Pricing Variable Annuity Guarantees in a Local Volatility framework

# Pricing Variable Annuity Guarantees in a Local Volatility framework

Griselda Deelstra and Grégory Rayée
Department of Mathematics, Université Libre de Bruxelles,
Boulevard du Triomphe, CP 210, Brussels 1050, Belgium
###### Abstract

In this paper, we study the price of Variable Annuity Guarantees, especially of Guaranteed Annuity Options (GAO) and Guaranteed Minimum Income Benefit (GMIB), and this in the settings of a derivative pricing model where the underlying spot (the fund) is locally governed by a geometric Brownian motion with local volatility, while interest rates follow a Hull-White one-factor Gaussian model. Notwithstanding the fact that in this framework, the local volatility depends on a particularly complicated expectation where no closed-form expression exists and it is neither directly related to European call prices or other liquid products, we present in this contribution different methods to calibrate the local volatility model. We further compare Variable Annuity Guarantee prices obtained in three different settings, namely the local volatility, the stochastic volatility and the constant volatility models all combined with stochastic interest rates and show that an appropriate volatility modelling is important for these long-dated derivatives. More precisely, we compare prices of GAO, GMIB Rider and barrier types GAO obtained by using local volatility, stochastic volatility and constant volatility models.

## 1 Introduction

Variable Annuities are insurance contracts that propose a guaranteed return at retirement often higher than the current market rate and therefore they have become a part of the retirement plans of many people. Variable Annuity products are generally based on an investment in a mutual fund composed of stocks and bonds (see for example Gao GaoJin () and Pelsser and Schrager Pelsser2 ()) and they offer a range of options to give minimum guarantees and protect against negative equity movement. One of the most popular type of Variable Annuity Guarantees in Japan and North America is the Guaranteed Minimum Income Benefit (GMIB). At her retirement date, a GMIB policyholder will have the right to choose between the fund value at that time or (life) annuity payments based on the initial fund value at a fixed guarantee rate. Similar products are available in Europe under the name Guaranteed Annuity Options (GAO). Many authors have already studied the pricing and hedging of GMIBs and GAOs assuming a geometric Brownian motion and a constant volatility for the fund value (see for example Boyle and Hardy Boyle01 (); Boyle (), Ballotta and Haberman Ballotta (), Pelsser Pelsser (), Biffis and Millossovich Biffis (), Marshall et al. Hardy2 (), Chu and Kwok Chu_Kwok ()).

GAO and GMIB can be considered as long-dated options since their maturity is based on the retirement date. When pricing long-dated derivatives, it is highly recommended that the pricing model used to evaluate and hedge the products takes into account the stochastic behavior of the interest rates as well as the stochastic behavior of the fund. Furthermore, the volatility of the fund can have a significant impact and should not be neglected. It has been shown in Boyle () that the value of the fund as well as the interest rates and the mortality assumptions influence strongly the cost of these guarantees. Some authors consider the evolution of mortality stochastic as well (see for example (Ballotta, ) and Biffis ()). In van_Haastrecht_GAO (), van Haastrecht et al. have studied the impact of the volatility of the fund on the price of GAO by using a stochastic volatility approach.

Another category of models able to fit the vanilla market implied volatilities are local volatility models introduced by Derman and Kani and Dupire in 1994 in resp. Derman () and Dupire () and recently extended to a stochastic interest rate framework by, among others, Atlan Atlan (), Piterbarg Piterbarg2005 () and Deelstra and Rayee Deelstra-Rayee (). The main advantage of local volatility models is that the volatility is a deterministic function of the equity spot and time which avoids the problem of working in incomplete markets in comparison with stochastic volatility models. Therefore local volatility models are more appropriate for hedging strategies. The local volatility function is expressed in terms of implied volatilities or market call prices and the calibration is done on the whole implied volatility surface directly. Consequently, local volatility models usually capture more precisely the surface of implied volatilities than stochastic volatility models.

Stochastic volatility models have the advantage that it is possible to derive closed-form solutions for some European derivatives. In (van_Haastrecht_GAO, ), van Haastrecht et al. have derived closed-form formulae for GAO prices in the Schöbel and Zhu stochastic volatility model combined with Hull and White stochastic interest rates. However, the GMIB Rider, one of the popular products traded by insurance companies in North America (see GMIB ()) has a more complicated payoff than a pure GAO and therefore there exists no closed-form solution for the price of a GMIB Rider, even not in the Schöbel and Zhu stochastic volatility model. The only way to evaluate a GMIB Rider is by using numerical approaches like for example Monte Carlo simulations.

