Fair Valuation of Participating Policies in Stochastic Interest Rate Models: Two-dimensional Cox-Ross-Rubinstein Approaches

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1 Fair Valuaion of aricipaing olicies in Sochasic Ineres Rae Models: Two-dimensional Cox-Ross-Rubinsein Approaches Liao, Szu-Lang Deparmen of Money and anking, Naional Chengchi Universiy, Taipei, Taiwan, R.O.C. Chang, Chi-Kai Deparmen of Saisics, Feng Chia Universiy, Taichung, Taiwan, R.O.C. Lin, Shih-Kuei Deparmen of Finance, Naional Universiy of Kaohsiung, Kaohsiung, Taiwan, R.O.C. Absrac acinello (003a) employed CRR model o numerically calculae he fair value of a paricipaing policy conaining a surrender opion. acinello assumed a consan rae of reurn on risk-free asses. Our sudy proposes insead a wo-dimensional CRR model in a sochasic ineres rae model as a means of providing a numerical mehod for conrac pricing. The wo-dimensional CRR model achieves accuracy wih respec o Mone Carlo simulaion in some simplified cases and converges rapidly. Two-dimensional CRR models are used o analyze he imporance and sensiiviy of a sochasic ineres rae model for he policy. Zero coupon bond volailiy is an essenial parameer in he surrender opion, and reference porfolio volailiy is imporan for pricing he paricipaing opion. The paricipaing and surrender opions are more valuable given upwards rending ineres raes han consan or downwards rending raes. Keywords: American-ype opion, arbirage pricing heory, ermudan-ype opion, Cox-Ross-Rubinsein model, endowmen, European-ype opion, fair valuaion, paricipaing opion, surrender opion.

2 . Inroducion To expand heir range of life insurance producs, life insurance companies have recenly begun offering complex conracs wih embedded opions: paricipaing policies wih ineres rae guaranees and surrender opions; and equiy-linked conracs wih a guaraneed payoff. Each opion conribues o firm liabiliies. According o riys and de Varenne (997), numerous life insurance companies neglec he significance of hese opions, hus exposing hemselves o he risk of insolvency. Consequenly, fair valuaion of life insurance liabiliies recenly has aroused considerable aenion. Owing o he pioneering work by rennan and Schwarz (976, 979a, 979b) and oyle and Schwarz (977), he valuaion of such producs recenly has advanced markedly. aricipaing policies enable policyholders o receive dividends (bonuses) in addiion o he promised paymens implied by he guaraneed ineres rae. ecause of he exisence of he ineres rae guaranee in he paricipaing policies, he paricipaing mechanism resembles ha of European call opions. rosen and Jørgensen (000), Jensen, Jørgensen, and rosen (00), and rosen and Jørgensen (00) analyzed a paricipaing policy wih a minimum ineres rae guaranee. European bonus opions are associaed wih he percenage of he posiive performance of he firm asse porfolio and priced by Mone Carlo simulaion. Milersen and ersson (003) invesigaed uaraneed Invesmen Conracs, which resemble deposi accouns wih a guaraneed rae of reurn. Milersen and ersson (003) designed a muli-period exension and also provided closed-form formulae under he Heah, Jarrow, and Moron (99) framework. acinello (00) analyzed life insurance endowmen paricipaing policies wih a guaraneed minimum ineres rae. Using lack and Scholes (973) and Meron (973), acinello obained he closed-form formulae of such policies in erms of one-year call opions. The presence of a surrender opion in a conrac implies ha he policyholders can sell he conrac back o he issuer before mauriy. The problem of valuing surrender opions embedded in life insurance conracs has been sudied wihin he framework of consan risk-free ineres rae.

3 rosen and Jørgensen (997) analyzed he valuaion of American-ype conracs wih ineres rae guaranee using he opional sampling heorem. rosen and Jørgensen (000) priced surrender opions embedded in bonus policies via a binomial ree approach. Meanwhile, Jensen, Jørgensen, and rosen (00) assessed such surrender opions using a finie difference approach. acinello (003a) also used he model of Cox, Ross and Rubinsein (979) o deermine he fair value of he conrac. Moreover, acinello (003a) performed sensiiviy analysis for he conracual parameers. Addiionally, acinello (003b) finished calculaing he periodic premiums for his policy. Tanskanen and Lukkarinen (003) numerically solved he parial differenial equaion for he fair value of paricipaing conracs. Life insurance policyholders may surrender heir conracs o exploi he higher yields available in financial markes. Surrender opion is a concern for life insurance companies, paricularly during high ineres rae volailiy. If he guaraneed reurn is no sufficienly high compared o oher forms of invesmen, mainly when ineres raes rise, policyholders may erminae heir exising policies early and chase higher yields offered in capial markes. Therefore, his work considers he sochasic ineres raes. The surrender opion complicaes he valuaion owing o he problems of he pah dependency and he opimal sopping ime, especially in siuaions involving sochasic ineres raes. Longsaff and Schwarz (00) developed he leas squares Mone Carlo (LSM) approach for valuing and opimally exercising American-ype opions. Moreover, ernard e. al. (005) sudied he valuaion of life insurance paricipaing conracs as inroduced by rosen and Jørgensen (00) in a sochasic ineres rae environmen. Meanwhile, ernard e. al. (005) considered early firm defaul and approximaed defaul ime disribuion and derived he semi-closed formulae for he policy. The main aim of his sudy is o price he paricipaing policies inroduced by acinello (00) embedded wih surrender opions under a sochasic ineres rae model. This work proposes a wo-dimensional Cox-Ross-Rubinsein (CRR) model capable of efficienly calculaing he embedded surrender opion in he policy. 3

4 Two-dimensional CRR approaches are applied o analyze he imporance and sensiiviy of a sochasic ineres rae model for he policy. The res of his paper is organized as follows. Secion discusses he srucure of he paricipaing policy and a sochasic ineres rae model. Two-dimensional CRR models are hen developed o assess he conrac in Secion 3. Nex, Secion 4 analyzes he accuracy and convergence of he wo-dimensional CRR model, and analyzes he sensiiviy of he fair value o he volailiy parameers. Finally, conclusions and furher research direcions are oulined in Secion 5. 4

