Fair Pricing of Variable Annuities with Guarantees under the Benchmark Approach
Jin Sun, Kevin Fergusson, Eckhard Platen, Pavel V. Shevchenko
FFair Pricing of Variable Annuities with Guarantees under theBenchmark Approach
Jin Sun a,c , Kevin Fergusson b, ∗ , Eckhard Platen a,e , Pavel V. Shevchenko d a Faculty of Science, University of Technology Sydney b Centre for Actuarial Studies, Department of Economics, University of Melbourne, Victoria, Australia c Data 61, CSIRO Docklands d Department of Actuarial Studies and Business Analytics, Macquarie University, Australia e UTS Business School, University of Technology Sydney
Abstract
In this paper we consider the pricing of variable annuities (VAs) with guaranteed minimumwithdrawal benefits. We consider two pricing approaches, the classical risk-neutral approachand the benchmark approach, and we examine the associated static and optimal behaviorsof both the investor and insurer. The first model considered is the so-called minimal marketmodel, where pricing is achieved using the benchmark approach. The benchmark approachwas introduced by Platen in 2001 and has received wide acceptance in the finance community.Under this approach, valuing an asset involves determining the minimum-valued replicatingportfolio, with reference to the growth optimal portfolio under the real-world probabilitymeasure, and it both subsumes classical risk-neutral pricing as a particular case and extendsit to situations where risk-neutral pricing is impossible. The second model is the Black-Scholesmodel for the equity index, where the pricing of contracts is performed within the risk-neutralframework. Crucially, we demonstrate that when the insurer prices and reserves using theBlack-Scholes model, while the insured employs a dynamic withdrawal strategy based on theminimal market model, the insurer may be underestimating the value and associated reservesof the contract.
JEL classification : C61, G22
Keywords : variable annuity guarantee, stochastic optimal control, stochastic reserve, bench-mark approach
1. Introduction
Variable annuities (VA) with guarantees of living and death benefits are offered by wealthmanagement and insurance companies worldwide to assist individuals in managing their pre- ∗ Corresponding author
Email addresses: [email protected] (Jin Sun), [email protected] (Kevin Fergusson), [email protected] (Eckhard Platen), [email protected] (Pavel V. Shevchenko) a r X i v : . [ q -f i n . P R ] J un etirement and post-retirement financial plans. These products take advantage of marketgrowth while providing a protection of the savings against market downturns. The VA contractcash flows received by the policyholder are linked to the choice of investment portfolio (e.g.the choice of mutual fund and its strategy) and its performance while traditional annuitiesprovide a pre-defined income stream in exchange for a lump sum payment.A variety of VA guarantees can be added by policyholders at the cost of additionalfees. Common examples of VA guarantees include guaranteed minimum accumulation benefit(GMAB), guaranteed minimum withdrawal benefit (GMWB), guaranteed minimum incomebenefit (GMIB) and guaranteed minimum death benefit (GMDB), as well as combinationsof these, e.g., guaranteed minimum withdrawal and death benefit (GMWDB), among others.These guarantees, generically denoted as GMxB, provide different types of protection againstmarket downturns, shortfall of savings due to longevity risk or assurance of stability of incomestreams. Precise specifications of these products can vary across categories and issuers; seeBauer et al. (2008); Ledlie et al. (2008) and Kalberer and Ravindran (2009) for an overviewof these products.Since the recent financial crisis, the need for accurate estimation of hedging costs of VAguarantees has become increasingly important. Such estimation includes the pricing of fu-ture cash flows that must be paid by the insurer to the policyholder in order to fulfill theliabilities of the VA guarantees, as well as the associated hedging strategy to deliver the lia-bility payments. The standard pricing approach follows the risk-neutral pricing theory, whereunder the condition of no-arbitrage, the VA product is priced as the expectation of the to-tality of discounted future cash flows offered by the product under the so-called equivalentrisk-neutral pricing measure. There have been a number of contributions in the academicliterature considering numerical methods for the pricing of VA guarantees. These includestandard and regression-based Monte Carlo, partial differential equation (PDE) and directintegration methods. A comprehensive overview of numerical methods for the pricing of VAguarantees is provided in Shevchenko and Luo (2016).The benchmark approach (BA) offers an alternative pricing theory. Under very generalconditions, there exists in a given investment universe a unique growth-optimal portfolio(GP). The BA takes the GP as the numeraire, or benchmark, such that any benchmarkednonnegative portfolio price process assumes zero expected instantaneous returns. The GPassumes the highest expected instantaneous growth rate among all nonnegative portfolios inthe investment universe, and maximizes the expected log-utility of the terminal wealth. It isa well-diversified portfolio that draws on all tradable risk factors and the corresponding riskpremia to achieve the growth optimality. Following the real-world pricing formula under theBA, real-world pricing of any given future cash flows identifies their minimal possible replica-tion cost. The BA is considered as a generalization of the risk-neutral pricing theory, in thatan equivalent risk-neutral pricing measure need not exist, yet it includes risk-neutral pricing2s a special case involving additional conditions that ensure the existence of an equivalentrisk-neutral probability measure. Another special case is the minimal market model (MMM),described in Appendix B, which permits mean-reversion of the GP around an exponentiallygrowing market trend and which captures both the leptokurtic behavior of logreturns, asshown in Fergusson and