ARTICLE IN PRESS Journal of Computational and Applied Mathematics ( )

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1 Journal of Compuaional and Applied Mahemaics ( ) Conens liss available a ScienceDirec Journal of Compuaional and Applied Mahemaics journal homepage: Pricing life insurance conracs wih early exercise feaures Anna Ria Bacinello a, Enrico Biffis b,, Piero Millossovich a a Deparmen of Applied Mahemaics, Universiy of Triese, Piazzale Europa 1, Triese, Ialy b Tanaka Business School, Imperial College London, Souh Kensingon Campus, SW7 2AZ, Unied Kingdom a r i c l e i n f o a b s r a c Aricle hisory: Received 1 December 2007 Received in revised form 10 April 2008 MSC: IE10 IE50 IB10 Keywords: Insurance conracs Surrender opions Leas squares Mone Carlo mehod American coningen claims In his paper we describe an algorihm based on he Leas Squares Mone Carlo mehod o price life insurance conracs embedding American opions. We focus on equiy-linked conracs wih surrender opions and erminal guaranees on benefis payable upon deah, survival and surrender. The framework allows for randomness in moraliy as well as sochasic volailiy and jumps in financial risk facors. We provide numerical experimens demonsraing he performance of he algorihm in he conex of muliple risk facors and exercise daes Elsevier B.V. All righs reserved. 1. Inroducion The main characerisic of life insurance conracs is o provide benefis coningen on survival or deah of individuals. While he provision of proecion agains he risk of deah is cerainly imporan, he savings componen plays a crucial role in many life policies. In he las decades, he compeiion wih alernaive invesmen vehicles offered by he financial indusry has generaed subsanial innovaion in he design of life producs and in he range of benefis provided. In paricular, equiy-linked policies have become more and more popular, offering policyholders exposure o financial indices as well as differen ways o consolidae invesmen performance over ime. While he inroducion of more appealing (and exoic) guaranees has ensured a saisfacory ake-up of insurance producs, a he same ime he managemen of life policies has become increasingly complex, requiring he proper undersanding and analysis of inegraed financial and insurance risks. One of he mos common opions available in policies wih a considerable savings componen 1 is he possibiliy o exi (surrender) he conrac before mauriy and o receive a lump sum (surrender value) reflecing he insured s pas conribuions o he policy, minus any coss incurred by he company and possibly some charges. The idea is o boos sales by ensuring ha he policyholder does no perceive insurance securiies as an illiquid invesmen. On he oher hand, surrenders are no welcomed by insurers, as hey imply a reducion in he asses under managemen and may generae imbalances in he exposure o he moraliy risk of remaining insureds (selecive surrenders). An addiional layer of risk is inroduced when surrender values allow for minimum guaranees. The objecive of he presen paper is o sudy producs embedding his ype of opion and guaranees. From he poin of view of asse pricing heory, surrender opions are nohing else han American claims wih a knockou feaure represened by he occurrence of deah. In oher words, a surrender opion is an American pu opion wrien Corresponding auhor. addresses: bacinel@unis.i (A.R. Bacinello), E.Biffis@imperial.ac.uk (E. Biffis), pierom@econ.unis.i (P. Millossovich). 1 An excepion is represened by immediae annuiies, because of high aniselecion risk /$ see fron maer 2008 Elsevier B.V. All righs reserved. doi: /j.cam

