Fair valuation of participating policies with surrender options and regime switching

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1 Insurance: Mahemaics and Economics Fair valuaion of paricipaing policies wih surrender opions and regime swiching Tak Kuen Siu Deparmen of Acuarial Mahemaics and Saisics, School of Mahemaical and Compuer Sciences, Herio-Wa Universiy, Edinburgh EH14 4AS, UK Received March 25; received in revised form May 25; acceped 2 May 25 Absrac We consider he fair valuaion of a paricipaing life insurance policy wih surrender opions when he marke values of he asse are modelled by Markov-modulaed Geomeric Brownian Moion GBM. We reduce he dimension of he opimal sopping problem for he policy by changing probabiliy measures. We also provide a decomposiion resul for he value of he policy. The Barone Adesi Whaley approximaion has been employed o approximae he soluion of he free boundary problem for he policy by second-order piecewise linear ordinary differenial equaions ODEs. The fair valuaion of paricipaing perpeual American conracs are also considered. 25 Elsevier B.V. All righs reserved. JEL classificaion: G13; G22 MSC: IM1; IE1 Keywords: Paricipaing American policies; Perpeual conracs; Change of measures; Second-order piecewise linear ODEs; Regime swiching 1. Inroducion Recenly, paricipaing wih-profis life insurance producs become more and more popular in he insurance and finance markes due o heir lower risk bu similar reurns compared o oher equiy-index producs. In erms of marke size, paricipaing life insurance producs are considered he mos imporan modern life insurance producs in major insurance and finance markes around he world, such as Unied Kingdom, Unied Saes and Japan, ec. see Balloa, 24. Paricipaing life insurance producs are invesmens plans wih associaed life insurance benefis, a specified benchmark reurn, a guaranee of an annual minimum rae of reurn and a specified rule of Tel.: ; fax: address:.k.siu@ma.hw.ac.uk /$ see fron maer 25 Elsevier B.V. All righs reserved. doi:1.116/j.insmaheco

2 534 T.K. Siu / Insurance: Mahemaics and Economics he disribuion of annual excess invesmen reurn above he guaraneed reurn. The policyholder has o pay a single lump sum deposi o he insurer o iniialize he conrac. The insurer plays he role of a fund manager o manage he invesmen of funds in a specified reference porfolio. One major feaure of hese invesmen plans is he sharing of profis from an invesmen porfolio beween he policyholder and he insurer. Typically, he insurer employs a specified rule of surplus disribuion, namely reversionary bonus, o credi ineres a or above a specified guaraneed rae o he policyholders every period, say per annum. If he erminal surplus of he fund is posiive, he policyholder can also receive a erminal bonus. Grosen and Jørgensen 2 and Balloa e al. 23 provided a comprehensive discussion on differen conracual feaures of paricipaing policies. Since here is a global rend of using he marke-based and fair valuaion accounancy sandards for he implemenaion of risk managemen pracice for paricipaing policies, i is of pracical imporance and relevance o develop appropriae and objecive models for he fair valuaion of hese policies. Wilkie 1987 pioneered he use of modern opion pricing heory o invesigae he embedded opions in bonuses on paricipaing life-insurance policies. Grosen and Jørgensen 2 developed a flexible coningen claims model o incorporae he minimum rae guaranees, bonus disribuions and surrender opions. They considered he surrender opions as he possibiliy of early exercise of an American-syle opion and invesigaed he pricing behaviors of he policy by varying he levels of ineres raes, he parameers of bonus policy and he volailiy of he marke value of he reference asse. Priel e al. 21 incorporaed he pah dependence associaed wih he rule of he bonus disribuion in heir coningen claims model. They reduced he dimension of he parial differenial equaion in heir model by similariy ransformaion of variables. Bacinello 23 adoped discree-ime binomial models for compuing he numerical soluions of he fair valuaion problem of paricipaing policies. Grosen and Jørgensen 22 adoped a barrier opion framework o invesigae he impac of regulaory inervenion rules for reducing he insolvency risk of he policies. Willder 24 considered he use of modern opion pricing approach for invesigaing he effecs of various bonus sraegies in uniized paricipaing policies. Chu and Kwok 25 consruced a coningen claims model for paricipaing policies ha can incorporae rae guarranee, bonuses and defaul risk. They obained an analyical approximaion soluion o he problem by perurbaion mehod and he numerical soluion by developing effecive finie difference algorihms. Balloa 24 developed a valuaion mehod based on he Esscher ransform for paricipaing policies under a jump-diffusion process for he dynamics of he marke values of he reference asse. In his paper, we consider he fair valuaion of a paricipaing life insurance policy wih surrender opions, bonus disribuions and rae guaranees when he dynamics of he marke values of he asse is driven by Markovmodulaed Geomeric Brownian Moion GBM. As in Buffingon and Ellio 22 and Siu 25, we suppose ha he parameers of he marke values of he asses, namely he marke ineres raes, he expeced growh rae and he volailiy of he risky asse, depend on unobservable saes of an economy which are modelled by a coninuous-ime hidden Markov chain process. The swiching behavior of he economic saes can be due o he srucural changes in economic condiions and business cycles. In pracice, many life insurance producs are relaively long-daed compared wih financial producs. There can be subsanial flucuaions in economic variables, which affec he dynamics of he marke values of he asses, over a long period of ime. Hence, i is of pracical imporance and relevance o incorporae he swiching behavior of he economic saes in modelling he dynamics of he marke values of he asses for he fair valuaion of insurance producs. Following Siu 25, we adop regime swiching Esscher ransform developed in Ellio e al. 25 o deermine an equivalen maringale measure for he fair valuaion of he policy in an incomplee marke described by he Markov-modulaed model. The seminal work by Gerber and Shiu 1994 pioneered he use of he Esscher ransform for opion valuaion. By considering he surrender opions as he early exercise privilege of an American-syle opion see Grosen and Jørgensen, 2, he fair valuaion of he policy can be formulaed as an opimal sopping problem from which he value of he policy and he opimal surrender sraegy can be deermined. We reduce he dimension of he opimal sopping problem for he policy by adoping he mehod of changing probabiliy measures in Hansen and Jørgensen 2. We also provide a decomposiion for he value of he policy ino he value of an idenical policy wihou surrender opions and he premium for surrender opions. The fair valuaion problem can also be formulaed as a free boundary