Platen (2006), and the leverage effect, manifest as a spike in volatil-ity in a rapidly falling market; see for example Campbell and Viceira (2005) and Shiller (2015).In contrast, the Black-Scholes model (BSM), also described in Appendix B, captures onlymesokurtic behavior of logreturns, and is unable to model the leverage effect because of itsconstant volatility.In this paper we consider a standard VA product with GMWB, which provides a guar-anteed withdrawal amount per year until the maturity of the contract, regardless of theinvestment performance. The total withdrawal amount is such that the initial investment isguaranteed to be returned over the life of the contract. Additional features, such as a deathbenefit, can be added straightforwardly if so desired, see, e.g., Luo and Shevchenko (2015).Two classes of withdrawal strategies of the policyholder are often considered in the liter-ature: a static withdrawal strategy under which the policyholder withdraws a predeterminedamount on each withdrawal date; or a dynamic strategy where the policyholder “optimally”decides the amount of withdrawal at each withdrawal date depending on the information avail-able at that date, where the optimality usually refers to the maximization of the value of thecurrent and future cash flows. By assuming an optimal policyholder’s withdrawal behavior,the pricing of the VA product corresponds to the hedging cost of the worst case scenario facedby the VA provider. In other words, the price of the VA product under the respective dynamicstrategy provides an upper bound of hedging cost from the VA provider’s perspective; see Sunet al. (2018). It should be noted that the actual policyholders’ withdrawal strategies could befar from optimal; see, e.g., Moenig and Bauer (2015).Assuming that the policyholder takes the dynamic withdrawal strategy, that is, the optimalstrategy that maximizes the present value of current and future cash flows of the VA product,the actual withdrawals still depend on the pricing method adopted by the policyholder. Onthe other hand, the VA provider, who maintains a hedging portfolio to deliver the liability cashflows of the VA product also faces the same choices for pricing and hedging for the portfolio.In this paper, we consider two pricing methods, the risk-neutral pricing approach and the BA.We investigate the outcomes and implications of different choices of withdrawal and hedgingstrategies by the policyholder and the VA provider. In particular, we study empirically thecases when the two parties take different pricing approaches. In the VA pricing literature itis most often the case that the same pricing model is adopted by both the policyholder andthe VA provider. The important situation where the policyholder and the VA provider holdfundamentally different valuation perspectives, as is described in this paper, has not beeninvestigated. This paper attempts to fill this vacancy and hopefully initiate more interest in3his direction.The paper is organized as follows. In Section 2 we present the contract details of theGMWB guarantee together with its pricing formulation under a stochastic optimal controlframework. Then, in Section 3 we empirically test these pricing and corresponding withdrawaland hedging strategies, respectively, for the policyholder and the VA provider, when the twoparties use the same or different modeling approaches. Section 4 concludes with remarksand discussion. The appendices contain background technical aspects of our VA modelingand pricing approaches. Appendix A provides an overview of the benchmark approach. InAppendix B we describe the pricing models under the risk-neutral approach and the BA.Finally, in Appendix C the VA pricing problem is formulated for optimal policyholder’swithdrawals under both pricing frameworks and models, and the numerical algorithm to solvethe problem is described.
2. Description of the VA Guarantee Product
We consider the VA product where a policyholder invests at time t = 0 a lump-sumof W (0) into a wealth account W ( t ) , t ∈ [0 , T ] that tracks an equity index S ( t ) , t ∈ [0 , T ],where t = T corresponds to the expiry date of the VA contract. We assume both W ( t ) and S ( t ) are discounted by the locally risk-free savings account, as are all other values of wealthencountered hereafter. The (discounted) equity index evolves under the real-world probabilitymeasure P according to the SDE dS ( t ) S ( t ) = µ ( t ) dt + σ ( t ) dB ( t ) , t ∈ [0 , T ] , (1)where µ ( t ) and σ ( t ) are the instantaneous market risk premium and volatility of the index,respectively. Here B ( t ), t ∈ [0 , T ], is a standard P -Brownian motion driving traded marketuncertainties.As mentioned in Appendix B, the diversified equity index approximates well the GP, andwe have, as a particular case of (1), the Black-Scholes model (BSM) of the GP, specified bythe SDE (B.1). Also, as another particular case of (1), we have the minimal market model(MMM) of the GP, specified by the SDE (B.5). Further details of both of these models aresupplied in Appendix B.The policyholder selects a GMWB rider in order to protect his wealth account W ( t ), t ∈ [0 , T ], over the lifetime of the VA contract. The GMWB contract allows the policyholderto withdraw from a guarantee account A ( t ), t ∈ [0 , T ], on a sequence of pre-determinedcontract event dates, 0 = t < t < · · · < t N = T . The initial guarantee A (0) matches theinitial wealth W (0). We assume here that the guarantee account stays constant over time,unless a withdrawal is made on one of the event dates, which reduces the guarantee account4alance. Other forms of guaranteed returns can be modeled similarly. For simplicity, wedo not include in our discussion features such as death or early surrender benefits. Under amore realistic setting, these additional features can be included straightforwardly within theframework described in this paper.To simplify notations, we denote by X ( t ) the vector of state variables at time t , given by X ( t ) = ( µ ( t ) , σ ( t ) , S ( t ) , W ( t ) , A ( t )) , t ∈ [0 , T ] . (2)Here, we assume that the variable S ( t ) follows a Markov process, so that X ( t ) contains allthe market and account balance information available at t . For simplicity, we assume thatthe state variable S ( t ) is continuous, and W ( t ) and A ( t ) are left-continuous with right-handlimit (LCRL).On event dates t n , n = 1 , . . . , N , a nominal withdrawal γ n from the guarantee accountis made. The policyholder may choose γ n ≤ A ( t n ) on t n < T . Otherwise, a liquidationwithdrawal of max( W ( t n ) , A ( t n )) is made. That is, γ n = Γ( t n , X ( t n )) , n < N , max ( W ( t n ) , A ( t n )) , n = N , (3)where Γ( · , · ) is referred to as the withdrawal strategy of the policyholder. The net cashflow received by the policyholder, which may differ from the gross amount, is denoted by C n ( γ n , X ( t n )). In our case this cash flow is set to C n ( γ n , X ( t n )) = γ n − β max( γ n − G n , , n < N, max ( W ( T ) , A ( T )) − β max( A ( T ) − G N , , n = N, (4)where G n is a pre-determined withdrawal amount specified in the GMWB contract, and β isthe penalty rate applied to the part of the withdrawal from the guarantee account exceedingthe contractual withdrawal G n . Here we assume the penalty also applies to the last withdrawalof the guarantee account A ( T ). The part of the wealth account balance in excess of theguarantee account balance is not subject to the penalty. Upon withdrawal by the policyholder,the guarantee account is reduced by the nominal withdrawal γ n , that is, A ( t + n ) = A ( t n ) − γ n , (5)where A ( t + n ) denotes the guarantee account balance “immediately after” the withdrawal. Notethat A ( t + n ) ≥
0. The wealth account is reduced by the amount min( γ n , W ( t n )) and remainsnonnegative. That is, W ( t + n ) = max( W ( t n ) − γ n , , (6)5here W ( t + n ) is the wealth account balance immediately after the withdrawal. It is assumedthat γ = 0, i.e., there are no withdrawals at the start of the contract. Both the wealth andthe guarantee account balance are 0 after contract expiration. That is, we have W ( T + ) = A ( T + ) = 0 . (7)Throughout the VA contract, the wealth account is charged an insurance fee continuouslyat rate α tot for the GMWB rider by the insurer to pay for the hedging cost of the guarantee.Discrete fees may be modeled similarly without any difficulty. The wealth account in turnevolves as dW ( t ) W ( t ) = ( µ ( t ) − α tot ( t )) dt + σ ( t ) dB ( t ) , (8)for any t ∈ [0 , T ] at which no withdrawal of wealth is made. Here, α tot ( t ) = α ins ( t ) + α m ( t ) isthe total fee rate, where α ins denotes the insurance fee and α m denotes the management fee.We denote the VA plus guarantee value function at time t by V ( t, X ( t )) , t ∈ [0 , T ], whichcorresponds to the present value under the respective pricing model of all future cash flowsentitled to the policyholder on or after the current time t . The remaining value after the finalcash flow is, obviously, 0, i.e., V ( T + , X ( T + )) = 0 . (9)
3. Backtesting the Reserving and Policyholder Strategies
In this section, we conduct backtests of the two proposed pricing models for VA products.We consider both from the policyholder’s perspective, where he or she decides the optimalwithdrawal amount based on the pricing model chosen by the policyholder, and from the VAprovider’s perspective, where the strategy of the hedging portfolio for the VA product is basedon the pricing model chosen by the provider.We take the S&P500 index as the equity index underlying the VA product, and runsimulated withdrawals and hedging strategies on the historical data of the underlying well-diversified index. In particular, we take the historically observed monthly prices of the S&P500index from 1871 to 2018, with all dividends reinvested, and discounted by the locally risk-free savings account. The S&P500 data after 1963 are obtained from Datastream, and theearlier part from 1871 until 1963 is reconstructed in Shiller (2015). The S&P500 total returnindex provides a good approximation of the dynamics of the market portfolio (MP) of theUS domestic stock market. We consider the pricing of a GMWB contract written on a VAaccount tracking the index over the 30-year period from Feb. 1988 to Feb. 2018, based on theunderlying models estimated from the historical index prices prior to this period. We considerboth the MMM and the BSM as the underlying dynamics, as specified by SDEs (B.5) and(B.1) respectively, and compare the evolution of the guarantee values under both models and6able 1: Estimated model parameters from the S&P500 historical prices. α η σ MMM 2.857 0.0435 -BSM - - 0.1441respective pricing rules.The log-prices of the S&P500 total return index are shown in Figure 1 (a). We take thefirst 97 years of the available data, from Jan. 1871 to Feb 1988 for estimations. The MMMparameters were estimated as follows: To estimate the overall trend of growth S ( t ) Y ( t ) = α η e ηt for the MMM, a straight line is fitted to the log-index prices and the slope is taken as theestimated total growth rate η , and the scaling factor α is determined from the intercept. Thenormalized index Y ( t ) thus obtained is shown in Figure 1 (b). The BSM volatility in (B.1)was estimated following the standard MLE estimator. The estimated parameters are shownin Table 1. /
01 1892 /
01 1913 /
01 1934 /
01 1955 /
02 1976 /
02 1997 /
02 2018 / S&P500trend /
01 1892 /
01 1913 /
01 1934 /
01 1955 /
02 1976 /
02 1997 /
