Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

Transcription

1 Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear Waer Bay, Hong Kong, China Absrac We consider he pricing of variable annuiies wih he Guaraneed Minimum Wihdrawal Benefi (GMWB under he Vasicek sochasic ineres raes framework. he holder of he variable annuiy conrac pays an iniial purchase paymen o he insurance company, which is hen invesed in a porfolio of risky asses. Under he GMWB, he holder can wihdraw a specified amoun periodically over he erm of he conrac such ha he reurn of he enire iniial invesmen is guaraneed, regardless of he marke performance of he underlying asse porfolio. he invesors have he equiy paricipaion in he reference invesmen porfolio wih proecion on he downside risk. he guaranee is financed by paying annual proporional fees. Under he assumpion of deerminisic wihdrawal raes, we develop he pricing formulaion of he value funcion of a variable annuiy wih he GMWB. In paricular, we derive he analyic approximaion soluions o he fair value of he GMWB under boh equiy and ineres rae risks, obaining boh he lower and upper bound on he price funcions. he pricing behavior of he embedded GMWB under various model parameer values is also examined. 1 Curren address: Deparmen of Saisics, Universiy of Wisconsin, Madison, Wisconsin, USA 2 Curren address: Deparmen of Sysems Engineering and Engineering Managemen, Chinese Universiy of Hong Kong, Hong Kong, China 3 Correspondence auhor; maykwok@us.hk 1

2 1 Inroducion A variable annuiy is a conrac beween a policyholder and an insurance company. A iniiaion of he conrac, he policyholder makes a lump-sum purchase paymen. In reurn, he insurer agrees o make periodic paymens o he policyholder ha sar immediaely or a some fuure dae. he policyholder can choose o inves he purchase paymen in a range of muual funds, hus he policyholder has he equiy paricipaion in a porfolio of risky asses. In oher words, he policyholder is exposed o he equiy risk of he reference invesmen porfolio. he value of he personal sub-accoun of he policyholder in a variable annuiy will depend on he performance of he reference invesmen porfolio. Various forms of guaranees are commonly embedded in variable annuiies. In recen years, variable annuiies wih he guaraneed minimum wihdrawal benefi (GMWB have araced significan aenion and sales. he GMWB allows he policyholders o wihdraw funds on an annual or semi-annual basis, and promises o reurn he enire iniial purchase paymen over he life of he policy. hus he guaranee can be viewed as an insurance opion. he provision of his opion is financed by he proporional fees paid o he insurer by he policyholder. he personal sub-accoun will be depleed by hese periodic parial wihdrawals and proporional insurance fees. he curren charges for his benefi ypically range from 35 o 75 bps per annum. he guaranee kicks in when he personal sub-accoun falls o zero prior o he policy mauriy dae. Under he clause of he benefi graned, he insurer coninues o provide he guaraneed wihdrawal amoun unil he enire original premium is paid ou. When he underlying invesmen porfolio performs well so ha he personal sub-accoun says posiive a mauriy, he whole remaining balance in he personal sub-accoun is paid o he policyholder a mauriy. In a ypical guaranee, wihdrawals are aken as a fixed percenage of he premium (say, 5% per annum unil he premium is exhaused. his is called he saic wihdrawal policy. In a more complicaed design of he GMWB ha allows dynamic wihdrawals, he policyholder may wihdraw a a higher rae or exercise complee wihdrawal prior o mauriy (usually wih penaly charges. he pricing models of variable annuiies embedded wih he GMWB have been sudied in several earlier papers. Milevsky and Salisbury (26 propose he pricing formulaions of GMWB wih saic and dynamic wihdrawals under consan ineres rae. hey analyze he fair proporional fees ha should be charged on he provision of he guaranee. 2

