Semi-static Hedging of Variable Annuities

Carole Bernarda, Minsuk Kwakb,∗

aUniversity of Waterloo, Canada

bDepartment of Mathematics, Hankuk University of Foreign Studies,81 Oedae-ro, Mohyeon-myeon, Cheoin-gu, Yongin-si, Gyeonggi-do, 449-791, Korea

Abstract

This paper focuses on hedging financial risks in variable annuities with guarantees. Wefirst review standard hedging strategies to identify pros and cons of each of them. Usingstandard dynamic hedging techniques, we show that best hedging performance is obtainedwhen the hedging strategy is rebalanced at the dates when the variable annuity fees arecollected. Thus, our results suggest that the insurer should incorporate the specificity ofthe periodic payment of variable annuities fees to construct the hedging portfolio and focuson hedging the net liability. Since standard dynamic hedging is costly in practice becauseof the large number of rebalancing dates, we propose a new hedging strategy based ona semi-static hedging technique and thus with fewer rebalancing dates. We confirm thatthis new strategy outperforms standard dynamic hedging as well as traditional semi-statichedging strategies that do not consider the specificity of the payments of fees in their opti-mization. We also find that short-selling or using put options as hedging instruments givesbetter hedging performance.

JEL classification : G22

Keywords: hedging variable annuities; semi-static hedging

1. Introduction

Life expectancies are steadily increasing and the population faces the need to guaranteea sustainable income after retirement. There has been a growing demand for variableannuities, which offer guaranteed income for post-retirement life: U.S. and Japan are thetwo first providers (Hardy (2003)). EIOPA (2011) illustrate a sharp increase in VA volume

∗Corresponding authorEmail addresses: carole.bernard@grenoble-em.com (Carole Bernard), mkwak@hufs.ac.kr (Minsuk

Kwak)

Preprint submitted to Elsevier July 19, 2015

in Europe around 2011. In China, the first VAs were marketed in 2011, it is still a verysmall volume. A recent state-of-the-art on the Variable Annuities market can be foundin Haefeli (2013) with figures and charts illustrating the Variable Annuities market in theU.S., Canada, Europe and Japan between 1990 and 2011.

A typical variable annuity contract consists of an accumulation phase, during which thepolicyholder invests an initial premium and possibly subsequent premiums into a basket ofinvested funds. At the end of the accumulation phase, the policyholder may receive a lumpsum or may annuitize it to provide retirement income. Variable annuities contain both in-vestment and insurance features. They improve upon traditional fixed annuity contractsthat offer a stream of fixed retirement income, as policyholders of variable annuities couldexpect higher returns in a bull market. Variable annuities also provide some protectionagainst downside risk with several kinds of optional benefit riders. Traditional annuitiescould simply be hedged by an appropriate portfolio of bonds. But the presence of pro-tection benefits require more sophisticated hedging strategies. In a variable annuity, thepolicyholder invests her premiums in a separate account, which is protected from downsiderisks of investing in the financial market through additional options that are sold with thevariable annuity and also called benefit riders. The simplest benefit is the Guaranteed Min-imum Accumulation Benefit (GMAB). It guarantees a lump sum on a specific future dateor anniversary. But it is not as popular as the following more complex benefits, GMIBs andGMWBs. Guaranteed Minimum Income Benefits (GMIB) guarantee a stream of lifetimeannuity income after policyholder’s annuitization decision is made. Guaranteed MinimumWithdrawal Benefits (GMWB) guarantee the ability to withdraw a specified percentage ofthe benefit base during a specified number of years or it could be a lifetime benefit (Kling,Ruez, and Russ (2011)). A Guaranteed Minimum Death Benefit (GMDB) guarantees aspecified lump sum benefit at the time of death of the policyholder. More details on eachof these guarantees can be found in Hardy (2003).

There are lots of studies on the pricing of variable annuities and how to find the fairfee: Milevsky and Posner (2001) and Bacinello (2003) investigate the valuation of GMDBin variable annuities using risk-neutral pricing. Lin, Tan, and Yang (2009) price simpleguarantees in VAs in a regime switching model. Marshall, Hardy, and Saunders (2010)study the value of a GMIB in a complete market, and the sensitivity of the value of GMIBto the financial variables is examined. They suggest that the fee rate charged by insurancecompanies for GMIB may be too low. GMWBs are studied intensively by several authorsincluding Milevsky and Salisbury (2006), Chen, Vetzal, and Forsyth (2008), Dai, Kwok,and Zong (2008), Kolkiewicz and Liu (2012) and Liu (2010). Milevsky and Salisbury(2006) show that GMWB fees charged in the market are too low and not sustainableand argue that the fees have to increase or the product design should change to avoidarbitrage. Chen, Vetzal, and Forsyth (2008) also conclude that normally charged GMWBfees are not enough to cover the cost of hedging. Bacinello, Millossovich, Olivieri, andPitacco (2011) and Bauer, Kling, and Russ (2008) propose general valuation frameworkfor variable annuities.

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There are fewer papers on hedging, although hedging embedded guarantees of variableannuities is a challenging and crucial problem for insurers. There are three main sourcesof risks, mortality risks, policyholders’ behavioral risks and financial risks. Let us discusshere these three sources, we will then focus on financial risks in the rest of the paper. Whenmortality risk is fully diversifiable, it is straightforward to hedge mortality risks by sellingindependent policies to a group of policyholders with similar risks of death. However,mortality risks cannot always be diversified and VAs are exposed to longevity risk (i.e. asystematic change in mortality risk affecting simultaneously all the population). It is atopic of research by itself and a thorough analysis of longevity risk in VAs can be foundin Ngai and Sherris (2011), Hanewald, Piggott and Sherris (2012), Gatzert and Wesker(2012), Fung, Ignatieva, and Sherris (2013) for example.

Behavioral risks in variable annuities come from the uncertainty faced by the insurerabout the policyholders’ decisions (choice of surrender, partial withdrawal, annuitization,reallocation, additional contributions, and so on). In general, they are hard to hedge.Under the assumption that investors do not act optimally and base their decisions on non-financial variables, behavioral risk can be diversified similarly as mortality risk by using adeterministic decision making process using historical statistics, which state for instancethat x% of the policyholders follow a given behaviour in a specific situation. However, thereis empirical evidence that policyholders may act optimally, or at least that their decisionis correlated with some market factors and depends on the moneyness of the guarantee, sothat all behavioral risks may not be diversified away (see for instance the empirical studyof Knoller, Kraut, and Schoenmaekers (2015)). Kling, Ruez, and Russ (2014) study theimpact of behavioral risks on the pricing as well as on the effectiveness of the hedgingof VAs. They consider various assumptions on behaviors (from deterministic to optimaldecision making). Interestingly, the impact of model misspecification on policyholders’behaviors depends highly on the design of the policy. For instance, the effect of thesurrender decision is more important in VAs without ratchets. MacKay, Augustyniak,Bernard, and Hardy (2015) design VAs that are never optimal to surrender. See alsoAugustyniak (2013) for a study of the effect of lapsations on the hedging effectiveness ofthe Guaranteed Minimum Maturity Benefit (GMMB).

In this paper, we ignore mortality and longevity risks, policyholder behavioral risksand focus on hedging financial risks. It can be done via delta hedging, semi-static hedgingor static hedging. Guarantees in VAs are similar to options (financial derivatives) on thefund value and the insurer plays the role of option writer. However, they are also verydifferent from standard derivatives. A crucial difference is that the costs of these optionsare not paid upfront like initial option premiums. By contrast, fees are paid periodicallyas a percentage of the fund value throughout the life of the contract. The fees collectedshould then be invested to hedge the provided guarantees. After finding the suitable levelof fees to cover the guarantees (fair pricing of VAs), hedging the guarantees consists ofinvesting these collected fees in an appropriate way so that at the time the guaranteesmust be paid the investment matches the guarantees. The main difficulty is that there

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are no certain premiums paid upfront in a VA so that uncertainty and risk are inherent inthe payoff and also in the premiums. Therefore, both components should be hedged. Inaddition, there is an unavoidable mismatch between the (random) value of fees collectedfrom the policyholder’s account and the hedging cost, because the value of collected feesand the cost of hedging move in opposite directions. When the policyholder’s accountvalue is high, the value of the fee is also high while the value of the embedded option inthe guarantee is low. If the account value is low, the value of the fee is low but the insurerneeds more money for the guaranteed benefits because the value of embedded option ishigh. This mismatch represents a real challenge for hedging guarantees in VAs. It is alsoexactly the reason why hedging guarantees in VAs differ from hedging standard options(for which fees are paid at inception only).