02 2018 / normalized indexmean (a) S&P500 index with the estimated trend (b) normalized index with the meanFigure 1: Estimated trend of growth and normalized index for the MMM.We consider a stylized VA contract with GMWB where the policyholder invests 1 Millionunits of the US dollar savings account in a mutual fund that tracks the S&P500 total returnindex. For illustrative purposes, we assume there are no mutual fund management fees, sothat α m ( t ) = 0 in (8), and the insurance fee rate α ins is set at 0. (The case with nonzerototal fees can be considered strictly analogously without affecting the main results from thecurrent discussion. ) The policyholder purchases a GMWB rider that guarantees equal annualpayments of the initial investment of 1 Million savings account units over a period of 30 years.The contracted withdrawals are, therefore, rated at 33,333 units of the savings account perannum. If the policyholder decides to withdraw more than the contracted amount, a penaltycharge of 10% should apply to the excess part of the withdrawal. As mentioned in Section 2,7e assume that the penalty charge also applies to thew last withdrawal. That is, if the balanceof the guarantee account exceeds the contracted withdrawal amount at maturity, withdrawalof this balance is mandatory and the same penalty rate applies to the excess part.We first consider the situation where the VA provider prices the product under the BSMwith the risk-neutral pricing approach (see Appendix B) assuming that the policyholdermakes optimal withdrawals under the same pricing rule. The VA provider maintains a nominalwealth account W ( t ) of the policyholder’s wealth, and a nominal guarantee account A ( t ) tokeep track of the remaining guaranteed withdrawal allowance. The VA provider maintains anactual hedging portfolio V ( t ) consisting of shares of the index-tracking mutual fund, or theindex for short, and the locally risk-free savings account. We refer to the hedging portfolioas the reserve account of the VA product, which is the only real investment account involved.The reserve account starts at value V (0), the initial price of the VA product, and maintainsa self-financing hedging strategy until a withdrawal is made on one of the withdrawal dates t n , when the net cash flow C n is paid out of this account to the policyholder. The strategymaintained by the reserve account between withdrawal dates is the respective delta-hedgingstrategy.Following Algorithm 1 described in Appendix C, we compute recursively the price processof the VA product, based on the historical index prices. The price process is shown inFigure 2 (a), with an initial value of 1.22 Million, and a terminal value of 7.11 Million beforethe final liquidation. The reserve account process, realized through delta-hedging, is shownin the same plot for comparison. The intial value of the reserve account is the same as theprice process, and the terminal value is 7.07 Million. After liquidation, the reserve accountended up with a small deficit of -0.0344 Million, possibly due to hedging errors from discretehedging.The nominal wealth account W ( t ) is shown in Figure 2 (b), the optimal withdrawalsmade by the policyholder are shown in Figure 2 (c), and the guarantee account balance isshown in Figure 2 (d). Note that both the wealth account and the guarantee account arenominal only, used for keeping track of the status of the policyholder’s VA contract. Noactual trading happens to these accounts. The optimal withdrawals are relatively uniform,except for no withdrawals in the beginning periods, and a large withdrawal in the last period.The relatively uniform withdrawal behavior is typical for the BSM and risk-neutral pricing,where the equity index dynamics is time-homogeneous. The only motivations to change thewithdrawal pattern are changing the maturity date and wealth / guarantee account ratio.For verification purposes we consider the alternative static withdrawal behavior from thepolicyholder. That is, we assume the policyholder makes a uniform withdrawal equal to 1 /N Million units of the savings account on any one of the N withdrawal dates. On the otherhand, the VA provider manages the reserve account in the same way as in the optimal caseirrespective of the policyholder’s withdrawal behavior, which is considered suboptimal in this8 Years V a l u e contract valuereserve account value Years V a l u e (a) contract value and reserve account (b) nominal wealth account V a l u e Years
Years V a l u e (c) withdrawals (d) nominal guarantee accountFigure 2: Value processes (in millions of units of the savings account) associated with the VAproduct when the pricing and hedging as well as optimal withdrawals are performed based onthe BSM and risk-neutral pricing. 9ase. The results are shown in Figure 3. The contract value process in this case is thesame as for the case with optimal withdrawals. The reserve account process as well as thenominal wealth account process differ from the previous case due to different (suboptimal)withdrawals. The reserve account ended up with a rather significant surplus of 1.39 Million.This is due to the loss made by the policyholder for withdrawing suboptimally. In particular,premature withdrawals led to less wealth accumulations in the nominal wealth account, leadingto significantly less liquidation cash flow entitled to the policyholder. Since the reserve accountmaintained the same hedging strategy as in the previous case, it ended up having a surplusafter paying the reduced liabilities. Years V a l u e contract valuereserve account value Years V a l u e (a) contract value and reserve account (b) nominal wealth account V a l u e Years
Years V a l u e (c) withdrawals (d) nominal guarantee accountFigure 3: Value processes associated with the VA product when the pricing and hedgingare performed based on the BSM and risk-neutral pricing assuming an optimal withdrawalbehavior, while the actual withdrawal behavior follows the static strategy.We next consider the situation where the policyholder makes withdrawals based on theMMM under the BA. That is, the policyholder’s withdrawals maximize the value of the VAcontract as priced by the MMM under the BA. The VA provider, believing in the BSMunder the risk-neutral pricing framework, manages the reserve account in the same way as inthe previous cases, and views the policyholder’s withdrawals as being suboptimal. The VAprovider thus expects to receive a surplus in the reserve account after maturity of the VAcontract. The outcomes of this scenario are shown in Figure 4.10 Years V a l u e contract valuereserve account value Years V a l u e (a) contract value and reserve account (b) nominal wealth account V a l u e Years