3 Dai, Kwok and Zong (28 develop a singular sochasic conrol model for pricing GMWB under dynamic wihdrawal. An efficien finie difference algorihm using he penaly approximaion approach is also proposed for solving he singular sochasic conrol model. Chen, Vezal and Forsyh (28 explore he effecs of various modeling assumpions on he opimal wihdrawal sraegy of he policyholder, and examine he impac on he guaranee value under sub-opimal wihdrawal behavior. Effecive numerical schemes for pricing various ypes of guaraneed minimum benefis in variable annuiies using he impulse conrol formulaion are also proposed by Chen and Forsyh (28. Bauer, Kling and Russ (28 adop a generalizaion of a finie mesh discreizaion approach in Mone Carlo mehod o price GMWB in variable annuiies under he opimal policyholder behavior. In all of hese papers, he guaranees are priced under he assumpion of consan ineres rae. Since variable annuiies are long-erm conracs, he assumpion of consan ineres rae becomes unrealisic in pricing. More reliable pricing models of variable annuiies should allow for sochasic ineres raes. Lin and an (23 and Kijima and Wong (27 consider he pricing of equiyindexed annuiies under sochasic ineres raes. In his paper, we consider he model formulaion of he variable annuiies embedded wih GMWB under saic wihdrawal and subjec o boh equiy and ineres rae risks. In paricular, we examine he various forms of decomposiion of he value of he GMWB. For numerical valuaion of he guaranee, we show how o obain he analyic approximaion soluions o he fair value of he GMWB by deriving boh he lower and upper bound on he value funcion of he variable annuiies conrac. he paper is organized as follows. In he nex secion, we propose he pricing formulaion of variable annuiies wih GMWB subjec o equiy and ineres rae risks. We illusrae how o decompose he value of he GMWB ino a cerain-erm annuiy and a pu opion on some pah dependen funcion of he value of he personal sub-accoun. Also, we show how he GMWB value is relaed o he insurer s liabiliies and iniial premium. Since he erminal payoff of he GMWB exhibis pah dependence of he value process of he sub-accoun due o saic wihdrawals, he pricing model does no admi a closed form soluion. In Secion 3, we apply Roger-Shi s mehod and hompson s mehod o deduce he lower and upper bound on he value of he GMWB, respecively. In Secion 4, we repor he numerical ess ha were performed for checking he accuracy of hese numerical bounds. We also examine he pricing behavior of he GMWB under various model parameer 3

4 values. Conclusive remarks are presened in he las secion. 2 Formulaions of he value funcion he pricing model of a variable annuiy conrac wih he GMWB under consan ineres rae has been formulaed by Milevsky and Salisbury (26. We exend he pricing formulaion of he GMWB under boh equiy and ineres rae risks. Assumpions on he underlying price process of he reference risky porfolio and he financial marke condiions in our coninuous pricing models are summarized as follows: he process of holder s wihdrawal and he paymen sream of proporional (insurance fees o he insurance company are assumed o be deerminisic and coninuous in ime. he financial marke is complee and free of arbirage opporuniies. here are no ransacion coss and no resricion on shor selling. he value process of he underlying reference porfolio of risky asses follows he Geomeric Brownian process wih deerminisic volailiy. he sochasic ineres rae process is characerized by he Vasicek shor rae model. Le S denoe he fund value process of he reference porfolio of risky asses underlying he variable annuiy policy before he deducion of he proporional fees. We assume he exisence of a risk neural probabiliy measure Q such ha all discouned asse price processes are Q-maringales. Under he risk neural measure Q, he join dynamics of he fund value process S and he shor rae process r is governed by ds = r S d + 1 ρ 2 σ S S db 1, + ρσ S S db 2, dr = k(θ r d + σ r db 2,, (2.1 where B 1, and B 2, are independen sandard Q-Brownian processes, ρ is he insananeous correlaion coefficien beween he sochasic processes S and r, k and θ are consan parameers in he Vasicek model, σ S and σ r are consan volailiy values of S and r, respecively. I is well known ha he discoun bond price funcion under he Vasicek model is given by D(, = a(, e b(,r, (2.2 4

