1 ( I e ()f e (, S l ()) I e () ) m ξ i,e ()S i (). C. The Asian opion. As an example of a pah-dependen claim, le us consider an Asian opion, which ( T +, essenially is a T -claim of he form H = S l(τ) dτ K) where K. The price process is ( T + π() = Ẽ e T r S l (τ) dτ K) F Y = ) I e ()f e (, S l (), S l (τ) dτ, e where f e (, s, u) = Ẽ e T The discouned price process is r i=1 ( T + S l (τ) + u K) Y () = j, S l() = s (8.12). n π() = e r I e () f e (, S l (), j=1 ) S l(s). We obain parial differenial equaions in hree variables. The special case K = is simpler, wih only wo sae variables. D. Ineres rae derivaives. A paricularly simple, bu sill imporan, class of claims are hose of he form H = h(y T ). Ineres rae derivaives of he form H = h(r T ) are included since r(t ) = r YT. For such claims he only relevan sae variables are and Y (), so ha he funcion in (8.3) depends only on and e. The equaion (8.6) reduces o d d f e() = r e f e () (f f () f e ()) λ ef, (8.13) f Y e and he side condiion is (pu h(e) = h e ) f e (T ) = h e. (8.14) In marix form, d d f() = ( R Λ)f(), 29

subjec o The soluion is f(t ) = h. f() = exp{( Λ R)(T )}h. (8.15) I depends on and T only hrough T. In paricular, he zero coupon bond wih mauriy T corresponds o h = 1. We will henceforh refer o i as he T -bond in shor and denoe is price process by p(, T ) and is sae-wise price funcions by p(, T ) = (p e (, T )) e=1,...,j Y ; p(, T ) = exp{( Λ R)(T )}1. (8.16) For a call opion on a U-bond, exercised a ime T (< U) wih price K, h has enries h e = (p e (T, U) K) +. In (8.15) (8.16) i may be useful o employ he represenaion shown in (H.2), say. exp{( Λ R)(T )} = Φ D j=1,...,j Y (e ρj (T ) ) Φ 1, (8.17) 9 Numerical procedures A. Simulaion. The homogeneous Markov process {Y ()} [,T ] is simulaed as follows: Le K be he number of ransiions beween saes in [, T ], and le T 1,..., T K be he successive imes of ransiion. The sequence {(T n, Y (T n ))} n=,...,k is generaed recursively, saring from he iniial sae Y () a ime T =, as follows. Having arrived a T n and Y (T n ), generae he nex waiing ime T n+1 T n as an exponenial variae wih parameer λ Y (n) (e.g. ln(u n )/λ Y (n), where U n has a uniform disribuion over [, 1]), and le he new sae Y (T n+1 ) be k wih probabiliy λ Y (n)k /λ Y (n). Coninue in his manner K+1 imes unil T K < T T K+1. B. Numerical soluion of differenial equaions. Alernaively, he differenial equaions mus be solved numerically. For ineres rae derivaives, which involve only ordinary firs order differenial equaions, a Runge Kua will do. For sock derivaives, which involve parial firs order differenial equaions, one mus employ a suiable finie difference mehod, see e.g. [35]. 1 Risk minimizaion in incomplee markes A. Incompleeness. The noion of incompleeness perains o siuaions where a coningen claim 3

canno be duplicaed by an SF porfolio and, consequenly, does no receive a unique price from he no arbirage posulae alone. In Paragraph??F we were dealing implicily wih incompleeness arising from a scarciy of raded asses, ha is, he discouned basic price processes are incapable of spanning he space of all maringales w.r.. (F Y, P) and, in paricular, reproducing he value (7.23) of every financial derivaive (funcion of he basic asse prices). Incompleeness also arises when he coningen claim is no a purely financial derivaive, ha is, is value depends also on circumsances exernal o he financial marke. We have in mind insurance claims ha are caused by evens like deah or fire and whose claim amouns are e.g. inflaion adjused or linked o he value of some invesmen porfolio. In he laer case we need o work in an exended model specifying a basic probabiliy space wih a filraion F = {F } conaining F Y and saisfying he usual condiions. Typically i will be he naural filraion of Y and some oher process ha generaes he insurance evens. The definiions and condiions laid down in Paragraphs??C-E are modified accordingly, so ha adapedness of η and predicabiliy of ξ are aken o be w.r.. (F, P) (keeping he symbol P for he basic probabiliy measure), a T -claim H is F T measurable, ec. B. Risk minimizaion. Throughou he remainder of he paper we will mainly be working wih discouned prices and values wihou any oher menion han he noaional ilde. The reason is ha he heory of risk minimizaion ress on cerain maringale represenaion resuls ha apply o discouned prices under a maringale measure. We will be conen o give jus a skechy review of some main conceps and resuls from he seminal paper of Föllmer and Sondermann [12]. Le H be a T -claim ha is no aainable. This means ha an admissible porfolio θ saisfying Ṽ θ (T ) = H canno be SF. The cos, C θ (), of he porfolio by ime is defined as ha par of he value ha has no been gained from rading: C θ () = Ṽ θ () ξ (τ)d S(τ). The risk a ime is defined as he mean squared ousanding cos, R() = [( Ẽ C θ (T ) C ] θ ()) 2 F. (1.1) By definiion, he risk of an admissible porfolio θ is [ ] T R θ () = Ẽ ( H Ṽ θ () ξ (τ)d S(τ)) 2 F, which is a measure of how well he curren value of he porfolio plus fuure rading gains approximaes he claim. The heory of risk minimizaion akes 31

