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Pricing and hedging variable annuities in a levy market: a risk management perspective.

INTRODUCTION

Variable Annuities (VAs) are life insurance contracts linked to financial markets. Companies design VAs so that policyholders can benefit from favorable movements in the markets, yet remain protected when prices plummet. They are often grouped under the acronym GMxB. The G refers to guarantee, M to minimum, B to benefit, and x to a particular contract type: for example, M refers to maturity, D to death, A to accumulation, I to income, and W to withdrawal (see Hardy, 2003; Kalberer and Ravindran, 2009). These contracts have had great success in the United States, the United

Kingdom, Japan, and, to a lesser extent, in continental Europe. This success stems from the fact that VAs offer the opportunity to manage long-term savings and to potentially provide postretirement income. Specific tax advantages also provide incentives for investing in these products. VAs, together with similar contracts such as Equity Index Annuities (EIAs), represent a huge market. Taking into account the world's aging population, and the contribution that VAs can bring to the financing of postretirement income, it is likely that these markets will continue to expand over the coming years.

The pricing, hedging, and risk management of these products are the main areas of concern for insurers and represent a challenge for researchers. VAs involve different risks whose methods of interaction are unknown (Bacinello et al., 2011). Furthermore, these contracts contain nonstandard embedded options, for example, the Guarantee Minimum for Death Benefit (GMDB), in which the optional rider is an option whose expiration date is random (the policyholder's death). Milevsky and Posner (2001) name these options Titanic, give the fair fees for this type of contract, and compare their calculations with the fees actually charged by insurers.

A key assumption in the theoretical analysis of these contracts concerns the modeling of financial prices, and particularly the dynamics of the value of the referenced financial portfolio or market index. Many studies assume a Gaussian hypothesis for financial returns. However, the research now widely accepts that the distribution of these returns is not normal. Asymmetries and leptokurticities have to be taken into account. To cope with these stylized facts, the research has developed models with regime-switching schemes (Hardy, 2003), or with stochastic volatility (Heston, 1993). Furthermore, recent studies, such as Cont (2001) and Ait-Sahalia and Jacod (2009), show that jumps are present in financial prices. The embedded options in VAs are very sensitive to the tails of the underlying distribution, so jumps and/or stochastic volatility have to be taken into account. Pricing and hedging thus become more complex and must be performed in incomplete markets. This is not an easy task. The aim of this article is to suggest a general methodology for both pricing and hedging a subclass of VAs in a general Levy context so that insurers can evaluate fair mortality and expense fees, and can hedge the risk contained in the embedded option. This analysis is developed from an operational risk management point of view.

Pricing these contracts is a familiar theme in the literature. After the seminal article of Brennan and Schwartz (1976) using the principle of arbitrage in continuous time, an impressive flow of articles has been devoted to the pricing of life insurance contracts, in particular VAs and EIAs. Many risks have been considered-mortality risk, market risk, stochastic interest rate risk, default risk, and surrender risk-in the usual Black and Scholes economy, and more recently in a Levy environment (Kassberger, Kiesel, and Liebmann, 2008; Ballotta, 2005). The issue of hedging, extremely important for risk management purposes, has been far less frequently analyzed in the insurance literature than the issue of valuation. The theoretical analysis of hedging in incomplete markets is a well-established area of study in the world of mathematical finance. Many solutions have been proposed, for example, the well-known methods of total risk minimization or local risk minimization (Schweizer, 1991, 2001); however, applications to the insurance field are less frequent. Moller (2001) considers hedging strategies for life insurance in a Gaussian setting. Riesner (2006) and Vandaele and Vanmaele (2008) develop analyses in Levy market environments. These two articles are mainly theoretical, however, and the interesting solutions they provide are difficult to implement. Coleman, Li, and Patron (2007) are close in spirit to our article because of their interest in providing hedging strategies that can actually be implemented. In this last article it is shown that delta hedging, often used in industry, and the static hedging of Carr and Wu (2013) are not appropriate for efficient hedging. Coleman, Li, and Patron (2007) work with a Merton jump-diffusion process and with stochastic volatility models, and compare hedging portfolios based on the underlying asset with hedging portfolios based on standard European options.

For the pricing of the VAs considered, we use the arbitrage principle and work with a risk-adjusted probability measure. Hence, we determine the fair mortality and expense fees in a risk-neutral world. As emphasized by Coleman, Li, and Patron (2007) and Vandaele and Vanmaele (2008), hedging must be conducted in the real world using the actual probability measure. The contributions of this article are, firstly, to calculate explicit fair fees; second, to provide explicit hedging strategies for GMDB, Guaranteed Minimum for Maturity Benefit (GMMB), and a Guaranteed Minimum for Accumulation Benefit (GMAB) with a ratchet feature; and, third, to extend the Coleman, Li, and Patron (2007) hedging strategy using standard options to a general Levy context. We introduce mortality and transaction costs. The whole procedure is illustrated with two jump processes: the first, a Kou process; the second, a CGMY process, both fitted to actual data. This illustration has many practical implications for VA providers. In particular, it shows that the geometric Brownian motion (GBM) hypothesis leads to undervalue by far the actual economic capital necessary to hedge and gives an illusion of safety.

PRODUCTS

Let T be the expiration date of the contract. We denote by [gamma] [less than or equal to] T the time at which the guarantee is triggered. The contract payoffs are of the following type:

max([F.sub.[gamma]], [G.sub.[gamma]]),

where [F.sub.[gamma]] is the account, or fund value, of the policyholder at time [gamma]. The quantity [G.sub.[gamma]] is the guarantee, which can take many expressions such as a constant equal to the premium, the premium accrued at a guaranteed rate g, or the highest account value recorded up to [gamma]. Ratchets and caps can also be incorporated. Because

max([F.sub.[gamma]], [G.sub.[gamma]]) = [F.sub.[gamma]] + [[[G.sub.[gamma]], [F.sub.[gamma]].sup.+],

the contract can be considered, from the policyholder's point of view, as a long position in the fund and a put option with the strike price of [G.sub.[gamma]]. The insurer is at risk because of the short position in the option. The insured's upfront single premium is invested in an equity fund whose value at time t is [S.sub.t]. Let m denote the management charge continuously deducted from S at the beginning of each period. We consider m as the sum of two rates: [m.sub.o], used to fund the guarantee, and [m.sub.a], for the other management expenses. In other words,

m = [m.sub.o] + [m.sub.a].