Years V a l u e (c) withdrawals (d) nominal guarantee accountFigure 4: Value processes associated with the VA product when the pricing and hedgingare performed based on the BSM and risk-neutral pricing assuming an optimal withdrawalbehavior, while the actual withdrawal behavior follows the MMM under the BA.11o the surprise of the VA provider, instead of having a surplus, the reserve account inthis case ended up with a deficit of 1 Million, as indicated in Figure 4 (a). The withdrawalbehavior of the policyholder is such that there are no withdrawals until the very end of thecontract term, where a number of small withdrawals were followed by a large withdrawal onthe maturity date. The nominal wealth account accumulated a high level of wealth due to nowithdrawals in the early stages. The liquidation of this large wealth led to the deficit of thereserve account, which followed a hedging strategy assuming more early withdrawals.The failure of the VA provider in hedging the VA product, when the policyholder behavedoptimally under a different pricing framework, indicates the potential inappropriateness ofthe BSM and risk-neutral pricing adopted by the VA provider. In particular, the policyholderbelieved in the long-term growth of the market and invested for this growth according to theMMM. The VA provider, from a risk-neutral perspective, did not recognize the long-termgrowth, and managed the reserve account with a short-term vision, leading to the failure ofmatching the performance of the policyholder’s wealth account. Note that the MMM and theassociated long-term growth rate were estimated from prior returns of the index. Thus, no“looking into the future” is associated with the policyholder’s withdrawal behavior.It is interesting to see what happens in a reversed scenario, where the VA provider pricesand hedges under the MMM and BA, and the policyholder makes optimal withdrawals ac-cording to the BSM and risk-neutral pricing. Without repeating the detailed descriptionof this scenario, the outcomes are shown in Figure 5, where the reserve account was man-aged recognizing the long-term growth under the MMM and BA, leading to a higher level ofwealth accumulation than the nominal wealth account, and a surplus of 1.05 Million. Thewithdrawal behavior of the policyholder is similar to the first case considered, with a ratheruniform withdrawal a few years into the contract term.Finally, to complete the empirical study, we consider the scenario when both the VAprovider and the policyholder follow the MMM under the BA. The outcomes are shown inFigure 6. It can be seen that the VA provider in this case successfully hedged the VA product,ending up with a small deficit of 0.0319 Million in the reserve account.When comparing the two optimal withdrawal strategies based on the two pricing models,then one realizes that the strategy based on the BSM and risk-neutral pricing realizes a totalwithdrawal of 7.41 Million (of the locally risk-free security) over time, more than the totalwithdrawal of 6.29 Million following the static strategy. On the other hand, the optimalstrategy based on the BA using the MMM realized a total withdrawal of 8.47 Million, whichis producing more than 1 Million units of the savings account, thus, doubling the initialinvestment when counted in units of the savings account. This is considerably more than therisk-neutral approach delivered using the BSM. The reason is that the BSM under risk-neutralpricing creates a significant model error, which neglects the significantly positive long-termgrowth rate of the S&P500 over that of the savings account and, thus, gives major investment12 Years V a l u e contract valuereserve account value Years V a l u e (a) contract value and reserve account (b) nominal wealth account V a l u e Years
Years V a l u e (c) withdrawals (d) nominal guarantee accountFigure 5: Value processes associated with the VA product when the pricing and hedging areperformed based on the MMM and BA assuming an optimal withdrawal behavior, while theactual withdrawal behavior follows the BSM under risk-neutral pricing.13otential away. By changing the production method in the area of VAs as demonstrated,significantly higher returns on investment can be achieved. Years -20246810 V a l u e contract valuereserve account value Years -20246810 V a l u e (a) contract value and reserve account (b) nominal wealth account V a l u e Years
Years V a l u e (c) withdrawals (d) nominal guarantee accountFigure 6: Value processes associated with the VA product when the pricing and hedging aswell as the optimal withdrawals are performed based on the MMM and the BA.
4. Conclusions
We considered the pricing of a variable annuity (VA) with GMWB under the benchmarkapproach (BA), where a classical equivalent risk-neutral pricing measure may not exist. Weemployed the real-world pricing formula to compute the value of the VA contract with GMWBunder the minimal market model (MMM), and the associated hedging strategy for the VAproduct. We compared our results with those under the classical Black-Scholes model (BSM)under the risk-neutral pricing framework, through empirical backtests on the historical pricesof the S&P500 total return index, which were taken as the underlying of the VA product.From the empirical studies, we found that the VA provider can successfully hedge theVA product when the VA provider and the policyholder both employ the same pricing modelto make hedging and withdrawal decisions. When the policyholder took a static withdrawalstrategy without optimization, the VA provider ended up with a surplus. When the VAprovider relied on the MMM and the BA to make hedging decisions and the policyholder took14he BSM and the risk-neutral approach, the VA provider ended up with a surplus. However,when the VA provider took the BSM and risk-neutral approach, and the policyholder relied onthe MMM and the BA, the VA provider ended up with a deficit. Our empirical studies showthat the BSM and risk-neutral approach to the VA pricing problem may not be appropriate,in that when a sophisticated policyholder armed with a more accurate model such as theMMM, the VA provider risks having a significant deficit in the hedging of the VA product.