his eniy as is objec funcion and proves he exisence of an opimal admissible porfolio ha minimizes he risk (1.1) for all [, T ]. The proof is consrucive and provides a recipe for how o acually deermine he opimal porfolio. One ses ou by defining he inrinsic value of H a ime as [ ] Ṽ H () = Ẽ H F. Thus, he inrinsic value process is he maringale ha represens he naural curren forecas of he claim under he chosen maringale measure. By he Galchouk-Kunia-Waanabe represenaion, i decomposes uniquely as Ṽ H () = Ẽ[ H] + ξ H ()d S() + L H (), where L H is a maringale w.r.. (F, P) which is orhogonal o S. The porfolio θ H defined by his decomposiion minimizes he risk process among all admissible sraegies. The minimum risk is [ ] T R H () = Ẽ d L H (τ) F. C. Uni-linked insurance. As he name suggess, a life insurance produc is said o be uni-linked if he benefi is a cerain predeermined number of unis of an asse (or porfolio) ino which he premiums are currenly invesed. If he conrac sipulaes a minimum value of he benefi, disconneced from he asse price, hen one speaks of uni-linked insurance wih guaranee. A risk minimizaion approach o pricing and hedging of uni-linked insurance claims was firs aken by Møller [23], who worked wih he Black-Scholes-Meron financial marke. We will here skech how he analysis goes in our Markov chain marke, which conforms well wih he life hisory process in ha hey boh are inensiy-driven. Le T x be he remaining life ime of an x years old who purchases an insurance a ime, say. The condiional probabiliy of survival o age x + u, given survival o age x + ( < u), is u p x+ = P[T x > u T x > ] = e where µ y is he moraliy inensiy a age y. We have u µx+s ds, (1.2) d u p x+ = u p x+ µ x+ d. (1.3) Inroduce he indicaor of survival o age x +, I() = 1[T x > ], and he indicaor of deah before ime, N() = 1[T x ] = 1 I(). 32

The process N() is a (very simple) couning process wih inensiy I() µ x+, ha is, M given by dm() = dn() I() µ x+ d (1.4) is a maringale w.r.. (F, P). Assume ha he life ime T x is independen of he economy Y. We will work wih he maringale measure P obained by replacing he inensiy marix Λ of Y wih he maringalizing Λ and leaving he res of he model unalered. Consider a uni-linked pure endowmen benefi payable a a fixed ime T, coningen on survival of he insured, wih sum insured equal o one uni of sock No. l, bu guaraneed no less han a fixed amoun g. This benefi is a coningen T -claim, H = (S l (T ) g) I(T ). The single premium payable as a lump sum a ime is o be deermined. Le us assume ha he financial marke is complee so ha every purely financial derivaive has a unique price process. Then he inrinsic value of H a ime is Ṽ H () = π() I() T p x+, where π() is he discouned price process of he derivaive S l (T ) g. Using Iô and insering (1.4), we find dṽ H () = d π() I( ) T p x+ + π() I( ) T p x+ µ x+ d + ( π T p x+ ) dn() = d π() I( ) T p x+ π T p x+ dm(). I is seen ha he opimal rading sraegy is ha of he price process of he sum insured muliplied wih he condiional probabiliy ha he sum will be paid ou, and ha dl H () = T p x+ π dm(). Consequenly, R H () = T T sp 2 x+s Ẽ [ π(s) 2 F ] s p x+ µ x+s ds T = T p x+ Ẽ [ π(s) 2 ] F T sp x+s µ x+s ds. (1.5) 11 Trading wih bonds: How much can be hedged? A. A finie zero coupon bond marke. Suppose an agen faces a coningen T -claim and is allowed o inves only in he bank accoun and a finie number m of zero coupon bonds wih mauriies T i, i = 1,..., m, all pos ime T. For insance, regulaory consrains may be imposed on he invesmen sraegies of an insurance company. The quesion is, o wha exen can he claim be hedged by self-financed rading in hese available asses? 33