5 where b(, = 1 [ ] 1 e k(, k (( a(, = exp θ σ2 r [(b(, ( ] σ2 r b(, 2k k We may wrie he join dynamics of S and D(, as follows: where B = ( B1, B 2, ds S = r d + σ S db, dd(, D(, = r d + σ D (db, (2.3, σ S = ( 1 ρ 2 σ S ρσ S and σ D = ( σ r b(,. Le F = {F : } be he filraion generaed by he Brownian processes B 1, and B 2,. Under our coninuous model framework, he holder s wihdrawal process and he paymen sream of he proporional insurance fees are assumed o be deerminisic and coninuous in ime. Le W denoe he value process of he personal annuiy sub-accoun, which is depleed by he coninuous saic wihdrawal a he deerminisic rae G and he paymen of he coninuous sream of insurance fees a he consan proporional rae α. Le w denoe he iniial purchase paymen of he annuiy conrac. he mauriy ime and he iniial invesmen amoun w are relaed o he deerminisic wihdrawal rae G by he relaion: G d = w. (2.4 ha is, he whole iniial lump sum paymen w by he policyholder is wihdrawn a he rae G hroughou he life of he conrac (no regard is paid o he ime value of money. We define τ o be he firs passage ime of he value process W hiing he zero value, ha is, τ = inf{ : W = }. Once W his he zero value, i remains o be zero forever aferwards. ha is, he zero value is considered o be an absorbing barrier of W. Under Q, he value process of he personal sub-accoun is given by 5

6 (i < τ dw = [(r αw G ] d + W σ S db (2.5a wih W = w ; (ii τ W =. (2.5b As here is an absorbing barrier a zero for W, i is convenien for our laer discussion o define he corresponding unresriced process W o be d W = [(r α W G ] d + W σ s db,, (2.6 wih W = w. I is seen ha W and W are relaed by W = W τ = W 1 {τ >}. (2.7 he soluion o W can be shown o have he form (Karazas and Shreve, 1992 W = max( W,, (2.8 where ( X = exp W = X (w G u ( r u α 1 2 σ Sσ S du, X u du + σ S db u. Here, w X gives he soluion o he sub-accoun value wih proporional fees G u paymen bu wihou saic wihdrawal. Noe ha du represens he w X u proporion of he sub-accoun wihdrawn over he differenial ime inerval (u, u + du, so he accumulaed depleion of he sub-accoun due o saic G u wihdrawal from ime zero o ime is given by X du. X u Le V (W, r, denoe he no-arbirage value of he variable annuiy wih he GMWB subjec o boh equiy and ineres rae risks. In erms of W, he variable annuiy value is given by [ V (W, r, = E Q e R ] r u du W τ + e R u rs ds G u du F [ = E Q e R ] r u du max( W, + e R u rs ds G u du F. (2.9 6

7 he mahemaical jusificaion of replacing W τ by he opionaliy payoff max( W, is shown in Appendix A. A =, we have V (W, r, = I is convenien o define G u D(, udu + w E Q [ e R ru du X max A = ( 1 ] G u du,. (2.1 w X u G u w X u du. (2.11 Hence, he ime- value of he variable annuiy wih he GMWB can be expressed as [ V (W, r, = G u D(, udu + w E Q e R ] ru du X max(1 A,.(2.12 his represenaion formula indicaes ha he variable annuiy wih he GMWB can be decomposed ino a erm-cerain annuiy paying G per annum over he life of he conrac and a generalized pu opion on some pah dependen funcion of he value of he personal sub-accoun. he pah dependen sae variable A capures he depleion of he personal sub-accoun due o he coninuous wihdrawal process. he valuaion of he above pu opion erm requires he join dynamics of r, X and A under Q. Forunaely, he expecaion calculaion procedure can be much simplified under he corresponding new measure Q S wih S as he numeraire. Le M denoe he money marke accoun process. he corresponding Radon-Nikodym derivaive associaed wih he change of measure from Q o Q S is given by dq S dq = S /S, F M /M so ha he value of he generalized pu opion is given by [ E Q e R ] ru du X max(1 A, [ ( / ] M M S = E QS X max(1 A, M M S = e α E QS [max(1 A, ]. (2.13 7