2 2 A.R. Bacinello e al. / Journal of Compuaional and Applied Mahemaics ( ) on he residual value of he policy and canceled upon deah, wih a srike price given by he surrender value. A number of approaches o American opion pricing have been proposed in he lieraure (see [13,15, and references herein]). The difficuly wih insurance conracs is ha realisic models pose formidable compuaional asks for wo main reasons. Firs, insurance conracs are usually very long erm and offer grea flexibiliy in early erminaion of he conrac, meaning ha we are faced wih muliple exercise daes. Second, a realisic model should ake ino accoun a leas he key drivers of raional surrender decisions, which in he equiy-linked case may be quie involved. Incorporaing irraional or exogenous facors as well as parameer uncerainy would make he ask even more dauning. Empirical evidence suggess ha surrenders are driven by several facors including disribuion channels, misselling, financial marke condiions and deerioraion/improvemen of policyholders healh. In his paper, we focus on financial and demographic drivers, namely ineres rae risk, invesmen performance and moraliy risk. High ineres raes as well as poor invesmen performance are usually associaed wih policyholders exiing he conrac o ener more rewarding invesmen opporuniies. This generaes ouflows ha reduce asses under managemen and increase per-policy coss associaed wih insurance business. Wih regard o demographic facors, policyholders surrender policies when hey perceive here is no need for he proecion offered by he policy. On he oher hand, policyholders do no surrender policies when proecion is perceived valuable, even if hey should or if he level of moraliy proecion is marginal in he overall design of he policy. This generaes aniselecion risk (selecive surrenders) and can lead o dramaic changes in he demographic profile of an insurance porfolio. Moraliy risk and surrender risk compound when sums a risk (deah benefis minus reserves) are large. This aspec may be relevan even for equiy-linked producs, for example when deah benefis are nominally fixed or provide subsanial minimum guaranees. Irrespecive of he specific risk facor considered, i is usually difficul o analyze all hese risks simulaneously, as some of he benefis may be linked, ohers nominally fixed, and all of hem may have subsanial minimum guaranees. Several simulaion mehods for pricing American opions are available (see [15]), bu hey are usually no very effecive in he presence of muliple sae variables and several exercise daes. Since our objecive is o ake ino accoun he risk facors described above and o allow for a range of real-world markes feaures (sochasic volailiy, jumps in asse prices, randomness in moraliy raes, ec.), we focus on he powerful Leas Squares Mone Carlo (LSMC) approach, which was proposed in [11,18,24] in he conex of purely financial American claims. Surrender opions have araced he ineres of many researchers. Saring wih he seminal papers in [1,16,17], a number of sudies have followed. Due o he high dimensionaliy of he problem, he vas majoriy of papers provide resuls in sylized siuaions. When moving o more realisic models, conribuions become scarce. As an example, he inroducion of moraliy risk is carried ou in only a few papers. For insance, we menion [3 5] in he conex of binomial rees, [19,22] in he conex of free boundary problems, [2,6] in he conex of Mone Carlo simulaion and he LSMC approach. We noe ha previous models ypically use deerminisic (or even consan) moraliy raes. This is usually jusified by adoping diversificaion argumens and invoking he law of large numbers for large enough insurance porfolios. However, he surrender decision is made by individual policyholders, who are faced wih heir own ime of deah only. This alone would jusify he analysis of randomness in moraliy raes. The issues of selecive wihdrawals and possibly large sums a risk make he case for random moraliy even more compelling. Finally, we noe ha he applicaion of he LSMC approach in he conex of demographic risk requires care. This moivaed he work in [6], which is furher developed and exended in his paper. The paper is srucured as follows. Secion 2 is devoed o he descripion of our valuaion framework. In paricular, we inroduce he life insurance conracs of ineres, define he dynamics of financial and demographic risk facors and finally presen he valuaion problem. In Secion 3, we briefly describe he LSMC mehodology and presen our valuaion algorihm. Secion 4 offers some numerical resuls, while Secion 5 concludes. 