An allowed SF porfolio has discouned value process Ṽ θ () of he form dṽ θ () = m ξ i () e i=1 ( p f (, T i ) p e (, T i ))d M ef Y () = d( M e ()) F e ()ξ(), f Y e e where ξ is predicable, MY e () = ( M ef Y ())f Ye is he n e -dimensional row vecor comprising he non-null enries in he j-h row of MY () = ( M ef Y ()), and where F e () = Y e F F = ( p e (, T i )) i=1,...,m e=1,...,j Y = ( p(, T 1 ),, p(, T m )), (11.1) and Y e is he J Y e J Y marix which maps F o ( p f (, T i ) p e (, T i )) i=1,...,m f Y e. If e.g. Y n = {1,..., p}, hen Y JY = (I p p, p (n p 1), 1 p 1 ). The sub-marke consising of he bank accoun and he m zero coupon bonds is complee in respec of T -claims iff he discouned bond prices span he space of all maringales w.r.. (F Y, P) over he ime inerval [, T ]. This is he case iff, for each e, rank(f e ()) = J Y e. Now, since Y e obviously has full rank J Y e, he rank of F e () is deermined by he rank of F in (11.1). We will argue ha, ypically, F has full rank. Thus, suppose c = (c 1,..., c m ) is such ha Recalling (8.16), his is he same as F c = JY 1. m c i exp{( Λ R)T i }1 =, i=1 or, by (8.17) and since Φ has full rank, m D j=1,...,j Y ( c e ρeti i ) Φ 1 1 =. (11.2) i=1 Since Φ 1 has full rank, he enries of he vecor Φ 1 1 canno be all null. Typically all enries are non-null, and we assume his is he case. Then (11.2) is equivalen o m c e ρeti i =, j = 1,..., J Y. (11.3) i=1 Using he fac ha he generalized Vandermonde marix has full rank, we know ha (11.3) has a non-null soluion c if and only if he number of disinc eigenvalues ρ e is less han m. 34

In he case where rank(f e ()) < n j for some j we would like o deermine he Galchouk-Kunia-Waanabe decomposiion for a given FT Y -claim. The inrinsic value process has dynamics d H = ζ ef ()d M ef Y () = d( M e ()) ζ e (). (11.4) e f Y e e We seek a decomposiion of he form dṽ () = ξ i () d p(, T i ) + ψ ef () d M ef Y () i e f Y e = ξ i () ( p f (, T i ) p e (, T i )) d M ef Y () + e e j Y e i = d( M e ()) F e ()ξ e () + d( M e ()) ψ e (), e e f Y e ψ ef ()d M Y ef () such ha he wo maringales on he righ hand side are orhogonal, ha is, I j (F e ()ξ e ()) Λe ψ e () =, e f Y e where Λ e = D( λ e ). This means ha, for each e, he vecor ζ e () in (11.4) is o be decomposed ino is, Λe projecions ono R(F e ()) and is orhocomplemen. From (H.3) and (H.4) we obain F e ()ξ e () = P e ()ζ e (), where hence P e () = F e ()(F e () Λe F e ()) 1 F e () Λe, ξ e () = (F e () Λe F e ()) 1 F e () Λe ζ e (). (11.5) Furhermore, ψ e () = (I P e ())ζ e (), (11.6) and he risk is T p Y ()e (s ) e f Y e λ ef (ψ ef (s)) 2 ds. (11.7) The compuaion goes as follows: The coefficiens ζ ef involved in he inrinsic value process (11.4) and he sae-wise prices p j (, T i ) of he T i -bonds are obained by simulaneously solving (8.6) and (8.13), saring from (8.9) and (8.13), respecively, and a each sep compuing he opimal rading sraegy ξ by (11.5) and he ψ from (11.6), and adding he sep-wise conribuion o he 35

variance (11.7) (he sep-lengh imes he curren value of he inegrand). B. Firs example: The floorle. For a simple example, consider a floorle H = (r r T ) +, where T < min i T i. The moivaion could be ha a ime T he insurance company will ascribe ineres o he insured s accoun a curren ineres rae, bu no less han a prefixed guaraneed rae r. Then H is he amoun ha mus be provided per uni on deposi and per ime uni a ime T. Compuaion goes by he scheme described above, wih he ζ ef () = f f () f e () obained from (8.13) subjec o (8.14) wih h e = (r r e ) +. C. Second example: The ineres guaranee in insurance. A more pracically relevan example is an ineres rae guaranee on a life insurance policy. Premiums and reserves are calculaed on he basis of a pruden so-called firs order assumpion, saing ha he ineres rae will be a some fixed (low) level r hroughou he erm of he insurance conrac. Denoe he corresponding firs order reserve a ime by V (). The (porfolio-wide) mean surplus creaed by he firs order assumpion in he ime inerval [, + d) is (r r()) + p x V () d. This surplus is currenly credied o he accoun of he insured as dividend, and he oal amoun of dividends is paid ou o he insured a he erm of he conracs a ime T. Negaive dividends are no permied, however, so a ime T he insurer mus cover H = T e T s r (r r(s)) + sp x V (s) ds. The inrinsic value of his claim is [ ] T H = Ẽ e s r (r r(s)) + sp xv (s) ds F = e s r (r r(s)) + sp x V (s) ds + e r e I e ()f e (), where he f e () are he sae-wise expeced values of fuure guaranees, discouned a ime, [ ] T f e () = Ẽ e s r (r r(s)) + sp xv (s) ds Y () = e. Working along he lines of Secion 8, we deermine he f e () by solving subjec o d d f e() = (r r e ) + p x V () + r e f e () f Y e (f f () f e ()) λ ef, f e (T ) =. (11.8) 36

