Optimal design of prot sharing rates by FFT.

Transcription

1 Optimal design of prot sharing rates by FFT. Donatien Hainaut May 5, 9 ESC Rennes, 3565 Rennes, France. Abstract This paper addresses the calculation of a fair prot sharing rate for participating policies with a minimum interest rate guaranteed. The bonus credited to policies depends on the performance of a basket of two assets: a stock and a zero coupon bond and on the guarantee. The dynamics of the instantaneous short rates is driven by a Hull and White model. Whereas the stocks follow a double exponential jump-diusion model. The participation level is determined such that the return retained by the insurer is sucient to hedge the interest rate guarantee. Given that the return of the total asset is not lognormal, we rely on a Fast Fourier Transform to compute the fair value of bonus and guarantee options. Keywords : Policies with prot, Fast Fourier Transform, fair pricing. 1 Introduction. Most of life insurance products oer a minimal rate of return: guaranteed interests are credited periodically usually, once a year to policies. This guarantee being relatively low compared to the market performance, the insurer grants an extra bonus a prot share depending on the return of his assets portfolio. Our purpose is to value a fair prot sharing rate and to show how it is linked to the guarantee and to the insurer's investment strategy. Our contribution with respect to the existing literature is to consider this issue under stochastic interest rates and for an insurer having in portfolio, zero coupon bonds and stocks driven by a jump diusion process. The recent studies on participating policies rely on the Briys and de Varenne 1997a, 1997b model. Their aim was the valuation of the market price of insurance liabilities in a single period model. The asset of the insurance company is ruled by a geometric Brownian motion and the costs of guarantee and bonus are calculated by the Black and Scholes formula. A multi-period extension has been proposed by Miltersen and Persson 3. Bacinello 1 has analysed in a contingent claims framework the pricing of participating policies and takes explicitly into account the mortality risk. Grosen & Jørgensen have proposed a dynamic model to value participating contracts and the properties of this model were explored numerically by Monte Carlo simulations. Jørgensen 1 and Grosen & Jørgensen have showed that a guaranteed participating policy may be split into four terms: a zero coupon bond, a bonus option and if any, a put option linked to the default risk and nally a rebate given to the policyholders in case of default prior to the maturity date. In papers of Bernard et al. 5, as in Grosen & Jørgensen, the possibility of an early payment is envisaged. The default mechanism is of structural type and the default barrier is exponential. Jensen et al. 1 have used a nite dierence approach to study a similar issue. In the existing literature on fair valuation of insurance liabilities, the insurer's asset is usually modelled by a single geometric Brownian motion, which is correlated to stochastic interest rates. Corresponding author. donatien.hainaut@esc-rennes.fr 1

2 In this paper, we price the bonus and guarantee options when the insurer's portfolio contains stocks and zero coupons. One assumes that the short term interest rates are ruled by a Hull and White model. Whereas stocks are ruled by a double exponential jump-diusion model, directly inspired from the one used by Kou to price options. This dynamics is particularly well adapted to model the discontinuities aecting the stocks market. Considering two assets rather than one allows us to emphasize the dependence between static investment strategies, the cost of the guarantee and the fair prot sharing rate. The main drawback of working with two assets is that the distribution of the whole portfolio return is unknown. One relies then on a numerical method to price the options embedded in the participating policy. We have opted for the Fast Fourier Transform approach denoted FFT in the sequel and in particular for the multi-factor setting of Dempster and Hong, initially developed to price spread options, in a faster way than any Monte Carlo methods. For an introduction over FFT, we refer the interest reader to papers of Carr and Madan 1999 and Cerny 4. The outline of the paper is as follows. Section presents the insurer's balance sheet and denes what we call a fair participating rate. Section 3 details the assets dynamics and the interest rates modelling. So as to calculate the Fourier transform of options linked to the guarantee and bonus, we determine in section 4, the characteristic functions of assets under real and forward measures. Section 5 develops the pricing by FFT and section 6 illustrates numerically our results. Product's balance sheet. We consider an insurance company having a time horizon T, selling a participating policy and guaranteeing a xed interest rate r g. At time t =, the premium paid in by the insured is denoted K. The insurer invests a fraction ρ of the received premium in stocks, S t, and the rest in zero coupon bonds of maturity T p T, whose prices are denoted P t, T p. The insurer 's initial balance sheet is then: Assets P, T p = 1 ρk S = ρk Liabilities K The investment strategy is assumed to be static: the asset manager doesn't reallocate his assets portfolio till maturity. For a short term planning horizon, this assumption ts relatively well the reality due to transaction costs. If the investments perform well, at maturity T, the total asset is higher than the liabilities and a positive surplus appears: Assets P T, T p S T Liabilities Surplus Ke rgt This surplus is partly redistributed to the policyholder as a prot share and to the shareholder as a dividend. If the return of asset is insucient, the mathematical reserve is equal to the guaranteed amount, Ke rgt, and the terminal surplus is negative. We don't insert the equity of the insurance company in our model and accordingly, we ignore the possible bankruptcy of the insurer. The main reason justifying this choice is that the most of the insurance products are managed in segregated accounts. The prot sharing rules must then be independent from the equity to avoid dumping Our rst purpose is to determine the bonus as a contractual fraction γ of terminal surplus such