2. Valuaion of equiy linked endowmens 2.1. The conrac Consider an individual aged x a ime 0 when enering an endowmen conrac, i.e. a life policy wih mauriy T > 0 providing a lump sum benefi F s T a ime T upon survival or a benefi F d a ime (0, T] in case coincides wih he individual s ime of deah, denoed by τ. If F s = T 0 he conrac reduces o a erm assurance policy, if F τ d = 0 he conrac reduces o a pure endowmen. We will focus in paricular on equiy-linked producs, meaning ha (some of) he benefis are linked o he performance of a reference fund. Besides providing deah and survival benefis, hese policies may allow policyholders o exi he conrac before mauriy. If no paymen is provided upon wihdrawal, he policy is said o be lapsed. If insead a lump sum F w is paid upon wihdrawal a ime, he policy is said o be surrendered agains provision of he surrender value F w. Endowmens can usually be surrendered, given he subsanial savings componen of he conrac. On he oher hand, erm assurances are usually lapsed, since premiums and reserves are small and mean o provide pure proecion.. Le us denoe by S = (S ) 0 he marke value of he reference fund o which policy paymens are linked. Benefis provided by equiy-linked conracs ypically embed minimum guaranees. A common example is represened by erminal guaranees of he form ( ) F e S = F 0 max, exp(κ e ), (2.1) S 0

3 A.R. Bacinello e al. / Journal of Compuaional and Applied Mahemaics ( ) 3 where e = s, d, w, depending on wheher we consider survival, deah or surrender benefis. In he above expression F 0 represens he iniial value of he reference fund, which is financed by he policyholder s iniial premium (ne of any charges), while κ e represens he minimum ineres guaraneed on he differen benefi paymens. In his paper, we limi our aenion o he case of single premium policies. We refer he reader o [4] for consideraions on muliple premium paymens. In expression (2.1) benefis depend only on he value of he reference fund a he relevan dae. Some conracs are insead characerized by pah-dependen guaranees. An example is represened by clique guaranees, where guaraneed amouns are re-se regularly, possibly aking ino accoun he performance of a reference fund a several pas daes. For insance, we may consider payoffs defined by ( F e = F 0 max 1 + η u y ( ) ) Su j+1 1, exp(κ e ), (2.2) u y S u j u=1 j=1 meaning ha he policyholder is awarded a proporion η (0, 1] of he fund s performance smoohed over he las y years, subjec o a minimum guaranee κ e. Guaranees such as (2.2) are very common in paricipaing conracs, i.e. policies where S represens he insurance company s invesmen porfolio raher han some exernal reference fund. In his paper we focus on benefis of he ype given in (2.1), alhough he analysis of pah-dependen surrender opions would require minor modificaions. Denoe now by θ he ime a which he policyholder decides o erminae he conrac. Early erminaion can clearly occur if he individual is sill alive and he policy is sill in force. This poses no problem if θ < τ T, while we disregard surrender whenever θ τ T. The ime θ is in general a random variable whose law depends on he evoluion of marke and demographic condiions, which a any given ime make he surrender value more or less aracive wih respec o saying in he conrac. We call herefore θ an exercise policy. For given θ and fixed ime, we define he cumulaed benefis paid by he conrac up o ha ime by G (θ). = F s T 1 τ>t,t θ + F d τ 1 τ T θ + F w θ 1 θ, θ<τ T, (2.3) where he summands on he righ hand side are coningen on he occurrence of hree muually exclusive evens. Our objecive is o inroduce an arbirage-free securiies marke and deermine a surrender policy θ ha is opimal for a raional policyholder Valuaion framework We ake as given (Ω, F, F, Q), a filered probabiliy space supporing all sources of