The inrinsic value has dynamics (11.4) wih ζ ef () = f f () f e (). From here we proceed as described in Paragraph A. D. Compuing he risk. Consrucive differenial equaions may be pu up for he risk. example, for an ineres rae derivaive he sae-wise risk is R e () = T p eg (τ ) λ gf (ψ gf (τ)) 2 dτ. g f;f g As a simple Differeniaing his equaion, we find d d R e () = f;f e and, using he backward version of (7.2), we arrive a References T λ ef (ψ ef ()) 2 d + d p eg(τ ) g d d p eg(s ) = d d R e () = f;f e h;h e f;f g λ eh p hg (s ) + λ e p eg (s ), λ ef (ψ ef ()) 2 f;f e (ψ gf (τ)) 2 dτ, λ ef Rf () + λ e Re (). [1] Aase, K.K. and Persson, S.-A. (1994). Pricing of uni-linked life insurance policies. Scand. Acuarial J., 1994, 26-52. [2] Andersen, P.K., Borgan, Ø., Gill, R.D., Keiding, N. (1993). Saisical Models Based on Couning Processes. Springer-Verlag, New York, Berlin, Heidelberg. [3] Bibby J.M., Mardia, K.V., and Ken J.T. Mulivariae Analysis. Academic Press, 1979. [4] Björk, T., Kabanov, Y., Runggaldier, W. (1997): Bond marke srucures in he presence of marked poin processes. Mahemaical Finance, 7, 211-239. [5] Björk, T. (1998): Arbirage Theory in Coninuous Time, Oxford Universiy Press. [6] Black, F., Scholes, M. (1973): The pricing of opions and corporae liabiliies. J. Poli. Economy, 81, 637-654. [7] Bremaud, P. (1981). Poin processes and queues. Springer-Verlag, New York, Heidelberg, Berlin. 37

[8] Cox, J., Ross, S., Rubinsein, M. (1979): Opion pricing: A simplified approach. J. of Financial Economics, 7, 229-263. [9] Delbaen, F., Schachermayer, W. (1994): A general version of he fundamenal heorem on asse pricing. Mahemaische Annalen, 3, 463-52. [1] Eberlein, E., Raible, S. (1999): Term srucure models driven by general Lévy processes. Mahemaical Finance, 9, 31-53. [11] Ellio, R.J., Kopp, P.E. (1998): Mahemaics of financial markes, Springer-Verlag. [12] Föllmer, H., Sondermann, D. (1986): Hedging of non-redundan claims. In Conribuions o Mahemaical Economics in Honor of Gerard Debreu, 25-223, eds. Hildebrand, W., Mas-Collel, A., Norh-Holland. [13] Ganmacher, F.R. (1959): Marizenrechnung II, VEB Deuscher Verlag der Wissenschafen, Berlin. [14] Gerber, H.U. (1995). Life Insurance Mahemaics, 2nd edn. Springer- Verlag. [15] Harrison, J.M., Kreps, D.M. (1979): Maringales and arbirage in muliperiod securiies markes. J. Economic Theory, 2, 1979, 381-48. [16] Harrison, J.M., Pliska, S. (1981): Maringales and sochasic inegrals in he heory of coninuous rading. J. Soch. Proc. and Appl., 11, 215-26. [17] Hoem, J.M. (1969): Markov chain models in life insurance. Bläer Deusch. Gesellschaf Vers.mah., 9, 91 17. [18] Hoem, J.M. and Aalen, O.O. (1978). Acuarial values of paymen sreams. Scand. Acuarial J. 1978, 38-47. [19] Jordan, C.W. (1967). Life Coningencies. The Sociey of Acuaries, Chicago. [2] Karlin, S., Taylor, H. (1975): A firs Course in Sochasic Processes, 2nd. ed., Academic Press. [21] Meron, R.C. (1973): The heory of raional opion pricing. Bell Journal of Economics and Managemen Science, 4, 141-183. [22] Meron, R.C. (1976): Opion pricing when underlying sock reurns are disconinuous. J. Financial Economics, 3, 125-144. [23] Møller, T. (1998): Risk minimizing hedging sraegies for uni-linked life insurance. ASTIN Bull., 28, 17-47. [24] Norberg, R. (1991). Reserves in life and pension insurance. Scand. Acuarial J. 1991, 1-22. 38