In this paper, we study the prices of GAO, GMIB Riders and barrier type GAOs in the settings of a two-factor pricing model where the equity (fund) is locally governed by a geometric Brownian motion with a local volatility, while interest rates follow a Hull-White one-factor Gaussian model. In this framework, the local volatility expression contains an expectation for which no closed-form expression exists and which is unfortunately not directly related to European call prices or other liquid products. Its calculation can be done by numerical integration methods or Monte Carlo simulations. An alternative approach is to calibrate the local volatility from stochastic volatility models by establishing links between local and stochastic volatility. A last calibration approach presented in this paper is by adjusting the tractable local volatility surface coming from a deterministic interest rates framework.

Furthermore, we compare Variable Annuity Guarantee prices obtained in three different settings, namely, the local volatility, the stochastic volatility and the constant volatility models all in the settings of stochastic interest rates. We show that using a non constant volatility for the volatility of the equity fund value can have significant impact on the value of these Variable Annuity Guarantees and that the impact generated by a local volatility model is not equivalent to the one generated by a stochastic volatility model, even if both are calibrated to the same market data.

This paper is organized as follows: Section 2 introduces the local volatility model with stochastic interest rates we use in this paper to price Variable Annuity Guarantees. In Section 3, we present different approaches for the calibration of the local volatility function. In Subsection 3.1, we derive an analytical expression for the local volatility. In Subsection 3.2, we explain in detail the different steps for applying a Monte Carlo method in the calibration procedure. Next, in Subsection 3.3 we give a link between the local volatility function derived in a two-factor local volatility model and the tractable one coming from the simple one-factor Gaussian model. Finally, in Subsection 3.4, a link is given between the stochastic volatility model and the local volatility model in a stochastic interest rates framework. In Section 4, we present the three types of Variable Annuity Guarantees we study in this paper, namely, the GAO, the GMIB Rider and barrier types GAO. In Subsection 4.1, we present the GAO, then, in Subsection 4.2, we define a GMIB Rider and finally in Subsection 4.3, we study two types of barrier GAO. Section 5 is devoted to numerical results. In Subsection 5.1, we present the calibration procedure for the Hull and White parameters and the calibration of the local volatility with respect to the vanilla market. In Subsection 5.2, we compare local volatility surfaces obtained in a stochastic and in a constant interest rates framework. Subsection 5.3, 5.4 and 5.5 investigate how the local volatility model behaves when pricing GAO, GMIB Rider and barrier types GAO (respectively) with respect to the Schöbel-Zhu Hull-White stochastic volatility model and the Black-Scholes Hull-White model. Conclusions are given in Section 6.

## 2 The local volatility model with stochastic interest rates

In this paper we consider a two-factor model where the volatility of the fund value is a deterministic function of both time and the fund itself. This function is known as “local volatility”. In this model, the fund value is governed by the following dynamics

 dS(t)=(r(t)−q)S(t)dt+σ(t,S(t))S(t)dWQS(t), (1)

where interest rates follow a Hull-White one-factor Gaussian model (HullandWhite, ) defined by the Ornstein-Uhlenbeck processes.

 dr(t)=[θ(t)−α(t)r(t)]dt+σr(t)dWQr(t), (2)

where and are deterministic functions of time. Equations (1) and (2) are expressed under the risk-neutral measure . We have chosen the very popular Hull-White model since it is a tractable nontrivial interest rate model, allowing closed-form solutions for many derivatives which is useful for the calibration.

We assume that the dynamics of the fund and the interest rates are linked by the following correlation structure:

 EQ[dWQSdWQr]=ρSrdt. (3)

In the following we will denote this model by LVHW since it combines a local volatility model with a Hull-White one-factor Gaussian model. When equals a constant, this model reduces to the Black-Scholes Hull-White model, denoted by BSHW.

## 3 Calibration

Before using a model to price any derivatives, practitioners are used to calibrate it on the vanilla market. The calibration consists of determining all parameters present in the different stochastic processes which define the model in such a way that all European option prices derived in the model are as consistent as possible with the corresponding market ones. More precisely, they need a model which, after calibration, is able to price vanilla options such that the resulting implied volatilities match the market-quoted ones.

The calibration procedure for the LVHW model can be decomposed in three steps: (i) Parameters present in the Hull-White one-factor dynamics for the interest rates, namely and , are chosen to match European swaptions. Methods for doing so are well developed in the literature (see for example (Brigo-Mercurio, )). (ii) The correlation coefficient is estimated from historical data. (iii) After these two steps, one has to find the local volatility function which is consistent with the implied volatility surface.

### 3.1 The local volatility function

In Deelstra-Rayee () we derived the local volatility expression associated to a three-factor model by differentiating the expression of a European call price () with respect to its strike and its maturity . Following the same idea we can derive the local volatility expression associated to the two-factor model presented in Section 2. This leads to,

 σ(T,K)=   ⎷∂C(K,T)∂T+qC(K,T)−qK∂C(K,T)∂K+KP(0,T)EQT[r(T)1% {S(T)>K}]12K2∂2C∂K2, (4)

where is the -forward measure.