financial and demographic randomness. The filraion F =. (F ) 0 (saisfying he usual condiions of righ coninuiy and compleeness, and such ha F 0 = {, Ω}) represens he flow of informaion available o he insurer and he policyholder. I is naural o assume ha boh τ and θ are F-sopping imes, meaning ha a any ime he informaion carried by F allows us o ell wheher deah or surrender have occurred or no by ime. The probabiliy Q is a risk-neural probabiliy measure, meaning ha under Q he marke value of any securiy is given by he expeced value of is cumulaed dividends deflaed a he riskfree rae. Under he assumpion of fricionless securiies markes, he exisence of such Q is essenially guaraneed by he absence of arbirage, a minimal requiremen avoiding he availabiliy of riskless gains a zero cos (see [13] for example). Before inroducing a model for he relevan risk facors, we give more srucure o he informaion flow F. We inroduce. he deah indicaor process N = 1τ, which equals zero as long as he individual is alive and jumps o one a deah. Denoing by H he filraion generaed by (N ) 0, we assume ha F =. G H for some filraion G no including H and such ha G 0 is rivial. The inuiion is ha G carries all relevan informaion abou demographic and financial facors (in paricular, securiy prices and likelihood of deah), bu does no yield knowledge of occurrence of τ. A possible way of defining he arrival of deah is by seing τ. = inf { : Γ > ξ}, (2.4) wih (Γ ) 0 a G-adaped nondecreasing process and ξ a random variable independen of G and exponenially disribued wih parameer one. If Γ can be expressed as Γ = µ 0 sds for all and some G-predicable nonnegaive process µ, hen consrucion (2.4) is equivalen o he so called condiionally Poisson seup. This means ha, condiionally on G and under he measure Q, τ is he firs jump ime of a Poisson inhomogeneous process wih inensiy (µ ) 0. This seup is appealing because i generalizes he usual formulas employed in demography, insurance or economics o a sochasic framework (e.g., [9]). The seing also ensures ha any G-maringale is an F-maringale, 2 yields considerable simplificaions in pricing formulas, as will be explained in Secion To see his, noe ha consrucion (2.4) implies Q(τ > G ) = Q(τ > G ) = exp( Γ ). In oher words, for each 0, H and G are condiionally independen, given G. For any G-maringale (M s ) s 0 and T 0 we can hen wrie E Q [M T F ] = E Q [M T F ] = M.

4 4 A.R. Bacinello e al. / Journal of Compuaional and Applied Mahemaics ( ) 2.3. Financial and demographic risk facors As explained in he inroducion, empirical evidence on insurance markes suggess ha surrender decisions are driven by several facors including disribuion channels, misselling, financial marke condiions and deerioraion or improvemen of policyholders healh. Here, we focus on financial and demographic drivers of surrender decisions, in paricular on ineres rae risk, sock marke performance and moraliy risk. We model he erm srucure of ineres raes by a sandard Cox Ingersoll Ross model, a square-roo process feauring mean revering and nonnegaive raes. Is dynamics is given by dr = ζ r (δ r r )d + σ r r dz r, wih ζ r, δ r, σ r > 0 and Z r a sandard Brownian moion. For he marke value of he reference fund, we consider he sochasic exponenial S = exp(y), where Y evolves according o dy = (r 12 ) K λ Y µ Y d + ( ) K ρ SK dz K + ρ Sr dz r + 1 ρ 2 SK ρ2 Sr dz S + dj Y. The componen J Y is a compound Poisson process wih jump arrival rae λ Y > 0 and lognormally disribued jump sizes wih mean µ Y and sandard deviaion σ Y > 0. The correlaion coefficiens ρ SK and ρ Sr saisfy ρ 2 + SK ρ2 Sr 1. The facor K generaes sochasic volailiy (indeed, K is he square of he insananeous non-jump volailiy of S) wih mean-revering dynamics dk = ζ K (δ K K )d + σ K K dz K, where ζ K, δ K, σ K > 0. We ake he process (Z r, Z S, Z K ) o be a hree dimensional Brownian moion independen of he pure jump process J Y. The above formulaion is essenially he one sudied by [8], who show ha i is quie effecive in