A unified pricing of variable annuity guarantees under the optimal stochastic control framework

Transcription

1 A uified pricig of variable auity guaratees uder the optimal stochastic cotrol framework Pavel V. Shevcheko 1 ad Xiaoli Luo 2 arxiv: v1 [q-fi.pr] 2 May 2016 Draft paper: 16 April CSIRO Australia, Pavel.Shevcheko@csiro.au 2 CSIRO Australia, Xiaoli.Luo@csiro.au Abstract I this paper, we review pricig of variable auity livig ad death guaratees offered to retail ivestors i may coutries. Ivestors purchase these products to take advatage of market growth ad protect savigs. We preset pricig of these products via a optimal stochastic cotrol framework, ad review the existig umerical methods. For umerical valuatio of these cotracts, we develop a direct itegratio method based o Gauss-Hermite quadrature with a oe-dimesioal cubic splie for calculatio of the expected cotract value, ad a bi-cubic splie iterpolatio for applyig the jump coditios across the cotract cashflow evet times. This method is very efficiet whe compared to the partial differetial equatio methods if the trasitio desity or its momets) of the risky asset uderlyig the cotract is kow i closed form betwee the evet times. We also preset accurate umerical results for pricig of a Guarateed Miimum Accumulatio Beefit GMAB) guaratee available o the market that ca serve as a bechmark for practitioers ad researchers developig pricig of variable auity guaratees. Keywords: variable auity, guarateed livig ad death beefits, guarateed miimum accumulatio beefit, optimal stochastic cotrol, direct itegratio method. 1

2 1 Itroductio May wealth maagemet ad isurace compaies worldwide are offerig ivestmet products kow as variable auities VA) with some guaratees of livig ad death beefits to assist ivestors with maagig pre-retiremet ad post-retiremet plas. These products take advatage of market growth while provide protectio of the savigs agaist market dowturs. Isurers started to offer these products from the 1990s i Uited States. Later, these products became popular i Europe, UK ad Japa ad more recetly i Australia. The VA cotract cashflows received by the policyholder are liked to the ivestmet portfolio choice ad performace e.g. the choice of mutual fud ad its strategy) while traditioal auities provide a pre-defied icome stream i exchage for the lump sum paymet. Accordig to LIMRA Life Isurace ad Market Research Associatio) reports, the VA market is huge: VA sales i Uited States were $158 billio i 2011, $147 billio i 2012 ad $145 billio i The types of VA guaratees referred i the literature as VA riders) offered for ivestmet portfolios are classified as guarateed miimum withdrawal beefit GMWB), guarateed miimum accumulatio beefit GMAB), guarateed miimum icome beefit GMIB) ad guarateed miimum death beefit GMDB). These guaratees, geerically deoted as GMxB, provide differet types of protectio agaist market dowturs ad policyholder death. GMWB allows withdrawig fuds from the VA accout up to some pre-defied limit regardless of ivestmet performace durig the cotract; GMAB ad GMIB both provide a guarateed ivestmet accout balace at the cotract maturity that ca be take as a lump sum or stadard auity respectively. Guarateed lifelog withdrawal beefit GLWB), a specific type of GMWB, allows withdrawig fuds at the cotractual rate as log as the policyholder is alive. GMDB provides a specified paymet if the policyholder dies. Precise specificatios of the products withi each type ca vary across compaies ad some products may iclude combiatios of these guaratees. A good overview of VA products ad the developmet of their market ca be foud i Bauer et al. 2008), Ledlie et al. 2008) ad Kalberer ad Ravidra 2009). There have bee a umber of papers i academic literature cosiderig pricig of these products. Most of these are focused o pricig VA riders uder the pre-determied static) policyholder behaviour i withdrawal ad surreder. Some studies iclude pricig uder the active dyamic) strategy whe the policyholder optimally decides the amout of withdrawal at each withdrawal date depedig o the iformatio available at that date. Stadard Mote Carlo MC) method ca easily be used to estimate price i the case of pre-defied withdrawal strategy but hadlig the dyamic strategy requires backward i time solutio that ca be doe oly via the partial differetial equatio PDE), direct itegratio or regressio type MC methods. I brief, pricig uder the static ad dyamic withdrawal strategies via PDE based methods has bee developed i Milevsky ad Salisbury 2006), Dai et al. 2008) ad Che ad Forsyth 2008). Bauer et al. 2008) develops a uified approach with umerical estimatio via MC ad direct itegratio methods. The direct itegratio method was developed further i Luo ad Shevcheko 2015a,b) usig Gauss-Hermite quadrature ad cubic iterpolatios. Baciello et al. 2011) cosider may VA riders uder stochastic iterest rate ad stochastic volatility if the policyholder withdraws at the pre-defied cotractual rate or completely surreders the 2

3 cotract. Their pricig is accomplished either by the ordiary MC or Least-Squares MC to accout for the optimal surreder. Typically, pricig of VA riders is cosidered uder the assumptio of geometric Browia motio for the risky asset uderlyig the cotract, though a few papers looked at extesios such as stochastic iterest rate ad/or stochastic volatility, see e.g. Forsyth ad Vetzal 2014), Luo ad Shevcheko 2016), Baciello et al. 2011), Huag ad Kwok 2015). Azimzadeh ad Forsyth 2014) prove the existece of a optimal bag-bag cotrol for GLWB cotract whe the cotract holder ca maximize cotract writer s losses by oly ever performig o-withdrawal, withdrawal at the cotract rate or full surreder. However, they also demostrate that the related GMWB cotract does ot satisfy the bag-bag priciple other tha i certai degeerate cases. Huag ad Kwok 2015) developed a regressio-based MC method for pricig GLWB uder the bag-bag strategy i the case of stochastic volatility. GMWB pricig uder the bag-bag strategy was studied i Luo ad Shevcheko 2015c). The difficulty with applyig the well kow Least-Squares MC itroduced i Logstaff ad Schwartz 2001) for pricig VA riders uder the optimal strategy is due to the fact that the paths of the uderlyig VA wealth accout are affected by the withdrawals. I priciple, oe ca apply cotrol radomizatio methods extedig Least-Squares MC to hadle optimal stochastic cotrol problems with cotrolled Markov processes recetly developed i Kharroubi et al. 2014), but the accuracy ad robustess of this method for pricig VA riders have ot bee studied yet. Oe commo observatio i the above metioed literature is that pricig uder the optimal strategy ofte leads to prices sigificatly higher tha observed o the market. These studies rely o the optio pricig risk-eutral methodology i quatitative fiace to fid a fair fee. Here, the fudametal idea is to fid the cost of a dyamic self-fiacig replicatig portfolio which is desiged to provide a amout at least equal to the payoff of the cotract. The cost of establishig this hedgig strategy is the o-arbitrage price of the cotract. This is uder the assumptio that the cotract holder adopts a optimal strategy exercise strategy maximisig the moetary value of the cotract). If the purchaser follows ay other exercise strategy, the cotract writer will geerate a guarateed profit if cotiuous hedgig is performed. Of course the strategy optimal i this sese is ot related to the policyholder circumstaces. I pricig VA with guaratees, it is reasoable to cosider alterative assumptios regardig the ivestor s withdrawal strategy. This is because a ivestor may follow what appears to be a sub-optimal strategy that does ot maximise the moetary value of the optio. This could be due to reasos such as liquidity eeds, tax ad other persoal circumstaces. Moreover, mortality risk is diversified by the cotract issuer through sellig may cotracts to may people while the policyholder caot do it. Also, there might be o liquid secodary market for VAs o which the policy could be sold or repurchased) at its fair value. The policyholder may act optimally with respect to his prefereces ad circumstaces but it may be differet from the optimal strategy that maximises the moetary value of the cotract. I this case we calculate a fair fee to be deducted i order to fiace a dyamic replicatig portfolio for the guaratees optios) embedded i the cotract uder the assumptio of a particular exercise strategy. The replicatig portfolio will provide sufficiet fuds to meet ay future payouts that arise from writig the cotract. However, the fair fee obtaied uder the assumptio that ivestors behave optimally to max- 3