This local volatility function is not easy to calibrate with respect to the vanilla market since there is no immediate way to link the expectation term () with vanilla option prices or other liquid products. However, we present in this section three different methods to calibrate it.

Note that when assuming constant interest rates , equation (4) reduces to the simple Dupire formula (see Dupire ()) corresponding to the one-factor Gaussian case:

 σ(T,K)=   ⎷∂C(K,T)∂T+(r−q)K∂C(K,T)∂K+qC(K,T)12K2∂2C(K,T)∂K2. (5)

Since the market often quotes options in terms of implied volatilities instead of option prices, it is more convenient to express the local volatility in terms of implied volatilities. The implied volatility of an option with price , is defined through the Black-Scholes formula () and therefore computing the derivatives of call prices through the chain rule and substituting in equation (5) leads to the following equation (see (Wilmott2, )),

 σ(T,K) =     ⎷VegaBS(σimp2T+∂σimp∂T+(r−q)K∂σimp∂K)12K2VegaBS(1σimpK2T+2d+σimpK√T∂σimp∂K+∂2σimp∂K2+d+d−σimp(∂σimp∂K)2) (6) =  ⎷σ2imp+2Tσimp∂σimp∂T+2(r−q)KTσimp∂σimp∂K(1+Kd+√T∂σimp∂K)2+K2Tσimp(∂2σimp∂K2−d+(∂σimp∂K)2√T),

with

 d±=logS(0)K+(r−q±σ2imp2)Tσimp√T,
 N(x)=∫x−∞1√2πe−z22dz,
 VegaBS=e−qTS(0)N′(d+)√T,
 N′(x)=1√2πe−x22.

Using the same approach, the local volatility expression (4) can be written in terms of implied volatilities ,

 σ(T,K)=    ⎷VegaBS(σimp2T+∂σimp∂T−qK∂σimp∂K)+Kr(0)e−r(0)TN(d−)+KP(0,T)EQT[r(T)1{S(T)>K}]12K2VegaBS(1σimpK2T+2d+σimpK√T∂σimp∂K+∂2σimp∂K2+d+d−σimp(∂σimp∂K)2).

### 3.2 The Monte Carlo approach

In this section we present a Monte Carlo approach for the calibration of the local volatility expression (4) or (LABEL:locvol_implied_C). More precisely, we use Monte Carlo simulations to calculate an approximation of the expectation . Therefore we have to simulate interest rates and the fund value up to time starting from the actual interest rate and fund value respectively. Note that in the remainder of the paper we concentrate on the Hull and White model where and are positive constants. In HullandWhite1995b (), Hull and White remarked that the future volatility structure implied by (2) are likely to be unrealistic in the sense that they do not conform to typical market shapes. We therefore assume, exactly as Hull and White in HullandWhite1994A (), and as positive constants. In that case, one can exactly fit the market term structure of interest rates if the parameter satisfies (see Brigo-Mercurio () and HullandWhite1994A ())

 θ(t)=∂fmkt(0,t)∂T+αfmkt(0,t)+σ2r2α2(1−e−2αt),

where denotes the market instantaneous forward rate at time 0 for the maturity and where denotes partial derivatives of with respect to its second argument.

Since the expectation is expressed under the measure , we use the dynamics of and under that measure, namely,

 dS(t)=(r(t)−q−σ(t,S(t))σrb(t,T)ρSr)S(t)dt+σ(t,S(t))S(t)dWQTS(t), (8) dr(t)=[θ(t)−αr(t)−σ2r(t)b(t,T)]dt+σrdWQTr(t), (9)

where .

Following the Monte Carlo principle, we simulate times (i.e. scenarios) the stochastic variables and up to time , using as time step of discretization and applying an Euler scheme for example. Therefore, the expectation is approximated by:

 EQT[r(T)1{S(T)>K}]≅1nn∑i=1ri(T)1{Si(T)>K}, (10)

where corresponds to the -scenario, .

Note that with a well-known change of variable, one can remove the need to calculate . The idea is to rewrite the stochastic interest rates as a sum of a stochastic and a deterministic part (see (Brigo-Mercurio, )):

 r(t)=x(t)+¯¯¯x(t), (11)

where the stochastic part obeys the following dynamics:

 dx(t)=−(αx(t)+σ2r(t)b(t,T))dt+σr(t)dWQTr(t) (12)

and where the deterministic part obeys the dynamics :

 d¯¯¯x(t)=(θ(t)−α¯¯¯x(t))dt, (13)

which yields to (see (Brigo-Mercurio, )):

 ¯¯¯x(t)=fmkt(0,t)+σ2r2α2(1−e−αt)2.