The most common approach to hedge financial risks is to perform a dynamic delta-hedging approach (Kling, Ruez, and Russ (2014) and Kling, Ruez, and Russ (2011)) toreplicate the embedded guarantees. When hedging a guarantee that depends on a tradablefund Ft (or at least a replicable fund Ft), the hedger makes sure that he holds at any timea number of shares of fund equal to the delta of the guarantee (i.e. the sensitivity of thevalue of the guarantee to a change in the underlying price). Assume that the market iscomplete, with no transaction costs and the hedge is continuously rebalanced over time ina self-financing manner. Then, a dynamic delta hedging strategy achieves a perfect hedgeof the guarantees at the time they must be paid out. An alternative popular dynamichedging is based on a mean variance criteria and is sometimes referred as “optimal dynamichedging” (Papageorgiou, Remillard, and Hocquard (2008), Hocquard, Papageorgiou, andRemillard (2015, 2012)). This latter technique can be applied in incomplete markets.

Instead of dynamic hedging, Hardy (2003, 2000) and Marshall, Hardy, and Saunders(2010, 2012) investigate static hedging and suggest replicating maturity guarantees with astatic position in put options. Static hedging consists of taking positions at inception in aportfolio of financial instruments that are traded in the market (at least over-the-counter)so that the future cash-flows of the VA match the future cashflows of the hedge as wellas possible. Static hedging strategies have no intermediary costs between the inceptionand the maturity of the benefits, and tend to be highly robust to model risk because norebalancing is involved. There are several issues with this approach, in particular thenon-liquidity (and non-availability) of the long-term options needed to match the long-term guarantees as well as their exposure to counterparty risk. Also, most of guaranteebenefits are path-dependent and therefore are hard to hedge with static hedging using theEuropean path-independent options available in the market. Finally, static hedging of VAstends to forget about the specificity of the options embedded in VAs. Their premiums arepaid in a periodic way affecting the fund value. Thus a more natural hedging strategyshould account for the periodicity of these premiums. In a static strategy, the insurermust borrow a large amount of money at the inception of the contract to purchase thehedge. This borrowed money will then potentially be offset by the future fees collectedas a percentage of the fund. But the insurer is subject to the risk that the fees collected

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in the future do not match the amount borrowed at the beginning to purchase the hedge.Typically, if the contract is fairly priced, only the expected value of the discounted futurefees will indeed match the initial cost needed to hedge the guarantees.

Several authors including Coleman, Li, and Patron (2006), Coleman, Kim, Li, and Pa-tron (2007), and Kolkiewicz and Liu (2012) have studied semi-static hedging for variableannuities using options as hedging instruments. In semi-static hedging, the hedging portfo-lio is rebalanced periodically by following an optimal hedging strategy for some optimalitycriterion. The hedging portfolio is not altered until the next rebalancing date. Coleman,Li, and Patron (2006) and Coleman, Kim, Li, and Patron (2007) investigate hedging ofembedded options in GMDB with ratchet features. By assuming that mortality risk can bediversified away, they reduce the problem of hedging of variable annuities to the hedging oflookback options with fixed maturity. They show that semi-static hedging with local riskminimization is significantly better than delta hedging. Kolkiewicz and Liu (2012) reachsimilar conclusions for GMWBs. Semi-static hedging outperforms delta hedge, especiallywhen there are random jumps in the underlying price. In this paper, we provide additionalevidence of the superiority of semi-static hedging strategies over dynamic delta hedging inthe context of hedging guarantees in VAs. We propose a new version of semi-static hedgingwhich utilizes the collected fees right away to construct the hedging portfolio.

One focus in this paper is to explain how to use the periodic fees paid for the guar-antees to develop an efficient hedge. Kolkiewicz and Liu (2012) and Augustyniak (2013)1

point out the importance of hedging the net liability taking into account the periodic feesthat are paid as a percentage of the underlying fund. We propose here a new semi-statichedging strategy, which rebalances the hedging portfolio at the exact dates when the feesare paid. We assume that put options written on the underlying market index related topolicyholder’s separate investment account as well as the underlying market index itselfcan be used as hedging instruments for semi-static hedging. In practice, there are sev-eral variable annuity products which allow policyholders to decide their own investmentdecision. In this case, the insurer faces basis risk arising from the fact that the fund inwhich the premiums are invested is not directly traded. It is thus not directly possibleto take long or short positions in the fund and neither to buy options on the fund. Forthe ease of exposition, we assume that the premium paid by policyholder is invested in amarket index, which is actively traded in the market, e.g. S&P 500, or highly correlated tothe market index. This assumption gives us a liquid underlying index market and optionmarket for our hedging strategies.2

1We refer to Chapter 5 of Augustyniak (2013) on “Measuring the effectiveness of dynamic hedges forlong-term investment guarantees”.

2A rigorous account for basis risk is beyond the scope of this paper. Refer for instance to Adam-Mullerand Nolte (2011) for a comparison of standard approaches to hedging basis risk. In the context of variableannuities, it is possible to mitigate basis risk by managing the fund such as to maintain a high correlationto traded indices. Note that we design a strategy that rebalances at each fee payment date. Between twofee payments, the dynamic of the fund is not affected by the fees and there is thus little basis risk if the

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We provide evidence of the following properties of the hedge of variable annuities. First,the effectiveness of the hedge is improved by focusing on the liability net of collected fees(see Section 3 for an illustration). Second, the insurer should generally start to borrowmoney at time 0 to hedge long term guarantees and anticipate that collected fees willcover this borrowed money in the future. Third, the insurer should be able to short-sellthe underlying to perform an efficient hedge of long term guarantees. Fourth, hedging isimproved significantly by including put options as hedging instruments in a semi-statichedge which is rebalanced at the dates fees are collected. We find that semi-static hedgingis well adapted to VAs due to the periodicity of fee payments. Semi-static hedging is ableto utilize short term options that are more liquid and traded at a wider range of strikes.Finally, semi-static hedging of VAs, when accounting for future collected fees, allows todecrease the initial borrowing significantly. We study the semi-static hedging strategies inthree environments, first without any constraints when all positions in stocks, bonds andoptions can be taken, then with short-selling constraints, and finally when put options arenot available. Note that the Solvency II regulation does not prevent insurers to short-sell orto trade options. However, variable annuity writers should use derivatives or short-sellingfor risk management purposes only, and not to speculate (Solvency II recommendations).

Our improved semi-static strategy outperforms the traditional semi-static strategieswith local risk minimization. Especially, if short-selling is allowed, our strategy gives muchbetter hedging performance than the other strategies with and without put options ashedging instruments. With our new strategy, the fee collected at time 0 is enough to con-struct the optimal hedging portfolio at time 0 and almost no borrowing is necessary. Wealso find that short-selling is necessary to hedge the options embedded in guarantees be-cause the hedging targets are decreasing functions of the underlying index. In the presenceof shortselling constraints, put options become more important hedging instruments. Thelarger the set of strike prices of the put options used for hedging, the better the hedgingperformance.

In Section 2, we describe a simple GMMB as the toy example to illustrate and comparethe different hedging strategies. We then use a standard delta-hedging strategy to show theimportance of hedging the net liability in Section 3. In Section 4, we show how to improvedelta-hedging strategies by developing a new semi-static hedging strategy. Section 4.3compares the hedging effectiveness of the optimal semi-static strategy to other hedgingstrategies. Section 5 concludes.

2. Description of a Variable Annuity Guarantee

To illustrate the hedging of variable annuities, we consider a single premium variableannuity contract sold at time 0. Let F0 be the single premium paid by the policyholder at

investment is invested in an index for which options are available.