4 imise the value of the guaratee does offer a importat bechmark because it is a worst case sceario for the cotract writer. Also, as oted i Hilpert et al. 2014), secodary markets for equity liked isurace products where the policyholder ca sell their cotracts) are growig. Thus, third parties ca potetially geerate guarateed profit through hedgig strategies from fiacial products such as VA riders which are ot priced uder the assumptio of the optimal withdrawal strategy. Koller et al. 2015) metios several compaies recetly sufferig large losses related to icreased surreder rates, idicatig that either charged fees were ot sufficietly large or that hedgig program did ot perform as expected. Oe way to aalyze the withdrawal behavior of VA holder ad evaluate the eed of these products is to solve the life-cycle utility model accoutig for cosumptio, housig, bequest ad other real life circumstaces. Developig a full life-cycle model with all prefereces ad required parameters is challegig but there are already several cotributios reportig some iterestig fidigs i this directio: Moeig 2012); Horeff et al. 2015); Gao ad Ulm 2012); Steiorth ad Mitchell 2015). This topic will ot be cosidered i this paper. It is also importat to ote a recet paper by Moeig ad Bauer 2015) cosiderig the pricig uder the optimal strategy i the presece of taxes via subjective risk-eutral valuatio methodology. They demostrated that icludig taxes sigificatly affects the value of the VA withdrawal guaratees producig results i lie with empirical market prices. I this paper we review pricig of livig ad death beefit guaratees offered with VAs, ad preset a uified optimal stochastic cotrol framework for pricig these cotracts. The mai ideas have bee developed ad appeared i some forms i a umber of other papers. However, we believe that our presetatio is easier to uderstad ad implemet. We also preset direct itegratio method based o computig the expected cotract values i a backward timesteppig through a high order Gauss-Hermite itegratio quadrature applied o a cubic splie iterpolatio. This method ca be applied whe trasitio desity of the uderlyig asset betwee the cotract cashflow evet dates or its momets are kow i closed form. We have used this for pricig specific fiacial derivatives ad some simple versios of VA guaratees i Luo ad Shevcheko 2014, 2015a). Here, we adapt ad exted the method to hadle pricig VA riders i geeral. As a umerical example, we calculate accurate prices of GMAB with possible aual ratchets reset of the guarateed capital to the ivestmet portfolio value if the latter is larger o aiversary dates) ad allowig optimal withdrawals. The cotract that we cosider is very similar i specificatios to the real product marketed i Australia, see for example MLC 2014) ad AMP 2014). Numerical difficulties ecoutered i pricig this VA rider are commo across other VA guaratees ad at the same time comprehesive umerical pricig results for this product are ot available i the literature. These results reported for a rage of parameters) ca serve as a bechmark for practitioers ad researchers developig umerical pricig of VA riders. I the ext sectio, a geeral specificatio of VA riders is give. I Sectio 3 we discuss stochastic models used for pricig these products. Sectio 4 provides precise specificatio for some popular VA riders. I Sectio 5 we outlie the calculatio of the fair price ad fair fee as a solutio of a optimal stochastic cotrol problem. Sectio 6 presets the umerical methods ad algorithms for pricig VA riders. I Sectio 7 we preset umerical results for the fair fees of GMAB rider. Cocludig remarks are give i Sectio 8. 4

5 2 VA rider cotract specificatio Cosider a VA cotract with some guaratees for livig ad death beefits purchased by a x-year old idividual at time t 0 = 0 with the up-frot premium ivested i a risky asset e.g. a mutual fud), deoted as St) at time t 0. The VA rider specificatio icludes dates whe evets such as withdrawal, ratchet step-up), bous roll-up), death beefit paymet, etc. may occur. Precise defiitios of these evets deped o the cotract ad correspodig examples will be provided i Sectio 4. We assume that the withdrawal ca oly take place o the set of the ordered evet times T = {t 1,..., t N }, where T = t N is the cotract maturity. Also, the set of policy aiversaries whe the ratchet may occur is deoted as T r ad is assumed to be a subset of T. For simplicity o otatio we assume that all other evets may oly occur o the withdrawal dates. The value of VA cotract with guaratees at time t is determied by the three mai state variables. Wealth accout W t), value of the ivestmet accout which is liked to the risky asset St) ad modelled as stochastic process. Guaratee accout At), also referred i the literature as beefit base. It is ot chagig betwee evet times but ca be stochastic via stochasticity i W t) at the evet times depedig o the cotract features. Discrete state variable I {1, 0, 1} correspodig to the states of policyholder is beig alive at t, died durig t 1, t ], or died before or at t 1 correspodigly. Deote the death probability durig t 1, t ] as q = Pr[I = 0 I 1 = 1], i.e. Pr[I = 1 I 1 = 1] = 1 q. Note that q depeds o the age of the cotract holder at t ad thus depeds o the age x at t 0 = 0. Other state variables are eeded if the iterest rate ad/or volatility are stochastic but these are ot affected at the cotract evet times ad typically do ot eter formulas for the cotract cashflows; these will ot be cosidered explicitly. Extra state variable is also required to track a tax free base to accout for taxes; this will be cosidered i Sectio 5.4. I priciple, differet guaratees icluded i VA may have differet beefit base state variables. For otatioal simplicity ad also from practical perspective, we assume that all guaratees i VA are liked to the same beefit base accout. Iitially, W 0) ad A0) are set equal to the upfrot premium. The cotract holder is allowed to take withdrawal γ at time t, = 1,..., N 1. Deote the values of the beefit base just before ad just after t as At ) ad At + ) respectively, ad similarly for the wealth accout W t ) ad W t + ). The cotract product specificatio determies: The cotractual guarateed) withdrawal amout G for the period t 1, t ] that may deped o the beefit base At ) ad/or W t ). Jump coditios at the evet times relatig state variables before ad after the evet, 5