To illustrate the algorithm, consider the case of using an Euler scheme for equations (8) and (12),

 S(tk+1) = S(tk)+(r(tk)−q−σ(tk,S(tk))σrb(tk,T)ρSr)S(tk)Δt+σ(tk,S(tk))S(tk)√ΔtZS, x(tk+1) = x(tk)−[αx(tk)+σ2rb(tk,T)]Δt+σr√Δt[ρSrZS+√1−ρ2SrZr], r(tk+1) = x(tk+1)+¯¯¯x(tk+1), (14)

where

 ¯¯¯x(tk)=fmkt(0,tk)+σ2r2α2(1−e−αtk)2 and b(tk,T)=1α(1−e−α(T−tk))

and and are two independent standard normal variables.

As one can see in equation (14), the local volatility function has to be known for the simulation of the path for and . Consequently, the only possible way to work is forward in time. To begin, we have to determine the local volatility function at the first time step for all strikes . At this first step we assume that the initial local volatility is equal to the deterministic local volatility given by equation (5). Note that this local volatility is directly obtained by using market data (see equation (6)). More precisely, by this choice, we assume that for a “small time period”, interest rates are constant and in this case, the local volatility expression (4) reduces to (5). Knowing that local volatility function, and can be simulated. Then, the expectation can be computed for all by using:

 EQT1[r(T1)1{S(T1)>K}]≅1nn∑i=1ri(T1)1{Si(T1)>K}. (15)

At this step we are able to build the local volatility function at time , for all strikes . Following the same procedure we can easily calibrate the local volatility at time by using the local volatility obtained at time and also the simulated paths until time . Following this procedure we are able to generate the local volatility expression up to a final date .

### 3.3 Comparison between local volatility with and without stochastic interest rates

Assuming that quantities are extractable from the market, it is possible to adjust the tractable Dupire local volatility function coming from the one-factor Gaussian model (see equation (5)) in order to obtain the local volatility surface which takes into account the effects of stochastic interest rates (i.e. equation (4)).

More precisely, the adjustment is given by111Details about the derivation of (16) can be received on request. Remark that a similar derivation can be found in (Atlan, ) or (Deelstra-Rayee, ) in other settings.

 σ22f(T,K)−σ21f(T,K)=P(0,T)CovQT[r(T),1{S(T)>K}]12K∂2C∂K2, (16)

where represents the covariance between two stochastic variables X and Y with dynamics expressed in the -forward measure .

### 3.4 Calibrating the local volatility by mimicking stochastic volatility models

In this subsection, we give the link between the local volatility model and a stochastic volatility one under the assumption that interest rates are stochastic in both models. This link gives a way to calibrate the local volatility from stochastic volatility models.

Consider the following risk neutral dynamics for the equity spot

 dS(t)=(r(t)−q)S(t)dt+γ(t,ν(t))S(t)dWQS(t), (17)

Common designs for the function are , and . The stochastic variable is generally modelled by a Cox-Ingersoll-Ross (CIR) process (as for example the Heston model (heston, )) or if the function allows for negative values for the stochastic volatility , by an Ornstein-Uhlenbeck process (OU) (as for example the Schöbel and Zhu (SchobelandZhu, ) stochastic volatility model).

In (Atlan, ) and (Deelstra-Rayee, ), the authors have shown that if there exists a local volatility such that the one-dimensional probability distribution of the equity spot with the diffusion (1) is the same as the one of the equity spot with dynamics (17) for every time when assuming that the risk neutral probability measure used in the stochastic and the local volatility framework is the same, then the local volatility function is given by the square root of the conditional expectation under the -forward measure of the instantaneous equity stochastic spot volatility at the future time , conditional on the equity spot level being equal to :

 σ(t,K)=√EQt[γ2(t,ν(t))∣S(t)=K]. (18)
###### Remark 1

If we assume independence between the spot equity and its volatility, the local volatility function is given by . Depending on the dynamics chosen for the stochastic variable , it is sometimes possible to derive closed-form solutions for this expectation. For example consider the following stochastic volatility model where interest rates have Hull and White dynamics and with Schöbel and Zhu dynamics for the equity spot volatility ():

 dS(t)=(r(t)−q)S(t)dt+ν(t)S(t)dWQS(t), (19) dr(t)=[θ(t)−αr(t)]dt+σrdWQr(t), (20) dν(t)=κ[ψ−ν(t)] dt+τdWQν(t). (21)

In this case, we have a closed-form solution for the local volatility function given by

 σ(T,K)= ⎷(ν(0)e−κT+(ψ−ρrνσrτακ)(1−e−κT)+ρrνσrτα(α+κ))(1−e−(α+κ)T))2+τ22κ(1−e−2κT).