5 . Model A paricipaing policy is characerized by allowing policyholders o paricipae in he upside reurns of a reference porfolio. This secion describes a life insurance paricipaing policy in deail, including is benefi paymens, policy paricipaion mechanism (bonus mechanism) and surrender opion. Addiionally, he dynamic processes of a reference porfolio and he risk-free ineres rae are inroduced.. Conrac Srucure Consider an endowmen policy issued a ime 0 wih mauriy a ime T and iniial benefi C. The insurer pays he benefi if he insured dies before mauriy T or survives unil policy mauriy. In he case of a single premium, if he policy is in force a he end of he -h year, he reserve is adjused by a rae δ, defined as follows, η g ig δ = max,0, =,,..., T, () + i where he parameer η, beween 0 and, denoes he paricipaion coefficien, represens he annual reurn rae of he reference porfolio and i g is he guaraneed ineres rae. According o acinello (003a), Eqn. () reveals ha he annual ineres rae credied o he reserve during he -h year reaches is maximum beween η g and i, namely, g { ηg i } max, = ( + i )( + δ ), =,,..., T. g g g Under he paricipaing mechanism oulined by acinello (003a), a he end of each year, an annual dividend (bonus) is used o purchase an addiional paid-up endowmen policy wih mauriy dae T. Following acinello (003a), suppose he benefi, denoed by C, is paid a he end of he -h year where he insured dies during he -h year; oherwise, he benefi C T is paid a mauriy T. The dividend credied o he reserve implies a proporional adjusmen a rae δ for he benefi According o he prospecive mehod of equivalence principle, he reserve can be conceived as he presen value of he fuure financial obligaions discouned a a consan ineres rae, see owers e. al. (986). 5

6 C ; ha is, C = C ( + + δ), =,,..., T. () The adjused benefi a he end of he -h year can be calculaed inducively, as follows: C = C ( + δ ), =,,..., T. (3) + j= j The inclusion of a surrender opion in a conrac means ha he policyholders can sell he conrac back before mauriy. acinello (003a) assumes ha he surrender opion can be exercised a he sar of he policy year immediaely afer he announcemen of he renewal benefi. The surrender cash value is he payoff o he policyholder upon exercising he surrender opion. enerally, he surrender cash value equals he reserve minus a surrender charge. The surrender charge gradually decreases o zero afer he policy has persised for a specified number of years. For simpliciy, his work adops he surrender cash value of acinello (003a), R = kc A, =,,..., T. (4) + x+ T : The parameer k denoes a consan parameer of policy and x+ T : T ( h ) ( T ) ( g ) ( ) h x+ g T x+,,,..., T h=+ A = + i q + + i p =, (5) where x denoes he issuing age of he insured. ased on acuarial noaion, q x represens he probabiliy of he insured dying wihin he -h year, while p x is he probabiliy ha he insured remains alive a ime. This work assumes ha A x + T : is discouned a guaraneed ineres rae i. g 6

7 . Sochasic Models The paricipaing policy wih surrender opions described above is subjec o financial risk and moraliy risk. This work follows acinello (003a) and assumes hese wo risks o be independen. The price of he coningen claim can be calculaed as he discouned expeced value wih respec o he risk-neural moraliy and financial measures, respecively. This secion primarily inroduces he sochasic ineres model o describe he financial risk. In he risk-neural economy (Nielsen and Sandmann, 995), he dynamic processes of he zero coupon bond wih mauriy ' and he reference porfolio can be wrien as d(, ') * = r() d + (, ') dw (), (6) (, ') and ds() * * = r() d + dw () + dw (), (7) S () where r () is he risk-free ineres rae, and W * () and W * () are independen Wiener processes under he risk-neural probabiliy measure * wih naural F * * = W (), s W (): s s. { } filraion ( ) Le π ( ) X denoe he no-arbirage price a ime for a coningen claim * X which is seled a ime. Under he risk-neural measure, he value of he coningen claim equals r( s) ds π ( X) = E * e X F. The fair valuaion framework assumes a perfec financial marke wih coninuous rading, no resricions on borrowing or shor-sales, and in which he zero coupon bonds and he socks are infiniely divisible. This sudy implemens he fair valuaion under he risk-neural probabiliy measure. 7

8 To solve he inegral problem of join disribuion of he ineres rae { rs (): s } < < and he coningen claim X, he risk-neural measure * is changed ino anoher equivalen measure heorem, shown as follows,, he forward measure, using irsanov's ( dw ) ( * * (), s dw() s dw () s (,) s ds, dw () s ) =. (8) Under he forward measure, he value of he coningen claim is rewrien as ( ) π ( X) (, ) E X = F. (9) Applying I ô 's lemma from Eqns. (6) and (7), he dynamic process of he reference porfolio wih numeraire, he zero coupon bond wih mauriy under he -measure can be obained as S ( 0) S(0) = exp (( ( s, )) ) ds ( ( s, )) dw ( s) dw( s) ( 0, ) (0, ) (0) for 0 <. iven 0 = and 0 = in Eqn. (0), he dynamics of he relaive price in successive years, denoed by (), are as follows: (( (, )) ) s + ds Δ S () () = = exp. () S ( ) (, ) + ( (,) s) dw () s + dw() s From Eqn. (6), he dynamic process of a zero coupon bond (, ) under he forward measure is (0, ) (,) = exp (, s ) (,) s ds (, s ) (,) s dw() s (0, ) ( ) ( ) 0 0 () 8