[25] Norberg, R. (1995): Differenial equaions for momens of presen values in life insurance. Insurance: Mah. & Econ., 17, 171-18. [26] Norberg, R. (1995): A ime-coninuous Markov chain ineres model wih applicaions o insurance. J. Appl. Soch. Models and Daa Anal., 245-256. [27] Norberg, R. (1998): Vasiček beyond he normal. Working paper No. 152, Laboraory of Acuarial Mah., Univ. Copenhagen. [28] Norberg, R. (1999): A heory of bonus in life insurance. Finance and Sochasics., 3, 373-39. [29] Norberg, R. (21): On bonus and bonus prognoses in life insurance. Scand. Acuarial J. 1991, 1-22. [3] Papariandafylou, A. and Waers, H.R. (1984). Maringales in life insurance. Scand. Acuarial J. 1984, 21-23. [31] Proer, P. (199). Sochasic Inegraion and Differenial Equaions. Springer-Verlag, Berlin, New York. [32] Pliska, S.R. (1997): Inroducion o Mahemaical Finance, Blackwell Publishers. [33] Ramlau-Hansen, H. (1991): Disribuion of surplus in life insurance. ASTIN Bull., 21, 57-71. [34] Sverdrup, E. (1969): Noen forsikringsmaemaiske emner. Sa. Memo. No. 1, Ins. of Mah., Univ. of Oslo. (In Norwegian.) [35] Thomas, J.W. (1995): Numerical Parial Differenial Equaions: Finie Difference Mehods, Springer-Verlag. [36] Widh, E. (1986): A noe on bonus heory. Scand. Acuarial J. 1986, 121-126. Appendices A Calculus A. Piecewise differeniable funcions. Being concerned wih operaions in ime, commencing a some iniial dae, we will consider funcions defined on he posiive real line [, ). Thus, le us consider a generic funcion X = {X } and hink of X as he sae or value of some process a ime. For he ime being we ake X o be real-valued. 39

In he presen ex we will work exclusively in he space of so-called piecewise differeniable funcions. From a mahemaical poin of view his space is iny since only elemenary calculus is needed o move abou in i. From a pracical poin of view i is huge since i comforably accommodaes any idea, however sophisicaed, ha an acuary may wish o express and analyse. I is convenien o ener his space from he ouside, saring from a wider class of funcions. We firs ake X o be of finie variaion (FV), which means ha i is he difference beween wo non-decreasing, finie-valued funcions. Then he leflimi X = lim s X s and he righ-limi X + = lim s X s exis for all, and hey differ on a mos a counable se D(X) of disconinuiy poins of X. We are paricularly ineresed in FV funcions X ha are righ-coninuous (RC), ha is, X = lim s X s for all. Any probabiliy disribuion funcion is of his ype, and any sream of paymens accouned as incomes or ougoes, can reasonably be aken o be FV and, as a convenion, RC. If X is RC, hen X = X X, when differen from, is he jump made by X a ime. For our purposes i suffices o le X be of he form X = X + x τ dτ + <τ (X τ X τ ). (A.1) The inegral, which may be aken o be of Riemann ype, adds up he coninuous incremens/decremens, and he sum, which is undersood o range over disconinuiy imes, adds up incremens/decremens by jumps. We assume, furhermore, ha X is piecewise differeniable (PD); A propery holds piecewise if i akes place everywhere excep, possibly, a a finie number of poins in every finie inerval. In oher words, he se of excepional poins, if no empy, mus be of he form {, 1,...}, wih < 1 <, and, in case i is infinie, lim j j =. Obviously, X is PD if boh X and x are piecewise coninuous. A any poin / D = D(X) D(x) we have d d X = x, ha is, he funcion X grows (or decays) coninuously a rae x. As a convenien noaional device we shall frequenly wrie (A.1) in differenial form as dx = x d + X X. (A.2) A lef-coninuous PD funcion may be defined by leing he sum in (A.1) range only over he half-open inerval [, ). Of course, a PD funcion may be neiher righ-coninuous nor lef-coninuous, bu such cases are of no ineres o us. B. Inegrals. Le X and Y boh be PD and, moreover, le X be RC and given by (A.2). The inegral over (s, ] of Y wih respec o X is defined as Y τ dx τ = Y τ x τ dτ + s s s<τ Y τ (X τ X τ ), (A.3) 4