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time 0 and FT be the account value of the policyholder’s fund at time T > 0. We furtherassume that the variable annuity contract guarantees K at time T , that is, the policyholderis guaranteed the payoff GT := max{FT , K} at time T . Our model can be used in thefollowing situations:

• A GMAB, which guarantees a lump sum benefit K at time T .

• A variable annuity contract with GMIB rider, which provides flat life annuity pay-ment b = ηmax{FT , K} with annuitization rate η > 0 (see Bacinello, Millossovich,Olivieri, and Pitacco (2011)).

• A variable annuity contract with GMDB rider which guarantees K with assumptionthat the mortality risk can be diversified away (see Coleman, Li, and Patron (2006)and Coleman, Kim, Li, and Patron (2007)).

Notice that the payoff GT of the variable annuity contract can be written as

GT = max{FT , K} = FT + (K − FT )+,

with x+ := max{x, 0}. Therefore, the payoff GT corresponds to the policyholder’s accountvalue at time T plus an embedded option with payoff VT := (K − FT )+ at time T , whereT can be interpreted as the maturity of a put option written on the fund.

2.1. Guarantee put option

The embedded option payoff VT must be funded by the fees collected from the policy-holder’s account. Let us consider the time step size δt := T/N corresponding to a numberof periods N > 0 and time steps tk := kδt for k = 0, 1, . . . , N , that is,

0 = t0, t1, . . . , tN = Nδt = T.

Then we assume that the fees εFtk are withdrawn periodically at each time tk, for k =0, 1, . . . , N − 1, from the policyholder’s account, where Ftk is the account value of thepolicyholder at time tk right before the withdrawal of the fee, and ε is the fee rate. At timetk−1, k = 1, . . . , N − 1, the remaining account value (1 − ε)Ftk−1

after the fee withdrawalstays invested in the fund whose value at time t > 0 is St. Therefore, at time tk, the fundvalue satisfies the following relationship:

Ftk = (1− ε)Ftk−1

StkStk−1

, k = 1, . . . , N − 1.

At time T = ttN , FT is given by

FT = FtN = (1− ε)FtN−1

StNStN−1

.

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Consequently, Ftk for k = 0, 1, . . . , N is given by

Ftk = (1− ε)kF0StkS0

.

As already discussed in the introduction, to simplify the exposition, we neglect basis risk,which arises from the mismatch between the policyholder’s investment account and theavailable hedging instruments and assume that the money is fully invested in a tradedindex St and such that put options written on St can also be used as hedging instruments.For instance we can assume that St is a market index, e.g. S&P 500 with a liquid marketof options written on this index.3

2.2. Fair valuation of the VA

Assume a constant risk-free interest rate r and let ZT be the value at time T of allaccumulated fees taken from inception to time T , then ZT is given by

ZT =N−1∑i=0

εFtier(T−ti) = εF0e

rNδt + εFt1er(N−1)δt + · · ·+ εFtN−1

erδt. (1)

The value of the payoff of the embedded option at time T is

VT = (K − FT )+.

Since the collected fees are used to hedge the embedded option, the fair value of the feerate is determined by solving the following equation

EQ [ZT ] = EQ [VT ] , (2)

where Q is a risk-neutral measure. Here, using the risk-neutral measure is essential becausewe assume that financial risks can be hedged using the financial market and that a risk-neutral probability gives rise to prices that are consistent with an arbitrage-free financialmarket.

2.3. Liability and net liability

The VA terminal payoff with GMMB is equal to

GT = FT + (K − FT )+.

Since FT is policyholder’s fund value at maturity time T , it will be liquidated and trans-ferred to the policyholder at T . Therefore, it does not need to be hedged per se and theremaining liability is the option payoff

VT = (K − FT )+. (3)

3In practice, liquid index futures are used for hedging, however, we use the index directly for simplicity.

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Using the accumulated fees ZT defined in (1), the liability at time T net of collected feesor net liability is

(K − FT )+ − ZT , (4)

and it should be distinguished from the liability in (3). The insurer should provide theguarantee (K−FT )+ but receives a random amount of fees at discrete dates. We will showthat considering the net liability instead of the sole guarantee plays a crucial role in theefficiency of the hedge.

3. Delta Hedging

We consider the VA contract described in the previous section. This section is dedicatedto delta hedging and to show how a delta hedging strategy of the net liability (4) outper-forms a naive delta hedging strategy of the liability (3) very significantly. We illustratethis point with the Black-Scholes setting but the result is robust to more general assump-tions on the market. The gap between the naive hedge and the hedge of the net liabilityis important and due to the hedging methodology and not due to the distribution of theunderlying. In fact, our experiments have convinced us that adding stochastic volatility,stochastic interest rates or jumps will not change the conclusion that the hedge of the netliability accounting for the periodicity of the payment significantly outperforms the hedgeof the liability. Thus, in what follows, the index follows a geometric Brownian motion

dStSt

= µdt+ σdWt (5)

with constant µ > 0 and σ > 0 and standard Brownian motion Wt under the real-worldprobability measure P. Without loss of generality, we may assume that S0 = 1. Recallthat in the absence of market frictions such as transaction costs, and when the underlyingasset is tradable in a complete market, the option can be hedged perfectly by rebalancingthe hedging portfolio continuously with a dynamic delta hedging strategy. In practice, wecannot rebalance the hedging portfolio continuously and frequent rebalancing causes hightransaction costs. Therefore, perfect hedging is not feasible and the hedging portfolio canbe rebalanced only at a discrete set of times. In the case of a VA, it is natural to assume thatthe portfolio is rebalanced at the dates when the fees are collected: t0 = 0, t1, t2, ..., tN−1and potentially at intermediary dates.

3.1. Profit and loss of the hedge

Recall that the only part that needs to be hedged in the VA is the maturity guaranteeVT = (K − FT )+. The collected fees taken periodically from the fund are utilized toconstruct the insurer’s hedging portfolio whose value at time T is denoted by XT .

To compare the performance of various hedging strategies, we consider the profit andloss of the hedge at time tN = T (surplus if it is positive and hedging error if it is negative).

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We denote it by ΠtN . It is equal to the difference between the hedging portfolio of theinsurer at T , XT , and the guarantee, VT , that needs to be paid to the policyholder at timeT

ΠtN = XT − VT .

3.2. No hedge case

As a benchmark, we consider the case when there is no hedge at all from the insurancecompany and that all collected fees are directly invested in the bank account. Then, theprofit and loss at time T is given by

ΠtN = ZT − (K − FT )+,

where ZT =∑N−1

i=0 εFtier(T−ti) (given in (1)). The expected value (under the real proba-

bility P) of the profit and loss can be computed explicitly as follows

EP[ΠtN ] = EP

[N−1∑i=0

εFtier(T−ti) − (K − FT )+

]=

N−1∑i=0

εer(T−ti)EP[Fti ]− EP[(K − FT )+]

=N−1∑i=0

εer(T−ti)(1− ε)iF0

S0

EP[Sti ]− eµTEP[e−µT (K − FT )+]

=N−1∑i=0

εer(T−ti)(1− ε)iF0eµti − Φ(−d2)K + eµTΦ(−d1)(1− ε)NF0.

which is easily derived from the Black-Scholes formula (using µ instead of r) with

d1 =ln(

(1−ε)NF0

K

)+ T (µ+ σ2

2)

σ√T

, d2 = d1 − σ√T .

This formula was checked by Monte Carlo simulations.

3.3. Delta Hedge of the Liability

A delta-hedge (∆-hedge) is a self-financing portfolio consisting of investment in theunderlying index, S, and in a bond to replicate a target payoff. Our goal is to implementa ∆-hedge with rebalancing at each dates tk for k = 0, 1, . . . , N − 1. We start fromX0 = εF0 at time 0, and at each tk, we construct a hedging portfolio consisting of πtk unitsof underlying index and Xtk − πtkStk risk-free bonds. Then, for time tk, we can representour self-financing replication relation as follows

Xtk+1= (Xtk − πtkStk)erδt + πtkStk+1

+ εFtk+1k = 0, 1, . . . , N − 2 (6)

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with X0 = εF0, and define the profit and loss at time T as

ΠtN = XtN − (K − FT )+

to investigate the hedging performance.