The account value process F then evolves according to:

d[F.sub.t]/[F.sub.t] = d[S.sub.t]/[S.sub.t] - mdt,

or in an equivalent definition, its value at time t is:

[F.sub.t] = [F.sub.0] [S.sub.t]/[S.sub.0] [e.sup.-mt] = [S.sub.t][e.sup.-mt]. (1)

The last equality follows from the convention [F.sub.0] = [S.sub.0]. To introduce mortality, let [bar.T] be the policyholder's age at death and [T.sub.x] = [bar.T] - x, the residual lifetime for the insured aged x at time t = 0. We denote by

t[p.sub.y] = 1 - t[q.sub.y] = P[[bar.T] > y + t|[bar.T] > y],

for any y [greater than or equal to] x and t [greater than or equal to] 0, the conditional survival probability under the physical measure P. In particular, t[p.sub.x] = P[[T.sub.x] > t] = exp {- [[integral].sup.t.sub.0] v(x + s)ds}, where v is the mortality intensity. For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [p.sub.x+t] = 1 - [q.sub.x+t], where [q.sub.x+t] is the probability that the insured aged x + t dies before the end of the year. For convenience, we first consider GMMB and GMDB with a constant guarantee K. If the insured is alive at time T, he or she receives [F.sub.T] + [[K - [F.sub.T]].sup.+] under the GMMB and, if death occurs before the contract expiration date, the beneficiary of the policy is entitled to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] under the GMDB. We then consider the GMAB contract, which adds the possibility of modifying the guarantee at certain specified dates by adding a ratchet effect. We detail these contracts in the "General Valuation Formulae" section. The investment period [0, T] is subdivided into T/h subintervals with the same duration h (e.g., h = [12.sup.-1] if monthly). If death occurs during a subinterval, the benefit is assumed to be paid at the end of that subperiod. In addition, we denote as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the initial price of the embedded guarantee in the VA contract. In this article, we rest on arbitrage theory and thus consider a risk-adjusted measure Q. As we focus on market risk, we assume that mortality risk can be diversified away and hence P and Q coincide on mortality-related events. We also assume independence of financial risk and mortality risk under Q.

Fair Fees

If [r.sub.u] is the instantaneous interest rate at time u in the economy, then the discount function is defined by

[[delta].sub.t] := exp {- [[integral].sup.t.sub.0] [r.sub.u] du}.

The insurer charges continuously the policyholder at rate m. A part of this rate, called the margin offset and denoted by [m.sub.o], is deducted from the fund in order to finance the guarantee. Then, the discounted fee process [bar.F] is such that

d[[bar.F].sub.t] = [m.sub.o][[delta].sub.t][F.sub.t]dt,

with [[bar.F].sub.0] = 0. We assume that the referenced equity portfolio value follows a geometric Levy process under a risk-neutral probability measure Q. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where Mt is defined by

[M.sub.t] = [[integral].sup.t.sub.0] [[integral].sub.R] z(N(ds, dz) - v(dz)ds) + [sigma][Z.sub.t], (2)

where N is the jump measure; v, the Levy measure; Z, a standard Brownian motion; and [sigma], the diffusion volatility coefficient. By construction, M is a martingale. Denoting by M&E the expected discounted value of all the mortality and expense fees paid as long as the contract is in force, that is, until death or maturity, whichever comes first, then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is shown (see Appendix A) that the mortality and expense M&E associated with the contract has the following market value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For a given management rate [m.sub.a], the equilibrium value or the fair price for the guarantee is the solution in [m.sub.o] of the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Otherwise stated, the fair cost is such that the value associated with the discounted continuous cash flows coming from the fees is equal to the contract's optional rider value. It is worth noting that this equilibrium equation gives a correspondence between the marginal offset rate [m.sub.o] and the management rate [m.sub.a]. Also, Equation (3) takes different expressions according to the particular VA contract. The related formulae in the "General Valuation Formulae" section are quite general, and to make them operational, choices regarding the modeling of the financial price and the mortality have to be made. For this last point, mortality tables or laws can be used. This determination of fair fees is a generalization of Milevsky and Posner (2001) and Quittard-Pinon and Randrianarivony (2011).

General Valuation Formulae

To price the guarantee, we resort to the arbitrage pricing theory in continuous time. Equation (1) suggests that the dynamics of F are the same as those of the risky equity process S distributing a continuous dividend rate equal to m. Let P([bar.S], d, K, [tau]) denote the price of a European put option written on S, with an initial value equal to [bar.S], with exercise price K, continuous dividend rate d, and time to maturity [tau]. With the independence assumption of financial risk and mortality risk under the risk-adjusted measure, the embedded guarantee option prices are:

* In case of life benefit (GMMB) paid at the contract's expiration date T, in case of survival:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

* In case of death benefit (GMDB) paid at the time of death [T.sub.x] [less than or equal to] T:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

* The third contract, a mix of life and death benefits, GMMB/GMDB, offers a guaranteed minimum benefit in the case of either death, or survival to maturity, whichever comes first:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

* The fourth contract, a GMAB is more complex. This contract incorporates a ratchet effect on the guarantee. To be more precise, a set I = ([t.sub.1], [t.sub.2], ..., [t.sup.-.sub.n-1], [t.sup.-.sub.n]} of [bar.n] rollover dates or tenor structure enables the guarantee to be modified at each of these times in the following way: let [F.sup.-.sub.t] and [F.sup.+.sub.t] be the fund values just before and just after the guarantee is reset at time t. First note that at each reset time [t.sub.i] [member of] I, the fund value immediately before renewal, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is related to the fund value brought forward from time [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], through the relationship in Equation (1). In other terms,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Immediately before each reset date [t.sub.i], the insurer compares the account value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to a guaranteed level (1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Provided that the contract is still in force, two scenarios are possible:

--If the guarantee is in the money, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the insurer cashes in the absolute difference [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and immediately after [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is set equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

--If the guarantee is out of the money, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there is no cash-in for the insurer, but [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is reset to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and immediately after, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is set equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In an equivalent way, at each [t.sub.i],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the starting guarantee is [G.sub.0], and the starting account value is [F.sub.0]. Let ([S.sub.0], [m.sub.o], [t.sub.i]) be the price at time 0 of the ith guarantee,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assuming a constant interest rate, note that for all n > 1, [t.sub.n] [member of] I, the value of the insurer's liability can be obtained from the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the starting step is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(see Appendix B). Introducing mortality, the initial price [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the GMAB optional rider can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

For all n > 1, [t.sub.n] [member of] I, the prices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] associated with the optional part [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are evaluated by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with the starting value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The formulae (4, 5, 6, 8) give the prices of the optional riders of the four contracts analyzed in this article. These formulae are quite general and are not dependent on any particular dynamics in the financial prices.

