ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT

Size: px
Start display at page:

Download "ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT"

Transcription

1 Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages S 94-9(5)634-4 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT UDC A. V. MELNIKOV AND M. L. NECHAEV Absrac. The paper deals wih he problem of pricing an equiy-linked insurance conrac based on sock prices. The sock prices are supposed o follow a sochasic exponen model wih respec o a given Gaussian maringale. The model gives a possibiliy o obain unified formulas for mean variance hedging and he corresponding premium for boh naural cases: Black Scholes and Gaussian discree ime models. 1. Inroducion Suppose ha an insurance company has a porfolio of l insurance conracs. Any conrac is associaed wih a random ime τ i, i = 1,...,l, which indicaes he ime of inciden occurrence. The corresponding premiums should be disribued beween financial asses o guaranee he bes correspondence beween liabiliies of he company and is capial V π. As a crierion of he qualiy of a financial porfolio π we shall use hemeanvariancedisance E (VT π f T ) 2, where f T represens he claim ha should be paid by he company a he erminal ime T. The insurance conrac based on he marke s price of a given asse S is called an equiy-linked life insurance conrac. An appropriae descripion of such a conrac was given, for insance, in 3, in he framework of he Black Scholes model for S. Thispaper is devoed o he pricing of such insurance conracs and in a more general case of a financial marke driven by Gaussian maringale. 2. Financial marke and insurance porfolio Suppose he company invess in he (B,S)-marke wih wo raded asses: bank accoun B 1andsockS, { S = S exp Y 1 } 2 Y, where Y is he righ coninuous Gaussian maringale (see 2) on a given sochasic basis (Ω 1, F 1,P 1,F 1 ) wih filraion F 1 generaed by Y. 2 Mahemaics Subjec Classificaion. Primary 6H3; Secondary 91B28, 91B3. Key words and phrases. Hedging, (B,S)-marke, coningen claim, insurance liabiliies, insurance premiums and claims. This paper was parially suppored by grans of NSERC and SSHRC. 15 c 25 American Mahemaical Sociey

2 16 A. V. MELNIKOV AND M. L. NECHAEV Any self-financing rading sraegy π =(β,γ) can be deermined by is capial V π = V π + γ u ds u, where γ is he number of socks in he porfolio a ime (see 4). The number of bank accoun unis β can be idenified from he balance equaion V = γ S + β. Using he explici form of S and he Kolmogorov Iô formula 2, we can derive he following expression for he capial V π (1) V π = V π + : γ u S u dy u + u γ u S u ( { exp Y u 1 } ) 2 Y u 1 Y u. The mauriy ime of his conrac will be denoed by T. We also assume a pure echnical condiion: Y T =. The corresponding coningen claim has he form f T = f(s T ), where he Borel funcion f saisfies he following condiion: ( f(x) c 1+x p1) x p2, c, p 1, p 2, x. In his case (see 1) we can define he funcion { ( 1 f(z) ln(x/z)+ 1 2 F (u, x) = exp ( Y T u)) } 2 dz. 2π Y T u R z 2( Y + T u) I is easy o prove ha his funcion is wice coninuously differeniable wih respec o boh argumen u and x and admis he represenaion (2) F ( Y,S )=E f(s T ) F 1 for any <T. Using he maringale convergence heorem 2 we can conclude ha F ( Y T,S T )=f(s T ). The Kolmogorov Iô formula gives direcly ha (3) F ( Y,S )=F (,S ) u F x ( Y u,s u )S u dy u F u ( Y u,s u )S u d Y u F xx ( Y u,s u )Su 2 d Y c u {,S u ),S u ) F x ( Y u,s u )S u Y u F u ( Y u,s u )S u Y u }, where F x, F u,andf xx are he corresponding derivaives. Taking ino accoun ha he process F ( Y,S ) should be a maringale, we can reduce (3) o he equaliy {u } F ( Y,S )=F (,S )+ F x ( Y u,s u )S u dy u (4) + {,S u ),S u ) F x ( Y u,s u )S u Y u }. The insurance porfolio of l conracs can be characerized by random imes τ i, i =1,...,l,