To replicate the payoff (K − FT )+at each tk using ∆-hedge, we compute Vtk the no-arbitrage value (risk-neutral expectation of discounted payoff conditional on Ftk) of VT attime tk. We have

Vtk = EQ [e−r(T−tk)(K − FT )+∣∣Ftk] = EQ

[e−r(T−tk)

(K − (1− ε)NF0

STS0

)+∣∣∣∣∣Ftk

].

Let α = (1− ε)N F0

S0. Then,

Vtk = αEQ

[e−r(T−tk)

(K

α− ST

)+∣∣∣∣∣Ftk

]= Φ(−d2(k))Ke−r(T−tk) − αΦ(−d1(k))Stk

which is also derived from the Black-Scholes formula with

d1(k) =ln(

αStkK

) + (T − tk)(r + σ2

2)

σ√T − tk

, d2(k) = d1(k)− σ√T − tk

and thus, it is then known that the number of shares to invest in shares to hedge theliability is

πtk = −αΦ(−d1(k)). (7)

3.4. Delta-Hedge of the Net Liability

Recall that the no-arbitrage value of VT at tk is Vtk . Then, the no-arbitrage value ofthe net liability (4) at time tk is

Vtk−N−1∑i=0

ε(1− ε)iF0

S0

EQ [Sti |Ftk ] e−r(ti−tk)

= Vtk −k−1∑i=0

ε(1− ε)iF0

S0

Stier(tk−ti) −

N−1∑i=k

ε(1− ε)iF0

S0

Stk .

By differentiating with respect to Stk , the new number of shares is

πtk := πtk −F0

S0

(1− ε)k(1− (1− ε)N−k

)(8)

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where πtk indicates the number of shares (7) that is needed to hedge the terminal liabilityignoring the intermediary payments of fees. The self-financing replication relation thatcorresponds to πtk can be obtained by replacing πtk by πtk (defined in (8)) in the expressionof the self-financing condition (6).

Note that it is obvious from (7) and (8) that both πtk and πtk are negative for allk = 0, 1, . . . , N − 1. In other words, we always need a short position in the underlyingindex when we delta-hedge guarantees in VAs. Consequently, if there is any limitation onshort-selling or the cost of short-selling is high, delta-hedging may not perform well.

Moreover, it can be seen from (8) that

πtk < πtk < 0, for all k = 0, 1, . . . , N − 1.

This implies that the delta-hedge of the net liability requires more short-selling than thedelta-hedge of the liability.

3.5. Numerical Illustration

We now illustrate and compare the hedging performance of the two proposed hedgingstrategies with numerical examples in the Black-Scholes framework as in (5). We generatesample paths of the underlying index from time 0 to T at the rebalancing dates, i.e. Stkfor k = 0, 1, . . . , N . Since we assume (5), we have

Stk+1= Stke

(µ−σ2

2)δt+σ(Wtk+δt

−Wtk) ∼ Stke

(µ−σ2

2)δt+σ

√δtz,

where z ∼ N(0, 1). We generate 10,000 simulated paths. The benchmark parameters forthe financial market and the variable annuity contract are given by

T = 10 (years), N = 20, δt = 0.5 (years), F0 = 100, K = 120

r = 2% (per year), σ = 20% (per year), µ = 6% (per year).

Under these parameters, the fair fee rate is ε = 0.0415 (solving (2) where the dynamics ofthe fund under Q is similar to (5) in which µ becomes r). This fee is slightly higher thanusual fee rates charged in the industry. However, if we consider a longer maturity T , asmaller volatility σ or a lower guaranteed value K, the fee rate becomes much lower thanthis example. Moreover, our findings do not hinge on the level of fee rate so we use thisset of parameters for our numerical experiments.

Hedging effectiveness is assessed from a risk management perspective and not usingstandard performance measures as hedging is not meant to be directly profitable. Haefeli(2013) note that dynamic hedging programs “are designed and applied according to strictrisk management rules to mitigate exposures to various market movements stemming fromthe guarantees provided to policyholders”. In Table 1 below, the expected return can belower when hedging. However, profit of a hedging strategy should be seen for instance from

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reduction in economic capital that is directly linked to the risk exposure of the company.We thus compare the hedging strategies based on risk measures and follow the methodologyof Cotter and Hanly (2006) by comparing their five hedging effectiveness (HE) metrics.

1. HE Metric 1 based on the variance

HE1 = 1−[

VarianceHedgedPortfolio

VarianceUnhedgedPortfolio

],

whereVariance = E

[(ΠtN − E[ΠtN ])2

].

HE1 is the percentage reduction in the variance of ΠtN with a hedging strategy ascompared with the variance of unhedged case. If HE1 = 1, it indicates that therisk is completely eliminated by the hedging strategy while HE1 = 0 means that thehedging strategy does not reduce the variance of ΠtN .

2. HE Metric 2 based on the semivariance

HE2 = 1−[

SemiVarianceHedgedPortfolio

SemiVarianceUnhedgedPortfolio

],

whereSemiVariance = E

[(max(0,−ΠtN ))2

].

The semivariance is a kind of downside risk measure, and generally defined byE [(max(0, τ −R))2], where R is the return on the hedged portfolio and τ is thetarget return. Our case is obtained by setting R = ΠtN and τ = 0.

3. HE Metric 3 based on the lower partial moment (LPM)

HE3 = 1−[

LPM3 HedgedPortfolio

LPM3 UnhedgedPortfolio

],

where the lower partial moment of order n is defined as

LPMn = E [(max(0,−ΠtN))n] .

The LPM is a more general concept than the semivariance, and the semivariance isa special case of LPM with n = 2. For HE3, we choose n = 3 which corresponds toa risk-averse investor as argued by Cotter and Hanly (2006).

4. HE Metric 4 based on the Value-at-Risk

HE4 = 1−[

VaR95% HedgedPortfolio

VaR95% UnhedgedPortfolio

].

5. HE Metric 5 - Conditional Value at Risk

HE5 = 1−[

CVaR95% HedgedPortfolio

CVaR95% UnhedgedPortfolio

].

13

Table 1: Comparison of Hedging Performance by Monte Carlo simulations with 105 simulations. Thecolumn corresponding to “No hedge” contains explicit computations (when available) or is obtained with1,000,000 simulations so that all digits are significant. π0 < 0 represents short-selling is necessary at time0.

∆-hedge ∆-hedgewith N rebalancing dates with 10N rebalancing dates

Characteristics ∆-hedge ∆-hedge ∆-hedge ∆-hedgeof ΠtN No hedge of Liability of Net Liability of Liability of Net LiabilityMean 31.9007 12.49 -1.88 13.22 -0.24

Median 22.1358 7.27 -0.04 7.61 -0.03Std 57.8022 28.09 5.81 29.78 1.29

VaR95% 47.2143 22.87 14.73 23.69 2.69CVaR95% 56.4590 27.80 20.65 28.50 3.93VaR90% 36.5202 17.68 9.35 18.45 1.71

CVaR90% 51.2175 23.95 16.23 24.69 3.04HE1 0 0.7624 0.9898 0.7438 0.9995HE2 0 0.7599 0.9018 0.7565 0.9965HE3 0 0.8840 0.9593 0.8814 0.9997HE4 0 0.5179 0.6895 0.5137 0.9439HE5 0 0.5134 0.6388 0.5095 0.9315π0 0 -36.01 -98.09 -36.12 -98.21

We illustrate the performance of a delta-hedging strategy using the computation of theshares as in (7) and (8) and report our results in Table 1.

From Table 1 and the two panels of Figure 1, we find the following properties of ∆-hedging:

• Hedging is necessary. In Table 1, the column “no hedge” means that the insurerdoes not use the collected fees for trading the underlying index or put options. Insteadhe invests all fees in the risk-free asset as if the guarantees of the VA were fullydiversifiable. All digits for the performance of “no hedge” are exact. Note thatthe no hedge performance is very poor: ΠtN has a very high standard deviationand the Value-at-Risk VaR95% and VaR90% are also very high. This implies thatthe insurer may encounter significant loss in unfavorable market conditions in theabsence of hedging. Thus a good hedging strategy is necessary for the insurer toavoid catastrophic losses.