reproducing he price behavior of equiy derivaives. For he inensiy of moraliy, we ake he lef coninuous version of he process dµ = ζ µ (m() µ )d + σ µ µ dz µ + dj µ, (2.5) where m( ), ζ µ, σ µ > 0, Z µ is a sandard Brownian moion and J µ is a compound Poisson process independen of Z µ, wih jump arrival rae λ µ 0 and exponenial jumps of mean γ µ > 0. Finally, he couple (Z µ, J µ ) is assumed o be independen of (Z r, Z S, Z K, J Y ), so ha financial and demographic facors are independen. The choice of risk facors oulined in his secion is aimed a providing a realisic seup o assess he performance of he LSMC mehodology. Alernaive dynamics could be analyzed, bu we hink ha he curren seup represens a valid es, given he number of facors and he presence of muliple exercise daes. To conclude, we se X = (N, µ, r, Y, K) and denoe by X = (µ, r, Y, K) he reduced sae variable process. We hen ake F o be he filraion generaed by X and G he one generaed by X. We observe ha boh X and X are affine processes. As a resul, when implemening he algorihm, we have a leas a benchmark for he case of European guaranees, since hey can be readily priced in he affine seing (see [10]) Valuaion Our financial marke is characerized by he invesmen fund S inroduced in he previous secion and a money marke accoun yielding he insananeous risk-free rae (r ) 0. For each 0, B = exp( 0 r sds) formalizes he proceeds from invesing one uni of money a ime 0 in risk-free deposis and rolling over he proceeds unil ime. Consider now he insurance conrac inroduced in Secion 2.1, where we assume ha he process F e is G-predicable for e = s, d, w: his always holds when S is defined as in Secion 2.3 and we consider he lef-coninuous versions of (2.1) or (2.2). In he presen framework and consisenly wih he classical reamen of American opions, i is naural o assume ha any exercise policy θ is an F-sopping ime, meaning ha he surrender decision is based on he enire informaion available over ime. Suppose ha he conrac is erminaed a he random ime θ: under no arbirage, he ime- value V (θ) of he conrac is hen given by he usual risk-neural formula [ ] V (θ) = B E Q B 1 u dg u(θ) F, (2.6) where G u (θ) is given by (2.3) and represens he cumulaed benefis provided by he conrac up o ime u if he exercise policy θ is adoped. Exploiing he srucure of he filraion F, in [21, p. 370] we know ha every F-sopping ime θ coincides wih a G- sopping ime θ up o ime τ. Assuming ha τ coincides wih he firs jump of a condiionally Poisson process wih inensiy µ, expression (2.6) reduces o (e.g., [14]) V (θ) = 1 τ> B E Q [ B 1 u dĝ u ( θ) G ], (2.7)

5 wih B. = exp(. 0 (r s + µ s )ds) and Ĝ u (θ) = F s T 1 T u θ + ARTICLE IN PRESS A.R. Bacinello e al. / Journal of Compuaional and Applied Mahemaics ( ) 5 u 0 F d s µ s 1 s T θ ds + F w θ 1 θ u,θ<t. As a resul, everyhing works as in he absence of he random ime of deah, provided we replace boh he money marke accoun and he deah benefi wih heir moraliy risk-adjused counerpars ( Bs ) s and (F d s µ s) s. Le us now focus on he pricing of he conrac a incepion. Denoing by T F (T G ) he se of finie valued F-sopping imes (G-sopping imes), he iniial price is given by he soluion of he opimal sopping problem V 0 = V 0(θ ) = sup V 0 (θ) = sup V 0 ( θ) = sup V 0 ( θ), (2.8) θ T F θ T G θ T G, θ τ where he las wo equaliies follow again by [21, p. 370] and he fac ha V 0 ( θ) = V 0 ( θ τ). Since θ and θ coincide up o τ, as seen from conrac incepion, from now on we simply use he noaion θ when dealing wih (2.8). A soluion θ o problem (2.8) is called a raional exercise policy, in he sense ha i maximizes he iniial arbirage-free value of he resuling claim. While (2.8) can be jusified by replicaion argumens when markes are complee, he case of incomplee markes is more delicae (e.g., [13]). This is exacly our siuaion, even if we disregard moraliy risk, because our financial marke is incomplee and perfec replicaion is in general no possible. We do no expand on his here and simply employ (2.8) under