6 subject to withdrawals γ belogig to a admissible space A : ) W t + ) : = h W W t ), At ), γ, 1) ) At + ) : = h A W t ), At ), γ, 2) γ A W t ), At ) ), 3) where h W ) ad h A ) are some fuctios that may also deped o the fee, pealty ad aual step-up parameters. For example, if oly a ratchet is possible at t T r ad o other cotract evets, the At + ) = max At ), W t ) 1 t T r ). I practice, several evets such as withdrawal, ratchet, bous, etc. may occur at the same time t, ad the cotract specificatio determies the order of this evets. The payout P T W, A) at the cotract maturity if policyholder is alive at t = T. The payout D W, A) to the beeficiary at t i the case of the policyholder death durig t 1, t ], = 1,..., N. The cashflow received by the policyholder f W t ), At ), γ ) at the evet times t, = 1,..., N 1, that might be differet from γ due to pealties. The specificatio details typically vary across differet compaies ad are difficult to extract from the very log product specificatio documets. Moreover, results for specific GMxB riders preseted i academic literature ofte refer to differet specificatios. Oce the above coditios, i.e. fuctios h W ), h A ), P T ), P D ), f γ ) ad admissible rage for withdrawal A are specified by the cotract desig, ad a specific stochastic evolutio of the state variables is assumed withi t 1, t ), = 1,..., N, the pricig of the cotract ca be accomplished by umerical methods. I particular, if withdrawals are optimal the pricig ca be accomplished by PDE, direct itegratio or regressio based MC methods. If withdrawals are determiistic, the stadard MC alog with PDE ad direct itegratio methods ca be used. The use of a particular umerical techique is determied by the complexity of the uderlyig stochastic model. 3 Stochastic Model Commoly i the literature, stochastic models for the fiacial risky asset St) uderlyig the VA rider assume that there is o arbitrage i the fiacial market which meas that there is a risk-eutral measure Q uder which paymet streams ca be valued as expected discouted values. Moreover, this meas that the cost of portfolio replicatig the cotract is give by its expected discouted value uder Q. Hece, the fair price of the cotract ca be expressed as a expectatio of the cotract discouted cashflows with respect to Q. Some models cosidered i the literature assume that the fiacial market is complete which meas that the risk-eutral measure Q is uique. It is also assumed that market has a risk-free asset that accumulates 6

7 cotiuously at risk free iterest rate. These are typical assumptios i the academic research literature o pricig fiacial derivatives, for a good textbook i this area we refer the reader to e.g. Björk 2004). Regardig the mortality risk, it is assumed that it is fully diversified via sellig the cotract to may policyholders. I the case of systemic udiversified) mortality risk, the risk-eutral fair value ca be adjusted usig a actuarial premium priciple, see e.g. Gaillardetz ad Lakhmiri 2011). Aother commo assumptio is that mortality ad fiacial risks are idepedet. A bechmark model commoly cosidered i the literature o pricig VA riders is the well-kow Black-Scholes dyamics for the referece portfolio of assets St) that uder the risk-eutral measure Q is kow to be dst) = rt)st)dt + σt)st)dbt). 4) Here, Bt) is the stadard Wieer process, rt) is the risk free iterest rate ad σt) is the volatility. Uder this model the fiacial market is complete. Without loss of geerality, the model parameters ca be assumed to be piecewise costat fuctios of time for time discretizatio 0 = t 0 < t 1 < < t N = T. Deote correspodig asset values as St 0 ),..., St N ) ad risk free iterest rate ad volatility as r 1,..., r N ad σ 1,..., σ N respectively. That is, σ 1 is the volatility for t 0, t 1 ]; σ 2 is the volatility for t 1, t 2 ], etc. ad similarly for the iterest rate. Pricig VA riders i the case of extesios of the above model to the stochastic iterest rate ad/or stochastic volatility have bee developed i e.g. Forsyth ad Vetzal 2014), Luo ad Shevcheko 2016), Baciello et al. 2011), Huag ad Kwok 2015). Regardig mortality modellig, the stadard way is to use official Life Tables to estimate the death probability q = Pr[I = 0 I 1 = 1] durig t 1, t ]. Life Tables provide aual death probabilities for each age ad geder i a give coutry; probabilities for time periods withi a year ca be foud by e.g. liear iterpolatio, see Luo ad Shevcheko 2015b). Istead of a Life Table, stochastic mortality models such as the bechmark Lee-Carter model itroduced i Lee ad Carter 1992) ca also be used to forecasts the required death probabilities accoutig for systematic mortality risk). For a give process of risky asset St), t 0, the value of the wealth accout W t) evolves as W t ) = W t+ 1) St 1 ) St )e αdt, W t + ) = maxw t ) γ, 0), = 1, 2,..., N, 5) where dt = t t 1 ad α is the aual fee cotiuously charged by cotract issuer for the provided guaratee. I the case of St) followig the geometric Browia motio process 4), we have St ) = St 1 )e r 1 2 σ2 )dt+σ dtz, where z 1,..., z N are idepedet ad idetically distributed stadard Normal radom variables. I practice, the guaratee fee is charged discretely ad proportioal to the wealth accout that ca easily be icorporated ito the wealth process 5). Deotig the discretely charged 7