3 T.K. Siu / Insurance: Mahemaics and Economics problem. Following Buffingon and Ellio 22, we employ he approximaion mehod due o Barone-Adesi and Whaley 1987 o approximae he soluion of he free boundary problem by second-order piecewise linear ordinary differenial equaions ODEs. We also consider he fair valuaion of paricipaing perpeual American conracs in order o provide some insighs in heir finie-mauriy counerpars. The value of a perpeual conrac can also serve as an approximaion o is finie-mauriy counerpar since mos of he paricipaing conracs are relaively long-daed compared wih oher financial conracs. This paper is oulined as follows. Secion 2 presens he opimal sopping problem associaed wih he fair valuaion of he paricipaing American policies under he Markov-modulaed GBM. We also adop he mehod of changing probabiliy measures o reduce he dimension of he problem. Secion 3 presens a decomposiion for he value of he policy. In Secion 4, we consider he free-boundary problem for he fair valuaion of he policy. In Secion 5, we consider he use of he approximaion mehod due o Barone-Adesi and Whaley 1987 and is exension by Buffingon and Ellio 22 o approximae he soluion of he free boundary problem by second-order piecewise linear ODEs. Secion 6 discusses he fair valuaion of paricipaing perpeual American conracs. The final secion summarizes he paper and suggess some poenial opics for furher invesigaion. 2. The opimal sopping problem 2.1. Inroducion In his secion, we consider a coninuous-ime perfec and fricionless financial marke in which here are no axes, ransacion coss and shor-sales consrains. We suppose ha here are a risk-free money marke accoun and risky asses backing a paricipaing life insurance conrac wih surrender opions. The dynamics of he marke values of he reference asse is governed by a Markov-modulaed Geomeric Brownian Moion. We consider he valuaion of he paricipaing life insurance conrac wih surrender opions paricipaing American-syle conrac when he marke ineres raes, he drif and volailiy of he underlying asse base depend on he saes of a coninuous-ime hidden Markov chain process. The saes of he hidden Markov chain process represens various saes of an economy. The regime swiching of he saes of he economy can be due o he srucural changes in he macro-economic condiions, he changes in poliical siuaions, he impac of some macro-economic news and business cycles. We furher assume ha here is no expense charges, lapses and moraliy risk. The marke described by he regime swiching model is incomplee in general see Guo, 21; Buffingon and Ellio, 22; Ellio e al., 25; Siu, 25. Hence, here are infiniely many equivalen maringale measures for he valuaion of a paricipaing conrac. As in Siu 25, we employ he regime swiching Esscher ransform described in Ellio e al. 25 o deermine an equivalen maringale measure for he fair valuaion of he paricipaing conrac in he incomplee marke seing. The paricipaing American-syle conrac is differen from is European counerpar in ha i provides he policyholder he righ o erminae exercise he conrac a any ime before he mauriy of he conrac. In addiion o he bonus opion and he ineres rae credied wih guaranee, he policyholder is graned wih an opion o sell back he policy o he issuer any ime before he mauriy of he policy. The opion ha allows he policyholder o sell back he policy o he issuer any ime is known as he surrender opion in he insurance markes. A paricipaing American-syle conrac consiss of hree componens, namely he risk-free bond, bonus opion and he surrender opion while a paricipaing European-syle conrac can be decomposed ino wo componens, namely he risk-free bond and he bonus opion. The presence of he surrender opion in a paricipaing American-syle conrac makes is valuaion problem more complicaed han ha of is European counerpar. As in he valuaion of American-syle opions, he valuaion of a paricipaing American-syle conrac can be formulaed as an opimal sopping problem, or equivalenly, a free boundary problem. For he valuaion of a paricipaing American-syle policy, we need o deermine he fair value of he policy and is opimal erminaion or exercise sraegy. In pracice, some numerical mehods, such as Mone-Carlo simulaion, finie-difference mehods and binomial models, are employed o obain numerical approximaions o he valuaion problem. Due o he facs ha Mone-Carlo simulaion is a forward

4 536 T.K. Siu / Insurance: Mahemaics and Economics mehod and ha he early exercise sraegy is no known in advance, Mone-Carlo simulaion was considered o be no suiable for he valuaion of American-syle opions in general. However, some recen sudies see, for example, Tilley, 1993; Broadie and Glasserman, 1997 indicaed ha Mone Carlo simulaion is a feasible way o solve he valuaion problem of American-syle opions numerically. The finie-difference mehods provide a naural way o solve he free-boundary problems for he valuaion of American-syle opions and he paricipaing Americansyle conracs. Grosen and Jørgensen 2 implemened he recursive binomial mehod for he valuaion of he paricipaing American-syle conracs. Much of he lieraure focuses on he valuaion of paricipaing American conracs under GBM. There is a relaively lile work on he fair valuaion of paricipaing American conrac under he regime swiching models for he asse price dynamics. Here, we provide a valuaion mehod of some paricular paricipaing American-syle conracs under he Markov-Modulaed GBM by modifying he mehod employed in Hansen and Jørgensen 2. In he sequel, we inroduce he se up of our model Definiion of he liabiliies In his secion, we describe he dynamics of he liabiliy side of he balance shee. For each ime T, le R and D denoe he book value of he policy reserve and he bonus reserve buffer, respecively. R can be considered he policyholder s accoun balance and is in general differen from he concurren fair value of he policy. The liabiliies of he insurance company consiss of wo componens, namely he policy reserve R and he bonus reserve D. Le A denoe he marke value of he asse backing he policy asse base. Then, A,R and D saisfy he following accouning ideniy: A = R + D, T, 2.1 where R := αa,α, 1], and R is he single iniial premium paid by he policyholder for acquiring he conrac and α is he cos allocaion parameer. Noe ha he α-porion of he iniial asse porfolio is financed by he policyholder. The funds are disribued o he wo componens of liabiliy over ime according o he bonus policy described by he coninuously compounded ineres rae credied o he policy reserve c R ; ha is, dr = c R R d. 2.2 In pracice, he bonus policy and c R are decided by he managemen level of an insurance company. The deerminaion of c R involves many issues, such as he poliical siuaions, legal issues and he sraegic consideraions wihin he insurance company. There is no consensus on a unified rule for he specificaion of c R.AsinGrosen and Jørgensen 2, we specify he ineres rae crediing mechanism in he form of a mahemaical funcion, which can provide an accurae approximaion and realisic descripion o he rue bonus policy as much as possible. In his way, as noed by Grosen and Jørgensen 2, we can adop arbirage pricing models in mahemaical finance for our analysis. Grosen and Jørgensen 2 menioned ha he acual ineres rae crediing mechanism c R can be specified as a funcion c R A, R of he asse base A and he policy reserve R. Hence, he ineres bonus can be disribued according o he acual invesmen performance and he curren financial condiion, such as he degree of solvency, of he insurance company. Typically, he insurance company has specified a consan long-erm arge buffer raio β of he bonus reserve D o he policy reserve R, where a realisic value of β should be beween 1 and 15%. The policyholder of he conrac can receive a cerain proporion, say δ, 1], of he excess of he raio of he bonus reserve D o he policy reserve R over β. The proporional consan δ is called he reversionary bonus disribuion rae or disribuion raio. In pracice, appropriae values of δ are chosen o achieve a sable smoohing of he surplus and ypical values for δ are around 2 3%. We also suppose ha here is a specified guaranee rae r g for he minimum ineres rae credied o he policyholder s accoun. This means ha he ineres rae c R A, R r g. Grosen and Jørgensen 2, Priel e al. 21 and Chu and Kwok 25 provided differen specificaions for he ineres rae crediing scheme. As in Chu and Kwok 25, we adop he coninuous compounding version of he