our given fixed risk-neural probabiliy measure Q. While expression (2.7) is exremely appealing from he poin of view of inerpreaion, i can be compuaionally more inensive han (2.6), as explained in [7]. This is why for he pricing algorihm described in he nex secion we use he las equaliy in (2.8), bu work direcly wih expression (2.6) Exension o exogenous surrender facors The model described so far can be exended o include exogenous surrender decisions, where by exogenous we mean ha he decision o wihdraw from he conrac may be riggered by somehing differen from coninuaion values falling below surrender benefis. A naural way of capuring his possibiliy is o inroduce anoher sopping ime ν wih G-predicable inensiy (φ ) 0, where we emphasize ha G could be generaed by non-financial and non-demographic facors. Inspecion of expression (2.8) shows ha he opimal sopping problem can be modified by simply replacing τ wih τ ν. Since τ and ν admi inensiies µ and φ, he sopping ime τ ν admis he inensiy µ+φ. If τ ν, raher han τ, is assumed o saisfy he condiions described in Secion 2.2, hen he valuaion framework can be applied wih no subsanial changes, apar from replacing µ wih µ + φ. Similarly, he algorihm described in he nex secion can be used wih no subsanial modificaion. 3. The algorihm The LSMC approach is based on he join use of Mone Carlo simulaion and Leas Squares regression in Markovian environmens. The mehod firs requires discreizaion of he ime dimension, in order o replace he original opimal sopping problem (2.8) wih is discreized version along a ime grid T. Denoing by n he number of periods in which we divide he inerval [0, T] and seing i = i n T for i = 0,..., n, we obain a ime grid T = { 0,..., n } and a discreized sopping problem sup E Q [g θ ], θ T F,T (3.1) wih T F,T denoing he family of T-valued F-sopping imes and g. = 0 B 1 0 dg s(θ), where G s (θ) is defined by (2.3). As is common when dealing wih American opions, one can inroduce he Snell envelope of (g ) 0 and apply he dynamic programming principle o develop a backward procedure involving a comparison, a each ime sep, beween he payoff provided by he surrender opion and he coninuaion value (i.e., he reward from no exercising). The LSMC mehod looks a such procedure in erms of opimal sopping imes, in he sense ha an opimal policy θ = θ 0 is compued according o he backward algorihm { θ = n n = T θ j = j 1 gj >U j + θ j+1 1 g j U j for j = n 1,..., 0,. = EQ [g θ j+1 F j ]. Since we work in a Markovian environmen, we can wrie U j where U j = E Q [g θ j+1 X j ] = u( j, X j ), for some Borel funcions u(, ), T. A second approximaion involves replacing each u( j, X j ) wih he orhogonal projecion from L 2 (Ω) ono he H-dimensional vecor space generaed by a finie se of funcions aken from a suiable basis {e 1,..., e H,...}. For fixed H

6 6 A.R. Bacinello e al. / Journal of Compuaional and Applied Mahemaics ( ) and each j, we define such projecion by ũ( j, X j ) =. β j e(x j ), where e( ) =. (e 1 ( ),..., e H ( )) and β. j = (β 1 j,..., β H j ) is he opimal vecor obained by leas squares regression. In paricular, we simulae he sae variable process X over he ime grid T (or over a finer grid) and hen se β j = arg min M β j R H m=1 ( g m θ j+1 β j e(x m j )) 2, (3.2) where M denoes he number of simulaions and X m j, g m j denoe he simulaed values of X j and g j in he m-h simulaion, for m = 1,..., M. Convergence resuls for he LSMC algorihm are provided in [12], while numerical resuls for several choices of basis funcions are repored in [20,23]. We now show how o apply he LSMC mehod o he seup of Secion Algorihm Assume ha M simulaed pahs have been generaed for he reduced sae variables process X and a uni exponenial random variable. For each simulaion m (m = 1,..., M), we le ξ m denoe he simulaed value of he exponenial random variable and by (µ m ) T a simulaed pah of he sochasic inensiy given in (2.5). According o (2.4), he simulaed ime of deah is obained by seing τ m = min{ T : Γ m > ξ m }, where