8 By he Girsanov heorem, B Q S relaed by db Q S D(, he dynamics of We wrie S ( D(, d = S is Q S -Brownian. Also, B and B Q S = db σ S d. under Q S can be shown o be D(, S and observe D(, = 1, so we obain σ QS (, = σ D (, σ S, are [σ D (, σ S ] db Q S. ( = e αs = D(, e α R e σ 1 2 Q S (u,σ QS (u, du+ R σ Q S (u,db Q S u. (2.15 X S Decomposiion of he value of he GMWB From he perspecive of he insurer, she receives he proporional fees unil he personal sub-accoun his he zero value a he random ime τ. Afer hen, under he GMWB, he insurer has o pay he guaraneed wihdrawal sream o he policyholder unil mauriy. herefore, he ime- value of he liabiliy o he insurer associaed wih he GMWB is given by ] L = E Q [ τ e R ru du G d τ αe R ru du W d where W is he ime- value of he sub-accoun. Ineresingly, i can be shown ha (see Appendix B L = V (W, r, w. (2.16 his agrees wih he financial inuiion ha he ime- value of he GMWB is equal o he iniial premium w plus he ime- value of he insurer s liabiliy. 3 Analyic approximaion formulas I has been shown in he las secion ha pricing of he GMWB wih boh equiy and ineres rae risks amouns o he evaluaion of he pu opion value: E QS [max(1 A, ], whose closed form soluion does no exis. Using 8,

9 similar echniques ha have been developed for finding he analyic approximaion soluions o Asian opion models, we show how o obain he lower and upper bound of he above pu opion value. Lower bound using Rogers-Shi s mehod We follow Rogers-Shi s mehod (1995 o deduce he lower bound of he expecaion erm: E QS [max(1 A, ]. By Jensen s inequaliy, we have E QS [max(1 A, ] = E QS [E QS [max(1 A, Z]] E QS [max (E QS [1 A Z], ], (3.1 where Z is a condiional variable. On he oher hand, we can deduce E QS [max(1 A, Z] E QS [max(e QS [1 A Z], ] 1 2 E Q S [ var(a Z]. he qualiy of he above lower bound l Z = E QS [max(e QS [1 A Z], ] (3.2 is highly dependen on he choice of Z. Here, we choose Z such ha E QS [ var(a Z] is minimized. Following a similar choice as shown in Rogers and Shi (1995, we choose Z = 1 ( σ QS (u, db Q S u d, (3.3 Σ where ( Σ 2 = var QS σ QS (u, db Q S u d. I can be shown ha Z is a sandard normal disribuion under Q S. heorem 1 A lower bound on he value of he generalized pu opion defined in Eq. (3.2 is given by l Z = N(z 2 1 G D(, e α N(z 2 m(d, w 9

10 where z 2 is he larger roo of he following equaion: Here, g(z = e mz m2 2 and F(z = 1 1 G D(, e α g(zd =. w m = 1 ( σ QS (u, σ QS (u, sds du. u he proof of heorem 1 is shown in Appendix C. Upper bound using hompson s mehod We follow hompson s mehod (1999 o deduce he upper bound on he expecaion erm: E QS [max(1 A, ]. Suppose f is a random funcion such ha 1 f d = 1 and he wihdrawal rae G is consan, hen an upper bound can be deduced as follows: E QS [max(1 A, ] = 1 ( ] E Q S [max (f 1X d, 1 ( E QS [max f 1 ], d. (3.4 X As suggesed by Lord (25, we choose [ f = µ + β σ QS (u, db Q S u 1 ( σ QS (u, db Q S u ] d, (3.5 where β is a deerminisic parameer and µ is a deerminisic funcion which saisfies 1 µ d = 1. Also, we choose η = σ QS (u, db Q S u 1 ( σ QS (u, db Q S u d (3.6 and a normal variable Ẑ = 1 Σ σ QS (u, db Q S u 1