5 T.K. Siu / Insurance: Mahemaics and Economics ineres rae crediing scheme as follows: c R A, R = max r g,δ ln AR β, 2.3 where he disribuion rae δ is assumed o be equal o one; ha is, here is a full paricipaion. When he policyholder erminaes he policy by exercising he surrender opion a ime, he value ha is paid o he policyholder by he insurer is called he inrinsic value of he policy. The inrinsic value of he policy depends on he marke value of he asse base A and he policy reserve R a ime and is given as follows: ga,r, = { R if A < R α, R + γαa R ifa R α, 2.4 where γ, 1 is he bonus disribuion rae. Le T denoe he mauriy of he policy. P T := maxαa T R T, is he erminal bonus opion, which can be considered a sandard European call opion ha grans he policyholder he righ o pay he policy value as a srike price o receive α-porion of he asse porfolio. The erminal bonus opion can also be viewed as a sandard European pu opion ha grans he insurer he righ o sell he asse for he policy value when he asse value falls below he policy value. The erminal payoff of he policy a ime T is given by: R T ga T,R T,T = R T + γαa T R T 2.3. Model dynamics = R T + γp T. if A T < R T α ifa T R T α We fix a complee probabiliy space Ω, F, P, where P is he real-world probabiliy measure. Le T denoe he ime index se [,T] of he model. Le {W } T denoe a sandard Brownian Moion on Ω, F, P wih respec o he P-augmenaion of is naural filraion F W :={F W } T. The saes of an economy are described by a coninuousime hidden Markov chain {X } T wih a finie sae space S := s 1,s 2,...,s N. As in Ellio e al. 1994,wecan idenify he sae space of {X } T o be a finie se of uni vecors {e 1,e 2,...,e N }, where e i =,...,1,...,. We suppose ha {X } T and {W } T are independen. Wrie Π for he generaor [π ij ] i,j of he hidden Markov chain model. Then, from Ellio e al. 1994, we have he following semi-maringale represenaion heorem for he process {X } T : X = X + ΠsX s ds + M. Here {M } T is a maringale incremen process wih respec o he filraion generaed by {X } T. Le {r } T denoe he insananeous marke ineres rae of he money marke accoun, which depends on he saes {X } T ; ha is, r := r, X = r, X, T, 2.7 where r := r 1,r 2,...,r N wih r i > for each i = 1, 2,...,N and, denoes he inner produc in R N. In his case, he dynamics of he price process {B } T for he bank accoun is described by: B = exp r s ds. 2.8

6 538 T.K. Siu / Insurance: Mahemaics and Economics Now, we assume ha he expeced growh rae {µ } T and he volailiy {σ } T of he marke value of he asse backing he paricipaing conrac asse base also depend on {X } T and are given by: µ := µ, X = µ, X, σ := σ, X = σ, X, 2.9 where µ := µ 1,µ 2,...,µ N and σ := σ 1,σ 2,...,σ N wih σ i > for each i = 1, 2,...,N. We focus on he asse side of he balance shee. As in Grosen and Jørgensen 2, we assume ha he insurer mainains he invesmen of he asse base in a well-diversified and well-specified reference porfolio a any ime. For each T, le A denoe he marke value of he reference porfolio a ime. We do no impose any assumpions on he porfolio composiions wih respec o various asses, such as bonds, equiies and real esaes. No assumpions are imposed on he dynamics of he individual asses in he reference porfolio. We analyze he dynamics of he marke value of he asses backing he paricipaing conrac in an aggregae level and assume ha he dynamics of he marke value of he reference porfolio {A } T is governed by he following Markov-modulaed GBM: da = µ A d + σ A dw, A = a Fair valuaion The fair value of he paricipaing American conrac can be decomposed ino he fair values of he risk-free bond, he bond opion and he surrender opion. Suppose he sae of he economy X a ime is X, where [,T]. Then, when A = A and R = R, he inrinsic value of he policy if i is exercised a ime is given by: { R if A< R ga, R, X, = α, R + γαa R ifa R α Le P := maxαa R, represen he payoff of he bonus opion. Then, ga, R, X, can be wrien in he following form: ga, R, X, = R + γp The fair valuaion of he risk-free par of he paricipaing conrac, namely he risk-free bond, is sraighforward see Grosen and Jørgensen, 2. For he fair valuaion of he risky par of he policy consising of he bond opion and he surrender opion, we need o deermine an equivalen risk-neural maringale measure o ensure ha here are no arbirage opporuniies in he marke described by he model see Harrison and Kreps, 1979; Harrison and Pliska, 1981, We employ he regime swiching Esscher ransform o deermine an equivalen maringale measure for he valuaion of he policy by he maringale approach. For more discussions on he maringale approach for he valuaion of paricipaing policies, see Bacinello 21. The choice of he equivalen maringale measure by he regime swiching Esscher ransform can be jusified by minimizing he relaive enropy of an equivalen maringale measure and he real-world probabiliy P see Ellio e al., 25. The Esscher ransform has been adoped by Balloa 24 and Siu 25 for he valuaion of paricipaing producs in incomplee marke seings. Le Y denoe he logarihmic reurn lna /A from he asse over he ime duraion [,]. Then, he dynamics of A can be wrien as: where A = A u expy Y u, Y = 2.13 µ s 1 2 σ2 s ds + σ s dw s Wrie {F X } T and {F Y } T for he P-augmenaion of he naural filraions generaed by {X } T and {Y } T, respecively. For each T, define G as he σ-algebra F X F Y. Define he regime swiching parameer θ :=