## 4 Variable Annuity Guarantees

In this section we present three different Variable Annuity products and in section 5.3, we discuss the price of all these products using local, stochastic and constant volatility models. We first define in Subsection 4.1 the Guaranteed Annuity Option and afterwards we define in Subsection 4.2 the Guaranteed Minimum Income Benefit (Rider). This last product has the particularity to be strongly dependent upon the path of the fund value. Finally, in subsection 4.3, we study two barrier type GAOs with a strong dependence upon the path of the interest rates.

### 4.1 Guaranteed Annuity Options

Consider an year policyholder who disposes at time of the payout of his capital policy which corresponds to an amount of money . A Guaranteed Annuity Option gives to the policyholder the right to choose either an annual payment of where is a fixed rate called the Guaranteed Annuity rate or a cash payment equal to the equity fund value at time which can be considered as an annual payment of , with being the market annuity payout rate defined by with where is the largest survival age, is the zero-coupon bond at time maturing at and is the probability that the remaining lifetime of the policyholder at time is strictly greater than . At time the value of the GAO is given by

 V(T) = max(S(T)g¨ax(T),S(T)) (22) = S(T)+gS(T)max((ω−(x+T)∑n=0.npx+T P(T,T+n))−K,0), (23)

where .

Assuming that the mortality risk is unsystematic and independent of the financial risk and applying the risk-neutral valuation procedure, we can write the value of a GAO entered by an -year policyholder at time as

 V(x,0,T) = EQ[e−∫T0r(s)dsV(T)1(τx>T)|F0] (24) = EQ[e−∫T0r(s)dsV(T)]EQ[1(τx>T)] = EQ[e−∫T0r(s)dsV(T)].Tpx,

where is a random variable which represents the remaining lifetime of the policyholder. Substituting equation (23) in (24) leads to

 V(x,0,T)=.TpxEQ[e−∫T0r(s)dsS(T)]+C(x,0,T). (25)

where

 C(x,0,T)=.TpxEQ⎡⎣e−∫T0r(s)dsgS(T)[max((ω−(x+T)∑n=0.npx+TP(T,T+n))−K,0)]⎤⎦. (26)

Under the usual assumption of absence of arbitrage opportunities, the discounted value of the risky fund is a martingale under the risk neutral measure and therefore the value of the GAO becomes

 V(x,0,T)=.TpxS(0)+C(x,0,T). (27)

The first term in equation (27) is a constant and therefore, in the literature, one studies generally only the second term . In Ballotta () and van_Haastrecht_GAO (), the authors define as the GAO total value. In this paper we keep the same terminology. More precisely, in section 5.3, when we compare GAO total values obtained in different models, we compare values obtained for .

To derive analytical expressions for in the Black-Scholes Hull-White and the Schöbel-Zhu Hull-White models it is more convenient to work under the measure (where the numeraire is the fund value ), rather than under the risque neutral measure (see Ballotta () and van_Haastrecht_GAO ()). By the density process a new probability measure equivalent to the measure is defined, see e.g. Geman et al. Geman (). Under this new measure , the value becomes

 C(x,0,T)=.TpxgS(0)EQS[(ω−(x+T)∑n=0.npx+TP(T,T+n)−K)+]. (28)

Under one has the following model dynamics

 dS(t) = [r(t)−q+σ2(t,S(t))]S(t)dt+σ(t,S(t))S(t)dWQSS(t), (29) dx(t) = [−αx(t)+ρrSσrσ(t,S(t))]dt+σrdWQSr(t). (30)

The zero-coupon bond in the Gaussian Hull and White one-factor model with and constant has the following expression (see e.g. (Brigo-Mercurio, ))

 P(T,T+n)=A(T,T+n)e−b(T,T+n)x(T), (31)

where

 A(T,T+n) = Pmkt(0,T+n)Pmkt(0,T)e−12[V(0,T+n)−V(0,T)−V(T,T+n)], b(T,T+n) = 1α(1−e−α(n)), V(t1,t2) = σ2rα2[t2−t1+2αe−α(t2−t1)−12αe−2α(t2−t1)−32α].

Substituting the expression (31) in equation (28) leads to the following pricing expression for under

 C(x,0,T)=.TpxgS(0)EQS[(ω−(x+T)∑n=0.npx+TA(T,T+n)e−b(T,T+n)x(T)−K)+]. (32)

Note that when pricing GAO using a local volatility model, one has to use some numerical methods like Monte Carlo simulations. The calculation of the price can be based on the equation (32) but it is also convenient to work under the -forward measure , where the dynamics of and are given by (8) and (9) respectively and where the GAO value is given by

 C(x,0,T)=.TpxgP(0,T)EQT[S(T)(ω−(x+T)∑n=0.npx+TA(T,T+n)e−b(T,T+n)x(T)−K)+]. (33)

Since in both cases one has to compute the local volatility value at each time step, both methods are equivalent in time machine consumption.