11 wih Σ 2 = σ QS (u, σ QS (u, du. Condiioning on Ẑ, f 1 X is known o be disribued as µ 1 X + β ( E QS [η Ẑ] + var QS (η Ẑǫ, where ǫ is a sandard normal variable. Le n( denoe he densiy funcion of a sandard normal random variable. As a resul, we manage o obain an esimae on he upper bound of he pu value, he resul of which is summarized in heorem 2. heorem 2 An upper bound on E QS [max(1 A, ], as defined in Eq. (3.4, is given by E QS [max(1 A, ] 1 [ ( a(, z a(, zn b(, z + b(, zn ( ] a(, z n(zdz d, b(, z where a(, z and b(, z are defined by a(, z = µ D(, e α Σ b bσ z + β b(, z = β var(η Ẑ = βσ 1 m2. Σ 2 ( Σ Σm Σ z Remark he ighness of he upper bound depends sensibly on he choice of he parameer β and he funcion µ. We would like o find β and µ such ha 1 E QS [max ( f 1 ], X is minimized, in addiion o he observaion of he consrain 1 µ d = d

12 Accordingly, we define he Lagrangian L(µ, β; λ = 1 ( E QS [max f 1 ] ( 1, d λ µ d 1. X (3.7 he firs order condiion for he opimaliy of µ gives Q S [ 1 X βη µ ] = λ. By applying he approximaion: e x 1 + x, we use 2 Ỹ = D(, e α Σ b 2 ( 1 + Σ Ẑ βη o approximae 1 X βη. Noe ha Ỹ is normally disribued wih mean D(, e α b Σ 2 2 and is variance is given by where We wrie so ha var(ỹ = c2 Σ 2 + 2βc Σm c = D(, e α Σ b 2 2 β. + β2 Σ 2 2, ζ = N 1 (λ = µ D(, e α var(ỹ bσ 2 2 µ = D(, e α Σ b ζ var(ỹ. (3.8 he consan ζ is deermined by he condiion: 1 µ d = 1. 12

13 Nex, we derive he opimal condiion for he deerminaion of he parameer β. Similarly, we apply he firs order derivaive condiion of L(µ, β; λ wih respec o β and obain = 1 = E QS ] E QS [η 1 {µ+βη 1 >} d X ( η N ln µ+βη E D(,e α QS [ξ η ] d, (3.9 varqs (ξ η where ξ is defined in Eq. (C.1 in Appendix C, and E QS [ξ η ] = Σ cov(ξ, η var(η η var QS (ξ η = Σ 2 cov(ξ, η 2 cov(ξ, η = Σ 2 Σm var(η var(η = Σ 2 2Σm + Σ Numerical performance of he analyic approximaion formulas and pricing behavior of he GMWB Firs, we presen he numerical experimens ha were performed o access he ighness of he lower and upper bound on he pu opion value embedded in he GMWB. Recall ha he pu opion value is given by E QS [max(1 A, ]. he basic se of parameer values employed in he numerical calculaions are: θ =.5, k =.349, α =.6. In our calculaions, we vary he insananeous correlaion coefficien ρ, mauriy, ineres rae volailiy σ r and fund value volailiy σ S. We also compue he pu opion value using direc Mone Carlo simulaion wih 1, simulaion rials. he sandard deviaion of he Mone Carlo simulaion resuls is ypically less han.1% of he opion value. he simulaion resuls are used o serve as he benchmark for comparing he numerical resuls obained from he analyic approximaion formulas of finding he lower and upper bound. he 13