8 fee with the aual basis as α, the wealth process becomes W t ) = W t+ 1) St 1 ) St ), W t + ) = max W t )1 αdt ) γ, 0 ), = 1, 2,..., N. 6) Typically, the differece betwee cotiuously ad discretely charged fees is ot material as observed i our umerical results give i Sectio 7. Aother popular fee structure correspods to fees charged as a proportio of the beefit base, so that W t ) = W t+ 1) St 1 ) St ), W t + ) = max W t ) At ) αdt γ, 0 ), = 1, 2,..., N. 7) Here, it is assumed that discrete fees are deducted before withdrawal but it ca be vice versa depedig o the cotract specificatios. For simplicity, we do ot cosider maagemet fees α m charged by a mutual fud for maagig the ivestmet portfolio. If maagemet fees α m is give exogeously, the it will have a impact o the fair fee α that should by charged by the VA guaratee issuer. This ca be accomplished as described i e.g. Forsyth ad Vetzal 2014) ad ca be easily icorporated i the framework outlied i our paper. Obviously, α will be larger for give α m > 0 comparig to the case α m = 0. The maagemet fees reduce the performace of the ivestmet accout thus icreasig the value of the guaratee as reported i e.g. Che et al. 2008) for GMWB or Forsyth ad Vetzal 2014) for GLWB. They commeted that isurers wishig to provide the cheapest guaratee could provide the guaratee o the correspodig iexpesive exchage traded idex fud rather tha o a maaged mutual fud accout with extra fees. 4 VA riders There are may differet specificatios for GMWB, GLWB, GMAB, GMIB ad GMDB i the idustry ad academic literature. I this sectio we provide a mathematical formulatio for some stadard VA rider setups. We assume that the guaratee fee α is charged cotiuously. If the fee is charged discretely ad before withdrawal ad other cotract evets), the oe should make the followig adjustmet to the formulas i this sectio: W t ) W t )1 αdt ), if the fee is proportioal to the wealth accout ad if the fee is proportioal to the beefit base. W t ) maxw t ) At ) αdt, 0), 8

9 4.1 GMWB A VA cotract with GMWB promises to retur at least the etire iitial ivestmet through cash withdrawals durig the policy life plus the remaiig accout balace at maturity, regardless of the portfolio performace. Ofte i academic literature, the studied GMWB type has a very simple structure, where the pealty is applied to the cashflow paid to the cotract holder, while the beefit base is reduced by the full withdrawal amout. Specifically, At + ) := h A W t ), At ), γ ) = At ) γ, 8) with γ A, A = [0, At )]; ad cashflow paid to the cotract holder is f W t ), At ), γ ) = { γ, if 0 γ G, G + 1 β)γ G ), if γ > G, 9) where β [0, 1] is the pealty parameter for excess withdrawal. The cotractual amout is defied as G = W 0)t t 1 )/T ad the maturity coditio is P T W t N ), At N )) = maxw t N ), f At N ))). Note that the above specificatio does ot allow early surreder which ca be icluded via extedig the withdrawal space A. Also, there is o death beefit; it is assumed that beeficiary will maitai the cotract if the case of policyholder death. This cotract has oly basic features facilitatig compariso of results from differet academic studies, such as Che ad Forsyth 2008), Dai et al. 2008), Luo ad Shevcheko 2015c), Luo ad Shevcheko 2015a). Specificatios commo i the idustry iclude cases where the cotractual amout G is specified to be differet from G = W 0)t t 1 )/T ad a pealty is applied to both the withdraw amout ad the beefit base. For example, specificatios used i Moeig ad Bauer 2015) to compare with the idustry products iclude: f W t ), At ), γ ) = γ δ excess δ pealty, δ excess = β e max γ miat ), G ), 0), δ pealty = β g γ δ excess ) 1 x+t<59.5, 10) where x is the age of the policyholder i years at t 0 = 0, β e ad β g are excess withdrawal ad early withdrawal pealty parameters that ca chage with time, ad γ A, A = [ 0, max W t ), miat ), G ) )]. Moeig ad Bauer 2015) also cosidered several specificatios for the beefit base jump coditios. Specificatio 1: { maxat At + ) γ, 0), if γ G, ) = ) ) max mi At ) γ, At ) W t+ ), 0, if γ W t ) > G. 11) 9

10 Specificatio 2: At + ) = { maxat ) γ, 0), if γ G, max mi At ) γ, W t + )), 0), if γ > G. 12) Specificatio 3: { maxat At + ) γ, 0), if γ G, ) = max At W t ) G, 0) + ), if γ maxw t )G > G.,0) 13) I additio, a ratchet reset of the beefit base to the wealth accout if the latter is higher) ca apply at aiversary dates. If it occurs before the withdrawal, the i the above formulas oe should make the followig adjustmet At ) maxat ), W t )), if t T r. If the reset is takig place after the withdrawal, the oe should have At + ) maxat + ), W t + )), if t T r. 4.2 GLWB GLWB is similar to GMWB but provides guarateed withdrawal for life; upo death the remaiig wealth accout value is paid to the beeficiary. The cotractual withdrawal amout G is typically based o a fixed proportio g of the beefit base At), i.e. G = g At )t t 1 ). The beefit base ca icrease via ratchet step-up) or bous roll-up) features. Bous feature provides a icrease of the beefit base if o withdrawal is made o a withdrawal date. Complete surreder refers to the withdrawal of the whole policy accout. The withdrawal ca exceed the cotractual amout ad i this case the et amout received by the policyholder is subject to a pealty. Uder the typical specificatio cosidered e.g. i Huag ad Kwok 2015), the cashflow received by the policyholder is f W t ), At ), γ ) = { γ, if 0 γ G, G + 1 β)γ G ), if γ > G, γ A, A = [0, maxw t ), G )], where β is the pealty parameter for excess withdrawal. icludig ratchets ad bous features, is give by At + ) = max At )1 + b ), W t )1 t T r ) 1γ=0 14) The beefit base jump coditio, + max ) At ), maxw t ) γ, 0) 1 t Tr 10<γ G + max At ) W ) t ) γ, W t W t ) γ ) 1 t T ) G r 1 G<γ W t ), 15) where b is the bous rate parameter that may chage i time. Fially, if the policyholder dies durig t 1, t ], the beeficiary receives a death beefit paymet D W t ), At )) = W t ) ad t N correspods to the maximum age beyod which survival is deemed impossible. 10