References
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Appendix A. A Brief Overview of the Benchmark Approach
In this section, we give a brief overview of the BA under a diffusive market model. Muchof this section follows from Platen and Heath (2006), to which interested readers are referredfor a more complete treatment.We consider a general diffusive financial market model with uncertainties driven by a d-dimensional Brownian motion W , with W ( t ) = { ( W ( t ) , ..., W ( t ) d ) (cid:62) , t ∈ [0 , T ] } , defined ona filtered probability space (Ω , F , F , P ), where T is some fixed time horizon, and the filtra-tion F = {F t , t ∈ [0 , T ] } satisfies the usual conditions of right continuity and completeness,and models the accumulation of information over time; see Karatzas and Shreve (1991). Weassume that there exists a locally risk-free savings account S ( t ) and m nonnegative risky pri-mary security accounts S ( t ) = ( S ( t ) , ..., S m ( t )) (cid:62) satisfying the vector stochastic differentialequation (SDE) d S ( t ) = S ( t ) ( a ( t ) dt + b ( t ) · d W ( t )) , t ∈ [0 , T ] , (A.1)where a ( t ) is the instantaneous drift vector and b ( t ) the instantaneous volatility matrix, whichboth are assumed to be predictable and such that a unique strong solution of the above systemof SDEs exists. We assume that all dividends and interests are reinvested. Without loss ofgenerality, we further assume that S ( t ) ≡
1. This means that we denominate all securityprices in units of the locally risk-free savings account. In practice, the locally risk-free savingsaccount may be approximated by the money market account that invests in short-term T-billsin a rolling manner. Thus in our notation, all primary security accounts are discounted bythe locally risk-free savings account. 16e denote by S π the value process of a strictly positive, self-financing portfolio withportfolio weights π ( t ) = ( π ( t ) , ..., π m ( t )) (cid:62) , t ∈ [0 , T ], which invests at time t a fraction π j ( t )of the total wealth in the j th primary security account, and the remaining wealth in thelocally risk-free savings account. The value process satisfies then the SDE dS π ( t ) S π ( t ) = π ( t ) (cid:62) ( a ( t ) dt + b ( t ) · d W ( t )) , t ∈ [0 , T ] . (A.2)By Ito’s formula, the SDE for the log-price is of the form d log S π ( t ) = π ( t ) (cid:62) (cid:18)(cid:18) a ( t ) − b ( t ) b ( t ) (cid:62) π ( t ) (cid:19) dt + b ( t ) · d W ( t ) (cid:19) , t ∈ [0 , T ] . (A.3)We consider the growth-optimal portfolio (GP) S π ∗ of this investment universe for which theinstantaneous expected growth rate, that is, the drift of (A.3), is maximized for all t . This isachieved by setting the optimal portfolio weights π ∗ ( t ) to π ∗ ( t ) = arg max π π (cid:62) (cid:18) a ( t ) − b ( t ) b ( t ) (cid:62) π (cid:19) , t ∈ [0 , T ] . (A.4)We assume that a solution to (A.4) exists a.s. for all t ∈ [0 , T ]. One potential such solutionis given by π ∗ ( t ) = (cid:16) b ( t ) b ( t ) (cid:62) (cid:17) + a ( t ) , t ∈ [0 , T ] , (A.5)where (cid:0) b ( t ) b ( t ) (cid:62) (cid:1) + denotes the Moore-Penrose generalized inverse of the self-adjoint matrix b ( t ) b ( t ) (cid:62) . Note that the value process of the GP is unique, however, the fractions may varydue to potential redundancies in the primary security accountsFor the market model to be viable, we assume that the GP process, denoted as S ( t ) := S π ∗ ( t ) , t ∈ [0 , T ], with π ∗ ( t ) given by (A.5), exists and is strictly positive. By substituting(A.5) into (A.2), we obtain the SDE dS ( t ) S ( t ) = (cid:107) θ ( t ) (cid:107) dt + θ ( t ) · d W ( t ) , t ∈ [0 , T ] , (A.6)where θ ( t ) = b ( t ) (cid:62) π ∗ ( t ). The above SDE can further be written as dS ( t ) = α ( t ) dt + (cid:112) α ( t ) S ( t ) dB ( t ) , t ∈ [0 , T ] , (A.7)where the drift α ( t ) = (cid:107) θ ( t ) (cid:107) S ( t ) is assumed to be strictly positive, and B ∗ ( t ), defined bythe SDE dB ( t ) = θ ( t ) (cid:107) θ ( t ) (cid:107) · d W ( t ) t ∈ [0 , T ] , (A.8)with B ∗ (0) = 0, forms a standard Brownian motion by Levy’s characterization theorem. So17ar, we only re-parametrized the GP dynamics different to the common volatility modelingspecification. Note that the above drift α ( t ) can be, at this stage, still very general. Lateron, we will make this drift more specific, which yields then a proper model.The GP is the unique portfolio which, when used as numeraire or benchmark, makesany benchmarked portfolio process ˆ S π , defined as ˆ S π ( t ) = S π ( t ) S ( t ) , a local martingale. If weassume the portfolio process to be nonnegative, the benchmarked portfolio process becomesa supermartingale by Fatou’s lemma. Given an F T -measurable nonnegative contingent claim H ( T ) ≥ T , its, so called, fair price process under the BA is given by thereal-world pricing formula as H ( t ) = E t (cid:18) S ( t ) S ( T ) H ( T ) (cid:19) , t ∈ [0 , T ] , (A.9)where E t ( · ) = E ( ·|F t ) denotes the F t -conditional expectation under the real-world probabilitymeasure P . The benchmarked fair price process, defined as ˆ H ( t ) = H ( t ) S ( t ) , forms then a non-negative ( F , P )-martingale. The benchmarked fair price process ˆ H , if replicable, representsthe least expensive portfolio among all benchmarked nonnegative self-financing replicationportfolios, which form supermartingales. Appendix B. Modeling the Underlying Equity Index
In this section we specify the model parameters of the SDE governing the underlyingequity index (1). The model parameters are described respectively under the risk-neutralpricing framework and the BA. As mentioned in Section 2, we take the locally risk-freesecurity account as the numeraire, and consider all prices denominated in units of the locallyrisk-free savings account.