### 4.2 Guaranteed Minimum Income Benefit (Rider)

Guaranteed Minimum Income Benefit (GMIB) is the term used in North America for an analogous product of a GAO in Europe (see Hardy ()). GAO and GMIB payoffs are usually slightly different but they have in common that they are both maturity guarantees in the form of a guaranteed minimum income on the annuitization of the maturity payout.

There exist many different guarantee designs for GMIB. The policyholders can for example choose between a life annuity or a fixed duration annuity; or choose an annual growth rate guarantee for the fund, or a Withdraw option, etc. (for more details see Hardy () and GMIB ()). In this paper we focus on the valuation of an “exotic GMIB”, namely a GMIB Rider.

A GMIB Rider (based on examples given in GMIB () and in Hardy2 ()) gives the year policyholder the right to choose at the date of annuitization between 3 guarantees: an annual payment of where is a guaranteed annual rate; an annual payment of , where , are the anniversary values of the fund or a cash payment equal to the equity fund value at maturity . Therefore, at time the value of the GMIB Rider is given by

 V(T)=max(S(0)(1+rg)Tg¨ax(T),maxn∈A(S(n))g¨ax(T),S(T)), (34)

where is the set of anniversary dates . The valuation of this product in the Black-Scholes Hull-White model has been studied by e.g. Marshall et al. in Hardy2 ().

Assuming that the mortality risk is unsystematic and independent of the financial risk, we can write the value of a GMIB Rider entered by an -year policyholder at time as

 V(x,0,T) = EQ[e−∫T0r(s)dsV(T)].Tpx (35) = P(0,T)EQT[V(T)].Tpx.

The GMIB Rider payoff is path-dependent and more complicated than a pure GAO. There is no closed-form expression in the BSHW neither in the SZHW model nor in the LVHW model. Consequently one has to use numerical methods to evaluate the expectation in equation (35) in all models we consider. In Subsection 5.4, we compare GMIB Rider values given by using the LVHW model, the SZHW model and the constant volatility BSHW model using a Monte Carlo approach.

### 4.3 Barrier GAOs

In this section, we introduce path-dependent GAOs which are barrier type options. The first one is a “down-and-in GAO” which becomes activated only if the interest rates reach a downside barrier level . This product is interesting for buyers since it protects against low interest rates at the retirement age and has a smaller price than the pure GAO price. The second exotic GAO we study here is the “down-and-out GAO” which becomes deactivated if the interest rates reach a downside barrier level . This barrier option gives protection to insurers against low market of interest rates. For example it allows to avoid situations as it occurs in the UK where insurers have sold products with guarantee rate of 11% while the market rates are now really low,222In United Kingdom in the 1970’s and 1980’s the most popular Guaranteed Annuity rate proposed by UK life insurers was about 11% (see Bolton et al. Bolton ()). resulting in considerable losses for insurance companies. To our knowledge, no closed-form solutions exist for the price of these path-dependent GAOs in the BSHW model nor in the SZHW and LVHW models. Therefore we will also price these path-dependent derivatives by using a Monte Carlo simulation approach.

Under the measure , the price of the “down-and-out GAO” is given by the following expression

 CDO(x,0,T)=.TpxgS(0)EQS[(ω−(x+T)∑n=0.npx+TA(T,T+n)e−b(T,T+n)x(T)−K)+1(x(t)>B, 0

When pricing barrier options with Monte Carlo simulations it is possible to miss some barrier hitting realization between two time steps. Consider the particular path realization of Figure 1. There are five time steps and none of the five underlying realizations have breached the barrier and therefore the path is going to count in the payoff sum. Following the idea presented in Shevchenko (), we should weight the actualized payoff by a certain factor accounting for the probability of breaching the barrier between the discrete time points. As a weighting factor we can use the product over all time intervals of the survival probability of the option,

 w=n∏i=1Psurv(Δit). (37)

Assuming that the volatility of the equity is constant over the interval , it is possible to derive an analytical expression for applicable in all different models treated in this paper. More precisely, under this assumption, is normally distributed with mean and variance given by

 μx = ρSrσrσSα(1−e−αΔit), σ2x = σ2r2α(1−e−2αΔit). (38)

Using the well-known reflection principle (see Shreve ()), we have the analytical expression for

 Psurv(Δit) = P(x(s)>B, t

The last assumption about the volatility of the equity is in contradiction with the nature of the LVHW and the SZHW models. However, as time intervals become smaller, this assumption is more and more justified.

The price of the “down-and-in GAO” can easily be computed from the price of the “down-and-out GAO” and the “pure GAO” by using the following relation

 CDI(x,0,T)=C(x,0,T)−CDO(x,0,T). (41)

Note that for knock-in type options, the survival probability is readily derived from the one coming from the corresponding knock-out type option by complementarity ().