9 Due o he long mauriy of life insurance policies and he limiaions on compuaional feasibiliy, his sudy assumes ( s, ) = ( s) (Nielsen and Sandmann, 995; Ho and Lee, 986), where is consan. 3 Therefore, Eqns. () and () are reduced o () = exp (( ( s)) + ) ds + ( ( s) ) dw ( s) + dw( s) (, ) (3) and (0, ) (, ) = exp ( ) + W ( ) (0, ). (4) The surrender opion complicaes he valuaion because of he problem of opimizing he sopping ime. The CRR model provides a powerful ool for dealing wih a pah-dependency opion such as he surrender opion. Therefore, based on he CRR model, he nex secion develops wo-dimensional CRR model for conrac pricing. 3. Fair Valuaion Using Two-dimensional CRR Model This secion firs devises wo-dimensional CRR models for he relaive price of a reference porfolio and a zero coupon bond price o fairly value he paricipaion conrac and early exercise of he surrender opion. A backwards algorihm is hen deveploed for conrac pricing. 3. Two-dimensional CRR Model A CRR (Cox, Ross & Rubinsein, 979) model wih a consan ineres rae provides a 3 Ho and Lee model adops he iniial ineres rae erm srucure as an inpu. The zero coupon bond prices are consisen wih hose observed in he marke and have he Markovian propery for he CRR binomial ree pricing approach. 9

10 simple and efficien numerical mehod for valuing opions in siuaions where premaure exercise may be opimal, for example American or ermudan opions. However, given a sochasic reference porfolio and zero coupon bond price, i becomes necessary o esablish a wo-dimensional CRR model for a join wo-dimensional log-normal disribuion of he relaive price of reference porfolio and he zero coupon bond price, for efficienly calculaing he fair value of he paricipaion policy wih a surrender opion. Therefore, his subsecion devises a wo-dimensional CRR model for he prices of a reference porfolio and a zero coupon bond o approximae he wo-dimensional log-normal densiy. The ree srucure of he wo-dimensional CRR model is: The roo of he ree is he bond price (0,). For =,,..., T, (see Fig. ) Sep:(The firs dimension) iven iniial bond price (, ), generae he nodes of relaive price (). Sep :(The second dimension) For each node of ( ) derived in Sep, generae nodes of bond price + (, ). Noably, under he Ho and Lee model, he join disribuion of bond price + (, ) and relaive price ( ) given informaion se F depends only on (, ). Therefore, he wo-dimensional CRR model follows a Markovian process. Tha is, given a curren value of bond price (, ), fuure developmen of he bond price + (, ) and relaive price ( ) are independen of pas movemen. Deails of proof are represened in Appendix A. The wo seps in he wo-dimensional CRR model are deailed as follows. In sep, from Eqn. (3), under he -measure, he dynamic process of relaive price ( ), given bond price (, ), is denoed by () = exp (( ( s)) + ) ds+ ( ( s) ) dw ( s) + dw() s (, ) d = exp + W () (, ) (5) 0

11 (0) () Q (0) () (0) + (, ) (, ) Q ( ) () i.... () () i Q (0) ( ) (, + ) () i Q ( j ) ( i ) (, + ) () ( ) + (, ) j Q ( n) ( i) (, + ) () Q ( ) () n ( ) + (, ) n ( ) () n Figure Two-dimensional CRR model for relaive price () and bond price + (, ) during ime o

12 where d = indicaes an equal disribuion, W is a new rownian Moion and = (( ( s)) + ) ds = (6) From he CRR model, his sudy divides each policy year ino n sub-period wih equal lengh, Δ= /n. Moreover, he drif erm equals 0. Furhermore, he up and down facors are U e Δ =, D =. (7) U Under he CRR model, he probabiliy of even { ( τ ) U ( τ )} +Δ = is given by Q = U D D, (8) whereas U Q = U D denoes he probabiliy of even { ( τ ) D ( τ )} +Δ =. Noably, given a drif erm r, o preven arbirage opporuniies, he volailiy parameer is fixed so ha rδ D < e < U, implying a sricly posiive value less han for boh Q and Q. rδ In sep, he drif erm equals 0, so he arbirage condiion, D < e < U, clearly holds. () i Le { ( ) : i 0,,..., n} = represen he possible values derived by () U D (, ) () i n i i =, i = 0,,..., n. (9) wih corresponding probabiliies

13 n Q Q Q i n i i ( i ) = ( ) (), i = 0,,..., n. (0) In sep, From Eqn. (), under he coupon bond price + (, ) is + -measure, he dynamic process of zero (0, + ) (, + ) = exp + W () (0, ) +. () y changing he measure, under he -measure, he bond price process + (, ) is (0, ) (, + ) = + exp + W () (0, ). () Deails of he compuaional procedure are presened in Appendix A. Dividing Eqn. () by Eqn. (4), yields he following resul, (0, ) (0, + ) ( ) (, + ) = (,) e exp dw( s) + (0, ). (3) Applying log-ransformaion o Eqns. (3) and (3), log + (, ) and log ( ) have a bivariae aussian disribuion. Using linear regression, he condiional disribuion of log + (, ) given log ( ) is a aussian disribuion wih condiional mean μ and condiional volailiy. Deails of he compuaion are lef in Appendix. Therefore, he condiional disribuion of log + (, ) given log ( ) is equivalen o he following process: d d log (, + ) = μ + ε = log α + r() + W (), (4) 3

14 where α depends on bond price (, ), r () depends on bond price (, ) and relaive price (), ε ~ N(0,), and W is a new rownian Moion. Deails of calculaion are lef in Appendix. To approximae he condiional disribuion of log + (, ) given log ( ), his work chooses α and r as he iniial value and drif erm, respecively, for he CRR model. Then, he up and down facors, for he firs dimension CRR model, are U e Δ =, D =. (5) U and he probabiliy of up movemen is given by Q e = U r ( ) Δ D D. (6) Significanly, ruling ou arbirage requires assuming ha r () saisfies he r ( ) no-arbirage condiion Δ D < e < U, so ha boh he risk-neural probabiliies, Q and Q, are sricly posiive and below. The no-arbirage condiion hen is obained by r () < Δ. Under he -measure and given bond price (, ) and relaive price (), he possible value of bond price + (, ) can be derived by + = U D, j = 0,,..., n, (7) ( ) (, ) j n α i i wih corresponding probabiliy n n j j Q ( j ) = Q ( ) (, ) () Q +, j = 0,,..., n. (8) j value Noably, his work possesses wo advanages. Firs, using Eqn. (7), he iniial α and he moving facors U and D only depend on bond price (, ), bu no on he relaive price (). Hence, he value of he zero coupon bonds + (, ) remains unchanged regardless of he relaive price (). However, 4