PRICING AND HEDGING IN A LEVY CONTEXT

As shown in Equation (1), the dynamics of the account value process F are the same as those of S, up to the continuous dividend rate d equal to m. We assume that the price of the equity portfolio is modeled by the exponential of a Levy process X. Therefore, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Except in the Gaussian case, Levy processes present jumps; consequently, the market is incomplete. The main problem with the above-mentioned formulae (4, 5, 6, 8) is therefore pricing and hedging European options in this incomplete market context. This is a complex and well-known problem that is often studied from a theoretical point of view. However, in this article we focus on efficient solutions from an implementation perspective. We choose the generalized Fourier approach (see Titchmarsh, 1937), in line with the work of Boyarchenko and Levendorskii (2000), Lewis (2001), and Eberlein, Glau, and Papapantoleon (2010). This method differs from the Carr and Madan (1998) solution and gives accurate and fast results. The key element in our analysis is the characteristic exponent of the process X denoted by and definable from the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Pricing

Assuming a constant interest rate r, it can be shown that the European put option defined in the "General Valuation Formulae" section can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

with b > 0, x' = ln ([bar.S]/K), and with the equivalent martingale measure (EMM) condition

r - d + [[psi].sub.Q](-i) = 0. (11)

It is worth noting that Equation (10) holds for a European call option, with b < -1. The expression in Equation (10) is very well suited for a Fast Fourier Transform (FFT) computation. Option prices can thus be obtained quickly and accurately once the characteristic exponent of X is known.

Hedging

The problem of hedging in incomplete markets has been extensively studied. These studies propose various solutions: the mean-variance approach, super hedging, quantile hedging, and the local risk-minimizing framework. For a survey see Cont and Tankov (2004). Although the theoretical solutions improve the efficiency of hedging, the research rarely takes a computational point of view, in contrast to the approach that we adopt in this section. We first introduce [theta]-optimal hedging and then compare it with [DELTA] hedging before finally considering hedging with a portfolio of standard European options.

Hedging Using the Underlying: The [theta] Approach. In this article, we follow the example of Boyarchenko and Levendorskii (2000) allowing to obtain elegant analytical results. Consider an investor with wealth W(t) at time t who takes a short position in the European put option with time to maturity [tau]. The investor also takes a long position in [theta]([S.sub.t], [tau]) units of the underlying S, and invests the residual wealth [bar.[omega]](t) in the riskless account with the constant return r, resulting in

[bar.[omega]](t) = W(t) - [[theta]([S.sub.t], [tau])[S.sub.t] - P([S.sub.t], d, K, [tau])].

At time t + [DELTA]t, [DELTA]t < [tau], the investor's wealth is

W(t + [DELTA]t) = -P([S.sub.t+[DELTA]t], d, K, [tau] - [DELTA]t) + [theta]([S.sub.t], [tau]) [S.sub.t+[DELTA]t] + [e.sup.r[DELTA]t][bar.[omega](t).

Following Boyarchenko and Levendorskii (2000), we choose the hedging ratio [theta] that minimizes the variance of the investor's wealth at t + [DELTA]t. After computations, it can be proven that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

with the EMM (11), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that Formula (12) is also valid for calls with the choice: b < -1. The hedging ratio is then obtained through a quadrature that can be performed using the FFT in exactly the same way as for the calculation of the option price, thereby unifying the pricing and the hedging methods. Formula (12) clearly shows that hedging must be thoroughly performed because the parameters in the optimal ratio pertain to different universes: namely, historical versus risk neutral. This distinction disappears if the local risk minimization is performed in the risk-neutral universe, as presented by Cont and Tankov (2004). The hedging ratio formula that they give is different but leads to the same numerical results (see Kelani and Quittard-Pinon, 2014). For a thorough discussion of quadratic hedging in a Levy context, we refer to Riesner (2006) and to Vandaele and Vanmaele (2008) who obtain Schweizer's local risk-minimization strategy for hedging exhibiting a Follmer-Scheiwzer decomposition under the real probability measure. Their formula, although different, has the same structure as those given by Cont and Tankov (2004) and Riesner (2006). All these formulae are more difficult to implement than the [theta] ratio in Equation (12). We emphasize that the solution supported by this article is first to model the dynamics of the associated financial portfolio in the historical universe, and second to choose a risk-neutral probability measure for valuation only. We make the convenient choice of using Esscher measures (see, e.g., Gerber and Shiu, 1994) to pass between the two universes. We choose the Esscher measure under which the discounted gain process is a martingale. Relationships between the Levy processes in the historical and in the Esscher risk-neutral universes exist. Appendix (C) provides general formulae for the bridge between process characteristics in these two universes.

[DELTA] Hedging Versus [theta] Hedging. Let [DELTA] be the derivative of the option price with respect to the underlying price, the so-called delta ratio. This ratio measures the sensitivity of the option to the underlying price. Using Formula (10),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

The difference between the [theta] in Equation (12) and the above delta ratio [DELTA] is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Figure 1 illustrates this difference in the CGMY setting defined in the "Financial Price Process" section. We let the time to expiry [tau] vary from 0.5 to 1 for the European put option. In this numerical example we see that the differences between [theta] and [DELTA] are almost null for the deeply out-of-the-money options and rise more slowly to zero when the option becomes increasingly in the money. When the option is at the money, the difference is at its highest. This behavior has practical implications for our contracts, especially for the GMAB as developed in the "Hedging Using the Underlying Asset" section.

Hedging Using Standard Options. Hedging strategy usually consists of a portfolio based on both the underlying and riskless assets. However, with the expansion of liquid option trading, especially for short-term maturities, standard options can be used as hedging instruments. VAs being long-term contracts, options with the same maturity as the embedded options may not exist. Thus, a potential solution is to use rolling positions in options with shorter maturities; see Carr and Wu (2013) who assume the existence of a continuum of standard options. Coleman, Li, and Patron (2006) suggest another approach involving the use of a finite number of options, which proves to be more realistic and which we extend to a general Levy context. We only sketch this extension here and refer to Coleman, Li, and Patron (2006, 2007) for their complete approach. We assume that the hedging instruments' trading dates form a partition of the considered hedging horizon [0, T] and that this partition is regularly spaced with the T/h subintervals. In other terms, define the hedging strategy ([[zeta].sub.k], [[eta].sub.k]) at trading dates [t.sub.k], for k = 1, ..., T/h, where [[zeta].sub.k] denotes the quantities of standard European options for the kth subperiod, and [[eta].sub.k] the amount invested in the riskless asset. Let H([t.sub.k]) be the value of the hedging portfolio at time [t.sub.k] and [U.sub.k]([t.sub.k+1]), the values of the risky assets at time [t.sub.k+1]. The usual local risk minimization involves the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the information revealed up to time [t.sub.k]. Coleman, Li, and Patron (2006) use a discretization of (14) and first impose the hedging portfolio, at the maturity date T, equal to the payoff of the embedded option. They then minimize, in the least square sense, the incremental hedging error incurred from adjusting the portfolio at each time back to the initial time t = 0. They compute the hedging positions in a uniformly spaced grid given the set of possible asset prices. The optimal holdings are then obtained for each grid node by solving a weighted least square problem; see Equation (10) in Coleman, Li, and Patron (2006, p. 220). The weights are the transition probabilities from the current stock price at each grid node to a feasible stock price set. Using interpolations, they obtain the hedging strategies corresponding to independent simulations. In the Black-Scholes and the Merton jump-diffusion frameworks they consider, analytic formulae exist both for the option prices and for the transition probabilities. In our Levy context, we use (10) to obtain the option prices and invert the characteristic function (9) to compute the transition probabilities. Hence, Coleman, Li, and Patron's (2006) algorithm can be extended to Levy processes with known characteristic exponents.