11 4.3 GMAB GMAB rider provides a certaity of capital till some maturity e.g. 10 or 20 years) ad the potetial for a capital growth. Typical GMAB products sold o the market do ot impose pealty o the policyholder withdrawal amout but ca pealise the beefit base protected capital balace) uder some coditios. It is also commo to have a ratchet feature, where the protected capital balace icreases to the wealth accout if the latter is higher o a aiversary date. Withdrawals from the accout are allowed subject to a pealty. For example, specificatios of the product marketed by MLC 2014) ad AMP 2014) i Australia are very close to the followig formulatio: f W t ), At ), γ ) = γ, 16) { max At At + ) := h A W t ), At ), γ ) = ), W t )) C γ ), if t T r, max At ) C γ ), 0), otherwise, where C γ ) is a pealty fuctio that ca be larger tha γ as defied below, ad γ A = [0, W t )]. The product is offered for the super ad pesio accout types. The super accout is desiged for a ivestor beig i a accumulatio phase, while the pesio accout is for a retired ivestor i a auitizatio phase. The differece betwee the accouts i terms of techical details is oly i the pealty applied to the protected capital after withdrawals; the super accout discourages withdrawals more tha the pesio accout. I both cases the pealty is i the form of a reductio of the protected capital beefit base) larger tha the withdraw amout. The pealty oly applies if the wealth accout balace is below the protected capital amout. A super accout pealizes ay amout of withdrawals, while the pesio accout oly pealize excessive withdrawals. Specifically, for a super accout, the fuctio C γ ) is give by 17) C γ ) = { γ, if W t ) At ), At )γ /W t ), if W t ) < At ), 18) ad for a pesio accout, the pealty is C γ ) = { γ, if W t ) At ) or γ G, At )γ /W t ), if W t ) < At ) ad γ > G. 19) That is, the pealty for the pesio accout applies oly if the wealth accout balace is below the protected capital amout ad the withdrawal is above a pre-determied amout G. Fially, the termial coditio is give by P T W t N ), At N )) = maxw t N ), At N )). A total withdrawal of the wealth accout balace effectively termiates the cotract, as the pealty mechaism esures the protected capital is always exhausted to zero by a complete withdrawal. 11

12 4.4 GMIB At maturity, the holder of GMIB rider ca select to take a lump sum of the wealth accout W T ) or auitise this amout at auitizatio rate ä T curret at maturity or auitize the beefit base AT ) at pre-specified auitizatio rate ä g. Auitizatio rate is defied as the price of a auity payig oe dollar each year. If the accout value is below the beefit base, the the customer caot take AT ) as a lump sum but oly as a auity at pre-specified rate. Thus, the payoff of VA with GMIB at time T is ) P T W t N ), At N )) = max W t N ), At N )ät. ä g The beefit base may iclude roll-ups ad ratchets. Agai, this rider ca be offered joitly with other riders. For example, it ca be part of GMWB or GMAB cotract maturity coditios. For discussio ad pricig of GMIB i academic literature, see Marshall et al. 2010) ad Bauer et al. 2008). 4.5 GMDB GMDB rider provides a death beefit if the policy holder death occurs before or at the cotract maturity. Assumig that if the policyholder dies durig t 1, t ], the the beeficiary will be paid a amout D ) at t, where some of the commo death beefit types are: D W t ), At )) = maxat ), W t )), death beefit type 0, W 0), death beefit type 1, maxw 0), W t )), death beefit type 2, W t ), death beefit type 3. Some providers adjust the iitial premium W 0) for iflatio i the death beefit. For some policies, the death beefit type may chage at some age, e.g. death beefit type 1 or type 2 may chage to type 0, effectively makig the death beefit expirig at some age e.g. at the age of 75 years). The death beefit ca be provided o top of some other guaratees ad the cotract may provide a spousal cotiuatio optio that allows a survivig spouse to cotiue the cotract. The cotract may have accumulatio phase where the death beefit may icrease, ad cotiuatio phase where the death beefit remais costat. Pricig GMDB has bee cosidered i e.g. Milevsky ad Poser 2001), Bélager et al. 2009), Luo ad Shevcheko 2015b). 20) 5 Fair Pricig Deote the state vector at time t before the withdrawal as X = W t ), At ), I ) ad X = X 1,..., X N ). Give the withdrawal strategy γ = γ 1,..., γ N1 ), the preset value of the overall payoff of the VA cotract with a guaratee is a fuctio of the state vector H 0 X, γ) = B 0,N H N X N ) + 12 N1 =1 B 0, f X, γ ). 21)

13 Here, H N X N ) = P T W T ), AT ) ) 1 I=1 + D N W T ), AT ) ) 1 I=0 22) is the cashflow at the cotract maturity ad f X, γ ) = f W t ), At ), γ ) 1 I=1 + D W t ), At ) ) 1 I=0 23) is the cashflow at time t. Also, B i,j is the discoutig factor from t j to t i tj ) B i,j = exp rt)dt, t j > t i. 24) 5.1 Pricig as Stochastic Cotrol Problem t i Let Q t W, A) be the price of the VA cotract with a guaratee at time t, whe W t) = W, At) = A ad policyholder is alive. For simplicity of otatio, if the policyholder is alive, we drop mortality state variable I = 1 i the fuctio argumets. Assume that fiacial risk ca be elimiated via cotiuous hedgig. Also assume that mortality risk is fully diversified via sellig the cotract to may people of the same age, i.e. the average of the cotract payoffs H 0 X, γ) over L policyholders coverges to E I t 0 [H 0 X, γ)] as L, where I is the real probability measure correspodig to the mortality process I 1, I 2,.... The the cotract price uder the give withdrawal strategy γ ca be calculated as Q 0 W 0), A0)) = E Q,I t 0 [H 0 X, γ)]. 25) Here, E Q,I t [ ] deotes a expectatio with respect to the state vector X, coditioal o iformatio available at time t, i.e. with respect to the fiacial risky asset process uder the risk-eutral probability measure Q ad with respect to the mortality process uder the real probability measure I. The the fair fee value of α to be charged for VA guaratee correspods to Q 0 W 0), A0)) = W 0). That is, oce a pricig of Q 0 W 0), A0)) for a give α is developed, the a umerical root search algorithm is required to fid the fair fee. The withdrawal strategy γ ca deped o time ad state variables ad is assumed to be give whe price of the cotract is calculated i 25). The withdrawal strategies are classified as static, optimal, ad suboptimal. Static strategy. Uder this strategy, the policyholder decisios are determiistically determied at the begiig of the cotract ad do ot deped o the evolutio of the wealth ad beefit base accouts. For example, policyholder withdraws at the cotractual rate oly. Optimal strategy. Uder the optimal withdrawal strategy, the decisio o the withdrawal amout γ depeds o the iformatio available at time t, i.e. depeds o the state variable X. The optimal strategy is calculated as γ X) = argsupe Q,I t 0 [H 0 X, γ)], 26) γ A where the supremum is take over all admissible strategies γ. Ay other strategy γx) differet from γ X) is called suboptimal. 13