• Short-selling is necessary. The ∆-hedging strategy does not require borrowing attime 0 but significant short-selling of the underlying index is needed.

14

• Hedging effectiveness is highest when hedging the net liability. First, weobserve a significant reduction in the standard deviation and hedge improvementwhen ∆-hedging the net liability instead of the liability. We find that the hedgingeffectiveness metrics for hedging the net liability are significantly higher than theeffectiveness of the hedging of the liability.

• Rebalancing at more dates is only useful for ∆-hedging the net liability. Itis expected that when rebalancing more often, the performance of ∆-hedging mustimprove. However, we observe that the uncertainty of the collected fees that is nottaken into account in the ∆-hedge of the liability prevents any significant improve-ment in performance of the ∆-hedge. On the other hand, we observe a significantimprovement in the case of ∆-hedging the net liability, when the number of rebal-ancing dates is multiplied by 10.

Figure 1: Comparison of Probability Densities of ΠtN

(a) Rebalancing 2 times a year

−100 −50 0 50 100 150 2000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Profit and loss (ΠtN

)

No hedgeΔ−hedge of liabilityΔ−hedge of net liability

(b) Rebalancing 20 times a year

−100 −50 0 50 100 150 2000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Profit and loss (ΠtN

)

No hedgeΔ−hedge of liabilityΔ−hedge of net liability

4. Semi-Static Hedging

In this section, we develop a semi-static hedging technique that minimizes the varianceof the hedge at a set of discrete dates by investing in stocks, risk-free bonds and someput options. For the ease of exposition, we assume that the hedging strategy utilizes thecollected fees at the time they are paid to construct the hedging portfolio. Thus we assumethat the rebalancing dates at which the portfolio is hedged coincide exactly with the datestk for k = 0, 1, . . . , N − 1 (time steps at which fees are taken from the policyholder’saccount). It is possible to include other time steps, but this assumption simplifies the

15

model and does not affect our conclusions.4

4.1. Two semi-static strategies

At time tk, k = 0, 1, . . . , N − 1, we construct a hedging portfolio Xtk consisting of βtkrisk-free bonds (in other words invested in a money market account), πtk units of underlyingindex (whose value at time tk is Stk), and αtk put options with strike price Ktk and maturitytk+1. Then the value of the hedging portfolio at time tk becomes

Xtk := πtkStk + βtk + αtkP (Stk , Ktk , δt), k = 0, 1, . . . , N − 1,

where P (St, K, τ) denotes the price of a put option at time t with current underlying assetvalue St, strike price K, and time to maturity τ . It is possible to add more put optionsbut the exposition would be more complicated. This simplest situation with put optionswith only one given strike is already insightful.

At time tk+1 before any rebalancing, the change in the value of hedging portfolio overδt due to the movement of the underlying index is

πtk(Stk+1− Stk) + βtk(e

rδt − 1) + αtk((Ktk − Stk+1)+ − P (Stk , Ktk , δt)).

Then we define the accumulated gain (possibly negative when it is a loss) at time tk as

Ytk :=k−1∑n=0

πtn(Stn+1 − Stn) + βtn(erδt − 1) + αtn((Ktn − Stn+1)+ − P (Stn , Ktn , δt))

for k = 1, 2, . . . , N with Y0 = 0, and the cumulative cost at time tk as

Ctk := Xtk − Ytk , k = 0, 1, . . . , N,

with XtN := VT = (K − FT )+.

Remark 1. The cost increment Ctk+1−Ctk can be interpreted as the variation in the value

of the hedging portfolio between tk and tk+1 adjusted by potential gains and losses. It canbe computed as

Ctk+1− Ctk = Xtk+1

− (Xtk + Ytk+1− Ytk)

with

Xtk+1=

πk+1Stk+1

+ βtk+1+ αtk+1

P (Stk+1, Ktk+1

, δt) if k = 0, 1, . . . , N − 2,

(K − FT )+ =(K − (1− ε)NF0

StNS0

)+if k = N − 1,

(9)

Xtk + Ytk+1− Ytk = πtkStk+1

+ βtkerδt + αtk(Ktk − Stk+1

)+, k = 0, 1, . . . , N − 1. (10)

4We have conducted analyses with one intermediary date and it does not change the results significantly.

16

Ultimately, we are interested in the profit and loss of the hedging portfolio at the dateT when the payoff is paid. To do so, we need to take into account the collected fees anddefine the profit and loss Πt at any time t. At time 0, the insurer receives the fee εF0 andconstructs a hedging portfolio whose value is C0 = X0. Therefore, the profit and loss attime 0 is

Π0 = εF0 − C0.

At time t1, the initial profit and loss (computed at time 0) becomes Π0erδt and the insurer

receives the fee εFt1 . However, additional cost Ct1 −C0 is needed to construct the hedgingportfolio at time t1. As a result, the profit and loss at time t1 is given by

Πt1 = Π0erδt + εFt1 − (Ct1 − C0) = (εF0 − C0)e

rδt + εFt1 − (Ct1 − C0).

A similar argument gives the following equation for the profit and loss at time T .

ΠT = ΠtN =N−1∑k=0

εFtker(N−k)δt − C0e

rkδt −N∑k=1

(Ctk − Ctk−1)er(N−k)δt. (11)

If ΠtN is positive, the insurer has positive surplus at T after paying (K − FT )+ to thepolicyholder. On the other hand, if ΠtN is negative, the insurer is short of money to paythe guaranteed value at time T to the policyholder. Thus ΠtN can be considered as theprofit and loss and the hedging performance of strategies can be investigated using ΠtN .

Liability Hedge: Traditional semi-static hedging

Consider a traditional semi-static hedging strategy (referred as liability hedge). Sinceperfect hedging is not feasible with semi-static hedging, some optimality criterion is nec-essary at each rebalancing date. In semi-static hedging of liability, the hedging strategyat each rebalancing date is determined to minimize local risk, which is defined as the con-ditional second moment of the cost increment during each hedging period. At each timestep tk for k = N − 1, N − 2, . . . , 0, starting from the last period, one solves the followingoptimization recursively

min(πtk ,βtk ,αtk ,Ktk )∈Atk

L1

k (πtk , βtk , αtk , Ktk ; s),

with

L1

k (πtk , βtk , αtk , Ktk ; s) := Etk

[(Ctk+1

− Ctk)2∣∣∣Stk = s

]= Etk

[{Xtk+1

−(Xtk + Ytk+1

− Ytk)}2∣∣∣Stk = s

], (12)

where Atk is some admissible set of controls at time tk and Etk [·] := Etk [·|Ftk ] is theconditional expectation on the information at time tk. Atk may be defined differently

17

depending on the constraints we consider. Equation (12) implies that, if we hedge theliability at time tk, Xtk+1

is the hedging target and our goal is to minimize the conditionalsecond moment of increment of cost.

This optimality criterion was applied to semi-static hedging of variable annuities byColeman, Li, and Patron (2006), Coleman, Kim, Li, and Patron (2007), and Kolkiewiczand Liu (2012). In these papers, the authors use several standard options with differentstrike prices in the hedging strategy, but as described above, we only consider a single putoption and the strike price is determined by the optimization. Our model gives insightsabout the moneyness of the options needed for hedging variable annuities.

In (12), note that(Xtk + Ytk+1

− Ytk)

is the amount available at time tk+1 from thehedging portfolio Xtk . However, the insurer receives the fee εFtk at time tk for k =0, 1, . . . , N − 1. If we hedge the liability, consistently with the previous literature, we onlyuse

(Xtk + Ytk+1

− Ytk)

to construct the hedging portfolio at time tk+1 and the fee is notconsidered in the hedging strategy. The strategy below shows how to take this fee intoaccount in the design of the semi-static hedging strategy.