RISK MANAGEMENT

In the previous sections, we suggested a general approach for pricing VA contracts, determining the fair cost of the guarantees, and defining the optimal hedging ratios. From a practical point of view, continuous monitoring is impossible, as monitoring can only be performed in discrete time. Thus, if the approach uses the theoretical hedge ratio formulae, a hedging error occurs that must then be computed. In addition, transaction costs have to be taken into account. We now present a method to compute this error inspired by Hardy (2003).

Hedging Life-Contingent Benefits

We illustrate the hedging procedure with mortality through a combined GMMB/GMDB contract. Using actuarial notations, we introduce [sub.[bar.y]|h][q.sub.y], for any y [greater than or equal to] x and [bar.y] [greater than or equal to] 0, as the probability that he or she survives during the next [bar.y] years and dies in the subsequent period of length h. Let t [member of] [0, T], and assume that the contract is still in force at that date. For the sake of simplicity, we denote by P([S.sub.t], [tau]) the price of the European put option with time to maturity [tau], and strike K, and by [xi](t), the contract value at time t. The corresponding value conditional on survival at t, denoted [[xi].sub.c](t), is

[[xi].sup.c](t) = [T - h.summation over (k=t)] [sub.k-t|h][q.sub.x+t] P([S.sub.t], k + h - t) + [sub.T-t][p.sub.x+t] P([S.sub.t], T - t).

The unconditional value at time t is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To construct the hedging portfolio at time t, we consider the above conditional and unconditional scenarios. In the first case, conditional on survival at age x + t, the number of risky asset [[PSI].sup.c.sub.t] is

[[PSI].sup.c.sub.t] = [T - h.summation over (k=t)] [sub.k-t|h][q.sub.x+t] [theta]([S.sub.t], k + h - t) + [sub.T-t][p.sub.x+t] [theta]([S.sub.t], T - t).

while the amount invested in the riskless asset [[??].sup.c.sub.t] is [[xi].sup.c](t) - [[PSI].sup.c.sub.t][S.sub.t]. Hence, the hedging portfolio value, denoted [H.sup.c](t), is [[PSI].sup.c.sub.t][S.sub.t] + [[??].sup.c.sub.t] = [[xi].sup.c](t). However, just before the rebalancing at time t, its value is [H.sup.c]([t.sup.-]) = [[PSI].sup.c.sub.t-h] [S.sub.t] + [[??].sup.c.sub.t-h] [e.sup.rh]. In the latter case, the number of units of the risky asset [[PSI].sub.t] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

while the amount invested in the riskless account [[??].sub.t] is [sub.t][p.sub.x] [[??].sup.c.sub.t] = [xi](t) - [[PSI].sub.t][S.sub.t]. The hedging portfolio value H(t) is [sub.t][p.sub.x][H.sup.c](t) = [[PSI].sub.t][S.sub.t] + [[??].sub.t] = [xi](t), and just before the rebalancing at time t, its value is worth H([t.sup.-]) = [[PSI].sub.t-h][S.sub.t] + [[??].sub.t-h][e.sup.rh].

Actual Hedging Error

Two scenarios determine the hedging error HE. First, if the insured survives at time t, the error is the difference between the value of the hedge portfolio required at t and the portfolio value before the rebalancing H([t.sup.-]). Second, if the insured dies between t - h and t, the HE is the difference between the guarantee in the case of death and H([t.sup.-]). Precisely, the hedging error HEC, conditional on survival at time t - h, is

[HE.sup.c](t) = [sub.h][p.sub.x+t-h]([H.sup.c](t) - [H.sup.c](t-)) + [sub.h][q.sub.x+t-h][((K - [F.sub.t]).sup.+] - [H.sup.c]([t.sup.-])).

Specifically, with [sub.h][p.sub.x+t-h] + [sub.h][p.sub.x+t-h] = 1, we obtain

[HE.sup.c](t) = [sub.h][p.sub.x+t-h][H.sup.c](t) + [sub.h][q.sub.x+t-h][(K - [F.sub.t]).sup.+] - [H.sup.c]([t.sup.-]).

Finally, the unconditional error is obtained by weighting the previous conditional hedging error by [sub.t-h][p.sub.x], hence:

HE(t) = [sub.t-h][p.sub.x][HE.sup.c](t) = H(t) - H([t.sup.-]) + [sub.t-h|h][q.sub.x][(K - [F.sup.t]).sup.+]. (15)

The quantity HE(t) could be either positive or negative depending on the market scenario. In the latter case, it is a source of profit because the hedge portfolio's value before rebalancing, H([t.sup.-]), is more than that required to hedge the guarantee at time t.

Losses

Future losses appear as the aggregation of three components. The first is the value of the residual errors from Equation (15), the second denoted by TC, is generated by transaction costs, and the third is the margin amount M that is deducted by the insurer to fund the guarantee. The value of the loss at time t, [L.sub.t], is thus given by

[L.sub.t] = HE(t) + [TC.sub.t] - [sub.t][p.sub.x][M.sub.t], (16)

with

[M.sub.t] = [m.sub.o][F.sub.t],

where [m.sub.o] is determined in equilibrium as a result of Equation (3). Furthermore, the transaction costs, denoted by C, are proportional to the absolute variation of the amount allocated to the hedging portfolio. In an equivalent way,

[TC.sub.t] = [CS.sub.t] [absolute value of [[PSI].sub.t] - [[PSI].sub.t-h]]. (17)

Formula (16) allows for the assessment of different measures for the risk management of the VA contracts. We consider the accumulated discounted values of losses (ADL) and apply some standard risk measures to them. We use the same formula for hedging with standard European options. However, it is worth noting that the resulting cumulative discounted residual error corresponds to the total risk cost, that is, the amount of money by which the hedging strategy is short of meeting the liability at the maturity, as in Coleman, Li, and Patron (2006) and Mercurio and Vorst (1996).

ILLUSTRATION

In this section, we numerically illustrate the complete procedure proposed in this article. We do so by using a Gompertz-Makeham mortality law, and the Kou and CGMY models for the Levy process X defined in the "Pricing and Hedging in a Levy Context" section. We also consider the GBM as a benchmark. All these processes are fitted to actual data.

Mortality

The mortality intensity is

v(x) = A + [Bc.sup.x] , with B > 0, A [greater than or equal to] -B, c > 1 and x [greater than or equal to] 0.