15 Eqns. (6) and (8) show ha he corresponding risk neural probabiliies differ wih he relaive price (). Second, he condiional volailiies remain consan for each ime period, 4 he ree srucure of he zero coupon bond price is recombined, and he node number increases only linearly wih he number of exercise daes. The proposed approach is less compuaionally inensive, hus making i appropriae for long erm policies. 3. Valuaion Framework This subsecion inroduces he fair valuaion of he policy and a recursive backwards algorihm based on he wo-dimensional CRR models described above. A he end of each year, he coninuaion value for he policyholder can be deermined by discouning fuure cash flows. The policyholder surrenders he conrac if coninuaion value is less han surrender value or keeps conrac alive unil he end of nex year. Hence, he opimal value for he policyholder is he maximum of coninuaion value and he surrender value. Le V and Y denoe he opimal and coninuaion values a ime, respecively. iven ime, he benefi is C + paid a ime + if he insured dies beween imes and ime +, oherwise, he conrac value for policyholders is V +. Consequenly, using Eqn. (9) and he independence of financial and moraliy risks, he coninuaion value Y a ime is presened as follows: Y = (, + ) q C + p E V ( ) x+ + x+ + p + F, = 0,,,..., T. (9) Specially, a ime T, if he policy remains in force, he policyholder is paid by C T a he end of he T -h year, regardless of wheher he insured dies during he T -h year or survives. Therefore, he conrac value a mauriy dae T equals C T. From Eqn. (9), he coninuaion value a ime T is Y = T CT( T, T). (30) 4 See Appendix. 5

16 Comparing he coninuaion value wih he surrender cash value, he opimal value V can be deermined by { } V = max Y, R, =,,..., T, (3) where R is calculaed from Eqn. (4). Repeaing he previous procedures such as Eqn. (9)-(3) from ime T o 0 recursively, he final coninuaion value Y 0 is he fair value of he conrac. The following lemma simplifies he calculaion of he fair value of conrac Y 0 and can be applied o he recursive backward algorihm for he wo-dimensional CRR model. Lemma Le S denoe he fair value of he policy wih basic benefi C and mauriy dae T. Define V =, and T { ( ) + + p + + : } V = ( + δ ) max (, + ) qx + px E V + F, ka,. (3) x T for = T back o. Then, ( ) S = C(0,) qx + pxe V p. (33) roof. Leing V = CV in Eqns. (9)-(3), Eqns (3) and (33) can be obained by inducion. ased on he wo-dimensional ree srucure and Lemma, he backward recursive procedure for he fair value of he conrac is as follows: For a wo-dimensional ree iniiaed from a node of bond price (, ) (see Fig. ), from Eqns. (3)-(33), he corresponding conrac values V can be calculaed for each node of bond price + (, ), ha is, 6

17 { ( ) : } V (, i j) ( () i )max (, ) ( j) (, ) ( j) = + δ + qx+ + px EQ V + + +, ka (34) x+ T and he expeced opimal conrac value is provided by V for he roo node, bond price (, ), ( ) ( n n E V (, ) = Q Q V, (35) i j i (, i j) Q ) ( ) ( ) () (, + ) () i= 0 j= 0 where E represens aking expecaion wih respec o he risk neural probabiliies Q such as hose in Eqns (0) and (8). Repeaing he previous procedure from ime T back o 0 recursively, he final value S for roo node (0,) is he fair conrac value. ( ) = C (0,) q + p E V S x x Q Noably, in Eqn. (34), his work derives he adjusmen rae δ via he relaive price ( ) raher han he reference porfolio price S, ( ) and hen doing so wihou reference o he price during he previous ime S ( ). Furhermore, if he coninuaion values are calculaed based on Eqns. (9)-(3) direcly raher han Lemma, i becomes necessary o calculae he benefis C or C T based on previous pah movemen which leads o he iniial node of bond price (, ). Conversely, Eqn. (34) demonsraes ha here are wo sochasic sources: adjusmen rae δ and bond price + (, ), which can be generaed from he bond price (, ) wihou reference o he previous pah. Therefore, his work devises a complee recursive algorihm for conrac pricing. 4. Numerical Analysis This secion firs verifies he accuracy of he wo-dimensional CRR model, and hen ess he ineres rae erm srucure and sensiiviy of he volailiy parameers. 7

18 (0) () ( δ ) (0) Q (0) () (, ) Q ( (, )) E V Q ( ) () i.... () () i () ( δ i ) Q (0) ( ) (, + ) () i Q ( j ) ( i ) (, + ) () (0) + (, ) (,0) i ( V ) ( ) + (, ) j (, ) ( V i j ) Q ( n) ( i) (, + ) () Q ( ) () n ( ) + (, ) n (, ) ( V in ) ( ) () n ( ) ( δ n ) Figure Cash flow in each node of a wo-dimensional CRR model during ime period o 8

19 4. Accuracy and Convergency To check he accuracy of he fair valuaion using he wo-dimensional CRR model, he fair values obained using he wo-dimensional CRR model are compared wih hose obained by Mone Carlo simulaion for a simple case wih T = 3. The simulaion procedure is as follows. From he fair valuaion framework described in subsecion 3., he fair value of he conrac is where ( ) Y0 = (0,) qxc+ pxe V p, { x x ( δ { : } + + p x+ ) x+ : } V = C( + δ) max (, ) q + p E ( + ) max (, 3), ka F, ka. (36) The expecaion value ( 0) E V p ) can be compued in wo differen ways: by Mone Carlo simulaion and by means of wo-dimensional CRR approximaion wih a variable number n of sub-inernal in each uniary period. Under he Mone Carlo approach, he simulaed value of V are obained by sampling M value of (,) and () from he exac join log-normal disribuion and using he exac closed-form expressions for E ( + δ ) max { (, 3), ka } F derived in Appendix C, ha is, x+ p EV F (or ( ) ( : ) M EV ˆ( ) = V., m M m = As a resul, he fair value of he conrac can be calculaed by ( ) Y ˆ 0 (0,) qxc+ pxe V. This sudy considers hree differen ineres rae erm srucure scenarios. Scenario I shows a normal erm srucure wih an upwards sloping yield curve, 9