14 Give that the state variable X = X 1,..., X N ) is a Markov process ad the cotract payoff is represeted by the geeral formula 21), calculatio of the cotract value 25) uder the optimal withdrawal strategy 26) is a stadard optimal stochastic cotrol problem for a cotrolled Markov process. Note that, the cotrol variable γ affects the trasitio law of the uderlyig wealth W t) process from t to t +1 ad thus the process is cotrolled. For a good textbook treatmet of stochastic cotrol problems i fiace, see Bäuerle ad Rieder 2011). This type of problems ca be solved recursively to fid the cotract value Q t x) at t whe X = x for = N 1,..., 0 via the backward iductio Bellma equatio ) Q t x) = sup f x, γ ) + B,+1 Q t+1 x )K t dx x, γ ), 27) γ A startig from the fial coditio Q T x) = H N x). Here, K t dx x, γ ) is the stochastic kerel represetig probability to reach state i dx at time t +1 if the withdrawal actio) γ is applied i the state x at time t. Obviously, the above backward iductio ca also be used to calculate the fair cotract price i the case of static strategy γ; i this case the space of admissible strategies A cosists oly oe pre-defied value ad sup ) becomes redudat. For clarity, deote Q t ) ad Q t + ) the cotract values just before ad just after the evet time t respectively. The, after calculatig expectatio with respect to the mortality state variable I +1 i 27), the required backward recursio ca be rewritte explicitly as [ Q t + W, A) = 1 q +1 )E Q B t +,+1 Q t +1 W t +1 ), At +1) ) ] W, A [ +q +1 E Q B t +,+1 D +1 W t +1 ), At +1) ) ] W, A 28) with the jump coditio Q t N Q t W, A) = max γ A f W, A, γ ) + Q t + h W W, A, γ ), h A W, A, γ ) )). 29) This recursio is solved for = N 1, N 2,..., 0, startig from the maturity coditio W, A) = P T W, A). 5.2 Alterative Solutio Give that the mortality ad fiacial asset processes are assumed idepedet, ad the withdrawal decisio does ot affect mortality process, oe ca calculate the expected value of the payoff 21) with respect to the mortality process, H0 W, A) = E I t 0 [H 0 X, γ)], ad the calculate the price uder the optimal strategy sup γ E Q t 0 [ H 0 W, A)] or uder the give strategy E Q t 0 [ H 0 W, A)]. It is easy to fid that H 0 W, A) = B 0,N p N P T W T ), AT ) ) + q N p N1 D N W T ), AT ) ) ) B 0, p f W t ), At ), γ ) + p 1 q D W t ), At ) )), 30) N1 + =1 14

15 where p = Pr[τ > t τ > t 0 ] ad q p 1 = Pr[t 1 < τ t τ > t 0 ] for radom death time τ, i.e. p = p 1 1 q ). Note that, previously we defied q = Pr[t 1 < τ t τ > t 1 ]. The payoff 30) has the same geeral form as the payoff 21). Thus, the optimal stochastic cotrol problem Ψ t0 W 0), A0)) = sup γ E Q t 0 [ H 0 W, A)] ca be solved usig Bellma equatio 27) leadig to the followig explicit recursio [ Ψ t + W, A) = E Q B t +,+1 Ψ t +1 W t +1 ), At +1) ) W, A], 31) Ψ t W, A) = max p f W, A, γ ) + p 1 q D W, A) γ A +Ψ t + h W W, A, γ ), h A W, A, γ ) ) ), 32) for = N 1, N 2,..., 0, startig from Ψ t W, A) = p N P T W, A) + p N1 q N D N W, A). N It is easy to verify that this recursio leads to the same solutio Ψ t0 W, A) = Q t0 W, A) ad the same optimal strategy for γ as obtaied from the recursio 28 29), otig that Ψ t W, A) = p Q t W, A) + p 1 q D W, A). The result is somewhat obvious because sup γ E Q,I t 0 [ [H 0 X, γ)] = sup E Q t 0 E I t0 [H 0 X, γ)] ]. 33) γ [ Note that, sup γ E Q,I t 0 [H 0 X, γ)] E I t supγ 0 E Q t 0 [H 0 X, γ)] ]. That is, oe caot fid the price uder the optimal strategy coditioal o the death time ad the average over radom death times, that would lead to the result larger tha Q t0 W, A), see Luo ad Shevcheko 2015b). 5.3 Remarks o Withdrawal Strategy The guaratee fare fee based o the optimal policyholder withdrawal is the worst case sceario for the issuer, i.e. if the guaratee is hedged the this fee will esure o losses for the issuer i other words full protectio agaist policyholder strategy ad market ucertaity). Of course this is uder the give assumptios about stochastic model for the uderlyig risky asset. If the issuer hedges cotiuously but ivestors deviate from the optimal strategy, the the issuer will receive a guarateed profit. Ay strategy differet from the optimal is sup-optimal ad will lead to smaller fair fees. Of course the strategy optimal i this sese is ot related to the policyholder circumstaces. The policyholder may act optimally with respect to his prefereces ad circumstaces but it may be differet from the optimal strategy calculated i 29). O the other had, as oted i Hilpert et al. 2014), secodary markets for equity liked isurace products where the policyholder ca sell their cotracts) are growig. Thus, fiacial third parties ca potetially geerate guarateed profit through hedgig strategies from fiacial products such as VA riders which are ot priced accordig to the worst case assumptio of the optimal withdrawal strategy. Thus the developmet of secodary markets for VA riders would lead to a icrease i the fees charged by the issuig compaies. Koller et al. 2015) udertakes a empirical study of policyholders behavior i Japaese VA market ad they show that the moeyess of the guaratee has the largest explaatory power for the surreder rates. 15