Hence, the survival probability is

[sub.t][p.sub.x] = exp {-At - [Bc.sup.x]([c.sup.t] - 1)/ln c}.

We use the U.S. mortality parameters obtained by Melnikov and Romaniuk (2006):

A = 9.5666 x [10.sup.-4], B = 5.162 x [10.sup.-5], c = 1.09369.

Financial Price Process

The GBM is the basic model in continuous time finance (see Black and Scholes, 1973), with v(z) = 0 in Equation (2) for all z. The characteristic exponent is reduced to:

[psi](u) = -iu [mu] + 1/2 [[sigma].sup.2] [u.sup.2].

Kou (2002) uses a jump-diffusion process where the jump component is a compound Poisson process and the random jump sizes follow an asymmetric double exponential distribution. The characteristic exponent is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with [[lambda].sub.1] > 1 and [[lambda].sub.2] > 0 such that - [[lambda].sub.1] < Im u < [[lambda].sub.2] and p, q [greater than or equal to] 0, such that p + q = 1, are, respectively, the probabilities of upward and downward jumps, and where [lambda] is the jump intensity. Empirical tests performed in Ramezani and Zeng (1998) suggest that this kind of model provides a better fit than the Merton jump-diffusion model. Although it is appropriate to use compound Poisson jumps to capture non-negligible probabilities of rare and large events such as market crashes, several authors observe that asset prices actually display many small jumps. These types of features are better explained by infinite activity processes, which generate an infinite number of jumps within any finite time interval. A popular example is the variance gamma model introduced by Madan and Seneta (1990), which is obtained by time-changing a Brownian motion with a gamma subordinator. Another popular example capable of generating different jump types is the CGMY model of Carr et al. (2002), with the following characteristic exponent:

[psi](u) = -iu [mu] - C [GAMMA](-Y) [[(M - iu).sup.Y] - [M.sup.Y] + [(G + iu).sup.Y] - [G.sup.Y]],

where C > 0, G [greater than or equal to] 0, M [greater than or equal to] 0, Y < 2 and where [mu] is a drift term added to the model (see Kim and Lee, 2007). This process is a generalization of the variance gamma model. The parameter C describes the intensity of the process; it plays a similar role to that of the variance of the Brownian motion. The parameters G and M control the rate of exponential decay on the right and on the left tail of the Levy density, respectively, leading to a skew when they are unequal, and so the case G < M is in line with empirical studies where the left tail of the process is heavier.

Fitting to the Data

Using the moment matching method, we fit the Kou and CGMY models to the monthly total return S&P 500 index prices, in U.S. dollars, observed from January 31, 1956 to May 30, 2014. The rounded annual parameter estimates are contained in Table 1, together with the four first central moments. We can note that the Kou and CGMY models are relatively close to each other in terms of moments. It is no surprise that one can hardly distinguish between them in Figure 2a. This observation is clearly due to the fact that parameters were estimated through moment matching and two distributions with the same four moments are approximately equal. Computations (e.g., expected values) using such distributions will always give very similar, if not indistinguishable, results. Alternatively, model parameters could be determined using a different criteria (i.e., fitting models to option prices). The goodness to fit can be seen in Figure 2 showing a two-sample QQ-Plot approximating a straight line in case the comparison distributions provide a good fit. Figure 2b-d shows the QQ-Plots from the historical log return distribution compared to the standard normal distribution and to a 100,000 simulated sample on the Kou and CGMY based estimates, respectively. As revealed by many empirical studies and we can see here, the jump models fit the data better than the GBM by providing the necessary adjustment to give a sufficient accurate fit in the left tail distribution. Furthermore, it is necessary to know the processes for the referenced portfolio in the risk-neutral universe for pricing and hedging. Following Kim and Lee (2007) and Le Courtois and Quittard-Pinon (2008), the correspondence between the historical and risk-neutral hatted parameters is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [k.sup.*] is the parameter such that the discounted gain process is a risk-neutral Esscher measure (see Equation (C1) in Appendix C). As pricing and hedging are in the end based on expected values, the estimates in Table 1 inevitably produce similar results for the CGMY and Kou models as far as we experiment. Therefore, the illustration will mainly use the Kou model.

Contracts

Unless otherwise stated, we choose the following values for the exogenous parameters. The referenced portfolio follows the Levy processes in the "Financial Price Process" section, with parameters in the historical universe given in Table 1. The transaction costs in Equation (17) are such that C = 0.2 percent. The insured age is x = 40 years at the contract inception. A 100 USD premium is invested in the referenced portfolio, which has an initial value of [F.sub.0] = 100. The interest rate is r = 6 percent. We consider four contracts: GMMB, GMDB, mixed GMMB/GMDB, and GMAB. The initial guarantees K and [G.sub.0] are assumed to be 100 percent of the initial account value [F.sub.0] for the first three contracts and 80 percent of [F.sub.0] for the last.

M&E Rates. The guarantee is funded endogenously per the equilibrium Equation (3). For a given management rate [m.sub.a], Figures 3 and 4 show the equilibrium value of the funding guarantee rate [m.sub.o]. As we can see, the margin offset rate increases with respect to [m.sub.a] and more rapidly for short-term contracts. This illustration highlights the high values of M&E charges noted by Milevsky and Posner (2001) in the VA market. Although this market has changed dramatically in the last decade, the market median charge of [m.sub.o] = 115 bp reported by the authors can be explained by management charges [m.sub.a] of 4.24 percent and of 4.26 percent for a 10-year joint GMMB/GMDB contract in the GBM and Kou economies, respectively. In what follows, we assume unless otherwise stated, a management rate [m.sub.a] to be equal to 3 percent for all the studied contracts.

Prices and Costs. Using Equation (3), Table 2 for GMMB, Tables 3 and 4 for GMDB, and Table 5 for joint GMMB/GMDB, give equilibrium prices for embedded options and fair costs for the margin offset rate [m.sub.o] necessary to fund the guarantee liability, while Table 6 gives the fair costs and fair prices for GMAB contracts. As expected, the following hierarchy is observed: GMAB is more expensive than joint GMMB/GMDB, which is more expensive than GMMB, which is in turn more expensive than GMDB, which is the cheapest. The GMAB, although attractive for investors, is particularly costly: for a 30-year maturity, the margin offset rate (in the case of either death or life) is 14.52 bp, while for a joint GMMB/GMDB, this value is 1.12 bp. The involved difference of 13.4 bp is the additional rate required to fund the attractiveness in the GMAB at its reset dates. In fact, at these dates of revision, the insurer may have either to reset a low policyholder account to a high value of guaranteed level brought forward, in case of poor market performance, or to reset this guaranteed level to a high account value in case of favorable movements. This accumulation benefit is financed by the additional rate of 13.4 bp with respect to the mixed GMMB and GMDB. In light of the estimates in Table 1, it is worth noting that the Kou jump-diffusion model results in fair fees higher than those in the GBM; nonetheless, the involved difference is not important as we can note in Tables 2-6. Thus, we conclude that the fair funding guarantee rates [m.sub.o] of GMMB, GMDB, GMMB/GMDB, and GMAB contracts are not very sensitive to the tails of the underlying distributions.