## 5 Numerical results

In this section we study the contribution of using a local volatility model with Hull and White stochastic interest rates (LVHW) (introduced in Section 2) to the pricing of Variable Annuity Guarantees. More precisely, we compare GAO, GMIB Rider and two barrier types GAO prices obtained by using the LVHW model to those obtained with the Schöbel-Zhu Hull-White (SZHW) stochastic volatility model and the Black-Scholes with Hull and White stochastic interest rates (BSHW). For a fair analysis, one first has to calibrate these three models to the same options market data. For this end, we have used the same data as in (van_Haastrecht_GAO, ). More precisely, the equity components (fund) of the Variable Annuity Guarantees considered are on one hand the EuroStoxx50 index (EU) and on the other hand the S&P500 index (US). In the following subsection we explain in details the calibration of the three models. We summarize the calibration of the interest rate parameters and the calibration of the SZHW and the BSHW made in (van_Haastrecht_GAO, ) and then we explain the calibration of the local volatility surface for the equity component in our LVHW model. In Subsection 5.3 we compare GAO values obtained by using the LVHW model with the SZHW and the BSHW prices studied in (van_Haastrecht_GAO, ). In Subsection 5.4 and Subsection 5.5, we do the same study for path-dependent Variable Annuity Guarantees namely GMIB Riders and barrier type GAOs.

### 5.1 Calibration to the Vanilla option’s Market

In order to compare LVHW GAO results to the ones obtained in the BSHW and the SZHW models presented in van Haastrecht et al. (van_Haastrecht_GAO, ) we are using Hull and White parameters and the implied volatility curve they have used333We would like to thank A. van Haastrecht, R. Plat and A. Pelsser for providing us the Hull and White parameters and interest rate curve data they used in van_Haastrecht_GAO ().. In (van_Haastrecht_GAO, ), interest rate parameters are calibrated to EU and US swaption markets using swaption mid prices of the 31st of July 2007. Moreover, the effective 10 years correlation between the log equity returns and the interest rates is determined by time series analysis of the 10-year swap rate and the log returns of the EuroStoxx50 index (EU) and the S&P500 index (US) over the period from February 2002 to July 2007 and turns out to be resp. and for the EU and the US market. The equity parameters in the SZHW and BSHW models are calibrated by using vanilla option prices on the EuroStoxx50 and S&P500 index obtained from the implied volatility service of MarkIT444A financial data provider, which provides (mid) implied volatility quotes by averaging quotes from a large number of issuers..

In (van_Haastrecht_GAO, ) the authors have calibrated the equity model to market option prices maturing in 10 years time. In the LVHW model, the equity volatility is a local volatility surface and the calibration consists in building this surface using equation (LABEL:locvol_implied_C). This calibration procedure uses the whole implied volatility surface and returns a local volatility for all strikes and all maturities. In this paper we consider three different cases. In a first case, we assume that the implied volatility is constant with respect to the maturity () and in that case we denote the local volatility model by LVHW1. The second case denoted by LVHW2 considers an increasing term structure () and finally we consider the case LVHW3 where the implied volatility is a decreasing function of the maturity (). Note that for the aid of a fair comparison between the models, we have always kept the same volatility smile at time . A plot of the three complete implied volatility surfaces and the resulting market call option prices can be found in Appendix C (see Figures 8(a), 8(b), 9(a), 9(b), 10(a), 10(b) for US data and 11(a), 11(b), 12(a), 12(b), 13(a), 13(b) for EUR data). Following the Monte Carlo approach given in Section 3.2, we have found the corresponding local volatility surface, see namely Figure 8(c), 9(c), 10(c) for the US data and 11(c), 12(c), 13(c) for the EUR data.

Stochastic volatility and local volatility models are both able to reproduce the market smile. For example, in Table 1, we compare the market implied volatility (for a range of seven different strikes and a fixed maturity ) with the calibrated volatility of each of the three models. The volatility curves of Table 1 generated by the market and these three models in the US and European 10 years maturity vanilla options market are presented in Figure 2. We notice that the local volatility model (LVHW) and the stochastic volatility model (SZHW) are both well calibrated since they are able to generate the Smile/Skew quite close to the market one. Note that, the LVHW1 implied volatilities are extracted from the European call values obtained by Monte Carlo simulations. Therefore, in the last column of Table 1, one also gives the corresponding 95% confidence interval. However, contrarily to the SZHW model, the LVHW model has to be calibrated over the whole implied volatility surface and as we will see in Subsection 5.3, 5.4 and 5.5, this fact has an impact to the price of Variable Annuity Guarantees.