Net Liability Hedge: Improved semi-static hedging strategy

For k = N − 1, N − 2, . . . , 0, solve the following optimization recursively

min(πtk ,βtk ,αtk ,Ktk )∈Atk

L2

k (πtk , βtk , αtk , Ktk ; s),

with

L2

k (πtk , βtk , αtk , Ktk ; s) := Etk

[{Ctk+1

−(Ctk + εFtk+1

1{06k6N−2})}2∣∣∣Stk = s

]= Etk

[{Xtk+1

−(Xtk + Ytk+1

− Ytk + εFtk+11{06k6N−2}

)}2∣∣∣Stk = s

].

The optimality criterion above enables us to utilize the collected fees right away in thehedging strategy. Notice that ΠtN in (11) can be written as

ΠtN = (εF0 − C0)erNδt +

N−1∑k=1

(εFtk + Ctk−1− Ctk)er(N−k)δt + (CtN−1

− CtN ), (13)

and the conditional second moments of every term of ΠtN except the first term in (13) areminimized when we hedge the net liability whereas the collected fees are not utilized whenwe hedge just the liability. Our numerical experiments show evidence that the net liabilityhedge gives a better hedge performance than the liability hedge.

We now illustrate the hedging performance of these two strategies (liability hedge andnet liability hedge) under various market environments. The effect of fees on hedgingperformance under different assumptions on the market environment is also discussed.

18

4.2. Description

We describe the semi-static hedging procedure based on hedging the net liability. Butsimilar explanations hold for the liability hedge because the only difference between thenet liability hedge and the liability hedge is that the fees do not appear in the optimalitycriterion of the liability hedge. Therefore, we can obtain the solution for the liability hedgeby removing the terms with the fees ε(1 − ε)k+1 F0

S0Stk+1

1{06k6N−2} wherever they appearin the net liability hedge.

One of the variables included in the optimization is the optimal level of the strike priceof the put options used in the hedging procedure. Since the strike prices of put optionsthat are actively traded in the market are limited and related to the underlying indexvalue at tk, we define a set Ktk(Stk) of strike prices available in the market at time tk withunderlying index value Stk . Specifically, we describe the available strikes as a percentageof the underlying value, so that a percentage equal to 100% represents an at-the-moneyoption, a percentage higher (respectively lower) than 100% corresponds to an out-of-the-money put option (respectively in-the-money). Mathematically, it amounts to restrictingthe set of possible strikes to the following set Ktk(Stk):

Ktk(Stk) = {ξStk |ξ ∈ {ξ1, ξ2, . . . , ξM}},

where ξ1 < ξ2 < · · · < ξM . Then we may examine the effect of the minimum availablestrike ξ1Stk at time tk and the maximum available strike ξMStk at time tk on the hedgingperformance. It is also straightforward to extend our approach with additional constraintsimposed on the other control variables πtk , βtk , and αtk .

We examine separately the effectiveness of the semi-static hedging strategy in threeenvironments that we call “UC”, “SC” and “w/o Put”.

• “UC” refers to the situation where hedging is done without any constraints. Shortselling of the underlying index is allowed and the index and put options on this indexare used as hedging instruments (see details in Appendix A.1).

• “SC” refers to the situation where hedging is done under short-selling constraints onthe underlying index. Put options are also used as hedging instruments (see detailsin Appendix A.2).

• “w/o Put” refers to the situation where hedging is done without any constraints onthe underlying index but put options are not used as hedging instruments: Only themoney market account and the underlying index can be used for hedging (see detailsin Appendix A.3).

4.3. Hedging Performance

For the ease of comparison with delta-hedging, we use the same parameter set as thenumerical illustration in Section 3.5. The theoretical results needed for the numerical

19

implementation are given in Appendix A.

In Figures 2 and 3, and in Table 2, we use ξ1 = 0.7 and ξM = 1.1. This choice isconsistent with market data of available strike prices of put options on the S&P 500 indexthat are actively traded in the market.

Figure 2: Comparison of Probability Densities of ΠtN

We compare the probability distribution functions of ΠtN for the two strategies described in Section 4when there are no constraints on the semi-static hedging strategy and when put options are not available.

(a) Unconstrained, ξ1 = 0.7, ξM = 1.1

−50 0 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Profit and loss (ΠtN

)

Pro

babi

lity

dens

ity

Liability Hedge:UCNet Liability Hege:UC

(b) Without Put Options, ξ1 = 0.7, ξM = 1.1

−50 0 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Profit and loss (ΠtN

)

Pro

babi

lity

dens

ity

Liability Hedge:w/o PutNet Liability Hedge:w/o Put

Figure 3: Hedging Targets with Short-selling Constraint Dk = 0, ξ1 = 0.7, ξM = 1.1

0.5 1 1.5 2 2.5−80

−60

−40

−20

0

20

40

60

80

100Hedging targets

St5

X5(S

t 5)

Liability Hedge:SCNet Liability Hedge:SC

From Figures 2 and 3, and Table 2, we make the following observations.

• The improved semi-static hedging strategy, the net liability hedge, out-performs the liability hedge. Figure 2a represents the probability distribution

20

of ΠtN of two strategies, the liability hedge and the net liability hedge, without anyconstraint. It can be directly compared with Figure 2b where the probability distri-butions of ΠtN of the liability hedge and net liability hedge are displayed when noput options are used in the hedging strategies. In both cases, we observe that thenet liability hedge gives significantly better hedging performance. This can also beobserved from the HE metrics in Table 2. In fact, this result is intuitive as betterhedging performance with the net liability hedge is expected from the equation (13)because the net liability hedge minimizes each term in ΠtN while the liability hedgeonly focuses on the additional cost part of ΠtN .

• Less borrowing is needed at time 0 with the net liability hedge Anotheradvantage of the net liability hedge instead of the liability hedge is that the initialprofit and loss Π0 = εF0−X0 with the net liability hedge is close to zero and greaterthan that with the liability hedge as we can see in Table 2. Negative Π0 means thatthe fee collected at time 0 is not enough to construct the optimal hedging portfolioX0 at time 0 so that the insurer needs to borrow to construct the optimal hedgingportfolio at time 0. Since the fee rate charged in this example is ε = 0.0415 andthe single premium paid by the policyholder at time 0 is F0 = 100, the fee collectedat time 0 is εF0 = 4.15. The insurer needs to borrow more than 50 which is morethan 1000% of εF0 to construct the optimal hedging portfolio for the liability hedgewithout constraints. On the contrary, the insurer only needs to borrow 0.02 which isless than 0.3% of εF0 to construct the optimal hedging portfolio for the net liabilityhedge. This is mainly due to the fact that we account for collected fees to constructthe hedging portfolio at each rebalancing date when hedging the net liability.

• Including put options as hedging instruments gives better hedging perfor-mance. In Table 2, the HE metrics of the hedging strategy without put options (w/oPut) are lower than the HE metrics of the hedging strategy involving put options(UC). This means that better hedging performance can be achieved by including putoptions as hedging instruments. This result is consistent with the findings of Cole-man, Li, and Patron (2006), Coleman, Kim, Li, and Patron (2007), and Kolkiewiczand Liu (2012).

• Short-selling is of utmost importance for hedging VA guarantees. FromTable 2, the hedging performance under short-selling constraint Dk = 0 (SC) appearssignificantly worse than that without short-selling constraint (UC) for each strategyunder study. Moreover, the hedging performance under short-selling constraints iseven worse than that without put options as hedging instruments (w/o Put). Thisfinding can be understood from the shape of the hedging target as a function of theunderlying index. Figure 3 depicts the hedging target at time t5 as an example.

Since the hedging targets of the two strategies are decreasing convex functions of theunderlying index (Figure 3), a short position of the underlying index is essential toconstruct an efficient hedging portfolio and the combination of a short position in

21

the underlying index and put options allows to match the hedging target well andthus for a good hedging performance. Therefore, if short-selling is not allowed, weget much worse hedging performance.