Hedging Using the Underlying Asset. The hedging strategy makes use of a Monte Carlo simulation with 20,000 sample paths and uses the [theta] ratio with a monthly rebalancing. For a given confidence level [alpha], we report the value at risk (VaR), that is, the minimum amount of money that the hedging portfolio is short of meeting the guarantee liability, with [alpha] probability. We also report for the given [alpha], the conditional tail expectation (CTE), that is, the expected loss conditional on the loss exceeding [VaR.sub.[alpha]]. Table 7 displays the results compared to the GBM. Although the GBM is not appropriate, it helps to concentrate on the impact of jumps in the actual loss. To be precise, note that the estimated volatilities on the GBM and Kou returns match in Table 1. The estimated quadratic variations being the same, the Kou jump-diffusion suggests that there are about 2.6 jumps per year with the average jump size (2) of -2.34 percent and jump volatility of 4.06 percent. Figures in Table 7 show the impact of this observation on the GMMB/GMDB and GMAB ADL. As we can see, the jump model entails an important part of loss, underestimated by the GBM. For the [alpha] = 99 percent probability, it requires an economic capital of ([VaR.sub.[alpha]] = 3.3929, [CTE.sub.[alpha]] = 3.9429) for the GMMB/GMDB and ([VaR.sub.[alpha]] = 6.5058, [CTE.sub.[alpha]] = 8.5835) for the GMAB, while these figures for the GBM are ([VaR.sub.[alpha]] = 1.9087, [CTE.sub.[alpha]] = 2.4515) and ([VaR.sub.[alpha]] = 2.7754, [CTE.sub.[alpha]] = 3.8086), respectively. In other words, for the 99 percent (VaR, CTE), jumps result in an additional reserve of (1.4842, 1.4914) for the mixed GMMB/GMDB and (3.7304, 4.7749) for the GMAB. This illustration warns that the actual accumulated discounted loss in VA guarantees is highly sensitive to the tails of the underlying distributions and may lead to an important mishedging as soon as less care is taken to extreme risks. Citing the recent episode of VA providers who had to leave the market as a consequence of mishedging their guarantees may support this observation. Also, we should highlight the following general remark: even for 10-year joint GMMB/GMDB and 22-year GMAB benefits, the [DELTA] hedging is less efficient than the optimal [theta] ratio allocation by requiring more economic capital (see Table 8). The involved difference becomes considerable in the management of the GMAB. For illustrative purposes, ignore mortality and assume [m.sub.o] = 0. In addition, consider a GMAB liability with a tenor I = {[t.sub.1], [t.sub.1] + 10}. Using the general Formula (8), the time t = 0 of the insurer's liability is therefore:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Observe that as t [right arrow] [t.sub.1], if the fund value is greater than [G.sub.0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In other terms, immediately after the renewal date [t.sub.1], the insurer position is similar to that of an at-the-money put option writer with a strike price of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where Figure 1 illustrates that the difference between the optimal [theta] and the [DELTA] ratios is at its highest. As noted by Hardy (2003), the GMAB provider is subject to a significant loss at each rollover date [t.sub.i]-and we should add in the light of Figure 1--at least enough if a [DELTA]-hedging allocation is used. This illustration also warns against the use of A ratio in the management of VA liabilities. Although the involved difference is not negligible for 100 USD premium contracts, it will undoubtedly be sizable if scaled to a large amount.

Hedging Using Standard Options. Due to liquidity considerations, standard European options with a maturity of 1 year are used, and we thus consider a yearly rebalancing period: h = 1. In other words, a hedging portfolio constructed at time t is liquidated at time t + 1 when a new portfolio is set up. We use three puts with strikes [90 percent, 100 percent, 120 percent] x [S.sub.t] where St is the current financial price. We numerically compute the ADL for the GMMB/GMDB mix using the Coleman, Li, and Patron (2006) method extended to the case of Levy processes. We analyze the strategy using a portfolio of standard options (HSO) in the same way we did for the strategy using the underlying asset (HUA). Using the same arguments as in the "Hedging Using the Underlying Asset" section about the identical quadratic variation estimates of returns, we note that non-Gaussian losses observed in the actual data and resulting in a heavier tail lead to a potential mishedging with respect to the (GBM framework. Following Table 9, we contrast the economic capital requirement of ([VaR.sub.99%] = 0.6022, [CTE.sub.99%] = 1.2849) in the jump setting to that of ([VaR.sub.99%] = -1.2456, [CTE.sub.99%] = -0.7603) in the GBM model. And we can note that the Gaussian setting claims almost no reserve for the VA provider. This illustration is again in line with the great attention that should be paid to extreme risks. The HSO strategy achieves better risk minimization on the future ADL loss: contrast ([VaR.sub.99%] = 0.6022, [CTE.sub.99%] = 1.2849) to ([VaR.sub.99%] = 3.3929, [CTE.sub.99%] = 3.9429) using the HUA in Table 7. Furthermore, the HSO entails a potential benefit for the VA provider in the left-tail distribution of the ADL. This observation brings back to the definition of the loss function in Equation (16) where the margin amount M is introduced. We can remark that the impact of the fees is more pronounced with the HSO than with the HUA: this is explained by the leverage effect coming from the holdings in the portfolio of put options used as hedging instruments.

CONCLUSION

Pricing, hedging, and assessing risk for VAs are major areas of concern for the insurance industry. Mishedging and mispricing can have severe consequences for VA providers. To address these questions, we suggest a unified methodology illustrated to the GMMB, GMDB, and GMAB. To take stylized facts into account, we choose to represent financial prices by geometric Levy processes leading to heavier tails, actually observed. Since our contracts are particularly sensitive to such fat tails, this allows us to incorporate extreme risks in the tails of the equity return distributions. We give general formulae that do not depend on a particular model. We obtain endogenously fair fees for mortality and expenses. Pricing is performed in a general Levy context, thus in an incomplete market. Assuming the case in which a risk-adjusted measure is chosen, we give very efficient pricing formulae for these contracts. For hedging, we consider two strategies, both using a local risk minimization performed in the real world. The first one, based on the underlying asset, uses the so-called [theta] ratio. Technically speaking, the formulae for pricing and hedging have the same structure and can be considered as quasi-closed-form solutions. The second strategy relies on a portfolio of standard European options. In particular, we extend the Coleman, Li, and Patron (2006, 2007) algorithm to a general Levy framework. For assessing risk, we use discounted accumulated losses including hedging errors, transaction costs, fees, and mortality. Then, we apply some standard risk measures, namely the VaR and the CTE. In our analysis, we consider the Makeham-Gompertz force of mortality. Mortality tables can also be used as well. We illustrate our whole procedure with a Kou and a CGMY process, both fitted to actual data. From this numerical analysis, we draw practical implications for VA providers. We emphasize that real returns do not conform to the Gaussian distribution. Alternative processes such as CGMY and Kou provide accurate fit, especially in the left tail. This leads to important consequences. The heavier tail results in higher fair fees and has even more important impact on hedging. Our illustration clearly proves that economic capital necessary to hedge is larger than those required in the GBM framework, and hedging VAs with this setting can be disastrous. On the contrary, CGMY and Kou processes incorporating extreme movements in the policyholder fund price constitute a convincing choice. Another important conclusion arising from this numerical analysis is that the A strategy should be avoided: the [theta] strategy gives better results. Our study also shows that the best performance is obtained by hedging with a portfolio of standard European options. This result is noted by Coleman, Li, and Patron (2007) for the Black-Scholes, Merton jump-diffusion and with stochastic volatility models. However, this strategy is more delicate to implement in practice.