### 5.2 A numerical comparison between local volatility with and without stochastic interest rates

In Section 3.3, equation (16) gives the difference between the tractable Dupire local volatility function coming from the one-factor Gaussian model and the one which takes into account the effects of stochastic interest rates .

Note that the difference between and can be derived in terms of implied volatilities () by using equation (LABEL:locvol_implied_C) and (6) and leads to

 σ22f(T,K)−σ21f(T,K)=Kr(0)e−r(0)TN(d−)−VegaBSr(0)K∂σimp∂K+KP(0,T)EQT[r(T)1{S(T)>K}]12K2VegaBS(1σimpK2T+2d+σimpK√T∂σimp∂K+∂2σimp∂K2+d+d−σimp(∂σimp∂K)2) (42)

In Figures 8(c), 9(c), 10(c), 8(d), 9(d), 10(d), 8(e), 9(e) and 10(e), we have plotted the local volatility coming from the stochastic interest rate framework (), the tractable Dupire local volatility () and the surface generated by the difference between these two local volatilities in the US case and in Figures 11(c), 12(c), 13(c), 11(d), 12(d), 13(d), 11(e), 12(e) and 13(e), the analogues in the EUR case.

### 5.3 GAO results

In this section, we study the impact generated to GAO values by using the LVHW model with respect to prices given by using the BSHW and the SZHW models computed in (van_Haastrecht_GAO, ). We make therefore the same assumptions as in (van_Haastrecht_GAO, ), namely that the policyholder is 55 years old and that the retirement age is 65 (i.e. the maturity of the GAO option is 10 years). The fund value at time , is assumed to be 100. The survival rates are based on the PNMA00 table of the Continuous Mortality Investigation (CMI) for male pensioners.

In Table 2 we give prices obtained for the GAO using the LVHW model case 1 (LVHW1), the LVHW model case 2 (LVHW2), the LVHW model case 3 (LVHW3), the SZHW and the BSHW models for different guaranteed rates . The results for the SZHW and BSHW models are obtained using the closed-form expression derived in resp. van_Haastrecht_GAO () and Ballotta (), which can be found in Appendix A for the convenience of the reader. Note that GAO prices presented in this section given by the BSHW and SZHW models are slightly different than those presented in van_Haastrecht_GAO (). These differences come from the interpolation method we use for constructing the zero coupon bond curve. As it was pointed out in van_Haastrecht_GAO (), GAO prices are sensitive to the interest rate curve and a small change in the zero coupon bond curve induces changes in GAO prices. The results for the LVHW model are obtained by using Monte Carlo simulations (100 000 simulations and 5000 steps). Table 3 and Figure 3 show the corrections induced by the LVHW and the SZHW models with respect to the BSHW model666More precisely, the LVHW correction is the difference between the LVHW price and the BSHW price. Similarly, the SZHW correction is the difference between the SZHW price and the BSHW price..

Table 4 presents the time value given by the difference between the GAO total value and its intrinsic value. The volatility of an option is an important factor for this time value since this value depends on the time until maturity and the volatility of the underlying instrument’s price. The time value reflects the probability that the option will gain in intrinsic value or become profitable to exercise before maturity. Figure 4 is a plot of the time value given by all considered models. For a deeper analysis about time value we refer the interested reader to van_Haastrecht_GAO ().

These results show that the use of a non constant volatility model such as the SZHW model and the LVHW model has a significant impact on the total value and the time value of GAOs. Furthermore, the term structure of the implied volatility surface has also an influence on the total value. The numerical results show that the LVHW3 prices tend to the SZHW prices, whereas the LVHW2 prices are the lowest of the three LVHW models, but still above the BSHW prices. In the LVHW1 model prices remain between BSHW and SZHW values.

The fact that GAO prices obtained in different implied volatility scenarios turn out to be significantly different, underlines that it is most important to take always into account the whole implied volatility surface. This impact in GAO value can be justified by equation (30) where you can see the influence of the level of the equity spot in the dynamics of under the measure .

In van_Haastrecht_GAO (), it is pointed out that GAO values are also particularly sensible to three different risk drivers namely the survival probabilities, the fund value and the interest rate market curve. From equation (32), we can easily deduce that an increase of of the equity fund value will induce an increase of the GAO value. It is also clear that a shift in the mortality table will induce a shift in the GAO value in the same direction. Finally, a shift down applied to the interest rates curve will increase the GAO value. The sensibility of GAO prices with respect to the implied volatility, the survival probabilities, the fund value and the interest rates market curve underline the fact that for all practical purposes, the market data used can be as important as the model used.

### 5.4 GMIB Rider

In this section, we analyze how the BSHW, the SZHW and the LVHW models behave in the pricing of a GMIB Rider. This product has a strong dependence on the path of the equity fund coming from the anniversary component in the payoff, namely