7 T.K. Siu / Insurance: Mahemaics and Economics θ, X as follows: θ = θ, X, 2.15 where θ := θ 1,θ 2,...,θ N R N. Then, as in Ellio e al. 25, he regime swiching Esscher ransform P θ P on G is: dp θ exp θ s dy s = dp ], T G E P [exp θ s dy s F X Hence, he Radon Nikodym derivaive of he regime swiching Esscher ransform can be wrien as: dp θ = exp θ s σ s dw s 1 θs 2 dp 2 σ2 s ds G Wrie { θ } T denoe a family of risk-neural regime swiching Esscher parameers. As in Ellio e al. 25,we assume ha { θ } T can be deermined from he following maringale condiion for he discouned marke values of he reference asse: A = E θ [ exp ] r s ds A F X, for any T, 2.18 where E θ denoe he expecaion operaor under P θ. Then, θ can be deermined uniquely by: θ = r µ σ 2, T. See Ellio e al. 25 for he proof. Now, we noice ha [ dp θ rs µ s = exp dp G σ s dw s rs µ 2 s ds]. 2.2 σ s By Girsanov s heorem, W := W + µs r s σ s ds is a sandard Brownian moion wih respec o {G } T under P θ. Hence, he dynamics of A under P θ is given by: da = r A d + σ A d W Suppose he rajecory of he hidden process X from ime o ime is known in advance, where T; ha is, a ime, he enlarged informaion se G is accessible o he marke s agen. Wrie T,T for he class of {G } T sopping imes aking values in [,T]. As in Grosen and Jørgensen 2, we deermine he fair value of he enire paricipaing American conrac based on he general heory of he fair valuaion of he American-ype coningen claims see Karazas, 1988, Then, he fair value of he paricipaing American policy V a ime can be deermined by solving he following opimal sopping problem wih respec o he enlarged informaion se G : V = ess sup τ T,T E θ [ exp τ r s ds ] ga, R, X, τ G As in he mehod of augmening an addiional sae variable in he valuaion problem of Asian opions, we consider an addiional sae variable R, which is a pah inegral of he process A, for deermining he fair value of

8 54 T.K. Siu / Insurance: Mahemaics and Economics he paricipaing policy. Since R is a pah inegral of A and A is a Markov process given ha he rajecory of X is known, A,R is a wo-dimensional Markov process given he rajecory of X. Due o he fac ha X is also a Markov process, A,R,X is a hree-dimensional Markov process wih respec o G.Now,ifA = A, R = R and X = X are given a ime, hen by he Markov propery of A,R,X, he fair value of he paricipaing policy V a ime is given by: V = V A, R, X, = ess sup τ T,T E θ [ e τ r s ds ga, R, X, τ A,R,X = A, R, X Buffingon and Ellio 22 adoped a similar mehod o deermine he price of an American opion Reducion of dimensionaliy ] We noice ha he opimal sopping problem for he fair valuaion of he paricipaing American policy involves hree sae variables. In order o simplify he problem, we employ he mehod of changing probabiliy measures in Hansen and Jørgensen 2 o reduce he dimension of he problem from hree sae variables o wo sae variables. Firs, we define a new sae variable Z := ln A R. We furher assume ha ga, R, X, c Z Z = c R A, R, g Z Z, X, = R Noe ha he inrinsic value a ime can be wrien as follows: g Z Z, X, = 1 + γ maxαe Z 1, By Iô s lemma, he dynamics of Z under P is given by: dz = µ c Z Z 1 2 σ2 d + σ dw Now, we define a P θ -maringale wih respec o G : A 1 ξ := exp r s ds = exp A 2 σ2 s ds + σ s d W s Then, we define a new equivalen measure ˆP as follows: d ˆP := ξ, T. G dp θ By Girsanov s heorem, Ŵ := W σ s ds, is a sandard Brownian moion under ˆP wih respec o G. Under ˆP, he dynamics of A can be represened by: da = r + σ 2 A d + σ A dŵ. 2.3 Now, by Iô s lemma, he dynamics of Z under ˆP is given by: dz = r σ2 c Z Z d + σ dŵ. 2.31

9 T.K. Siu / Insurance: Mahemaics and Economics Hence, given knowledge of he values of he hidden Markov chain process X, he new sae variable Z is a Markov process on is naural filraion. Wrie Ê for he expecaion operaor under ˆP. By Bayes rule, he opimal sampling heorem and he Markov propery of he sae variables, V = ess sup τ T,T E θ [ exp Ê [Ê = ess sup τ T,T τ [ 1 = ess sup Ê ξê τ T,T [ ξ = ess sup Ê τ T,T = ess sup τ T,T A Ê ] r s ds ga, R, X, τ G dp θ G d ˆP τ exp τ r s ds ] ga, R, X, τ G Ê dp θ G d ˆP τ ] ξt G τ exp r s ds ga, R, X, τ G τ ] ξτ exp r s ds ga, R, X, τ G = ess sup A Ê τ T,T [ ] Rτ ga, R, X, τ A,R,X = A, R, X A τ R τ [ Rτ A τ ] ga, R, X, τ G = ess sup A Êe Z τ g Z Z, X, τ Z,X = Z, X, 2.32 τ T,T when he values of he new sae variable Z and he hidden sae variable X a ime are Z and X, respecively. Due o he fac ha Z is also a Markov process on is naural filraion, he new opimal sopping problem wih wo sae variables is much easier o solve han he original opimal sopping problem wih hree sae variables. By Øksendal 23, he opimal sopping ime for his problem belongs o he se of sopping imes T,T Z :={τω, u T,T τ := f Z u,u,f is measurable}. Then, he opimal sopping problem becomes: V := V Z, X, = ess sup A Êe Z τ g Z Z, X, τ Z,X = Z, X τ T,T Z As in Hansen and Jørgensen 2, le Ṽ Z Z, X, denoe he value of he paricipaing American conrac denominaed by he asse price A. We also call Ṽ Z Z, X, he A-denominaed value of he conrac. Tha is, Ṽ Z Z, X, := ess sup Êe Z τ g Z Z, X, τ Z,X = Z, X τ T,T Z In he sequel, we provide he analysis for he A-denominaed value of he conrac Ṽ Z Z, X, insead of V Z, X,. Wrie g Z Z, X, for e Z g Z Z, X,. Then, R τ { e g Z Z, X, = e Z + γ e Z maxα e Z Z if Z ln1/α, 1, = γα + 1 γe Z if Z>ln1/α, 2.35 which is a decreasing funcion of Z. Le τ denoe he opimal sopping rule for he opimal sopping problem Then, Ṽ Z Z, X, = ess sup Ê g Z Z, X, τ Z,X = Z, X = Ê g Z Z, X, τ Z,X = Z, X τ T,T Z