APPENDIX A: MORTALITY AND EXPENSE

The policyholder's account-value dynamics under Q are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using the product rule,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Integrating the above equation, we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A1)

Note that the expression in Equation (A1) is true even if t [less than or equal to] T is random. In particular,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Under the risk-adjusted measure, the expected discounted value of all the fees paid until time T [conjunction] [T.sub.x] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because the discounted gain process of the fund is a Q-martingale, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using the optional sampling theorem (see Karatzas and Shreve, 1991), the last expectation on the right-hand side of the above equation is null. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A discrete approximation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and so, the expected discounted value of all the fees paid as long as the contract is in force, that is, until death or maturity, whichever comes first, has the following market value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

APPENDIX B: THE GMAB OPTION PRICE

Assuming the starting guarantee [G.sub.0] and the starting account value [F.sub.0] = [S.sub.0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

what we want to prove is the following result:

For all n > 1, [t.sub.n] [member of] I,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (B1)

with the starting value

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (B2)

Proof: By definition,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Up to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], note that the proof of the statement in (B1) together with (B2) is ended by the result in Lemma 1.

Lemma 1: For all n > 1, [t.sub.n] [member of] I, the expected discounted value of the fund just after the guarantee is reset at time [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (B3)

together with (B2).

Proof: First, (B3) holds for n = 2:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

leading to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, for n = 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B2).

Assuming that the above formula holds for some n > 2, Equation (B3) is also true for n + 1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

leading to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B2),

and which ends the proof. Also, note that the above valuation could be extended to random rates.

APPENDIX C: RELATIONSHIPS BETWEEN THE REAL WORLD AND THE RISK-NEUTRAL WORLD

Given the Levy process X, the Esscher measure [Q.sub.k] with the parameter k is defined by the Radon-Nikodym derivative:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The parameter [k.sup.*] that satisfies the martingale restriction defines the risk-neutral measure. Therefore, the martingale condition is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The chosen risk-neutral measure [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] comes from the parameter [k.sup.*] such that

r - d + [[psi].sub.p](-i([k.sup.*] + 1)) - [[psi].sub.p](-[ik.sup.*]) = 0, (C1)

which is the EMM relation. The characteristic triplet of a Levy measure ([bar.B], [bar.C], v) in the historical world with the probability measure P and under the equivalent Esscher measure [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(see Shiryaev, 1999, p. 685).

DOI: 10.1111/jori.12087

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Abdou Kelani and Francois Quittard-Pinon are at the CEFRA (Center For Financial Risk Analysis), EMLYON Business School, France. Kelani and Quittard-Pinon can be contacted via e-mail: abdou.kelani@gmail.com, quittardpinon@em-lyon.com. The authors would like to thank the participants of the AFFI 2013, in particular Yue-Kuen Kwok, and the participants of the MAF 2014, in particular Pietro Millossovich. We are indebted to two anonymous referees whose detailed comments have improved the paper significantly. In addition, we would like to thank Areski Cousin, Jean-Paul Decamps, Monique Jeanblanc, and Olivier Le Courtois for many fruitful discussions and the editor Keith J. Crocker for his constructive recommendations.

(1) Namely, that brought forward from time [t.sub.i-1].

(2) The expected jump size is [p/[[lambda].sub.1]] - [q/[[lambda].sub.2]] with the variance of pq[([1/[lambda].sub.1]] + [1/[lambda].sub.2]]).sup.2] + (p/[[lambda].sup.2.sub.1]] + q/[[lambda].sup.2.sub.2]]).

Caption: FIGURE 1 Difference [theta] - [DELTA] for the European Put Option, With d = 0: Dependence on Moneyness and Time to Expiry

Caption: FIGURE 2 Densities of 1-Year Log Return Obtained via Fourier Inversion (2a) and the S&P 500 Log Return QQ-Plots to the (2b) GBM, (2c) CGMY, and (2d) Kou-Based Returns

Caption: FIGURE 3 GMMB/GMDB: Endogenous Relation Between [m.sub.o] and [m.sub.a] per the Equilibrium Formula (3)

Caption: FIGURE 4 GMAB: Endogenous Relation Between [m.sub.o] and [m.sub.a] per the Equilibrium Formula (3)
TABLE 1
Annual Parameters of the Kou and CGMY Models Obtained by Fitting
to the Total Return S&P 500 Index Prices, From January 31, 1956
to May 30, 2014

Models                Kou                  CGMY             GBM

Estimates        [mu] = 0.1572         [mu] = 0.2799   [mu] = 0.0962
                [sigma] = 0.1264        C = 0.6235    [sigma] = 0.1473
               [lambda] = 2.6116        G = 21.0775
           [[lambda].sub.1] = 80.2741   M = 39.5137
           [[lambda].sub.2] = 25.8004     Y = 0.8
                    p = 0.3

Mean                 0.0961               0.0962
Std                  0.1474               0.1473
Skewness            -0.1968               -0.1969            0
Excess               0.2110               0.2111             0
Kurtosis

Note: Source of the data: Bloomberg[R].