14 lower bound is easier and more efficien o be compued since he calculaions involve one-dimensional inegrals only. I is less compuaionally efficien o compue he upper bound since wo-dimensional inegrals are involved in he calculaions. Also, he formulaion of he hompson s upper bound is limied o uniform wihdrawal rae. In able 1, we lis he numerical values of he pu opion value obained from he analyic approximaion formulas and Mone Carlo simulaion wih varying values of he differen parameers in he pricing model. he wihdrawal rae is assumed o be uniform hroughou he life of he conrac. he lower bound values are seen o be highly accurae wih percenage error less han 1%. he upper bound values are less igh when compared o he corresponding lower bound values. he percenage error may increase as high as 4% when = 15 and σ S =.4. In general, he accuracy of he analyic approximaion values decreases wih increasing fund value volailiy σ S, ineres rae volailiy σ r and mauriy. We conclude ha he Rogers-Shi approach of compuing he lower bound generaes sufficienly accurae approximaion soluions for pracical valuaion of he fair value of he annuiies. he valuaion of Rogers-Shi s approximaion o he pu value is compuaionally efficien compared o he Mone Carlo simulaion. he valuaion of opion values in our subsequen analysis of pricing properies of he GMWB had been performed using Rogers-Shi s lower bound approximaion. Nex, we explore he pricing behavior of he GMWB wih respec o differen parameer values in he pricing model. In paricular, we would like o examine he dependence of he pu opion value on varying values of he ineres rae volailiy σ r and insananeous correlaion coefficien ρ. In Figure 1, we plo he pu opion value agains σ r wih ρ =.2, ρ = and ρ =.2, respecively. he wihdrawal rae is uniform and he oher model parameer values are: σ S =.2, = 1, θ =.5, k =.349, α =.6. Wih non-negaive insananeous correlaion coefficien (ρ = and ρ =.2, he pu opion value is an increasing funcion of σ r. However, when he insananeous correlaion coefficien is negaive (ρ =.2, he pu opion value firs decreases wih increasing σ r unil a minimum value is reached, hen subsequenly increases wih increasing σ r. A simple explanaion o he above phenomenon can be offered by examining he dependence of σ QS (u, on ρ and σ r. Recall ha σ QS (u, σ QS (u, = σ QS (u, 2 = σ 2 S + 2ρb(u, σ S σ r + b(u, 2 σ 2 r. 14

15 When ρ is non-negaive, σ QS (u, 2 is always an increasing funcion of σ r. However, when ρ is negaive, σ QS (u, 2 is a decreasing funcion of σ r if σ r < ρσ S b(u,, and i becomes an increasing funcion if σ r > ρσ S b(u,. Since σ QS (u, 2 is an increasing funcion of ρ, so we expec ha he pu opion value is an increasing funcion of ρ. his propery is confirmed by he plo of he pu opion value agains ρ in Figure 2. he opion value under consan ineres rae (corresponds o σ r = is independen of ρ. I is seen ha when ρ is negaive, he pu opion value decreases wih increasing value of σ r. he pu opion value increases wih increasing σ r when ρ becomes posiive. We analyze he impac of varying saic wihdrawal policies on he fair value of he pu opion value agains ineres rae volailiy σ r. We choose he following wihdrawal policies in our calculaions (i uniform wihdrawal over he life of a 9-year annuiy conrac, (ii zero wihdrawal in he firs 3 years and uniform wihdrawal in he remaining 6 years, (iii seady increase in he wihdrawal rae, where G(1 = 3%, G(2 = 5%, G(3 = 7%,, G(8 = 17% and G(9 = 2%. I is revealed in Figure 3 ha he pu opion value has a higher value and becomes less dependen on σ r under he hird wihdrawal policy. When more is wihdrawn a he laer life of he conrac, he chance ha he pu opion being in-he-money is higher and so suscepibiliy o ineres rae flucuaions become less. Lasly, we compue he fair rae of proporional fees α o be charged o cover he embedded pu opion in he GWMB. he hree curves in Figure 4 show he plo of α agains mauriy wih consan ineres rae (solid curve and sochasic ineres raes (he upper doed curve corresponds o σ r =.3 and he middle dashed curve corresponds o σ r =.2.. When faced wih higher ineres rae risk, we expec ha α should be higher. Under consan ineres rae, he insurer charges a lower rae of proporional fees o cover he embedded pu opion when he life of he annuiy policy is lenghened. However, when he ineres rae volailiy is sufficienly high, he rae of proporional fees may increase wih increasing mauriy. 5 Conclusion We have considered he pricing of he embedded Guaraneed Minimum Wihdrawal Benefi (GMWB in variable annuiies wih boh equiy and ineres 15