Γ m represens he (approximaed) value of he inegral 0 µm s ds. If he insured survives a mauriy T (he se above is empy), we se τ m = + by convenion. Of course, here and in he sequel, one can simulae random processes over a grid finer han T o improve he approximaions. For each T such ha τ m, we denoe by r m, K m and S m he simulaed values of he shor rae, sochasic volailiy and reference fund. We can hen compue he simulaed discoun facors, denoed by v m.,s = B m (Bm s ) 1 (wih < s and, s T) and he simulaed benefis payable on deah (F d,m ), survival (F s,m ) and surrender (F w,m ). The valuaion algorihm requires execuion of he following seps: sep 1. (Iniializaion) For m = 1,..., M, if τ m T se θ,m = τ m and Pθ m d,m,m = F θ,m, oherwise se θ,m = T and Pθ m sep 2. (Backward ieraion) For j = n 1, n 2,..., 1: (1) (Coninuaion values) Le I j = {1 m M: τ m > j } and, for each m I j, se C m j = Pθ m,m vm j,θ,m,m = F s,m θ,m. (2) (Regression) Regress he coninuaion values C j = (C m j ) m Ij agains (e(x m j )) m Ij o obain C m j = β j e(x m j ) for each m I j. If F w,m j > C m j hen se θ,m = j and P m j = F w,m j. sep 3. (Iniial value) Compue he single premium of he conrac by V 0 = 1 M M m=1 P m θ,m vm 0,θ,m. The inroducion of exogenous facors in he surrender decision, as oulined in Secion 2.5, can be done as follows. We essenially need o focus on ι =. (τ ν) raher han τ, and replace µ wih µ + φ. For example, he simulaed realizaion ι m can be compued by using: { } ι m = min T : (µ m s + φ m s )ds > ξ m. (3.3) 0 For he iniializaion in Sep 1, we need of course o disinguish beween deah and surrender in case ι m T, unless deah and surrender benefis coincide. This can be done by drawing a Bernoulli for he condiional deah even, wih probabiliy [ E Q e ] ι m 0 (µ s+φ s )ds µ ι m Q(τ = ι ι = ι m ) = [ E Q e ], (3.4) ι m 0 (µ s+φ s )ds (µ ι m + φ ι m) where we noe ha boh numeraor and denominaor can be compued in closed form in he affine seing, or approximaed numerically using he already available pahs µ m and φ m. Alernaively, one could draw wo uni exponenials ξτ m and ξν m and simulae τ and ν individually insead of using (3.3) and (3.4). The simples way o obain he iniial value of he surrender opion (raher han he enire conrac) is o compue he iniial value V 0 of he European version of he conrac, and hen recover he surrender opion price from he difference V 0 V 0. When V 0 is no available in closed form, we can compue i by execuing only Sep 1 and Sep 3 of our algorihm. The nex secion offers numerical resuls on he implemenaion of he above procedure.

7 A.R. Bacinello e al. / Journal of Compuaional and Applied Mahemaics ( ) 7 Fig. 1. Solid line: sample pah of µ. Dashed line: Weibull inensiy m. Table 1 Parameers used in he simulaion r K S µ BDS = 1.00 r 0 = 0.05 K 0 = 0.04 S 0 = µ 0 = m(0) FDS = 0.01 ζ r = 0.60 ζ K = 1.50 ρ SK = 0.70 ζ µ = 0.50 δ r = 0.05 δ K = 0.04 ρ Sr = 0.00 σ µ = 0.03 σ r = 0.03 σ K = 0.40 λ Y = 0.50 λ µ = 0.10 µ Y = 0.00 γ µ = 0.01 σ Y = 0.07 c 1 = c 2 = 8.30 x = 40 Table 2 Value of he conrac κ (%) κ w (%) (0.031) (0.033) (0.033) (0.031) (0.031) (0.033) (0.031) (0.031) (0.032) (0.031) (0.030) (0.030) 6 V (0.047) (0.045) (0.041) 4. Numerical examples We consider he single premium equiy-linked endowmen described in Secion 2.1. The reference insured is a male aged x = 40 a ime 0. The conrac has mauriy T = 15 years and provides erminal guaranees on survival, deah and surrender benefis, as defined in (2.1). We apply he valuaion algorihm wih polynomial basis funcions of order 3, yielding H = 34. To replace he American claim wih a Bermudan claim, we discreize he ime dimension by using a ime sep (in years) which we call Backward Discreizaion Sep (BDS). To simulae he reduced sae variable process X, we employ a ime grid finer han T and call Forward Discreizaion Sep (FDS) he lengh in years of each ime inerval in he finer grid. All he relevan parameers used for our simulaions are repored in Table 1. Wih regard o moraliy dynamics, we noe