16 Oe way to itroduce a reasoable suboptimal withdrawal model is to assume that the policyholder follows a default strategy withdrawig a cotractual amout G at each evet time t uless the extra value from optimal withdrawal is greater tha θ G, θ 0. Settig θ = 0 correspods to the optimal strategy, while θ 1 leads to the strategy of withdrawals at the cotract rate. This is the approach cosidered e.g. i Forsyth ad Vetzal 2014) ad Che et al. 2008). More complicated approach would specify a life-cycle utility model to determie the strategy optimal for the policyholder with respect to his circumstaces ad prefereces, this is the approach studied i Moeig 2012); Horeff et al. 2015); Gao ad Ulm 2012); Steiorth ad Mitchell 2015). I ay case, oce the strategy is specified estimated empirically or by aother model), oe ca use 29) to calculate the fair price ad fair fee with the admissible strategy space A restricted to the specified strategy. 5.4 Tax cosideratio Withdrawals from the VA type cotracts may attract coutry ad idividual specific govermet taxes. Moeig ad Bauer 2015) demostrated that icludig taxes sigificatly affects the value of VA withdrawal guaratees. They developed a subjective risk-eutral valuatio methodology ad produced results i lie with empirical market prices. Followig closely to Moeig ad Bauer 2015), we itroduce a extra state variable Rt) to preset the tax base which is the amout that may still be draw tax-free, ad assume that all evet times t T are the policy aiversary dates. The iitial premium is assumed to be post-tax ad taxes are applied to future ivestmet gais ot the iitial ivestmet). Deote a margial icome tax rate as κ ad margial capital gai tax from ivestmet outside of VA cotract as κ. It is assumed that earigs from VA are treated as ordiary icome ad withdrawals are taxed o a last-i first-out basis. Thus if the wealth accout W t ) exceeds the tax base Rt ), ay withdrawal up to W t ) Rt ) will be taxed at the rate κ ad will ot affect the tax base; larger withdraws will ot be subject to tax but will reduce the tax base. Specifically, the tax base will be chaged at withdrawal time t as Rt + ) = Rt ) max γ t maxw t ) Rt ), 0), 0 ). The cashflow received by the policyholder will be reduced by taxes ) tax = κ mi f W t ), At ), γ ), maxw t ) Rt ), 0), i.e. oe has to make the followig chage i the cotract specificatios listed i Sectio 4 f W t ), At ), γ ) f W t ), At ), γ ) tax. Usig argumets for replicatig pre-tax cashlows at t with post-tax cashflows at t +1, it was show i Moeig ad Bauer 2015) that Q t + W, A, R) should be foud ot as the direct expectatio 28) but should be foud as the solutio of the followig oliear equatio [ V t +1 ) W, A, R ] Q t + W, A, R) = E Q t + + κ 1 κ EQ t + [ max [ V t +1) B,+1 Q t + W, A, R), 0 ] W, A, R 16 ], 34)

17 where V t +1) = 1 q +1 )B,+1 Q t +1 W t +1 ), At +1), Rt +1) ) +q +1 B,+1 D +1 W t +1 ), At +1), Rt +1) ). 35) This is referred to as subjective valuatio from the policyholder perspective ad depeds o the ivestor curret positio icludig possible offset tax resposibilities) ad tax rates. Numerical examples i Moeig ad Bauer 2015) show that the VA guaratee prices accoutig for taxes i the above way are lower tha igorig the taxes ot surprisigly, because it lead to the suboptimal strategy), makig the prices overall more aliged with those observed i the market. 6 Numerical Valuatio of VA riders I the case of realistic VA riders with discrete evets such as ratchets ad optimal withdrawals, there are o closed form solutios ad fair price has to calculated umerically, eve i the case of simple geometric Browia motio process for the risky asset. I geeral, oe ca use PDE, direct itegratio or regressio type MC methods, where the backward recursio 28 29) is solved umerically. Of course, if the withdrawal strategy is kow, the oe ca always use stadard MC to simulate state variables forward i time till the cotract maturity or policyholder death ad average the payoff cashflows over may idepedet realizatios. This stadard procedure is well kow ad o further discussio is eeded. I this sectio, we give a brief review of differet umerical methods that ca be used for valuatio of VA riders. The, we provide detailed descriptio of the direct itegratio method that ca be very efficiet ad simple to implemet, whe the trasitio desity of the uderlyig asset or it s momets betwee the evet times are kow i closed form. Fially, i Sectio 6.5 we preset calculatio of hedgig parameters referred i the literature as Greeks). 6.1 Numerical algorithms Simulatio based Least-Squares MC method itroduced i Logstaff ad Schwartz 2001) is desiged for ucotrolled Markov process problems ad ca be used to accout for the cotract early surreder, as e.g. i Baciello et al. 2011). However, it caot be used if a optimal withdrawal strategy is ivolved. This is because dyamic withdrawals affect the paths of the uderlyig wealth accout ad oe caot carry out a forward simulatio step required for the subsequet regressio i the backward iductio. However, it should be possible to apply cotrol radomizatio methods extedig Least-Squares MC to hadle the optimal stochastic cotrol problems with cotrolled Markov processes, as was recetly developed i Kharroubi et al. 2014). The idea is to first simulate the cotrol withdrawals) ad the state variables forward i time, where the cotrol is simulated idepedetly from other variables. The, use regressio o the simulated state variables ad cotrol to estimate expected value 28) ad fid the optimal withdrawal usig 29). However, the accuracy ad robustess of this method for pricig withdrawal beefit type products have ot bee studied yet. As usual, it is expected that the choice of the basis fuctios for the required regressio step will have sigificat impact o 17