Table 2: Comparison of Hedging Performances. ξ1 = 0.7, ξM = 1.1

Liability Hedge Net Liability HedgeCharacteristics of ΠtN No hedge UC SC w/o Put UC SC w/o Put

Mean 31.9007 12.99 14.68 13.13 0.10 3.05 0.24Median 22.1358 7.83 7.50 7.71 0.03 0.45 0.12

Std 57.8022 28.56 33.89 28.73 1.24 21.09 3.64VaR95% 47.2143 22.38 26.20 22.60 1.51 26.84 5.24

CVaR95% 56.4590 27.36 30.20 27.65 2.84 33.10 9.53VaR90% 36.5202 17.16 21.54 17.36 0.90 21.34 2.85

CVaR90% 51.2175 23.48 27.01 23.72 2.01 28.44 6.71HE1 0 0.7521 0.6508 0.7492 0.9995 0.8648 0.9960HE2 0 0.7618 0.6526 0.7570 0.9980 0.6110 0.9807HE3 0 0.8857 0.8200 0.8821 0.9997 0.7826 0.9950HE4 0 0.5330 0.4402 0.5171 0.9678 0.4265 0.8880HE5 0 0.5167 0.4665 0.5116 0.9498 0.4153 0.8318π0 0 -31.40 0 -35.56 -84.44 0 -88.60Π0 -53.06 -51.63 -52.94 -0.02 2.46 0.10

UC: Unconstrained, SC: Short-selling constraint Dk = 0, w/o Put: Without put option

The next observations relate to Figure 4 and 5 and Table 3.

• A larger set of strike prices gives a better hedge. In Figure 4 and Table 3,we examine the hedging performance with ξ1 = 0.5 and ξM = 1.5. This change givesthe insurer a wider range of strike prices of put options that can be used to constructthe hedging portfolio. It is clear that the hedging performance of strategies that useput options as hedging instruments, is improved when a wider range of strike pricesis available. If we compare Table 2 and Table 3, the hedging performance withoutshort-selling constraint is slightly improved. Noticeably, the hedging performanceis significantly improved in the presence of the short-selling constraint (SC). Forexample, the standard deviation of ΠtN with the net liability hedge under short-selling constraint is 21.09 if ξ1 = 0.7 and ξM = 1.1. By contrast, if ξ1 = 0.5 andξM = 1.5, the standard deviation is reduced dramatically and becomes 2.87. Thissignificant improvement of hedging performance under short-selling constraint canalso be observed in Figure 4. This is because the put option is the only hedging

22

instrument available to match the shape of the hedging target in the presence of short-selling constraints. Thus, put options become more useful hedging instruments whenthey are offered with a wider range of strike prices (which implies that extreme strikesare needed either deep out-of-the-money options or deep in-the-money options).

Figure 5 illustrates why the hedging performance under short-selling constraint isimproved significantly when a larger set of strike prices is available. In Figure 5b, ifξ1 = 0.5 and ξM = 1.5, the optimal strike prices without any constraints (UC) are farbelow the upper bound of available strike prices. However, the optimal strike pricesunder short-selling constraint (SC) are on the upper bound of available strike pricesin Figure 5b. Therefore, in Figure 5a with ξM = 1.1, the upper bound of availablestrike prices is much lower and the optimal strike prices under short-selling constraintare far below the optimal levels attained in Figure 5b with ξM = 1.5. As a result,the hedging performance under short-selling constraint is affected significantly by therange of strike prices.

Figure 4: Comparison of Probability Densities of ΠtN

We compare the probability distribution functions of ΠtN for the two strategies described in Section 4when there are short-selling constraints on the semi-static hedging strategy and when put options are notavailable for a wider range of strikes.

(a) Short-selling constraintDk = 0, ξ1 = 0.5, ξM = 1.5

−50 0 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Profit and loss (ΠtN

)

Pro

babi

lity

dens

ity

Liability Hedge:SCNet Liability Hedge:SC

(b) Without Put Option, ξ1 = 0.5, ξM = 1.5

−50 0 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Profit and loss (ΠtN

)

Pro

babi

lity

dens

ity

Liability Hedge:w/o PutNet Liability Hedge:w/o Put

23

Figure 5: Comparison of Optimal Strike Prices at Time t1We compare the optimal strike prices for the net liability hedge at time t1 for different set of strike prices.

(a) ξ1 = 0.7, ξM = 1.1

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Opt

imal

str

ike

pric

e

St1

out−of−the−money

in−the−money

↑ξ

1 S

t1

↓

ξM

St1

Net Liability Hedge:UCNet Liability Hedge:SCat−the−money

(b) ξ1 = 0.5, ξM = 1.5

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Opt

imal

str

ike

pric

eS

t1

out−of−the−money

in−the−money

↑ξ

1 S

t1

↓

ξM

St1

Net Liability Hedge:UCNet Liability Hedge:SCat−the−money

Table 3: Comparison of Hedging Performances. ξ1 = 0.5, ξM = 1.5

Liability Hedge Net Liability HedgeCharacteristics of ΠtN UC SC w/o Put UC SC w/o Put

Mean 12.91 13.02 13.05 0.07 0.17 0.21Median 7.51 7.66 7.36 0.04 0.01 0.10

Std 29.29 29.47 29.49 1.12 2.87 3.72VaR95% 23.05 23.19 23.21 1.50 3.88 5.65

CVaR95% 27.69 27.93 27.96 2.67 6.22 9.80VaR90% 17.76 17.99 17.99 0.91 2.45 2.96

CVaR90% 23.95 24.16 24.17 1.91 4.65 6.94HE1 0.7462 0.7430 0.7427 0.9996 0.9976 0.9959HE2 0.7582 0.7539 0.7534 0.9984 0.9913 0.9804HE3 0.8839 0.8808 0.8805 0.9999 0.9987 0.9948HE4 0.5196 0.5116 0.5162 0.9688 0.9192 0.8822HE5 0.5147 0.5106 0.5101 0.9532 0.8910 0.8283π0 -31.18 0 -35.56 -84.22 0 -88.60Π0 -53.06 -52.99 -52.94 -0.02 0.05 0.10

UC: Unconstrained, SC: Short-selling constraint Dk = 0, w/o Put: Without put option

5. Conclusions

This paper focuses on the hedging of financial guarantees in variable annuities. Wereview existing methods and introduce a new semi-static hedging strategy that outperforms

24

delta hedging and existing static and semi-static strategies presented in the literature. Weshow that it is crucial to take into account the fees collected periodically in the design ofthe hedging strategy. Another advantage of our new strategy is that almost no borrowingis needed at the beginning and the cost to construct the initial hedging portfolio can bematched to the fee collected initially.

We also show that short-selling and adding put options as additional hedging instru-ments gives better hedging performance of variable annuities. The larger the set of putoptions available in the market (wider range of strikes), the better the hedging performancewith put options.

Finally, we want to point out that our study has been performed in the Black-Scholessetting. It already allows us to get good intuition. Semi-static strategies with local riskminimization are known to be more robust to model risk than delta-hedging strategies.Therefore, we expect that our qualitative results hold under more general market condi-tions.

25

Appendix A. Optimal Semi-Static Hedging Strategies

We here provide full details of the implementation of the semi-static strategy. Wefirst consider the unconstrained optimization problem “UC” as a benchmark (AppendixA.1), and then we consider constrained problems “SC” (Appendix A.2) and “w/o Put”(Appendix A.3) to examine the effect of short-selling constraints and the absence of optionsin the hedging portfolio.

Appendix A.1. Unconstrained Semi-Static Hedging Strategy: “UC”

At each time step tk, k = 1, 2, . . . , N , consider a discretization {Sitk}Ii=1 of the underlying

index. Then we can find the optimal hedging strategies recursively in backward fromk = N − 1 to k = 0. The following are the steps to find the optimal strategy.

1. At time tk, the hedging target Xtk+1= Xtk+1

(Stk+1) is already known as a function

of Stk+1from the previous optimization problem at time tk+1. For each Sitk , there is

a corresponding set of strike prices Ktk(Sitk) of put options. Then do the followingsteps 2 and 3 for all i = 1, 2, . . . , I.