3 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS 17 of inciden occurrence and claim paymen values. We shall assume ha τ i are i.i.d. random variables on some probabiliy space (Ω 2, F 2,P 2 ). The paymen funcion g( ) for he conrac indicaes he value g(s T ) which should be paid if no inciden occurs during he insured period. Suppose ha he disribuion of τ i admis he following represenaion: { } P 2 (τ ) =1 exp µ s ds, where µ is called a force of moraliy. Denoe I k = I {τk }, andn = I I. l The couning process N indicaes he number of incidens on,. We shall equip (Ω 2, F 2,P 2 ) wih a filraion F 2 =(F 2 ) generaed by (N ). The process is a compensaor of N and M = N is a maringale wih respec o F 2. (l N s )µ s ds (l N s )µ s ds 3. Main resuls and examples I is quie naural o hink ha he financial marke and he lives of insured are independen. Hence he general probabiliy space for he model can be defined as a produc of (Ω 1, F 1,F 1,P 1 )and(ω 2, F 2,F 2,P 2 ) wih he general filraion F generaed by F 1 and F 2 : Ω=Ω 1 Ω 2, F = F 1 F 2, P = P 1 P 2, F = F 1 F 2 = { F = F 1 } F2. The paymen funcion of such a conrac has he form f T = g(s T )(l N T ), where he Borel funcion g( ) saisfies he condiions menioned above, g(x) c(1 + x p1 )x p2, c, p 1, p 2, x >. To opimize is liabiliies he company should choose an iniial capial ˆv and a selffinancing rading sraegy ˆπ such ha E (VT ˆπ (ˆv) f T ) 2 E (VT π (v) f T ) 2 for any v and any self-financing sraegy π. Denoe by V = Ef T F =Eg(S T )(l N T ) F he so-called racking process V ; using he independence of S and N we ge V = E 1 g(s T ) F 1 E 2 (l N T ) F 2. I is easy o check ha (5) E 2 (l N T ) F 2 l = E 2 (1 I {τi T }) F 2 =(l N ) p T, i=1 where p T = E 2 1 I {<τi T } F =exp { T µ s ds } is he probabiliy of inciden occurrence afer expiraion dae T. Using he represenaion (2) we have V = F ( Y,S )(l N ) p T.

4 18 A. V. MELNIKOV AND M. L. NECHAEV we obain he inegral repre- Applying he Kolmogorov Iô formula o he process V senaion (6) V = V + + u (l N u ) u p T d,s u )+,S u )(l N u ) u p T µ u du,s u ) u p T dn u,s u )(l N u ) u p T,S u )(l N u ) u p T Theequaliy(6)canberewrienas (7) V = V + + u (l N u ) u p T,S u )+,S u ) u p T N u. (l N u ) u p T d,s u ),S u ) u p T dm u,s u )(l N u ) u p T,S u )(l N u ) u p T where (l N u ) u p T,S u )+,S u ) u p T N u, M = N (l N u )µ u du. Taking ino accoun he represenaion (4) for F ( Y,S )wehavefrom(7)ha V = V + (l N u ) u p T F x ( Y u,s u )S u dy u,s u ) u p T dm u +,S u )(l N u ) u p T,S u )(l N u ) u p T u +,S u ) u p T N u (l N u ) u p T F x ( Y u,s u ) Y u. Noe ha any capial V π conrolled by a self-financing sraegy can be represened in he form (1). Consider he mean variance disance beween VT π and V T = f T : R(π, v) =E (VT π VT ) 2. Because of he maringale properies of V π and V, i is clear ha he iniial capial ˆv for he opimal rading sraegy should be equal o V : ˆv = le 1 g(s T )p(,t)=lf(,s ) P({τ >T}). Le he iniial capial v of he self-financing sraegy π be equal o ˆv; hen ( R(π, v) =E ((l N u ) u p T F x ( Y u,s u ) γ u )S u dy u (8) + u,s u ) u p T dm u (l N u ) u p T,S u ) (l N u ) u p T F x ( Y u,s u )S u Y u γ u S u + γ u S u Y u ) 2.