10 542 T.K. Siu / Insurance: Mahemaics and Economics Le C and S denoe he coninuaion region and he sopping region of he above opimal sopping problem, respecively. Then, and C ={Z, X, s s [, T ] Ṽ Z Z, X, s > g Z Z, X, s}, 2.37 S ={Z, X, s s [, T ] Ṽ Z Z, X, s = g Z Z, X, s} In he region C, he A-denominaed value of he policy is greaer han is A-denominaed inrinsic value. Hence, i is no opimal for he policyholder o surrender he policy. The region S specifies he hreshold curve on which i is opimal for he policyholder o surrender he policy. The hreshold curve of opimal surrendering can incorporae he dependency of he opimal surrendering decision of he policyholder on he saes of he economy. The opimal sopping rule τ is given by he firs exi ime of Z, X, from C; ha is, τ = inf{s [, T ] Z, X, s C}, 2.39 where C is he complemen of C. Suppose he number of regimes N for he hidden process X is wo. Then, we consider he case ha here exiss wo hreshold curves Z1 s,z 2 s ln1/α, for s [, T ], such ha he coninuaion region C can be represened in he following form: C = C 1 C 2, 2.4 where C i :={Z, e i,s s [, T ] Z Zi s, }, for i = 1, 2. By Karazas 1989, Øksendal 23 and Ellio and Kopp 24, we require ha Zi s is a coninuous funcion of ime s [, T ], for i = 1, 2 and T. Noe also ha Zi T = ln1/α. There are hree possible cases, namely Z1 s <Z 2 s,z 1 s = Z 2 s and Z 1 s >Z 2 s, for any given s [, T. 3. Decomposiion of he value In his secion, we provide a decomposiion for he A-denominaed value of he paricipaing American conrac ino he A-denominaed value of is European counerpar and he A-denominaed early exercise premium. For he sake of generaliy, we suppose ha he number of regimes of he economy is N when we derive he decomposiion resul in his secion. Le V E Z, X, denoe he A-denominaed value of a paricipaing European conrac wih idenical conracual feaures wih is American counerpar in Secion 2, excep wihou surrender opions. Following Siu 25, i can be shown ha V E Z, X, = Ê g Z Z, X, T Z,X = Z, X. 3.1 Wrie Vi E for V E Z, e i,, where i = 1, 2,...,N, and V E := V1 E,VE 2,...,VE N. Then, i has been shown in Siu 25 ha V E saisfies he following N coupled PDEs: L Z,ei V E i + V E,Πe i =, i = 1, 2,...,N. 3.2 Le ɛz, X, denoe he A-denominaed premium of surrender opion or early exercise premium A-denominaed a ime when Z = Z and X = X. The following proposiion provides a decomposiion of Ṽ Z Z, X, ino he wo componens V E Z, X, and ɛz, X,, as in Hansen and Jørgensen 2.

11 T.K. Siu / Insurance: Mahemaics and Economics Proposiion 3.1. The A-denominaed value of he paricipaing American conrac a ime is given by: Ṽ Z Z, X, = V E Z, X, + ɛz, X,, 3.3 where T ɛz, X, = Ê e Z u c Z Z u r u I {Zu S} du Z,X = Z, X, 3.4 and I A is he indicaor funcion of an even A. Proof. Following Hansen and Jørgensen 2, we consider he dynamics of he values of he paricipaing conrac separaely on he coninuaion region C and he sopping region S. Firs, since Z,X is a wo-dimensional Markov process wih respec o he enlarged filraion G, V E Z, X, = Ê g Z Z, X, T Z,X = Z, X = Ê g Z Z, X, T G, 3.5 which is a ˆP-maringale wih respec o G. By applying Iô s rule, [ V V E Z, X, = V E E Z, X, + u + V E + Z σ u dŵ u + and dx = ΠX d + dm. r u + 1 V E 2 σ2 u c ZZ u Z σ2 u 2 V E ] Z 2 du V E, dx u, 3.6 Since V E is a ˆP-maringale wih respec o G, V E Z, X, = V E V E Z, X, + Z σ u dŵ u, 3.8 or equivalenly, dv E = V E Z σ dŵ. 3.9 On he coninuaion region C, he dynamics of Ṽ Z and V E are idenical. Hence, dṽ Z = Ṽ Z Z σ dŵ. On he sopping region S, Ṽ Z is equal o is inrinsic value; ha is, Ṽ Z Z, X, = g Z Z, X, = e Z, 3.11 since Z ln1/α ins. This implies ha dṽ Z = Ṽ Z Z dz σ2 2 Ṽ Z d = e Z r Z σ2 c Z Z 1 2 σ2 d e Z σ dŵ = Ṽ Z Z r c Z Z d + Ṽ Z Z σ dŵ. 3.12