Appendix B.1. The Black-Scholes model
The classical BSM is probably the most widely-known model to describe the price dynam-ics of a risky security within the framework of risk-neutral pricing theory. Under the BSM,the drift α ( t ) in the general formulation (A.7) is modelled as α ( t ) = σ S ( t ), where σ is theconstant volatility parameter, and the equity index follows under the real-world probabilitymeasure P the SDE dS ( t ) = σ S ( t ) dt + σS ( t ) dB ( t ) , (B.1)where B is a standard P -Brownian motion. The equity index thus follows the geometricBrownian motion S ( t ) = S (0) exp (cid:18) σ t + σB ( t ) (cid:19) , t ∈ [0 , T ] . (B.2)18ollowing the standard procedures of Girsanov’s theorem, the BSM admits a unique equiv-alent risk-neutral pricing measure Q , with the Radon-Nikodym derivative given by Z ( t ) = e − σB ( t ) − σ t = S (0) S ( t ) , t ∈ [0 , T ] , (B.3)which is seen to be of the same form as the discount factor in (A.9), as expected. Under therisk-neutral measure Q , the index S is driftless and satisfies dS ( t ) = σS ( t ) dB Q ( t ) , t ∈ [0 , T ] , (B.4)where B Q ( t ) = B ( t ) + σt is a standard Q -Brownian motion. Appendix B.2. The minimal market model
The minimal market model (MMM), see Platen (2001), is a stylized model under the BA.The MMM is incompatible with the risk-neutral pricing framework, in that an equivalent risk-neutral probability measure cannot exist. When we apply the MMM, we make two importantassumptions. First, the GP of a given investment universe is, generally, difficult to construct.However, it is shown by Platen and Rendek (2012) that the GP of a stock market can beapproximated by a respective well-diversified equity index. The MMM assumes that theequity index is a good proxy for the GP. Second, the drift coefficient α ( t ) in the general GPmodel (A.7) is theoretically a complicated process depending on the instantaneous marketprices of risks, and is, thus, difficult to specify. The MMM makes a critical simplification byassuming that α ( t ) is a simple deterministic exponential function α ( t ) = α e ηt . As a result,under the MMM, the GP S ( t ) follows a time-transformed squared Bessel process of dimensionfour with a deterministic time transformation and satisfies the SDE dS ( t ) = (cid:0) α e ηt (cid:1) dt + (cid:112) α e ηt S ( t ) dB ( t ) , t ∈ [0 , T ] (B.5)under the real-world probability measure P ; see Revuz and Yor (1999). Here η is the long-termexpected growth rate of the equity index, and α is a constant representing the initial scaleof the index.We define in the model the normalized GP as Y ( t ) = ηα e ηt S ( t ) , t ∈ [0 , T ], which satisfiesthe SDE dY ( t ) = (1 − Y ( t )) ηdt + (cid:112) Y ( t ) ηdB ( t ) , t ∈ [0 , T ] . (B.6)The normalized GP is seen to be mean-reverting around the level 1. The mean-reversion ofthe normalized index implies a “trend reversion” of the GP around its long-term exponentialgrowth. It is well documented that a well-diversified index such as the S&P500 index movesin the long-run in a trend reverting pattern, where the trend is usually interpreted as a slowly19oving “fundamental value” process; see Shiller (2015). By decomposing the GP into thenormalized GP index Y ( t ) and a simple exponential fundamental value function α e ηt η , theMMM parsimoniously captures this important stylized fact. Furthermore, the instantaneoussquared volatility of the normalized GP equals that of the GP and is inversely proportionalto the value of the normalized GP, generating the so-called leverage effect.The normalized GP described by (B.6) is a time transformed square-root process of di-mension four, with the transition law of a noncentral Chi-squared (NCX ) distribution, givenby Y ( u ) L = 1 − e − η ( u − t ) χ (cid:32) e − η ( u − t ) − e − η ( u − t ) Y ( t ) (cid:33) , ≤ t < u ≤ T, (B.7)where χ ( ζ ) denotes a NCX random variable of degree 4 and noncentrality parameter ζ ; see,e.g., Broadie and Kaya (2006). The χ ( ζ ) random variable has finite moments of all positiveorders. The probability density function (PDF) is given by f ( ζ, x ) = 12 e − ζ + x (cid:18)(cid:114) xζ (cid:19) I (cid:16)(cid:112) ζx (cid:17) , x > , (B.8)where I ( · ) is the first order modified Bessel function of the first kind, see, e.g., Revuz andYor (1999). The transition density function of the normalized index is, thus, given by p ( t, Y ( t ); u, Y ( u )) = 41 − e − η ( u − t ) f (cid:32) e − η ( u − t ) − e − η ( u − t ) Y ( t ) , − e − η ( u − t ) Y ( u ) (cid:33) . (B.9)It is worth mentioning that the normalized GP is dimensionless, and serves as a nontrivialstate variable in that the transition density over the time period ( t, u ) depends nonlinearly onthe current state Y ( t ). This is in marked contrast to the BSM, where the current price servesas a scaling factor of a time-homogeneous geometric Brownian motion. An implication of thisdependence is that, unlike the BSM, (B.5) is neither scale nor time invariant. Note however,the GP, as a time transformed squared Bessel process, has some self-similarity property. Appendix C. Pricing of the VA with Guarantee