5 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS 19 Since he difference M = V V π is a maringale wih respec o F, here is a unique represenaion of he form M c + M d, where M c and M d are respecively he purely coninuous and disconinuous pars of M, which are orhogonal. Consequenly E M 2 = E (M c ) 2 + E (M d ) 2. Le us rewrie M c in he form M c ( ) = (l Nu ) u p T F x ( Y u,s u ) γ u Su dyu c,s u ) u p T dmu. c Since Y u and M u are independen, we ge E (M c ) 2 ( ) 2S = E (l 2 Nu ) u p T F x ( Y u,s u ) γ u u d Y c u + E (,S u )p T ) 2 d M c u. In he case of he purely disconinuous par M d, we have he formula M d = u u p T,S u ) N u +(l N u ) u p T,S u ) γ u S u, and herefore E (M d ) 2 = E ( u u p T,S u ) N u +(l N u ) u p T,S u ) γ u S u ) 2. I is well known (see 2) ha he imes of he jumps of he Gaussian maringale are deerminisic. Denoe he corresponding se by A. Define he processes γ and γ by he following formulas: γ s = γ s χ A, γ s = γ s χ A, where χ A is he indicaor funcion of he se A. Take γ s = γ s + γ s. Since A is a counable se, E (M c ) 2 = E ((l N u ) u p T F x ( Y u,s u ) γ u ) 2 Su 2 d Y c u (9) + E (,S u ) u p T ) 2 d M c u.

6 11 A. V. MELNIKOV AND M. L. NECHAEV On he oher hand i is clear ha E ( (M d ) 2 = E u p T,S u ) N u (1) = E u A ( + E u A ( +(l N u ) u p T,S u ) γ u S u u p T, S u ) N u u/ A u/ A ) 2 u p T,S u ) N u +(l N u ) u p T,S u ) γ u S u ) 2 ) 2 u p T,S u ) N u. Equaions (9) and (1) give us he explici forms of γ and γ: (11) γ =(l N ) p T F x ( Y,S )χ A and (12) γ = p T (l N ) E F ( Y,S ) S F E, (S ) 2 F where / is supposed o be equal o. So, we ge he following main resul of he paper: Theorem 1. For he (B,S)-marke conrolled by a Gaussian maringale and a porfolio of l homogeneous uni-linked pure endowmen insurance conracs, here is an opimal mean-variance hedging sraegy ˆγ = γ + γ, where he coninuous par γ and he purely disconinuous par γ are defined by (11) and (12), respecively. The iniial capial ˆv of he sraegy can be calculaed by ˆv = l Eg(S T ) P(τ >T), where g(s ) is a paymen funcion for one insurance conrac, T is a erminal ime, and τ is a random ime wih disribuion of he inciden occurrence. Example 1. The model invesigaed above includes he model considered by T. Møller (see 3) when Y is Brownian moion. Taking, in our seing, γ, ˆγ = γ =(l N ) P T F x ( Y,S ) gives he model and he resul presened in 3. Example 2. Anoher ineresing example is he discree model { S n = S exp M n 1 } 2 M n, B n 1, where M n = h h n, h i = σɛ i, ɛ i N(, 1), and ɛ i are i.i.d. random variables on he probabiliy space (Ω 1, F 1,P 1 ) wih filraion F 1 = { F n}, 1 F 1 n = σ{ɛ i,i n}. I is clear ha M n = σ 2 n and } S n = S exp {h h n σ2 2 n.

7 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS 111 Using he independence of ɛ i,wehave ES n F n 1 =S n 1 E exp { h n σ 2 /2 } F n 1 = Sn 1. We can embed he discree model o our general model by he following sandard way:,, 1), Y = M n, n, n +1),n<N, M N, N,N + δ), N+ δ = T, F 1 = F 1, F 1 = {F 1 }. I is clear ha Y is a Gaussian maringale on he sandard sochasic basis ( Ω 1, F 1,F 1,P 1) saisfying he echnical condiion Y T =. Inviewofheheorem γ n = because here is no coninuous par of Y. Regarding he oher par of a hedging sraegy γ n we have E F ( Y n,s n ) S n F n 1 =E(F ( Y n,s n ) F ( Y n 1,S n 1 )) S n F n 1 = EF ( Y n,s n ) S n F n 1 =Eg(S T S n ) F n 1. Hence we ge where S T = S N. γ = γ n = n p T (l N n 1 ) E g(s T S n ) F n 1 E ( S n ) 2, F n 1 Bibliography 1. O. Gloni and Z. Khechinashvili, Financial (B, S)-marke wih Gaussian maringale mean square opimal hedging sraegies, Proc. A. Razmadze Mah. Ins. 115 (1997), MR16391 (99j:91) 2. R. Lipser and A. N. Shiryaev, Theory of Maringales, Nauka, (Russian) MR (88h:691) 3. T. Møller, Risk-minimizing hedging sraegies for uni-linked life insurance conracs, Asin Bullein 28 (1998), no A. N. Shiryaev, Essenials of Sochasic Financial Mahemaics, World Sci., River Edge, NJ, MR (2e:9185) Seklov Mahemaical Insiue, Gubkina 8, Moscow , Russia Curren address: Deparmen of Mahemaical and Saisical Sciences, Universiy of Albera, Edmonon, Albera T6G 2G1, Canada Seklov Mahemaical Insiue, Gubkina 8, Moscow , Russia Received 12/MAR/23 Originally published in English