12 544 T.K. Siu / Insurance: Mahemaics and Economics Hence, in he enire sae space, dṽ Z = Ṽ Z Z r c Z Z I {Z S} d + Ṽ Z Z σ dŵ Therefore, he resul follows by inegraing 3.13 from o T and aking expecaion. 4. The free boundary problem We provide a characerisaion for he fair valuaion of he paricipaing American conrac based on a free boundary problem in he conex of Markov-modulaed GBM. Buffingon and Ellio 22 provided a formulaion of he free boundary problem for an American opion under Markov-modulaed GBM. As in Buffingon and Ellio 22, we suppose ha he number of regimes N for he hidden process X is wo. In his case, we assume ha he generaor Π of he hidden Markov process X is [π ij ] i,j=1,2, where π ii < i = 1, 2,π 12 = π 22 and π 21 = π 11. Define a wo-dimensional vecor Ṽ Z Z, as: Ṽ Z Z, := Ṽ Z Z, e 1,, Ṽ Z Z, e 2,. 4.1 For simplifying he noaions, we wrie Ṽ i Z for Ṽ Z Z, e i,, where i = 1, 2, and Ṽ Z for Ṽ 1 Z, Ṽ 2 Z. For X = e i,i= 1, 2, wrie C i and S i for he coninuaion region and he sopping region, respecively. Then, and C i ={Z, s s [, T ] Z>Zi s}, 4.2 S i ={Z, s s [, T ] Z = Zi s}. 4.3 For each s [, T ], le C i s denoe he s-secion of Ci. Then, by Buffingon and Ellio 22 and Ellio and Kopp 24, C i s is an inerval of he form Z i s,. As in Buffingon and Ellio 22, we consider he case ha Z1 s <Z 2 s, for s [, T ]. Firs, we suppose Z s >Z2 s. Then, Z, s C2 and Z, s C 1 ; ha is, Z, s is in he coninuaion region for boh saes. Now, we define he following parial differenial operaor L Z,ei, for i = 1, 2: L Z,ei Ṽ Z = Ṽ Z + r i + 1 ṼZ 2 σ2 i c Z Z Z σ2 i Since c Z Z = maxr g,z β, Ṽ Z + r i r g + 1 ṼZ 2 σ2 i Z Ṽ Z 2 σ2 i Z L Z,ei Ṽ Z = 2 if Z r g + β Ṽ Z + r i + β + 1 ṼZ 2 σ2 i Z Z Ṽ Z 2 σ2 i Z 2 if Z>r g + β = Ṽ [ Z + r i r g σ2 i I {Z rg +β} + r i + β σ2 i Z 2 Ṽ Z Z I {Z>rg +β} ] ṼZ Z σ2 i 2 Ṽ Z Z Then, by Buffingon and Ellio 22 and Siu 25, Ṽ Z := ṼZ 1, Ṽ Z 2 saisfies he following pair of coupled PDEs: L Z,ei Ṽ i Z + Ṽ Z,Πe i =, i = 1, Noe ha Zi ln1/α, for i = 1, 2.

13 When Z Z 1 and Z = Z, ṼZ 1 = e Z. When Z Z2 and Z = Z, T.K. Siu / Insurance: Mahemaics and Economics ṼZ 2 = e Z. 4.8 By he high conac principle of an opimal sopping problem, we require ha ṼZ i saisfies boh he coninuiy condiion and he smooh pasing condiion on he early exercise boundary Zi. Then, for i = 1, 2, he coninuiy condiion is described by: Ṽ Z Zi,e i, = e Z i, 4.9 and he smooh pasing condiion is given by: Z ṼZZi,e i, = e Z i. 4.1 When Z Z1,Z 2, which represens he ransiion region beween Z 1 and Z 2, Ṽ Z 1 saisfies he following parial differenial equaion: L Z,e1 ṼZ 1 + π 11Ṽ Z 1 π 11e Z =, 4.11 where π 11 < Second-order piecewise linear ODEs We adop he approximaion due o Barone-Adesi and Whaley 1987 o provide an approximae soluion o he valuaion of he paricipaing American conrac by second-order piecewise linear ODEs. The Barone-Adesi- Whaley approximaion has been adoped by Buffingon and Ellio 22 for obaining an approximae soluion o he valuaion of American opions under regime swiching models. Following he analysis in Buffingon and Ellio 22, we consider he cases of he common coninuaion region and he ransiion region Case I Suppose Z, is in he common coninuaion region CR := {Z, Z >Z2 } for boh saes. Hence, Ṽ Z saisfies he following pair of coupled PDEs: L Z,ei ṼZ i + Ṽ Z,Πe i =, i = 1, For Z Z2, Ṽ Z 2 saisfies he following boundary condiion: ṼZ 2 = e Z. Noe ha he coninuiy condiion is given by: Ṽ Z Z2,e 2, = e Z 2, and he smooh pasing condiion is given by: Z ṼZZ2,e 2, = e Z 2. Wrie ɛ i for ɛz, e i, and Vi E := V E Z, e i,, for i = 1, 2. Then, by Proposiion 3.1, ɛ i = ṼZ i V i E

14 546 T.K. Siu / Insurance: Mahemaics and Economics Le ɛ := ɛ 1,ɛ 2. Since boh Ṽ Z and V E saisfy he coupled PDEs in he common coninuaion region CR, ɛ also saisfies he same coupled PDEs in CR as follows: L Z,ei ɛ i + ɛ, Πe i =, i = 1, Now, we adop he Barone Adesi Whaley approximaion o find an approximae soluion o he valuaion of he paricipaing American conrac in a separaed form. For i = 1, 2, we assume ha ɛ i can be approximaed as follows: ɛ i := ɛz, e i, HZ, e i F. 5.7 Wrie H := HZ for HZ, e 1,HZ, e 2. Le H i := HZ, e i for i = 1, 2 and F := F. Then, H i saisfies: F H i + r i σ2 i c Z Z F H i Z σ2 i F 2 H i Z 2 + F H,Πe i =. 5.8 We assume ha F is given by he following form: [ T ] F = Ê 1 exp r u du. 5.9 Then, F = r F 1. Hence, for i = 1, 2, 1 2 H i 2 σ2 i Z 2 + r i σ2 i c Z Z Hi 5.1 Z + H,Πe i = r ih i 1 F, 5.11 F where he erm r ih i 1 F F is sill a funcion of ime. Now, in he coninuaion region CR, H saisfies he following wo coupled second-order ordinary differenial equaions ODEs: 1 2 H 1 2 σ2 1 Z H 2 2 σ2 2 Z 2 + r σ2 1 c H1 ZZ r σ2 2 c ZZ Z + π 11H 1 π 11 H 2 = r 1H 1 1 F, F H2 Z π 22H 1 + π 22 H 2 = r 2H 2 1 F F Since c Z Z = maxr g,z β, H saisfies he following wo second-order piecewise linear ODEs: 1 2 [ H 1 2 σ2 1 Z 2 + r 1 r g σ2 1 I {Z rg +β} + r 1 + β + 12 ] σ21 Z H1 I {Z>rg +β} Z + π 11H 1 π 11 H 2 = r 1H 1 1 F, F 1 2 H 2 2 σ2 2 Z 2 + = r 2H 2 1 F. F [ r 2 r g σ2 2 I {Z rg +β} + r 2 + β + 12 ] σ22 Z H2 I {Z>rg +β} Z π 22H 1 + π 22 H