provided ha he individual erms on he righ and also heir sum are well defined. Considered as a funcion of he inegral is iself PD and RC wih coninuous incremens Y x d and jumps Y (X X ). One may hink of he inegral as he weighed sum of he Y -values, wih he incremens of X as weighs, or vice versa. In paricular, (A.1) can be wrien simply as X = X s + s dx τ, (A.4) saying ha he value of X a ime is is value a ime s plus all is incremens in (s, ]. By definiion, Y τ dx τ = lim Y τ dx τ = Y τ dx τ Y (X X ) = Y τ dx τ, r s s s a lef-coninuous funcion of. Likewise, Y τ dx τ = lim Y τ dx τ = Y τ dx τ + Y s (X s X s ) = r s a lef-coninuous funcion of s. r s s (s,) [s,] Y τ dx τ, C. The chain rule (Iô s formula). Le X = (X 1,..., Xm ) be an m-variae funcion wih PD and RC componens given by dx i = x i d + (X i X ). i Le f : R m R have coninuous parial derivaives, and form he composed funcion f(x ). On he open inervals where here are neiher disconinuiies in he x i nor jumps of he X i, he funcion f(x ) develops in accordance wih he well-known chain rule for scalar fields along recifiable curves. A he excepional poins f(x ) may change (only) due o jumps of he X i, and a any such poin i jumps by f(x ) f(x ). Thus, we gaher he so-called change of variable rule or Iô s formula, which in our simple funcion space reads df(x ) = or, in inegral form, f(x ) = f(x s ) + s m i=1 m i=1 f x i (X ) x i d + f(x ) f(x ), x i f(x τ ) x i τ dτ + s<τ Obviously, f(x ) is PD and RC. A frequenly used special case is (check he formulas!) {f(x τ ) f(x τ )}. d(x Y ) = X y d + Y x d + X Y X Y = X dy + Y dx + (X X )(Y Y ) (A.5) (A.6) = X dy + Y dx. (A.7) 41

If X and Y have no common jumps, as is cerainly he case if one of hem is coninuous, hen (A.7) reduces o he familiar d(x Y ) = X dy + Y dx. The inegral form of (A.7) is he so-called rule of inegraion by pars: s Y τ dx τ = Y X Y s X s s X τ dy τ. (A.8) (A.9) B Probabiliy and expecaion Taking basic measure heoreic probabiliy as a prerequisie, we represen he relevan par of he world and is uncerainies by a probabiliy space (Ω, F, P). Here Ω is he se of possible oucomes ω, F is a sigmaalgebra of subses of Ω represening he evens o which we wan o assign probabiliies, and P : F [, 1] is a probabiliy measure. A se A F such ha P[A] = is called a nullse, and a propery ha akes place in all of Ω, excep possibly on a nullse, is said o hold almos surely (a.s.). If more han one probabiliy measure are in play, we wrie nullse (P) and a.s. (P) whenever emphasis is needed. Two probabiliy measures P and P are said o be equivalen, wrien P P, if hey are defined on he same F and have he same nullses. Le G be some sub-sigmaalgebra of F. We denoe he resricion of P o G by P G ; P G [A] = P[A], A G. Noe ha also (Ω, G, P G ) is a probabiliy space. A G-measurable random variable (r.v.) is a funcion X : Ω R such ha X 1 (B) G for all B R, he Borel ses in R. We wrie X G in shor. The expeced value of a r.v. X is he probabiliy-weighed average E[X] = X dp = Ω X(ω) dp(ω), provided his inegral is well defined. The condiional expeced value of X, given G, is he r.v. E[X G] G saisfying E{E[X G] Y } = E[XY ] (B.1) for each Y G such ha he expeced value on he righ exiss. I is unique up o nullses (P). To moivae (B.1), consider he special case when G = σ{b 1, B 2,...}, he sigma-algebra generaed by he F-measurable ses B 1, B 2,..., which form a pariion of Ω. Being G-measurable, E[X G] mus be of he form k b k1 Bk. Puing his ogeher wih Y = 1 Bj ino he relaionship (B.1) we arrive a E[X G] = B 1 j X dp Bj, P[B j j ] as i ough o be. In paricular, aking X = I A, we find he condiional probabiliy P[A B] = P[A B]/P[B]. One easily verifies he rule of ieraed expecaions, which saes ha, for H G F, E {E[X G] H} = E[X H]. (B.2) 42

C Change of measure If L is a r.v. such ha L a.s. (P) and E[L] = 1, we can define a probabiliy measure P on F by P[A] = L dp = E[1 A L]. (C.1) If L > a.s. (P), hen P P. The expeced value of X w.r.. P is A Ẽ[X] = E[XL] (C.2) if his inegral exiss; by he definiion (C.1), he relaion (C.2) is rue for indicaors, hence for simple funcions and, by passing o limis, i holds for measurable funcions. Spelling ou (C.2) as X d P = XL dp suggess he noaion d P = L dp or d P dp = L. The funcion L is called he Radon-Nikodym derivaive of P w.r.. P. Condiional expecaion under P is formed by he rule Ẽ[X G] = E[XL G] E[L G] To see his, observe ha, by definiion, (C.3). (C.4) Ẽ{Ẽ[X G] Y } = Ẽ[XY ] (C.5) for all Y G. The expression on he lef of (C.5) can be reshaped as The expression on he righ of (C.5) is E{Ẽ[X G] Y L} = E{Ẽ[X G] E[L G] Y }. E[XY L] = E{E[XL G] Y }. I follows ha (C.5) is rue for all Y G if and only if which is he same as (C.4). For X G we have Ẽ[X G] E[L G] = E[XL G], Ẽ G [X] = Ẽ[X] = E[XL] = E {X E[L G]} = E G {X E[L G]}, (C.6) showing ha d P G dp G = E[L G]. (C.7) 43