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

Term Structure of Prices of Asian Options

Term Structure of Prices of Asian Options Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

Dependent Interest and Transition Rates in Life Insurance

Dependent Interest and Transition Rates in Life Insurance Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

More information

2.5 Life tables, force of mortality and standard life insurance products

2.5 Life tables, force of mortality and standard life insurance products Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Option Pricing Under Stochastic Interest Rates

Option Pricing Under Stochastic Interest Rates I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres

More information

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b

LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.

More information

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his

More information

Research Article Optimal Geometric Mean Returns of Stocks and Their Options

Research Article Optimal Geometric Mean Returns of Stocks and Their Options Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics

More information

Time Consistency in Portfolio Management

Time Consistency in Portfolio Management 1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

An Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price

An Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

More information

Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing

Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364-765X eissn 526-547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion

More information

Life insurance cash flows with policyholder behaviour

Life insurance cash flows with policyholder behaviour Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK-2100 Copenhagen Ø, Denmark PFA Pension,

More information

DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR

DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

A general decomposition formula for derivative prices in stochastic volatility models

A general decomposition formula for derivative prices in stochastic volatility models A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

More information

A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES.

A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. DONATIEN HAINAUT, PIERRE DEVOLDER. Universié Caholique de Louvain. Insiue of acuarial sciences. Rue des Wallons, 6 B-1348, Louvain-La-Neuve

More information

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619 econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;

More information

ARCH 2013.1 Proceedings

ARCH 2013.1 Proceedings Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

More information

A martingale approach applied to the management of life insurances.

A martingale approach applied to the management of life insurances. A maringale approach applied o he managemen of life insurances. Donaien Hainau Pierre Devolder 19h June 2007 Insiu des sciences acuarielles. Universié Caholique de Louvain UCL. 1348 Louvain-La-Neuve, Belgium.

More information

ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION

ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed

More information

Stochastic Calculus and Option Pricing

Stochastic Calculus and Option Pricing Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes

More information

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear

More information

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical

More information

UNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.

UNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment. UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL

More information

Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach

Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach Opimal Consumpion and Insurance: A Coninuous-Time Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems

More information

DETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUN-SHAN WU

DETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUN-SHAN WU Yugoslav Journal of Operaions Research 2 (22), Number, 6-7 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUN-SHAN WU Deparmen of Bussines Adminisraion

More information

LIFE INSURANCE MATHEMATICS 2002

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.

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction

RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction RISK-SHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION AN CHEN AND PETER HIEBER Absrac. In a ypical paricipaing life insurance conrac, he insurance company is eniled o a

More information

T ϕ t ds t + ψ t db t,

T ϕ t ds t + ψ t db t, 16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in

More information

Longevity 11 Lyon 7-9 September 2015

Longevity 11 Lyon 7-9 September 2015 Longeviy 11 Lyon 7-9 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univ-lyon1.fr

More information

INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES

INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals

More information

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical

More information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

INVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE

INVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE INVESMEN UARANEES IN UNI-LINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN

More information

Analysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer

Analysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer Recen Advances in Business Managemen and Markeing Analysis of Pricing and Efficiency Conrol Sraegy beween Inerne Reailer and Convenional Reailer HYUG RAE CHO 1, SUG MOO BAE and JOG HU PARK 3 Deparmen of

More information

Black Scholes Option Pricing with Stochastic Returns on Hedge Portfolio

Black Scholes Option Pricing with Stochastic Returns on Hedge Portfolio EJTP 3, No. 3 006 9 8 Elecronic Journal of Theoreical Physics Black Scholes Opion Pricing wih Sochasic Reurns on Hedge Porfolio J. P. Singh and S. Prabakaran Deparmen of Managemen Sudies Indian Insiue

More information

A Probability Density Function for Google s stocks

A Probability Density Function for Google s stocks A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural

More information

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process, Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ

More information

PRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES

PRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES NADINE GAZER HAO SCHMEISER WORKING PAPERS ON RISK MANAGEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAGEMEN

More information

Dynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract

Dynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium

More information

A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul

A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul universiy of copenhagen Universiy of Copenhagen A Two-Accoun Life Insurance Model for Scenario-Based Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:

More information

The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts

The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts The Uncerain Moraliy Inensiy Framework: Pricing and Hedging Uni-Linked Life Insurance Conracs Jing Li Alexander Szimayer Bonn Graduae School of Economics School of Economics Universiy of Bonn Universiy

More information

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives

Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives Sochasic Calculus, Week 10 Term-Srucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Term-srucure models 3. Ineres rae derivaives Definiions and Noaion Zero-coupon

More information

Optimal Time to Sell in Real Estate Portfolio Management

Optimal Time to Sell in Real Estate Portfolio Management Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr

More information

Foreign Exchange and Quantos

Foreign Exchange and Quantos IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in

More information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Unit-linked life insurance in Lévy-process financial markets

Unit-linked life insurance in Lévy-process financial markets 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

More information

Valuation of Credit Default Swaptions and Credit Default Index Swaptions

Valuation of Credit Default Swaptions and Credit Default Index Swaptions Credi Defaul Swapions Valuaion of Credi Defaul Swapions and Marek Rukowski School of Mahemaics and Saisics Universiy of New Souh Wales Sydney, Ausralia Recen Advances in he Theory and Pracice of Credi

More information

Difficulties in pricing of real options

Difficulties in pricing of real options Difficulies in pricing of real opions Francis Asu U.U.D.M. Projec Repor 7:5 Examensarbee i maemaik, poäng Handledare och examinaor: Maciej Klimek Januari 7 Deparmen of Mahemaics Uppsala Universiy Acknowledgemen

More information

nonlocal conditions.

nonlocal conditions. ISSN 1749-3889 prin, 1749-3897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.3-9 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed

More information

Fair valuation of participating policies with surrender options and regime switching

Fair valuation of participating policies with surrender options and regime switching Insurance: Mahemaics and Economics 37 25 533 552 Fair valuaion of paricipaing policies wih surrender opions and regime swiching Tak Kuen Siu Deparmen of Acuarial Mahemaics and Saisics, School of Mahemaical

More information

On the Role of the Growth Optimal Portfolio in Finance

On the Role of the Growth Optimal Portfolio in Finance QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 1441-8010 www.qfrc.us.edu.au

More information

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se

More information

SPEC model selection algorithm for ARCH models: an options pricing evaluation framework

SPEC model selection algorithm for ARCH models: an options pricing evaluation framework Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,

More information

ARTICLE IN PRESS Journal of Computational and Applied Mathematics ( )

ARTICLE IN PRESS Journal of Computational and Applied Mathematics ( ) Journal of Compuaional and Applied Mahemaics ( ) Conens liss available a ScienceDirec Journal of Compuaional and Applied Mahemaics journal homepage: www.elsevier.com/locae/cam Pricing life insurance conracs

More information

On Valuing Equity-Linked Insurance and Reinsurance Contracts

On Valuing Equity-Linked Insurance and Reinsurance Contracts On Valuing Equiy-Linked Insurance and Reinsurance Conracs Sebasian Jaimungal a and Suhas Nayak b a Deparmen of Saisics, Universiy of Torono, 100 S. George Sree, Torono, Canada M5S 3G3 b Deparmen of Mahemaics,

More information

Optimal Longevity Hedging Strategy for Insurance. Companies Considering Basis Risk. Draft Submission to Longevity 10 Conference

Optimal Longevity Hedging Strategy for Insurance. Companies Considering Basis Risk. Draft Submission to Longevity 10 Conference Opimal Longeviy Hedging Sraegy for Insurance Companies Considering Basis Risk Draf Submission o Longeviy 10 Conference Sharon S. Yang Professor, Deparmen of Finance, Naional Cenral Universiy, Taiwan. E-mail:

More information

IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **

IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß ** IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include so-called implici or embedded opions.

More information

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES?