TABLE 2
Fair Values of the GMMB, Obtained Using Formula (4),
With [m.sub.a] = 0

                    Kou Model                      GBM Model

T (years)       m (bp)     Value    T (years)   m (bp)     Value

10               17.05    1.6616       10        16.68    1.6256
20                2.29    0.4363       20         2.21    0.4208
30                0.39    0.1058       30         0.37    0.1008

TABLE 3
Fair Values of the GMDB, Obtained Using Formula (5),
With [m.sub.a] = 0

                   Kou Model                       GBM Model

T (years)       m (bp)     Value    T (years)   m (bp)     Value

10                0.99    0.0975       10         0.98    0.0962
20                0.84    0.1613       20         0.83    0.1582
30                0.73    0.1993       30         0.71    0.1946

TABLE 4
Fair Values of the GMDB w.r.t Purchase Ages, Obtained Using
Formula (5), With Maturity of 10 Years and [m.sub.a] = 0

                    Kou Model       Purchase        GBM Model
Purchase Ages                         Ages
(years)         m (bp)     Value     (years)    m (bp)     Value

30                0.55    0.0548       30         0.55    0.0541
40                0.99    0.0975       40         0.98    0.0962
45                1.41    0.1378       45         1.39    0.1359

TABLE 5
Fair Values of the Mixed GMMB/GMDB, Obtained Using Formula (6),
With [m.sub.a] = 0

                   Kou Model                       GBM Model

T (years)       m (bp)     Value    T (years)   m (bp)     Value

2               245.59    4.7863        2       246.96    4.8123
5                67.77    3.3114        5        67.03    3.2760
10               18.14    1.7705       10        17.76    1.7331
20                3.15    0.6014       20         3.05    0.5827
30                1.12    0.3062       30         1.08    0.2964

TABLE 6
Fair Values of the GMAB, Obtained Using Formula (8),
With [m.sub.a] = 0

GMAB in Case of Death (Only)

Rollover dates                  Kou Model
([t.sub.1]/[t.sub.2]/
[t.sub.3]) (years)          m (bp)     Value

(2/12/22)                    1.63     0.3387
(5/15/25)                    1.81     0.4226
(10/20/30)                   2.23     0.6057

GMAB in Case of Death (Only)

Rollover Dates                  GBM Model
([t.sub.1]/[t.sub.2]/
[t.sub.3]) (Years)          m (bp)     Value

(2/12/22)                    1.61     0.3344
(5/15/25)                    1.79     0.4168
(10/20/30)                   2.20     0.5971

GMAB in Case of Death or Life

Rollover Dates                  Kou Model
([t.sub.1]/[t.sub.2]/
[t.sub.3]) (years)          m (bp)     Value

(2/12/22)                    19.33    3.9486
(5/15/25)                    18.11    4.1430
(10/20/30)                   14.52    3.8782

GMAB in Case of Death or Life

Rollover Dates                   GBM Model
([t.sub.1]/[t.sub.2]/
[t.sub.3]) (Years)          m (bp)     Value

(2/12/22)                    18.64    3.8109
(5/15/25)                    17.57    4.0210
(10/20/30)                   14.14    3.7794

TABLE 7
Probability Distribution Functions and Risk Measures of the ADL
for 100 USD Premium VAs With GMMB/GMDB and GMAB Benefits:
Hedging Using the Underlying Asset

                       GMMB/GMDB

                   Maturity: 10 Years

                       Kou Model

[alpha]   [VaR.sub.[alpha]]   [CTE.sub.[alpha]]

50%            0.5768              1.3627
90%            1.8198              2.5293
95%            2.3315              3.0087
97.5%          2.8943              3.4455
99%            3.3929              3.9429

                       GMMB/GMDB

                   Maturity: 10 Years

                       GBM Model

[alpha]   [VaR.sub.[alpha]]   [CTE.sub.[alpha]]

50%           -0.1077              0.4075
90%            0.6931              1.2213
95%            1.0524              1.5968
97.5%          1.4595              1.9744
99%            1.9087              2.4515

                         GMAB

             Rollover Dates: (2/12/22) Years

                       GBM Model

[alpha]   [VaR.sub.[alpha]]   [CTE.sub.[alpha]]

50%           -5.7630             -0.9353
90%            1.6703              3.7880
95%            3.2209              5.2228
97.5%          4.5468              6.6367
99%            6.5058              8.5835

                           GMAB

             Rollover Dates: (2/12/22) Years

                        GBM Model

[alpha]   [VaR.sub.[alpha]]   [CTE.sub.[alpha]]

50%           -4.3349             -1.5308
90%            0.0340              1.1964
95%            0.8962              2.0090
97.5%          1.6091              2.7927
99%            2.7754              3.8086

TABLE 8
Risk Measures of the ADL, for 100 USD Premium VAs With GMMB/GMDB
and GMAB Benefits, Obtained via the Optimal [theta] Ratio (12)
and Via the [DELTA] Ratio (13)

                          GMMB/GMDB

                      Maturity: 10 Years

                       [DELTA] Hedging
                         Std = 0.8912

[alpha]    [VaR.sub.[alpha]]    [CTE.sub.[alpha]]

50%              0.5832               1.3752
90%              1.8218               2.5472
95%              2.3664               3.0719
97.5%            2.9927               3.4598
99%              3.4172               4.0501

                          GMMB/GMDB

                      Maturity: 10 Years

                        [theta] Hedging
                         Std = 0.8801

[alpha]    [VaR.sub.[alpha]]    [CTE.sub.[alpha]]

50%              0.5768               1.3627
90%              1.8198               2.5293
95%              2.3315               3.0087
97.5%            2.8943               3.4455
99%              3.3929               3.9429

                           GMAB

             Rollover Dates: (2/12/22) Years

                      [DELTA] Hedging
                       Std = 8.5162

[alpha]    [VaR.sub.[alpha]]    [CTE.sub.[alpha]]

50%             -5.5646              -0.6647
90%              2.0382               4.2751
95%              3.6991               5.7751
97.5%            5.1405               7.2599
99%              7.1116               9.2713

                          GMAB

              Rollover Dates: (2/12/22) Years

                       [theta] Hedging
                         Std = 8.6158

[alpha]    [VaR.sub.[alpha]]    [CTE.sub.[alpha]]

50%             -5.7630              -0.9353
90%              1.6703               3.7880
95%              3.2209               5.2228
97.5%            4.5468               6.6367
99%              6.5058               8.5835

TABLE 9
Probability Distribution Functions and Risk Measures of the ADL
for a 100 USD Premium VA With GMMB/GMDB Benefits: Hedging Using
Standard Options

                         GMMB/GMDB

                     Maturity: 10 Years

                          Kou Model

[alpha]    [VaR.sub.[alpha]]    [CTE.sub.[alpha]]

50%             -5.7083              -2.9877
90%             -1.8361              -0.7471
95%             -0.9556              -0.0407
97.5%           -3.8312               0.3846
99%              0.6022               1.2849

                         GMMB/GMDB

                     Maturity: 10 Years

                         GBM Model

[alpha]    [VaR.sub.[alpha]]    [CTE.sub.[alpha]]

50%             -8.4690              -4.6815
90%             -3.0010              -2.2383
95%             -2.4271              -1.7346
97.5%           -1.9096              -1.2798
99%             -1.2456              -0.7603
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Author:Kelani, Abdou; Quittard-Pinon, Francois
Publication:Journal of Risk and Insurance
Article Type:Report
Geographic Code:9JAPA
Date:Mar 1, 2017
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