20 Scenario II involves a erm srucure wih consan yield rae, and Scenario III describes a reverse erm srucure wih a downwards sloping yield curve. The hree erm srucures are as follows: () Scenario I (Normal erm srucure): () Scenario II (Fla erm srucure): (0, ) = (.035) (3) Scenario III (Reverse erm srucure): 0.8 (0, ) = ( ) e 0.8 (0, ) = ( ) e All hree ineres rae erm srucures share he same one-year spo ineres rae of The moraliy probabiliy daa used in his sudy was exraced from he 00 Taiwan Sandard Ordinary Experience Moraliy Table (995~999). Suppose he volailiy parameers are fixed using = 8%, = 0%, and = 5%, implying ha insananeous covariance relaionship beween wo underlying asses is dsd(, ') = S ( ' ) d. For each scenario, he fair conrac value obained using he wo-dimensional CRR model is compared wih ha from Mone Carlo simulaion for 00,000 pahs in Table. When n 30, he difference in he resuls beween he wo-dimensional CRR model and Mone Carlo simulaion is less han one dollar (basic benefi amoun C = 000 ). From Figs. 3, 4, and 5, he fair conrac value converges given large subinerval number in Scenarios I, II, and III, respecively. Consequenly, his sudy seleced n = 30 hroughou he remainder of his secion. Table compares he resuls obained from he wo-dimensional CRR model wih hose from Mone Carlo simulaion using 00,000 pahs under hree differen scenarios and various volailiies parameers. The able shows ha he difference beween he wo-dimensional CRR mehod and Mone Carlo simulaion is less han one dollar (basic benefi amoun C = 000 ). 0

21 Table The difference of Mone Carlo simulaion and wo-dimensional CRR models wih he subinerval number n under T = 3 n Scenerio I Scenerio II Scenerio III Simulaion a. Three ineres rae erm srucure Scenerio I, II, and III, number of subinerval n. b. x = 40, C = 000, i g = 0.0, η = 0.5, k = 0.985, = 8%, = 0%, = 5%. c. Mone Carlo simulaion pahs M = d. The sandard deviaion of simulaion is abou % o 6% of basic benefi C. To calculae he fair values of he paricipaing and surrender opions embedded in he paricipaing policy, his work uses some noaions for he fair values of he paricipaing and surrender opions. 0,, and S denoe he fair values of he basic conrac, he non-surrenderable paricipaing policy, and he surrenderable paricipaing policy, respecively. Furhermore, O represens he fair value of he paricipaing opion, which is he difference beween he fair value of he non-surrenderable paricipaing policy and ha of he basic conrac; ha is, =. O 0 The fair value of he surrender opion S embedded in he surrenderable paricipaing conrac is represened by S = S. Moreover, he fair value of he basic conrac is T 0 = C (0, ) q (0, ) x + T T p x. (37) =

22 Figure 3 Converge of he fair value S in Scenario I Figure 4 Converge of he fair value S in Scenario II Figure 5 Converge of he fair value S in Scenario III

23 Table The difference of he fair value S using Mone Carlo simulaion and wo-dimensional CRR model for he volailiy parameers,, and Scenerio I Scenerio II Scenerio III CRR Simulaion CRR Simulaion CRR Simulaion a. ond price volailiy, relaive price volailiy and. b. x = 40, C = 000, i g = 0.0, η = 0.5, k = c. Mone Carlo simulaion pahs M = d. The sandard deviaion of simulaion is abou % o % for basic benefi C. Consider he benefis associaed wih he non-surrenderable paricipaing policy. Using Eqn. (9), he fair value of he non-surrenderable paricipaing policy can be deermined as follows: T = (0, ) E ( C ) q + (0, T) E ( C ) p. (38) T p x T T x = The expeced value of benefi C can be approximaed using he Mone Carlo simulaion mehod, as described by rosen and Jørgensen (000). Inuiively, if he surrender cash value is zero ( k = 0 ), he fair value of he non-surrenderable policy equals ha of he whole conrac S. Hence, wo mehods are used o calculae : a Mone Carlo simulaion mehod he fair value of he non-surrenderable policy and a ree model. Table 3 reveals a minimal difference in fair value beween he Mone Carlo simulaion and he wo-dimensional CRR model. Subsequenly, he wo-dimensional CRR models are adoped for pricing he non-surrenderable paricipaing policy. 3

24 Table 3 The difference of he fair value using Mone Carlo simulaion and wo-dimensional CRR model for he volailiy parameers,, and Scenerio I Scenerio II Scenerio III CRR Simulaion CRR Simulaion CRR Simulaion a. ond price volailiy, relaive price volailiy and. b. x = 40, C = 000, i g = 0.0, η = 0.5, k = 0. c. Mone Carlo simulaion pahs M = d. The sandard deviaion of simulaion is abou % o % for basic benefi C. Once he accuracy of he CRR approach wih respec o Mone Carlo simulaion has been checked in he case of T = 3 and judged saisfacory, his approach is now used o value conracs wih longer mauriy and o obain some comparaive resuls. Table 3 shows ha he fair value of he conrac increases wih mauriy dae. Addiionally, when he mauriy increases by year, he CU-ime of wo-dimensional CRR model increases by approximaely 3 seconds. 4. Sensiiviy Analysis The influence of conracual values is analyzed for he siuaion in which one volailiy parameer increases by % while he ohers remain fixed in all of he previously described scenarios. Tables 5-7 lis he resuls of he sensiiviy analysis, specifically he resuls for conracs for which x = 40, C = 000, i g = 0.0, η = 0.5, k = 0.985, n = 30, T = 5, = 8%, = 0%, and = 5% in all scenarios of he erm srucure. 4