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and Black-Meron-Scholes

More information

Distributing Human Resources among Software Development Projects 1

Distributing Human Resources among Software Development Projects 1 Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources

More information

Risk Modelling of Collateralised Lending

Risk Modelling of Collateralised Lending Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies

More information

New Pricing Framework: Options and Bonds

New Pricing Framework: Options and Bonds arxiv:1407.445v [q-fin.pr] 14 Oc 014 New Pricing Framework: Opions and Bonds Nick Laskin TopQuark Inc. Torono, ON, M6P P Absrac A unified analyical pricing framework wih involvemen of he sho noise random

More information

SHB Gas Oil. Index Rules v1.3 Version as of 1 January 2013

SHB Gas Oil. Index Rules v1.3 Version as of 1 January 2013 SHB Gas Oil Index Rules v1.3 Version as of 1 January 2013 1. Index Descripions The SHB Gasoil index (he Index ) measures he reurn from changes in he price of fuures conracs, which are rolled on a regular

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Why we have always used the Black-Scholes-Merton option pricing formula

Why we have always used the Black-Scholes-Merton option pricing formula Why we have always used he Black-Scholes-Meron opion pricing formula Charles J. Corrado Deakin Universiy Melbourne, Ausralia April 4, 9 Absrac Derman and aleb (he Issusions of Dynamic Hedging, 5 uncover

More information

PRICING and STATIC REPLICATION of FX QUANTO OPTIONS

PRICING and STATIC REPLICATION of FX QUANTO OPTIONS PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of

More information

An accurate analytical approximation for the price of a European-style arithmetic Asian option

An accurate analytical approximation for the price of a European-style arithmetic Asian option An accurae analyical approximaion for he price of a European-syle arihmeic Asian opion David Vyncke 1, Marc Goovaers 2, Jan Dhaene 2 Absrac For discree arihmeic Asian opions he payoff depends on he price

More information

Differential Equations and Linear Superposition

Differential Equations and Linear Superposition Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

More information

Modeling a distribution of mortgage credit losses Petr Gapko 1, Martin Šmíd 2

Modeling a distribution of mortgage credit losses Petr Gapko 1, Martin Šmíd 2 Modeling a disribuion of morgage credi losses Per Gapko 1, Marin Šmíd 2 1 Inroducion Absrac. One of he bigges risks arising from financial operaions is he risk of counerpary defaul, commonly known as a

More information

OptimalCompensationwithHiddenAction and Lump-Sum Payment in a Continuous-Time Model

OptimalCompensationwithHiddenAction and Lump-Sum Payment in a Continuous-Time Model Appl Mah Opim (9) 59: 99 46 DOI.7/s45-8-95- OpimalCompensaionwihHiddenAcion and Lump-Sum Paymen in a Coninuous-Time Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business

More information

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

More information

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100... Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The Black-Scholes Shl Ml Moel... pricing opions an calculaing

More information

Niche Market or Mass Market?

Niche Market or Mass Market? Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

A NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS

A NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.

More information

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance

Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance Finance Leers, 003, (5), 6- Skewness and Kurosis Adjused Black-Scholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance

More information

Optimal Investment, Consumption and Life Insurance under Mean-Reverting Returns: The Complete Market Solution

Optimal Investment, Consumption and Life Insurance under Mean-Reverting Returns: The Complete Market Solution Opimal Invesmen, Consumpion and Life Insurance under Mean-Revering Reurns: The Complee Marke Soluion Traian A. Pirvu Dep of Mahemaics & Saisics McMaser Universiy 180 Main Sree Wes Hamilon, ON, L8S 4K1

More information

On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations

On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen

More information

The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies

The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies 1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz- und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany

More information

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility Scand. Acuarial J. 00; 4: 46 79 ORIGINAL ARTICLE Pricing Dynamic Insurance Risks Using he Principle of Equivalen Uiliy VIRGINIA R. YOUNG and THALEIA ZARIPHOPOULOU Young VR, Zariphopoulou T. Pricing dynamic

More information

Debt Portfolio Optimization Eric Ralaimiadana

Debt Portfolio Optimization Eric Ralaimiadana Deb Porfolio Opimizaion Eric Ralaimiadana 27 May 2016 Inroducion CADES remi consiss in he defeasance of he Social Securiy and Healh public deb To achieve is ask, he insiuion has been assigned a number

More information

On the degrees of irreducible factors of higher order Bernoulli polynomials

On the degrees of irreducible factors of higher order Bernoulli polynomials ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

Hedging with Forwards and Futures

Hedging with Forwards and Futures Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

More information