15 T.K. Siu / Insurance: Mahemaics and Economics Case II Now, we suppose ha Z, is in he ransiion region TR := {Z, Z1 Z Z 2 } beween he wo early exercise boundaries Z1 and Z 2. In he ransiion region TR, we derive an approximae soluion o he valuaion of he paricipaing American conrac by applying he Barone Adesi Whaley approximaion only o he sae X = e 1. We suppose ha in he ransiion region TR, he dynamics of he marke values of he asse A under P is given by: da = µ 1 A d + σ 1 A dw, 5.14 where µ 1 and σ 1 are he consan drif and volailiy parameers in he sae X = e 1. We furher suppose ha he marke ineres rae for he bank accoun is r 1, which is he marke ineres rae in he sae X = e 1. Then, B = expr 1. Under hese assumpions abou he marke values of he asse A and he marke ineres rae, when Z>Z 1, he A-denominaed value of he paricipaing American conrac Ṽ 1 Z saisfies he following PDE: L Z,e1 Ṽ 1 Z =, since his corresponds o he case of no regime swiching. For Z Z1, Ṽ Z 1 saisfies he following boundary condiion: Ṽ 1 Z = e Z. Noe ha he coninuiy condiion is given by: Ṽ Z Z 1, = e Z 1, and he smooh pasing condiion is given by: Z ṼZZ1, = e Z The regime swiching Esscher ransform in Secion 2 reduces o he coninuous-ime Esscher ransform wih consan Esscher parameer. Le F W :={F W } T denoe he P-augmenaion of he naural filraion generaed by he process W. Following he same procedure as in Secion 2, we can define he risk-neural Esscher ransform Q 1 P as follows: [ r1 dq 1 µ 1 = exp W 1 r1 µ 2 1 ] dp 2 F W σ 1 σ 1 By Girsanov s heorem, W := W + µ 1 r 1 σ 1 is a sandard Brownian moion wih respec o F W under Q 1. Hence, he dynamics of A under Q 1 is governed by: da = r 1 A d + σ 1 A d W. 5.2 Now, we define a Q 1 -maringale wih respec o F W : ξ 1 := e r 1 A = exp 12 A σ21 + σ 1 W Then, we define a new probabiliy measure ˆQ 1 Q 1 as follows: d ˆQ 1 := ξ dq 1 1, T F W

16 548 T.K. Siu / Insurance: Mahemaics and Economics By Girsanov s heorem, Ŵ := W σ 1, 5.23 is a sandard Brownian moion under ˆQ 1 wih respec o F W. Under ˆQ 1, he dynamics of A can be represened by: da = r 1 + σ 2 1 A d + σ 1 A dŵ By Iô s lemma, he dynamics of Z under ˆQ 1 is: dz = r σ2 1 c ZZ d + σ 1 dŵ In he ransiion region TR, wrie V 1 E A, R, for he A-denominaed value of he paricipaing European conrac. Then, V E 1 A, R, = e r 1T 1 A E 1 ga, R, e 1,T A,R = A, R = 1 A E 1 e r 1T ga, R, e 1,T A,R = A, R, 5.26 where E 1 is he expecaion operaor under Q 1. By using he mehod of change of measures in Secion 2, we can show ha E 1 e r 1T ga, R, e 1,T A,R = A, R = A Ê 1 g Z Z, e 1,T Z = Z, 5.27 where Ê 1 is he expecaion operaor under ˆQ 1. Hence, V E 1 A, R, = Ê1 g Z Z, e 1,T Z = Z Le V 1 EZ, := Ê1 g Z Z, e 1,T Z = Z. Then, V 1 E := V 1 E Z, saisfies: V 1 E + r σ2 1 ZZ c V 1 E Z V 2 σ2 1 E 1 Z 2 =, 5.29 wih auxillary condiion: V E 1 Z, T = e Z T + γe Z T maxα e Z T 1,. 5.3 Now, we define ɛ 1 := Ṽ 1 Z V E 1. Then, ɛ 1 + r σ2 1 c ɛ1 ZZ Z ɛ 1 2 σ2 1 = Z2 As in Case I, we suppose ha ɛ 1 H 1 Z F, 5.32 Then, H 1 F + r σ2 1 c ZZ F H 1 Z σ2 1 F 2 H 1 = Z2

17 We furher assume ha F = 1 e r 1T. Then, F = r 1 F 1. Hence, H 1 saisfies he following second-order ODE: 1 2 H 1 2 σ2 1 Z 2 + T.K. Siu / Insurance: Mahemaics and Economics r σ2 1 c ZZ H 1 Z + r F 5.34 H 1 =, 5.35 where he erm r 1 1 F 1 H 1 is sill a funcion of ime. By noicing ha c Z Z = maxr g,z β, H 1 saisfies he following second-order piecewise linear ODE: 1 2 [ H 1 2 σ2 1 Z 2 + r 1 r g σ2 1 I {Z rg +β} + r 1 + β + 12 ] σ21 Z H 1 I {Z>rg +β} Z +r H 1 = F 6. Paricipaing perpeual american conracs Perpeual means no final mauriy dae. The policyholder of a paricipaing perpeual American conrac can erminae he conrac any ime indefiniely. The absence of a finie-mauriy dae for he conrac makes he valuaion of a perpeual conrac less complicaed han a finie-mauriy conrac. The paricipaing perpeual American policy is only a mahemaical idealisaion and i is no acually raded in he insurance markes see Ellio and Kopp, 24 for relaed discussions on perpeual American opions. However, he invesigaion of he valuaion of paricipaing perpeual American conracs can provide some insighs in heir finie-mauriy counerpars. Since paricipaing conracs are relaively long-daed compared wih oher financial producs, he valuaion of paricipaing perpeual American conracs can also serve as a reasonable approximaion o is finie-mauriy counerpar. Guo and Zhang 24 provided closed-form soluions for he valuaion of perpeual American pu opions wih regime swiching. Here, we derive a se of second-order piecewise linear ordinary differenial equaions ODEs for he valuaion of paricipaing perpeual American conracs under he Markov-modulaed asse price process. As in Guo and Zhang 24, we consider a wo-sae hidden Markov chain process wih he generaor Π given in Secion 4. Firs, le V p A, R, X denoe he value of he paricipaing perpeual American conrac. Then, he opimal sopping problem for he perpeual conrac can be obained by aking T in he corresponding problem for he finie-mauriy conrac and is given as follows: Wrie V p A, R, X = ess sup E θ τ [ exp τ r s ds ] ga, R, X, τ A,R,X = A, R, X. 6.1 T Z, :={τω, u [, τ := f p Z u,u,f p is measurable}. 6.2 By adoping he mehod of changing probabiliy measures as in Secion 2, he opimal sopping problem for he perpeual conrac can be wrien as follows: V p A, R, X = ess sup A Ê g Z Z, X, τ Z,X = Z, X. 6.3 τ T, Z