18 the performace. We also ote that i some simple cases of the withdrawal strategy admissible space such as bag-bag o withdrawal, withdrawal at the cotractual rate, or full surreder), it is possible to develop other modificatios of Least-Squares MC such as i Huag ad Kwok 2015) for pricig of the GLWB rider. The expected value 28) ca also be calculated usig PDE or direct itegratio methods. I both cases, the modeller discretizes the space of the state variables ad the calculates the cotract value for each grid poit. The PDE for calculatio of expected value 28) uder the assumed risk-eutral process for the risky asset St) is easily derived usig Feyma-Kac theorem; for a good textbook treatmet of this topic, see e.g. Björk 2004). However, the obtaied PDE ca be difficult or eve ot practical to solve i the high-dimesioal case. I particular, i the case of geometric Browia motio process for the risky asset 4), the goverig PDE i the period betwee the evet times is the oe-dimesioal Black-Scholes PDE, with jump coditios at each evet time to lik the prices at the adjacet periods. Sice the beefit base state variable At) remais uchaged withi the iterval t i1, t i ), i = 1, 2,..., N, the cotract value Q t W, A) satisfies the followig PDE with o explicit depedece o A, Q t + σ2 2 W 2 2 Q Q + r α)w rq = 0. 36) W 2 W This PDE ca be solved umerically usig e.g. Crak-Nicholso fiite differece scheme for each A backward i time with the jump coditio 29) applied at the cotract evet times. This has bee doe e.g. i Dai et al. 2008) ad Che ad Forsyth 2008) for pricig GMWB with discrete optimal withdrawals. Of course, if the volatility or/ad iterest rate are stochastic, the extra dimesios will be added to the PDE makig it more difficult to solve. Forsyth ad Vetzal 2014) used PDE approach to calculate VA rider prices i the case of stochastic regime-switchig volatility ad iterest rate. Uder the direct itegratio approach, the expected value 28) is calculated as a itegral approximated by summatio over the space grid poits, see e.g. Bauer et al. 2008). More efficiet quadrature methods requirig less poits to approximate the itegral) exist. I particular, i the case of a geometric Browia motio process for the risky asset, it is very efficiet to use the Gauss-Hermite quadrature as developed i Luo ad Shevcheko 2014) ad applied for GMWB pricig i Luo ad Shevcheko 2015a). Sectio 6.3 provides detailed descriptio of the method for pricig VA riders i geeral. This method ca be applied whe the trasitio desity of the uderlyig asset betwee the evet times or it s momets are kow i closed form. It is relatively easy to implemet ad computatioally faster tha PDE method because the latter requires may time steps betwee the evet times. I Luo ad Shevcheko 2016), this method was also used to calculate GMWB i the case of stochastic iterest rate uder the Vasicek model. I both PDE ad direct itegratio approaches, oe eeds some iterpolatio scheme to implemet the jump coditio 29), because state variables located at the grid poits of discretized space do ot appear o the grid poits after the jump evet. This will be discussed i detail i Sectio 6.4. Of course, if the uderlyig stochastic process is more complicated tha geometric Browia motio 4) ad does ot allow efficiet calculatio of the trasitio desity or its momets, oe ca always resort to PDE method. 18

19 I our umerical examples of GMAB pricig i Sectio 7, we adapt a direct itergatio method based o the Gauss-Hermite itegratio quadrature applied o a cubic splie iterpolatio, hereafter referred to as GHQC. For testig purposes, we also implemeted Crak- Nicholso fiite differece FD) scheme solvig PDE 36) with the jump coditio 29). 6.2 Overall algorithm descriptio Both PDE ad direct itegratio umerical schemes start from a fial coditio for the cotract value at t = T. The, a backward time steppig usig 28) or solvig correspodig PDE gives solutio for the cotract value at t = t + N1. Applicatio of the jump coditio 29) to the solutio at t = t + N1 gives the solutio at t = t N1 from which further backward i time recursio gives solutio at t 0. For simplicity assume that there are oly W t) ad At) state variables. The umerical algorithm the takes the followig key steps. Algorithm 6.1 Direct Itegratio or PDE method) Step 1. Geerate a auxiliary fiite grid 0 = A 1 < A 2 < < A J base balace A. to track the beefit Step 2. Discretize wealth accout balace W space as W 0 < W 1 < < W M to geerate the grid for computig the expectatio 28). Step 3. At t = t N, apply the fial coditio at each ode poit W m, A j ), j = 1, 2,..., J, m = 1, 2,..., M to get Q t W, A). N Step 4. Evaluate itegratio 28) for each A j, j = 0,..., J, to obtai Q t + W, A) either N1 usig direct itegratio or solvig PDE. I the case of direct itegratio method, this ivolves oe-dimesioal iterpolatio i W space to fid values of Q t W, A) at the N guadrature poits differet from the grid poits. Step 5. Apply the jump coditio 29) to obtai Q t A) for all possible values of γ N1W, N1 ad fid γ N1 that maximizes Q t W, A). I geeral, this ivolves a two-dimesioal N1 iterpolatio i W, A) space. Step 6. Repeat Step 4 ad 5 for t = t N2, t N3,..., t 1. Step 7. Evaluate itegratio 28) for the backward time step from t 1 to t 0 to obtai solutio Q 0 W, A) at W = W 0) ad A0), or may be at several poits if these are eeded for calculatio of some hedgig sesitivities such as Delta ad Gamma discussed i Sectio 6.5. I our implemetatio of the direct itegratio method based o the Gauss-Hermite quadrature for umerical examples i Sectio 7, we use a oe-dimesioal cubic splie iterpolatio required to hadle itegratio i Step 4 ad bi-cubic splie iterpolatio to hadle jump coditio Step 5. 19

20 If the model has other stochastic state variables similar to W ) chagig stochastically betwee the cotract evet times, such as stochastic volatility ad/or stochastic iterest rate, the grids for these extra dimesios should be geerated ad the required itegratio or PDE to evaluate 28) will have extra dimesios. Also, extra auxiliary state variables similar to A) uchaged betwee the cotract evet times, such as tax base ad/or extra beefit base, will require extra dimesios i the grid ad iterpolatio for the jump coditio at the evet times. We have to cosider the possibility of W t) goes to zero due to withdrawal ad market movemet, thus oe has to use the lower boud W 0 = 0. The upper boud W M should be set sufficietly far from the iitial wealth at time zero W 0). A good choice of such a boudary could be based o the high quatiles of distributio of ST ). For example, i the case of geometric Browia motio process 5), oe ca set coservatively W M = W 0)e mealst )/S0))) +5 stdevlst )/S0))). Ofte, it is more efficiet to use equally spaced grid i l W space. I this case, W 0 caot be set to zero ad istead should be set to a very small value e.g. W 0 = ). Also, for some VA riders, usig equally spaced grid i l A space is also more efficiet. 6.3 Direct itegratio method To compute Q 0 W 0), A0)), we have to evaluate the expectatios i the recursio 28). Assumig the coditioal probability desity of W t ) give W t + 1) is kow i closed form p w W t + 1)), the required expectatio 28) ca be calculated as where Q t + 1 W t + 1 ), A ) + = p w W t + 1)) Q t w, A)dw, 37) 0 Q t w, A) = B 1, 1 q )Q t w, A) + q D w, A) ). The above itegral ca be estimated usig various umerical itegratio quadrature) methods. Note that, oe ca always fid W t ) as a trasformatio of the stadard ormal radom variable Z as W t ) = ψz) := F 1 ΦZ)), where Φ ) is the stadard ormal distributio, ad F ) ad F 1 ) are the distributio ad its iverse of W t ). The, the itegral 37) ca be rewritte as Q t + 1 W t + 1 ), A ) = 1 + 2π e 1 2 z2 Qt ψ z), A)dz. 38) This type of itegrad is very well suited for the Gauss-Hermite quadrature that for a arbitrary fuctio fx) gives the followig approximatio + e x2 fx)dx q i=1 λ q) i fξ q) i ). 39) 20