25 Table 4 Compuaional Time for differen mauriy dae T T S CU ime (seconds) a. mauriy dae T, fair value of conrac S b. x = 40, C = 000, i g = 0.0, η = 0.5, k = 0.985, = 8%, = 0%, = 5%. The resuls lised in Table 5 are derived by seing he zero coupon bond volailiy beween % and 0%. Table 5 shows ha he fair value of he conrac is highly sensiive and increases monoonically wih zero coupon bond volailiy. When he zero coupon bond volailiy increases by %, he fair value of he whole conrac increases by approximaely 0 o 3 dollars (he basic benefi C = 000 ). Also sensiive, he fair value of he surrender opion monoonically increases wih he zero coupon bond volailiy, and increases from around 9 o 6 dollars (he basic benefi C = 000 ) for every % increase in he volailiy of a zero coupon bond. However, he fair value of he non-surrenderable paricipaing conrac and he paricipaing opion appears relaively insensiive o he zero coupon bond volailiy, because he zero coupon bond volailiy does no influence he fair value of he non-surrnderable paricipaing conrac and ha of he paricipaing opion, as shown in Eqn. (38). As expeced, Tables 6 and 7 demonsrae ha European-ype paricipaing opions are exremely sensiive o reference porfolio volailiy. Noably, he araciveness of he paricipaing opion increases wih reference porfolio volailiy. Meanwhile, he fair value of he surrender opion decreases wih increasing reference porfolio volailiy. Accordingly, since policyholders are inclined o keep heir policies alive o ake advanage of he paricipaing opion, he value of he surrender opions decreases comparaively. 5

26 Tables 5-7 show ha he paricipaing opion and he surrender opion in Scenario I are more valuable han hose in Scenario II and III. ecause he erm srucures in Scenarios I, II and III are respecively upwards sloping, consan, and downwards sloping, he fuure ineres rae in Scenario I is larger han in Scenarios II and III. Meanwhile, he resuls of surrender opion are he same wih hose of paricipaing opion. 6

27 Table 5 Sensiiviy of he zero coupon bond volailiy Scenerio I 0 O S S Scenerio II 0 O Scenerio III 0 O Relaive price volailiy = 0% and = 5% are fixed. S S S S 7

28 Table 6 Sensiiviy of he zero coupon bond volailiy Scenerio I 0 O Scenerio II 0 O Scenerio III 0 O ond price volailiy = 8% and relaive price volailiy = 5% are fixed. S S S S S S 8

29 Table 7 Sensiiviy of he zero coupon bond volailiy Scenerio I 0 O Scenerio II 0 O Scenerio III 0 O ond price volailiy = 8% and relaive price volailiy = 0% are fixed S S S S S S 9

30 5. Conclusion acinello (003a) employed CRR model o calculae he fair value of a paricipaing insurance policy. acinello assumed a consan risk-free ineres rae. The main conribuion of his sudy is o price he paricipaing policy under sochasic ineres rae. This work proposes wo-dimensional CRR models capable of efficienly pricing he embedded surrender (American-ype) opion for a long-erm policy. Under cerain volailiy parameer resricions, a no-arbirage condiion is mainained for he wo-dimensional CRR model. The numerical resuls of wo-dimensional CRR models are very close o hose of Mone Carlo simulaion under he simplified case, T = 3. The wo-dimensional CRR models can be exended o he paricipaing insurance policies wih longer mauriy dae. The wo-dimensional CRR models have an exremely rapid convergence speed. Zero coupon bond volailiy is an essenial parameer in he surrender opion, and reference porfolio volailiies are imporan for pricing he paricipaing opion. The paricipaing and surrender opions are more valuable when he ineres rae is rending upwards han when i is consan or rending downwards. Noably, he wo-dimensional CRR model and he resuls i yields can provide suggesions for insurance companies regarding he issue of paricipaing insurance policies. Fuure sudies are sill required regarding wo-dimensional CRR models. Firs, he models can be applied o he periodic-premium case described by acinello (003b). u i is necessary o solve a numerical equaion based on an acuarial equivalen principle. Second, wo-dimensional CRR models incorporaing he relaive price of a reference porfolio and a zero coupon bond price are also used for he non-linear volailiy assumpion, for example he Vasicek model (977), bu hese models exclude he he Markovian propery of zero coupon bond prices and he recombinaion of ree srucure. However, i encouners he difficuly of ime consuming. Finally, he proposed model can also be exended for pricing financial conracs embedding surrender or ermudan-ype opions ha could be exercised on specified daes. 30

31 References Albizzai, M.-O., and eman, H. (994). Ineres rae managemen and valuaion of he surrender opion in life insurance policies. Journal of Risk and Insurance, 6, acinello, A. R. (00). Fair pricing of life insurance paricipaing policies wih a minimum ineres rae guaraneed. Asin ullein, 3, acinello, A. R. (003a). Fair valuaion of a guaraneed life insurance paricipaing conrac embedding a surrender opion. Journal of Risk and Insurance, 70, acinello, A. R. (003b). ricing guaraneed life insurance paricipaing policies wih annual premiums and surrender opion. Norh American Acuarial Journal, 7, -7. ernard, C., Le Courois, O., and Quiard-inon, F. (005). Marke value of life insurance conracs under sochasic ineres raes and defaul risk. Insurance: Mahemaics and Economics, 36, lack, F., and Scholes, M. (973). The pricing of opions and corporae liabiliies. Journal of oliical Economy, 8, owers, N. L., erber, H. U., Hickman, J. C., Jones, D. A., and Nesbi, C. J. (986). Acuarial mahemaics. Sociey of Acuaries. oyle,.., and Schwarz, E. S. (977). Equilibrium prices of guaranees under equiy-linked conracs. Journal Risk and Insurance, 44, rennan, M. J., and Schwarz, E. S. (976). The pricing of equiy-linked life insurance policies wih an asse value uaranee. Journal of Financial Economics, 3, rennan, M. J., and Schwarz, E. S. (979a). Alernaive invesmen sraegies for he issues of equiy-linked life insurance policies wih an asse value guaranee. Journal of 3