In this section we consider the pricing of the VA product described in Section 2. For thepricing of a given set of cash flows, we adopt the concept of a stochastic discount factor (SDF),where the present value of the cash flows are given by the sum of their expected values, afterdiscounting by the SDF, conditional on all current information. Both risk-neutral pricing andpricing under the BA can be formulated in terms of an appropriate SDF. For more informationon risk-neutral pricing theory, see Delbaen and Schachermayer (2006) for an account. For abrief description of the BA, see Appendix A.20o price the VA with guarantee value function V (0 , X (0)), we note that no withdrawal ismade between any withdrawal dates ( t n − , t n ), and that the wealth account is self-financingwithin this period. This leads to the following recurrence relation for the (left-continuouswith right-hand limits) guarantee value function, V ( t + n − , X ( t + n − )) = E t + n − ( D ( t n − , t n ) V ( t n , X ( t n ))) , (C.1)where E t ( · ) = E ( ·| X ( t )) is the expectation under the real-world probability measure P , condi-tional on the current information represented by X ( t ), and D ( t, u ) , ≤ t < u ≤ T is the SDFover ( t, u ). Under the BA, the SDF is given by D ( t, u ) = S ( t ) S ( u ) , i.e., the ratio of the inverseGP. Under the risk-neutral pricing framework, the SDF D ( t, u ) = Z ( u ) Z ( t ) , with Z ( t ) := E t (cid:16) d Q d P (cid:17) being the Radon-Nikodym derivative of the measure change from the real-world probabilitymeasure P to the equivalent risk-neutral measure Q , conditional on all available informationat t .Upon withdrawal at time t n , 0 < n < N , the left-hand limit of the value function satisfiesthe following jump condition V ( t n , X ( t n )) = V ( t + n , X ( t + n )) + C ( γ n , X ( t n )) . (C.2)In other words, the guarantee value immediately before the withdrawal is the sum of thevalue immediately after the withdrawal and the cash flow of the withdrawal. The activepolicyholder follows a dynamic strategy, which we obtain as the solution of the respectivestochastic control problem. That is, for 0 < n < N , the withdrawal amount γ n is chosenaccording to the following total value maximizing strategy, γ n = Γ( t n , X ( t n )) = arg max ≤ γ ≤ A ( t n ) (cid:8) V ( t + n , X ( t n ) \ γ ) + C ( γ, X ( t n )) (cid:9) , (C.3)where X ( t n ) \ γ denotes the state variables X ( t + n ) after withdrawal γ is made, given thevalue of the state variables X ( t n ) before the withdrawal. On the other hand, the passivepolicyholder follows a static strategy of pre-determined withdrawal values. The contractvalue V (0 , X (0)) can, thus, be computed recursively from (9), (5), (6) (C.2) and (C.1), alongwith the chosen strategy. These procedures are summarized in Algorithm 1. To evaluate(C.1), we discretize the underlying risk factors and approximate the conditional expectationin (C.1) using a finite sum. For the BSM, the risk factor is taken as the scaled Brownianmotion B ( t ) √ t . For the MMM, the risk factor is the normalized GP Y ( t ). Both risk factorshave closed-form transition densities to facilitate the numerical computations. The contractvalue under the dynamic strategy is bounded from below by the corresponding value from anysimple strategy such as the static one. If a closed-form pricing formula for the simple strategy21 lgorithm 1 Recursive computation of V (0 , X (0)) initialize V ( T + , X ( T + )) = 0 set n = N while n > do compute the withdrawal amount γ n with the optimal strategy (C.3) or a pre-determinedstatic strategy compute V ( t n , X ( t n )) by applying jump condition (C.2) with appropriate cash flows compute V ( t + n − , X ( t + n − )) by computing (C.1) with terminal value V ( t n , X ( t n )) andthe appropriate SDF D ( t n − , t n ) set n = n − end while return V (0 , X (0)) = V (0 + , X (0 ++