D Sochasic processes: general conceps To describe he evoluion of random phenomena over some ime inerval [, T ], we inroduce a family F = {F } T of sub-sigmaalgebras of F, where F represens he informaion available a ime. More precisely, F is he se of evens whose occurrence or non-occurrence can be ascerained by ime. If no informaion is ever sacrificed, we have F s F for s <. We hen say ha F is a filraion, and (Ω, F, F, P) is called a filered probabiliy space. A sochasic process is a family of r.v.-s, {X } T. I is said o be adaped o he filraion F if X F for each [, T ], ha is, a any ime he curren sae (and also he pas hisory) of he process is fully known if we are currenly provided wih he informaion F. An adaped process is said o be predicable if is value a any ime is enirely deermined by is hisory in he sric pas, loosely speaking. For our purposes i is sufficien o hink of predicable processes as being eiher lef-coninuous or deerminisic. E Maringales An adaped process X wih finie expecaion is a maringale if E[X F s ] = X s for s <. The maringale propery depends boh on he filraion and on he probabiliy measure, and when hese need emphasis, we shall say ha X is maringale (F, P). The definiion says ha, on he average, a maringale is always expeced o remain on is curren level. One easily verifies ha, condiional on he presen informaion, a maringale has uncorrelaed fuure incremens. Here are some useful general resuls: Abbreviae P = P F, inroduce and pu L = L T. By (C.7) we have L = d P dp, L = E[L F ], (E.1) which is a maringale (F, P). Le X be a real-valued random variable such ha E X <. process M defined by M() = E[X F ] Then he is a maringale. This follows by he rule of ieraed expecaion and he filraion propery, F s F for s < : E[M() F s ] = E{E[X F ] F s } = E[X F s ] = M(s). A maringale M wih pahs ha are (almos surely) coninuous and of finie variaion in every finie inerval is consan as a funcion of ime; M() = M() 44

for all. This is seen as follows. Since M has finie variaion, i obeys he rules of ordinary calculus and, in paricular, M 2 () = M 2 () + 2 M(s) dm(s). Since M is coninuous, i is also predicable so ha he inegral 2M(s) dm(s) is a maringale. I follows ha E [ M 2 () ] = M 2 (). Since E[M()] = M(), we conclude ha Var[M()] =, hence M is consan. F Couning processes As he name suggess, a couning process is a sochasic process N = {N } T ha commences from zero (N = ) and hereafer increases by isolaed jumps of size 1 only. The naural filraion of N is F N = {F N } T, where F N = σ{n s ; s } is he hisory of N by ime. This is he smalles filraion o which N is adaped. The sric pas hisory of N a ime is denoed by F N. An F N -predicable process {Λ } T is called a compensaor of N if he process M defined by M = N Λ (F.1) is a zero mean F N -maringale. If Λ is absoluely coninuous, ha is, of he form Λ = λ s ds, hen he process λ is called he inensiy of N. We may also define he inensiy informally by λ d = P [dn = 1 F ] = E [dn F ], and we someimes wrie he associaed maringale (F.1) in differenial form, dm = dn λ d. (F.2) A sochasic inegral w.r.. he maringale M is an F N -adaped process H of he form H = H + h s dm s, (F.3) where H is F N -measurable and h is an F N -predicable process such ha H is inegrable. The sochasic inegral is also a maringale. 45

A fundamenal represenaion resul saes ha every F N maringale is a sochasic inegral w.r.. M. I follows ha every inegrable F N measurable r.v. is of he form (F.3). If H (1) = H (1) + h(1) s dm s and H (2) = H (2) + h(2) s dm s are sochasic inegrals wih finie variance, hen an easy heurisic calculaion shows ha [ ] T Cov[H (1) T, H(2) T F ] = E h (1) s h(2) s λ s ds F, (F.4) and, in paricular, [ ] T Var[H T F ] = E h 2 s λ s ds F. H (1) and H (2) are said o be orhogonal if hey have condiionally uncorrelaed incremens, ha is, he covariance in (F.4) is null. This is equivalen o saying ha H (1) H (2) is a maringale. The inensiy is also called he infiniesimal characerisic if he couning process since i enirely deermines i probabilisic properies. If λ is deerminisic, hen N is a Poisson process. If λ depends only on N, hen N is a Markov process. The change of variable rule (A.6) becomes paricularly simple when he argumen is a couning funcion. In fac, for f : R R and for N a couning process, we have f(n ) = f(n s ) + {f(n τ ) f(n τ )} (F.5) s<τ = f(n s ) + = f(n s ) + s<τ Wha hese expressions sae, is jus s f(j) = f() + {f(n τ + 1) f(n τ )}(N τ N τ ) {f(n τ + 1) f(n τ )} dn τ. j {f(i) f(i 1)}. i=1 (F.6) (F.7) Sill hey are useful represenaions when we come o sochasic couning processes. A comprehensive exbook on couning processes in life hisory analysis is [2]. G The Girsanov ransform Girsanov s heorem is celebraed in sochasics, and i is basic in mahemaical finance. We formulae and prove he couning process variaion: 46