32 usiness, 5, rennan, M. J., and Schwarz, E. S. (979b). ricing and Invesmen Sraegies for Equiy-Linked Life Insurance olicies," (hiladelpha: The S.S. Huebner Foundaion for Insurance Educaion, Wharon School, Universiy of ennsylvania). riys, E., and de Varenne, F. (997). On he risk of life insurance liabiliies: debunking some common pifalls. Journal of Risk and Insurance, 64, Cox, J. C., Ross, S. A., and Rubinsein, M. (979). Opion pricing: a simplified approach. Journal of Financial Economics, 7, rosen, A., and Jørgensen,. (997). Valuaion of early exercisable ineres rae guaranees. Journal of Risk and Insurance, 64, rosen, A., and Jørgensen,. (000). Fair valuaion of life insurance liabiliies: he impac of ineres rae guaranees, surrender opions, and bonus policies. Insurance: Mahemaics and Economics, 6, rosen, A., and Jørgensen,. (00). Life insurance liabiliies a marke value: an Analysis of insolvency risk, bonus policy, and regulaory inervenion rules in a barrier opion framework. Journal of Risk and Insurance, 69, Heah, D. C., Jarrow, R. A., and Moron, A. (99). ond pricing and he erm srucure of ineres raes: A new mehodology for coningen claim valuaion. Economerica, 60, Ho, T. S. Y. and Lee, S. (986). Term srucure movemens and pricing ineres rae coningen claims. Journal of Finance, 4, Jensen,., Jørgensen,., and rosen, A. (00). A finie difference approach o he valuaion of pah dependen life insurance liabiliies. The eneva apers on Risk and 3

33 Insurance Theory, 6, Longsaff, F. A. and Schwarz, E. S. (00). Valuing American opions by simulaion: a simple leas-square approach. The Review of Financial Sudies, 4, Meron, R. C. (973). Theory of Raional Opion ricing. ell Journal of Economics and Managemen Science, 4, Milersen, K., and ersson, S. (003). uaraneed invesmen conracs: disribued and undisribued excess reurn. Scandinavian Acuarial Journal, 4, Nielsen, J. A., and Sandmann, K. (995). Equiy-linked life insurance: a model wih sochasic ineres raes. Insurance: Mahemaics and Economics, 6, Tanskanen, A., and Lukkarinen, J. (003). Fair valuaion of pah-dependen paricipaing life insurance conracs. Insurance: Mahemaics and Economics, 33, Vasicek, O. (977). An equilibrium characerizaion of erm srucure. Journal of Financial Economics, 5,

34 Appendix Appendix A. (i) Changing of measure from + -measure o -measure: Since he pricing for coningen claim is performed under -measure, from Eqn. (8), for s<, and dw () s = dw () s ( + s) ds, + * dw () s = dw () s ( s) ds. * The following equaion can be obained by subracing he second equaions from he firs one; ha is, dw () s = dw () s ds. + From Eqn. (), under he -measure, he bond price process + (, ) is (0, ) (, + ) = + exp + W () (0, ). (ii) Markovian ropery Le L ( X ) denoe he condiional disribuion of random variable X given { } informaion se ( (), F (): = W s W s ) s condiional disribuion of ( log ( ), log (, ) ). From Eqns. (5) and (3), he + given informaion se F is L log (, ) log ( ), = N log (, + ) (0, ) (0, + ) log (, ) ( ) (0, ) (A.) 34

35 where Cov( log ( ),log (, + ) ) = ( ( s)) ds = ( ). Noably, he condiional disribuion of ( log ( ), log + (, ) ) given informaion se F depends on bond price (, ) which is F -measurable. Consequenly, he dynamics of bond price + (, ) and relaive price () are Markovian sochasic processes Appendix. The parameers for CRR model From Eqn. (A.), under -measure, he condiional disribuion of bond price + (, ) given informaion se F (or bond price (, ) ) and relaive price () is where d L ( log (, + ) ( )) = N ( μ, ) = log α + r( ) + W () ( log ( ),log (, + ) ) Var ( log ( ) ) Cov μ = E( log (, + ) ) + log ( ) E( log ( ) ) ( ) ( ) (0, + ) = log (, ) ( ) log ( ) log (, ) (0, ) Here, (0, + ) (0, ) α = (, ), (.) (0, ) ( ) r = , () ( ) (log ( ) log (, ) ) (.) 35

36 and = ( ρ ), (.3) ρ =. (.4) Appendix C Mone Carlo simulaion for T = 3 This Appendix shows he compuaion of (( + δ ) max { (, 3), x+ : } F ) E ka p which is used in Mone Carlo simulaion mehod for T = 3. The above expecaion value can be divided ino four pars; ha is, p (( + δ) max { (, 3), : } F x+ ) { } F E ka p ( max (,3), ) ( δmax : { (,3), : } F x+ p x+ ) = E ka + E ka + + (( (,3) ) F ) ( δ F ) ( δ ( (,3) ) F ) = A + E A + A E + E A p p p, where A = ka x +. Lemmas, 3, and 4 are applied o he second, hird and fourh : erms, respecively. Le Φ denoe he cumulaive disribuion funcion of sandard gaussian disribuion. Lemma Consider T = 3, given (, ), le A = ka +, under : x -measure, ( ) + (0,) (0,3) E ( (,3) A) F = (,) e Φ( d ) AΦ( d), p (0,) where d (0,) (0,3) log (, ) log A (0,) = +, and d = d. 36