18 55 T.K. Siu / Insurance: Mahemaics and Economics Le V p Z Z, X denoe he A-denominaed value of he perpeual conrac; ha is, V p Z Z, X = ess sup Ê g Z Z, X, τ Z,X = Z, X. 6.4 τ T, Z Le g Z Z, X := e Z + γ e Z maxα e Z 1,. Following Guo and Zhang 24, he coninuaion region C p and he sopping region S p are given by: and C p ={Z, X V p Z Z, X > g ZZ, X}, 6.5 S p ={Z, X V p Z Z, X = g ZZ, X}. 6.6 Wrie τ p for he firs exi ime of Z, X from C p ; ha is, τ p := inf{s Z s,x s C p }, where C p is he complemen of C p. Due o he fac ha Z,X is a wo-dimensional Markov Chain process wih respec o he enlarged filraion G, he opimal sopping rule is given by τ p. Then, V p Z Z, X = Ê g Z Z, X, τ p Z,X = Z, X. 6.7 Noe ha boh V p Z Z, X and g ZZ, X are decreasing funcions of Z for fixed X. Then, as in Guo and Zhang 24, we suppose ha here exiss wo hresholds Z p 1,Zp 2 ln1/α such ha he coninuaion region Cp can be represened by: C p ={Z, e 1 Z Z p 1, } {Z, e 2 Z Z p 2, }. 6.8 There are hree possible cases, namely Z p 1 <Zp 2,Zp 1 = Zp 2 and Zp 1 >Zp 2. As in Secion 4, we consider he case ha Z p 1 <Zp 2. Following he analysis of Guo and Zhang 24, we consider he following hree cases: Case I: Z [Z p 1,Zp 2 ] Le V p i := V p Z Z, e i, for i = 1, 2; V p := V p 1,Vp 2. Then, by Iô s differeniaion rule see Guo and Zhang, 24, V p saisfies: = r p V 2 σ2 1 c 1 ZZ Z V p 2 σ2 1 1 Z 2 + π 11V p 1 π 11e Z, V p 2 = e Z. 6.9 Hence, V p 1 saisfies he following second-order piecewise linear ODE: = 1 2 V p [ 2 σ2 1 1 Z 2 + r 1 r g σ2 1 I {Z rg +β} + r 1 + β + 12 ] p V σ21 Z 1 I {Z>rg +β} Z +π 11 V p 1 π 11e Z. 6.1 Case II: Z [Z p 2, Again, by Iô s differeniaion rule, V p saisfies: = = r p V 2 σ2 1 c 1 ZZ r σ2 2 c ZZ Z σ2 1 p V 2 Z σ2 2 2 V p 1 Z 2 + π 11V p 1 π 11V p 2, 2 V p 2 Z 2 + π 22V p 2 π 22V p

19 T.K. Siu / Insurance: Mahemaics and Economics Hence, V p saisfies he following wo coupled second-order piecewise linear ODEs: = 1 2 V p [ 2 σ2 1 1 Z 2 + r 1 r g σ2 1 I {Z rg +β} + r 1 + β + 12 ] p V σ21 Z 1 I {Z>rg +β} Z +π 11 V p 1 π 11V p 2, = 1 2 V p [ 2 σ2 2 2 Z 2 + r 2 r g σ2 2 I {Z rg +β} + r 2 + β + 12 ] p V σ22 Z 2 I {Z>rg +β} Z +π 22 V p 2 π 22V p Case III: Z,Z p 1 ] In his case, V p 1 = V p 2 = e Z Summary and furher invesigaion We have considered he fair valuaion of a paricipaing life insurance policy wih surrender opions, bonus disribuions and rae guaranees when he dynamics of he marke values of he asse is driven by a Markovmodulaed Geomeric Brownian Moion GBM. The mehod of changing probabiliy measures in Hansen and Jørgensen 2 has been employed o reduce he dimension of he opimal sopping problem for he valuaion of he paricipaing American conrac. We have provided a decomposiion for he value of he policy ino he value of an idenical conrac wihou surrender opions and he premium of surrender opions. The valuaion problem has also been formulaed as a free boundary problem. We have employed he approximaion mehod due o Barone- Adesi and Whaley 1987 o approximae he soluion of he free boundary problem by second-order piecewise linear ODEs. We have also considered he fair valuaion of paricipaing perpeual American conracs. For furher invesigaion, i is worh invesigaing he fair valuaion of oher modern insurance producs wih opion-embedded feaures and early exercise feaure or surrender opion, in he conex of Markov-modulaed diffusion processes. We may also invesigae he fair valuaion of paricipaing American conracs when he dynamics of he marke values of he reference porfolio is governed by oher ypes of regime swiching models, such as he Markov swiching jump-ype models and he Markov-modulaed Lévy processes. I is of pracical imporance and relevance o develop some mehods and echniques for measuring and managing risk inheren from rading he paricipaing conracs. The risk measuremen and managemen of paricipaing American conracs are more challenging han heir European counerpars due o he presence of surrender opions. I would be ineresing o invesigae he risk measuremen and managemen of paricipaing perpeual American conracs in order o gain some insighs in he risk measuremen and managemen of heir finie-mauriy counerpars. Acknowledgemen The auhor would like o hank he referee for many valuable and helpful commens and suggesions. References Bacinello, A.R., 21. Fair pricing of life insurance paricipaing policies wih a minimum ineres rae guaraneed. ASTIN Bull. 31 2, Bacinello, A.R., 23. Fair valuaion of a guaraneed life insurance paricipaing conrac embedding a surrender opion. J. Risk Insur. 7 3,

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