16 rae risks under saic wihdrawal policies. he value of he GMWB can be decomposed ino a erm-cerain annuiy and a pu opion. Also, he fair value of he conrac is shown o be equal o he sum of he insurer s liabiliies and he iniial premium. We apply he exension of Rogers-Shi s echnique and hompson s mehod o deduce he respecive lower and he upper bound of he opion value, respecively. he numerical accuracy of hese analyic approximaion formulas is found o be sufficienly accurae even under long mauriy and high volailiy. he pricing properies of he GMWB value under varying values of ineres rae volailiy and insananeous correlaion coefficien (beween equiy and ineres rae risks are examined. Ineresingly, when he insananeous correlaion coefficien is negaive, he value of he embedded pu opion may firs decrease wih increasing ineres rae volailiy and hen becomes increasing a sufficienly high level of ineres rae volailiy. Also, we have shown ha he GMWB value is highly dependen on he wihdrawal policies adoped by he policyholder. In addiion, we analyze he impac of sochasic ineres raes on he fair value of he proporional fees o be charged for he provision of he benefi. References Bauer, D., A. Kling and J. Russ (28, A universal pricing framework for guaraneed minimum benefis in variable annuiies, Asin Bullein, vol. 38(2, Chen, Z. and P.A. Forsyh (28, A numerical scheme for he impulse conrol formulaion for pricing variable annuiies wih a guaraneed minimum wihdrawal benefi (GMWB, Numerische Mahemaik, vol. 19, Chen, Z., K. Vezal and P.A. Forsyh (28, he effec of modelling parameers on he value of GMWB guaranee, Insurance: Mahemaics and Economics, vol. 43(1, Dai, M., Y.K. Kwok and J. Zong (28, Guaraneed minimum wihdrawal benefi in variable annuiies, Mahemaical Finance, vol. 8(6, Karazas, I. and S.E. Shreve (1992, Brownian moion and sochasic calculus, second ediion, Springer, New York. Kijima, M. and. Wong (27, Pricing of rache equiy-indexed annuiies under sochasic ineres raes, Insurance: Mahemaics and Economics, vol. 41(3,

17 Lin, X.S. and K.S. an (23, Valuaion of equiy-indexed annuiies under sochasic ineres raes, Norh American Acuarial Journal, vol. 6, Lord, R. (26, Parially exac and bounded approximaions for arihmeic Asian opions, Journal of Compuaional Finance, vol. 1(2, Milevsky, M.A. and.s. Salisbury (26, Financial valuaion of guaraneed minimum wihdrawal benefis, Insurance: Mahemaics and Economics, vol. 38(1, Rogers, L.C.G. and Z. Shi (1995, he value of an Asian opion, Journal of Applied Probabiliy, vol. 32, hompson, G.W.P. (1999, Fas narrow bounds on he value of Asian opions, Working paper of Universiy of Cambridge. 17

18 Appendix A proof of Eq. (2.1 Inuiively, once he unresriced process W becomes negaive, i will never reurn o he posiive region again. his is because once W his he zero value, he drif becomes negaive and he random erm becomes zero, so W is pulled back ino he negaive region immediaely. I suffices o show ha τ > if and only if W >. = par Suppose τ >, by he definiion of he firs passage ime, we hen have W >. = par Recall ha so ha W > G u W = X (w du X u if and only if Suppose W >, his implies ha Since X u, so for any <, we have G u du X u G u X u du < w. G u X u du < w. G u X u du < w. herefore, if W >, hen W > for any <. 18