37 roof. From Eqn. (3), bond price (,3) given (, ) under -measure is (0,) (0,3) (,3) = (,) e exp + dw ( s) (0,). Applying he lack-scholes mehod o calculae he call opion price of (,3) wih srike price A. This complees he proof of he lemma. Lemma 3 Consider T = 3, given (, ), under -measure, η ig E ( δ F ) = ( d) ( d) p Φ + Φ, + ig (,) η ig log (, ) log + η where d = +, and d = d. roof. From Eqn. (3), he process () given (, ) under -measure is () = exp (( ( s)) ) ds ( ( s) ) dw ( s) dw ( s) (, ) From Eqn. (.), + η ig E ( δ F) = E () ( ) p p + F. + ig η Taking he -expecaion of () ( + ) η lemma. i g +, his complees he proof of he 37

38 Le now Φ ( x, x; ρ) denoe he cumulaive disribuion funcion of wo-dimensional aussian disribuion wih correlaion coefficien ρ. Lemma 4 Consider T = 3, given (, ), le A = ka x +, under : -measure, i + η i+ j i j (0,) (0,3) i ig i= 0 j= 0 (0,) ig E ( δ ( (,3) A ) F ) = ( ) (,) A + p + η, i + ij( ) e Φ( x, x ; ρ) i, j i, j j where (0,) (0,3) log (, ) log A i, j (0,) x = + j ( i), and ig log (, ) log + i, j η x = + + i ( j). equals Eqn. (6) and ρ equals Eqn. (E.4). + roof. From Eqn. (), he condiional expeced value E ( δ( (,3) A) F p ) can be divided ino four pars as follows: 38

39 + η i + g + E ( δ( (,3) A) F) = E () ( ) ( (,3) ) p p + A F + ig η ig E () (,3) IA E (,3) I p F + p A η η F, (C. ) = + ig ig AE () IA + + AE I p F p A η F where A= () > ( + ) { (,3) > A } i g η. (i) The firs erm in Eqn. (C.) is derived as follows. y Eqns. (3) and (3), () (,3) (0,) (0,3) + (( ( s)) ) = e exp (0,) + dw () s + ( ( s) ) dw () s + dw () s ds Leing ( ( ) ) dw = dw s + ds, (C.) dw = dw ds where W and W are wo independen sandard Wiener processes under Q -measure. dq d = exp ( ) ( ) + + ( s ) ds ( s ) dw dw ( ( s) ) ds ( ( s) ) ds + = exp + ( ( s) + ) dw + dw 39

40 y changing of measure, (0,) (0,3) ( ) dq (0,) (0,3) + ( ) Q E () (,3) IA e E e I A e ( A ) p F = = p F F (0,) d (0,) From Eqn. (C.), (0,) (0,3) (, ) e exp + dw ( s) A (0,) > Q Q ( A F ) = (( ( s)) + ) ds i > + (, ) g exp ( ) η + ( ( s) ) dw ( s) + dw ( s) F = Q (0,) (0,3) (, ) e exp + + dw ( s) A (0,) > (( ( s)) ) ds + ig exp ( ) > + (, ) η + ( ( s) ) dw ( s) + dw ( s) F =Φ( x, x ; ρ),, where (0,) (0,3) log (, ) log A, (0,) x = +, and x, ig log (, ) log + η = + +, 40

41 ρ = = ( ( s) ) ds. (ii) The second erm in Eqn. (C.) is derived as follows. y Eqn. (3), (0,) (0,3) (,3) = (,) e exp + dw ( s) (0,). Leing dw = dw ds, (C.3) dw = dw where W and W are wo independen sandard Wiener processes under Q -measure. dq d = exp + ds dw. y changing of measure, (0,) (0,3) dq (0,) (0,3) Q E (,3) IA (,) e E I A (,) e ( A ) p F = = p F F (0,) d (0,) From Eqn. (C.3), 4

42 (0,) (0,3) (, ) e exp + dw ( s) A (0,) > Q Q ( A F ) = F (( ( s)) ) ds + ig exp ( ) > + (, ) η + ( ( s) ) dw ( s) + dw ( s) (0,) (0,3) (, ) e exp + dw ( s) A (0,) > F + (( ( s)) ) ds + ig exp ( ) > + (, ) η + ( ( s) ) dw ( s) + dw ( s) Q = =Φ( x, x,0 ; ρ),0 where x,0 (0,) (0,3) log (, ) log A (0,) =, and x,0 ig log (, ) log + η = + +, ρ = = ( ( s) ) ds. 4

43 (iii) The hird erm in Eqn. (C.) is derived as follows. y Eqn. (3), () = exp (( ( s)) ) ds ( ( s) ) dw ( s) dw ( s) (, ) Leing ( ( )) dw = dw s ds, (C.4) dw = dw ds where W and W are wo independen sandard Wiener processes under Q -measure. dq d = exp ( ) + + ( ) + ( s ) ds ( s ) dw dw y changing of measure, dq Q E () IA E I A ( A ) p F = p F = F (, ) d (, ) From Eqn. (C.4), (0,) (0,3) (, ) e exp + dw ( s) A (0,) > Q Q ( A F ) = F (( ( s)) ) ds + ig exp ( ) > + (, ) η + ( ( s) ) dw () s + dw () s 43

44 (, ) e exp dw ( s) A (( ( )) ) s + ds ig exp ( ) > + (, ) η + ( ( s) ) dw ( s) + dw ( s) (0,) (0,3) + + > (0,) Q = F 0, 0, =Φ( x, x ; ρ) where (0,) (0,3) log (, ) log A 0, (0,) x = +, and x 0, ig log (, ) log + η = +, ρ = = ( ( s) ) ds. (iv) The forh erm in Eqn. (C.) is derived as follows. (0,) (0,3) (, ) e exp + dw ( s) A (0,) > Q Q ( A F ) = F (( ( s)) ) ds + ig exp ( ) > + (, ) η + ( ( s) ) dw () s + dw () s =Φ( x, x ; ρ) 0,0 0,0 44

45 where (0,) (0,3) log (, ) log A 0,0 (0,) x =, and x 0,0 ig log (, ) log + η = +, ρ = = ( ( s) ) ds. Combining he formulas of (i)-(iv), hen, his complees he proof of he lemma. 45

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