ha he funcion m in (2.5) is obained by fiing a Weibull inensiy, given by m() = c c 2 1 c 2 (x + ) c2 1 (wih c 1 > 0, c 2 > 1), o he survival probabiliies implied by able SIM2001, commonly used in he Ialian endowmens marke. A sample pah of µ is reproduced in Fig. 1. We have considered differen values for he minimum raes guaraneed upon boh deah and survival (κ =. κ d = κ s ), as well for he surrender guaranee (κ w ). Of course, he price of he European conrac does no depend on κ w. The case of k w k is wha one encouners in pracice, for oherwise conrac design would favor wihdrawals. We consider he case of k w > k for numerical purposes only. We ran simulaions wih 140 differen seeds o obain he resuls repored in Tables 2 and 3. The las row of Table 2 repors he value of he European conrac (V 0 ), while he oher rows repor he values

8 8 A.R. Bacinello e al. / Journal of Compuaional and Applied Mahemaics ( ) Fig. 2. Value of he American conrac. Table 3 Surrender opion value κ (%) κ w (%) Table 4 Surrender opion values and percenage changes of he higher moraliy model wih respec o he base case of Table 3 κ (%) κ w (%) (+0.47%) (+9.02%) (+46.77%) ( 1.53%) (+2.42%) (+4.35%) ( 3.24%) ( 1.77%) (+2.08%) ( 6.56%) ( 6.27%) ( 4.46%) 6 of he American conrac wih sandard errors in parenhesis. Finally, Table 3 repors he values for he surrender opion, while Fig. 2 depics he value of he American conrac agains boh κ and κ w. As expeced, he value of he European conrac is increasing wih he minimum ineres rae guaraneed upon deah or survival, κ. The value of he American conrac is increasing wih boh κ and he minimum ineres rae guaraneed upon surrender, κ w. The value of he surrender opion insead increases wih κ w and decreases wih κ. The opion value is negligible when κ is large relaive o κ w, because sepping ou of he conrac is hen less aracive. [7] propose an alernaive algorihm ha can be used as a benchmark for our case and indeed confirms he validiy of our resuls. To undersand he impac of moraliy on our resuls, we compue surrender opion values for a moraliy model yielding a 9.67% decrease in expeced lifeime (from o years). This is obained by raising insananeous volailiy o σ µ = 0.10 and mean jump size o γ µ = We repor he resuls in Table 4, ogeher wih he percenage changes wih respec o he base case of Table 3. We see ha surrender opions become more valuable in he higher moraliy case as long as κ κ w. To undersand why, we noe ha while higher moraliy involves a lower probabiliy of receiving paymens upon survival (i.e., surrender and mauriy benefis), a he same ime i can be shown o increase he overall value of our sandard endowmen conracs. As a resul, here are wo effecs a play when considering he opimal sopping problem (3.1): on he one hand, higher moraliy increases he argumen of he supremum, on he oher hand, i reduces he probabiliy of being able o exercise before deah. When κ < κ w, surrender guaranees are more valuable relaive o deah and survival

9 A.R. Bacinello e al. / Journal of Compuaional and Applied Mahemaics ( ) 9 guaranees, bu he lower probabiliy of exercising before deah generaes a decrease in opion values. When insead κ κ w, surrender benefis are less valuable relaive o deah and survival benefis, bu he increase in value for he sopped conrac (induced by higher moraliy) makes opion values increase wih respec o he base case. 5. Conclusions In his paper we have described an algorihm for he valuaion of a life insurance conrac embedding surrender opions based on he Leas Squares Mone Carlo mehod. The approach is effecive in ha we can avoid imposing rigid simplifying assumpions on he valuaion model. We have hen shown how o implemen he model in he conex of equiylinked endowmen conracs wih minimum guaranees on deah, surrender and survival benefis. We have considered differen sources of uncerainy, such as random ineres raes, sochasic volailiy and jumps in asse prices, as well as random moraliy raes. 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