Theorem (Girsanov). Le N be a couning process wih (F, P)-inensiy λ, and le λ be a given non-negaive F-adaped process such ha λ = if and only if λ =. Then here exiss a probabiliy measure P such ha P P and N has (F, P)-inensiy λ. The likelihood process (E.1) is ( ) L = exp (ln λ s ln λ s ) dn s + (λ s λ s ) ds. Proof: We shall give a consrucive proof, saring from a guessed L in (C.3). Since L mus be sricly posiive a.e. (P), a candidae would be L = L T, where ( ) L = exp φ s dn s + ψ s ds wih φ predicable and ψ adaped. In he firs place, L should be a maringale (F, P). By Iô s formula, dl = L ψ d + L (e φ 1) dn = L ( ψ + ( e φ 1 ) λ ) d + L ( e φ 1 ) dm. The represenaion resul (F.3) ells us ha o make L a maringale, we mus make he drif erm vanish, ha is, ψ = ( 1 e φ) λ, (G.1) whereby ( dl = L e φ 1 ) dm, In he second place, we wan o deermine φ such ha he process M given by d M = dn λ d (G.2) is a maringale (F, P). Thus, we should have Ẽ[ M F s ] = M s or, by (C.4), [ ] E M L F s = E [L F s ] M s. Using he maringale propery (E.1) of L, his is he same as [ ] E M L F s = M s L s i.e. M L should be a maringale (F, P). Since d( M L ) = ( λ d)l + M (e φ 1)L ( λ d) ( + ( M + 1)L e φ M ) L ) dn ) ( = L d ( λ + e φ λ + ( M + 1)L e φ M ) L ) dm. 47

we conclude ha he maringale propery is obained by choosing φ = ln λ ln λ. The mulivariae case goes in he same way; jus replace by vecor-valued processes. H Some useful vecor and marix resuls Vecors and marices are denoed by in bold leers, lower and upper case, respecively. They may be equipped wih opscrips indicaing dimensions, e.g. A n m has n rows and m columns. We may wrie A = (a jk ) k K j J o emphasize he ranges of he row index j and he column index k. The ranspose of A is denoed by A. Vecors are invariably aken o be of column ype, hence row vecors appear as ransposed. The ideniy marix is denoed by I, he vecor wih all enries equal o 1 is denoed by 1, and he vecor wih all enries equal o is denoed by. By D j=1,...,n (a j ), or jus D(a), is mean he diagonal marix wih he enries of a = (a 1,..., a n ) down he principal diagonal. The n- dimensional Euclidean space is denoed by R n, and he linear subspace spanned by he columns of A n m is denoed by R(A). A diagonalizable square marix A n n can be represened as n A = Φ D j=1,...,n (ρ j ) Φ 1 = ρ j φ j ψ j, (H.1) j=1 where he φ j are he columns of Φ n n and he ψ j are he rows of Φ 1. The ρ j are he eigenvalues of A, and φ j and ψ j are he corresponding righ and lef eigenvecors, respecively. Eigenvecors (righ or lef) corresponding o eigenvalues ha are disinguishable and non-null are muually orhogonal. These resuls can be looked up in e.g. [2]. The exponenial funcion of A n n is he n n marix defined by exp(a) = p= 1 p! Ap = Φ D j=1,...,n (e ρj ) Φ 1 = n e ρj φ j ψ j, j=1 (H.2) where he las wo expressions follow from (H.1). The marix exp(a) has full rank. If Λ n n is posiive definie symmeric, hen ζ 1, ζ 2 Λ = ζ 1 Λζ 2 defines an inner produc on R n. The corresponding norm is given by ζ Λ = ζ, ζ 1/2. If Λ F n m has full rank m ( n), hen he, Λ -projecion of ζ ono R(F) is ζ F = P F ζ, where he projecion marix (or projecor) P F is P F = F(F ΛF) 1 F Λ. The projecion of ζ ono he orhogonal complemen R(F) is ζ F = ζ ζ F = (I P F )ζ. (H.3) (H.4) 48

Is squared lengh, which is he squared, Λ -disance from ζ o R(F), is ζ F 2 Λ = ζ 2 Λ ζ F 2 Λ = ζ Λ(I P F )ζ. (H.5) The cardinaliy of a se Y is denoed by Y. For a finie se i is jus is number of elemens. Ragnar Norberg Deparmen of Saisics London School of Economics Houghon Sree London WC2A 2AE Tel: 2-7955-63 Fax: 2-7955-7416 e-mail: R.Norberg@lse.ac.uk 49