19 Appendix B proof of Eq. (2.16 From he dynamics of W, we have αw d = r W d dw G d + W σ S db, < τ. Muliplying by he discoun facor e R ru du and inegraing from o τ, we obain τ e R ru du αw d = + τ τ d(e R τ ru du W e R ru du W σ S db = w W τ e R τ r u du + τ Rearranging he above erms and observing we obain τ τ e R ru du W σ S db. W τ e R τ r u du = W e R ru du, e R τ ru du G d = W e R ru du w + αe R ru du W d e R ru du G d e R ru du G d e R τ ru du G d e R ru du W σ S db. Lasly, by aking he expecaion under Q, we obain he resul in Eq. (

20 Appendix C proof of heorem 1 We define ξ = 1 2 σ QS (u, σ QS (u, du + σ QS (u, db Q S u, (C.1 which is also normal. For a pair of join normal random variables Z 1 and Z 2, he well known Projecion heorem saes ha and E[Z 1 Z 2 ] = E[Z 1 ] + cov(z 1, Z 2 (Z 2 E[Z 2 ] var(z 2 var(z 1 Z 2 = var(z 1 cov(z 1, Z 2 2. var(z 2 Applying he above relaions for condiional expecaion of normal random variables, we obain E QS [ξ Z] = 1 2 var QS (ξ Z = σ QS (u, σ QS (u, du + Zm σ QS (u, σ QS (u, du m 2, where m = E QS [ξ Z] = 1 [ ] σ QS (u, σ QS (u, sds du. Σ u [ ] 1 We hen obain E QS Z in erms of m and Z as follows: X [ ] 1 E QS Z X = D(, e α E QS [ e ξ Z ] = D(, e α e Zm m 2 2. As a resul, we obain l Z = E QS [max(e QS [1 A Z], ] ] = E QS [max (1 1w G D(, e α g(zd,, 2

21 where g(z = e mz m 2 2. he funcion g(z is an increasing (decreasing convex funcion for m > (m <. Since he sum of convex funcions remains o be convex, so he following equaion: F(z = 1 1 G D(, e α g(zd = w eiher has zero, unique or wo soluions. Le z 1 and z 2, z 1 < z 2, denoe he wo possible soluions of F(z =. We hen have F(z > for z 1 < z < z 2. For noaional convenience, if F(z = has no soluion, we se z 1 = z 2 = ; and if only one soluion exiss, hen eiher we se z 1 = or z 2 = depending on he sign of F (z. he roos of F(z can be found easily using any roo-finding algorihm. In erms of z 1 and z 2, he lower bound l Z can be evaluaed as follows: ]] [1 1w G D(, e α g(zd l Z = E QS [1 {z1 <z<z 2 } = N(z 2 N(z 1 1 z2 G D(, e α e (z m 2 2 dz d w 2π z 1 1 = N(z 2 N(z 1 1 G D(, e α [N(z 2 m( N(z 1 m(] d, w where N( is he sandard normal disribuion funcion. Suppose m > for [, ], hen F(z has unique roo and z 1 is se o be. his propery on m is commonly observed since B and Z are posiively correlaed. Under his condiion, we obain he following lower bound E QS [max(1 A, ] N(z 2 1 G D(, e α N(z 2 m(d. w 21

22 able 1 Comparison of numerical accuracy of he lower bound and upper bound on he pu opion values. he numerical resuls obained from Mone Carlo simulaion serve as he benchmark for comparison. he basic se of parameer values used in he pricing model are: θ =.5, k =.349, α =.6. he percenage errors of he lower bound values are less han 1% while he percenage errors of he upper bound values may reach as high as 4% under long mauriy and high volailiy values. 22