2. For each strike price Ki,jtk∈ Ktk(Sitk), j = 1, 2, . . . ,M , let us define (πi,jtk , β

i,jtk, αi,jtk ) as

(πi,jtk , βi,jtk, αi,jtk ) := argmin

(πtk ,βtk ,αtk )∈R3

L2

k (πtk , βtk , αtk , Ki,jtk

;Sitk) (A.1)

and find (πi,jtk , βi,jtk, αi,jtk ) for given Ki,j

tk. Notice that (πi,jtk , β

i,jtk, αi,jtk ) is the optimal choice

of rebalancing decision for a given value of underlying index Sitk and a given strike

price Ki,jtk

of the put option. If we choose Ki,jtk∈ Ktk(Sitk) such that it gives the

minimum value of L2

k (πi,jtk , βi,jtk, αi,jtk , K

i,jtk

;Sitk), it must be the optimal strike price.

3. Let us define Kitk

as

Kitk

:= Ki,j∗

tk= argmin

Ki,jtk∈Ktk (S

itk)

L2

k (πi,jtk , βi,jtk, αi,jtk , K

i,jtk

;Sitk),

then Kitk

is the optimal strike price of the put option for given time tk and underlyingindex Sitk . The corresponding optimal rebalancing decisions are given by

πitk := πi,j∗

tk, βitk := βi,j

∗

tk, αitk := αi,j

∗

tk.

4. Repeat the above procedures 1, 2, and 3 from k = N − 1 to k = 0. Then we obtainthe optimal rebalancing decisions for the net liability hedge without any constraint.

Proposition 1. From (9) and (10),

L2

k (πtk , βtk , αtk , Ki,jtk

;Sitk)

= Ek

[{Xtk+1

− πtkStk+1− βtkerδt − αtk(K

i,jtk− Stk+1

)+ −ε(1− ε)k+1F0Stk+1

S0

1{06k6N−2}

}2 ∣∣∣Stk = Sitk

].

26

Thus, the optimization problem in (A.1) is a least squares problem so that we can derive(πi,jtk , β

i,jtk, αi,jtk ) as follows.

πi,jtk = −1

2

4bkckgk − 2bkekik − 2ckdkhk + dkfkik + ekfkhk − f 2kgk

4akbkck − akf 2k − bke2k − ckd2k + dkekfk

,

βi,jtk = −1

2

4akckhk − 2akfkik − 2ckdkgk + dkekik + ekfkgk − e2khk4akbkck − akf 2

k − bke2k − ckd2k + dkekfk,

αi,jtk = −1

2

4akbkik − 2akfkhk − 2bkekgk + dkekhk + dkfkgk − d2kik4akbkck − akf 2

k − bke2k − ckd2k + dkekfk,

where

ak := Etk[S2tk+1

∣∣∣Stk = Sitk

]bk := e2rδt

ck := Etk[{

(Ki,jtk− Stk+1

)+}2 ∣∣∣Stk = Sitk

]dk := 2Etk

[Stk+1

erδt∣∣∣Stk = Sitk

]ek := 2Etk

[Stk+1

(Ki,jtk− Stk+1

)+∣∣∣Stk = Sitk

]fk := 2Etk

[erδt(Ki,j

tk− Stk+1

)+∣∣∣Stk = Sitk

]gk := 2Etk

[Stk+1

(ε(1− ε)k+1F0

Stk+1

S0

1{06k6N−2} −Xtk+1

) ∣∣∣Stk = Sitk

]hk := 2Etk

[erδt

(ε(1− ε)k+1F0

Stk+1

S0

1{06k6N−2} −Xtk+1

) ∣∣∣Stk = Sitk

]ik := 2Etk

[(Ki,j

tk− Stk+1

)+(ε(1− ε)k+1F0

Stk+1

S0

1{06k6N−2} −Xtk+1

) ∣∣∣Stk = Sitk

]jk := Etk

[(ε(1− ε)k+1F0

Stk+1

S0

1{06k6N−2} −Xtk+1

)2 ∣∣∣Stk = Sitk

].

To calculate the above conditional expectations, we need the hedging target Xtk+1as

a function of underlying index Stk+1. However, we only have the values of Xtk+1

whichcorrespond to the discretized values {Sitk+1

}Ii=1. Therefore, the hedging target Xtk+1corre-

sponding to Stk+1other than {Sitk+1

}Ii=1 must be obtained by interpolation or extrapolation.

Appendix A.2. With a Short-selling Constraint: “SC”

Consider the following short-selling constraint

πtk > −Dk, Dk > 0, k = 0, 1, . . . , N − 1. (A.2)

Then the optimal hedging strategies under short-selling constraint can be obtained byfollowing a similar procedure as we used for the unconstrained problem.

27

1. At time tk, suppose that the hedging target Xtk+1= Xtk+1

(Stk+1) is known as a

function of Stk+1from the previous optimization at time tk+1. For each Sitk , there is

a corresponding set of strike prices Ktk(Sitk) of put options. Do the following steps 2and 3 for all i = 1, 2, . . . , I.

2. For each Ki,jtk∈ Ktk(Sitk), j = 1, 2, . . . ,M , let us define (πi,jtk , β

i,jtk, αi,jtk ) as

(πi,jtk , βi,jtk, αi,jtk ) := argmin

(πtk ,βtk ,αtk )∈R3

L2

k (πtk , βtk , αtk , Ki,jtk

;Sitk)

subject to πtk > −Dk. (A.3)

3. Let us define Kitk

as

Kitk

:= Ki,j∗

tk= argmin

Ki,jtk∈Ktk (S

itk)

L2

k (πi,jtk , βi,jtk, αi,jtk , K

i,jtk

;Sitk),

then Kitk

is the optimal strike price of put option, and corresponding optimal rebal-ancing decisions are given by

πitk := πi,j∗

tk, βitk := βi,j

∗

tk, αitk := αi,j

∗

tk.

4. Repeat the above steps 1, 2, and 3 from k = N − 1 to k = 0 to obtain the optimalrebalancing decisions for the net liability hedge under short-selling constraints (A.2)on π.

Proposition 2. Let (πi,jtk , βi,jtk, αi,jtk ) be the solution to the unconstrained optimization (A.1).

Then the solution (πi,jtk , βi,jtk, αi,jtk ) of constrained optimization (A.3) can be determined as

follows

If πi,jtk > −Dk,

πi,jtk = πi,jtk , βi,jtk = βi,jtk , αi,jtk = αi,jtk .

If πi,jtk < −Dk,

πi,jtk = −Dk,

βi,jtk =(−2ckhk + fkik) + (2ckdk − ekfk)Dk

4bkck − f 2k

,

αi,jtk =(−2bkik + fkhk) + (2bkek − dkfk)Dk

4bkck − f 2k

,

where bk, ck, dk, ek, fk, hk, ik are obtained in the unconstrained problem “UC”.

28

Appendix A.3. Without Put Options in the Semi-static Hedging Strategy: “w/o Put”

In the last market environment considered in this document, we assume that put optionsare not available and not used in the hedging procedure. Then the optimization problemfor given Sitk at time tk becomes the following simpler optimization problem.

(πi,jtk , βi,jtk

) := argmin(πtk ,βtk )∈R2

L2

k (πtk , βtk , 0, 0;Sitk), (A.4)

where

L2

k (πtk , βtk , 0, 0;Sitk) (A.5)

= Ek

[{Xtk+1

− πtkStk+1− βtkerδt − ε(1− ε)k+1F0

Stk+1

S0

1{06k6N−2}

}2 ∣∣∣Stk = Sitk

].

It can be verified that the optimal rebalancing decision without put option defined by (A.4)is

πi,jtk =−2bkgk + dkhk

4akbk − d2k,

βi,jtk =−2akhk + dkgk

4akbk − d2k,

where ak, bk, dk, gk, and hk are obtained in the unconstrained problem.

Note that the third argument of L2

k in (A.5) is zero, i.e. αtk = 0, because we do not

use put options in this case. The fourth argument of L2

k in (A.5) corresponds to the strike

price of the put option, it does not need to be zero because L2

k is no longer affected by thestrike price.

Acknowledgment

C. Bernard acknowledges support from the Natural Sciences and Engineering ResearchCouncil of Canada and from the Society of Actuaries Centers of Actuarial Excellence Re-search Grant. M. Kwak acknowledges that this work was supported by Hankuk Universityof Foreign Studies Research Fund of 2015.

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