3 that the pricing is fair both for the policyholder and the shareholder. Mathematically, the contract is fair at time t = if the price of the guarantee is equal to the fair value of the surplus kept by the insurer: E Q e rsds [ Ke rgt P T, T p + S T ] + F } {{ } P ut = 1 γ E Q e rs.ds [ P T, T p + S T Ke rgt ] + F } {{ } Call where Q is the pricing measure and F is the initial ltration. The left and right sides of the equality.1 are respectively an European put option and an European call option, of strike price Ke rgt, written on a basket of stocks and bonds. Our second purpose is to calculate the expected real bonus for a given participating rate: Expected Bonus = γe P [ P T, Tp + S T Ke rgt ] F +. where P is the real measure. This measure is particularly interesting for the marketing of such policies. Before any further developments, we describe the dynamics of assets..1 3 Assets dynamics and interest rate modelling. As mentioned earlier, the insurer invests in two assets: stocks and zero coupon bonds. The real nancial probability space is noted Ω, F, P on which is dened a -dimensional Brownian motion Wt P = W r,p t, W S,P t and a jump process that is detailed in the sequel of this section. Those processes generate the ltration F = F t t. The nancial market is incomplete due to the presence of jumps. The set of risk neutral measures under which the discounted asset prices are martingales counts then more than one element. However, one assumes that the market prices securities under the same risk neutral measure noted Q. The instantaneous risk-free rate r t is modelled by a Hull & White model which has the following dynamic under P : dr t = abt r t dt + σ r dw r,p t + λ r dt. 3.1 } {{ } dw r,q t Under the risk neutral measure, the dynamics of interest rates becomes: dr t = abt r t dt + σ r dw r,q t 3. where W r,q t is a Brownian motion under Q. The constants a, σ r and λ r are respectively the speed of mean reversion, the volatility of r t and the cost of the risk coupled to r t. The level of mean reversion, bt, is chosen to t the initial yield curve and depends on instantaneous forward rate f, t 1 : where bt = 1 σ f, t + f, t + a t a 1 e at f, t = log P, t t 1 If Rt, T, T + is the forward rate as seen at time t for the period between time T and time T +, the instantaneous forward rate ft, T is equal to the following limit ft, T = lim Rt, T, T +. If P, t is the price of a zero coupon bond maturing at time t, one has that f, t = log P, t see Hull 1997, for further t details. 3

4 The market value P t, T p of a zero coupon bond of maturity T p, obeys to the next SDE: dp t, T p P t, T p where Bt, T p is dened as follows: = r t dt σ r Bt, T p dw r,p t + λ r dt = r t dt σ r Bt, T p dw r,q t, Bt, T p = 1 a 1 e atp t. In appendix A, the analytical formula of P t, T p is remembered. The second asset available on markets are stocks S t driven by a jump diusion model, correlated to interest rates. This class of model has recently received much attention given its ability to capture the asymmetry and leptokurticity of stocks returns. We refer the interested reader to Kou & Wang 4 for more details about this process. The number of jumps observed in stocks returns is a process noted N t. The amplitude of jumps depends on a positive random variable V such that its logarithm Y = log V has a double exponential distribution. For the sake of simplicity, one assumes that the jump process is identical under the real and the risk neutral measures. The intensity of N t is a positive constant, noted η N. This parameter represents the expected number of jumps on a small interval of time: E P,Q dn t = η N dt. The density function of Y = log V is the following: f Y y = pη 1 e η1y 1 {y } + qη e ηy 1 {y<}. where p, q, η 1, η are positive constants. The parameters p and q are such that p + q = 1 and represent respectively the probability of observing upward and downward exponential jumps. The expectations of Y under P and Q are equal to: whereas the expectations of V are given by: E P Y = E Q Y = p 1 η 1 q 1 η, η E P V = E Q V = q η p η 1 η 1 1. Under P, the dynamic of stocks is given by the following SDE: ds t = r t dt+σ Sr dw r,p t + λ r dt +σ S dw S,P t + λ S dt +V 1dN t E P V 1 η N dt, 3.3 S t where constants σ Sr, σ S and λ S are respectively the correlation between stocks and interest rates, the intrinsic volatility of stocks and the market price of risk. The risk premium coupled to stocks is therefore equal to: σ Sr λ r + σ S λ S. Under the risk neutral measure, S t is ruled by: ds t S t = r t dt + σ Sr dw r,q t + σ S dw S,Q t + V 1dN t E Q V 1 η N dt Change of measure and characteristic functions. The options present in eq..1 don't have any closed form expression given that the distribution of the total asset is unknown. So as to simplify future calculation, we perform a change of measure toward the forward F T measure. Given that the expectation of a discounted payo under Q is This assumption may be relaxed. In this case, one has two sets of parameters dening the frequency and the amplitude of jumps one under P and one under Q. 4

5 equal to the price of a zero coupon bond times the expected payo under F T, the price of the call option present in relation.1 is rewritten as follows: E Q e rsds [ P T, T p + S T Ke rgt ] + F = P, T E F T [P T, Tp + S T Ke rgt ] + F. } {{ } computed by F F T Once that the market value of the call is valued by FFT, the price of the put option involved in equation.1 is directly inferred by the put call parity: Call + P, T Ke rgt = P ut + P, T p + S. 4.1 The next step required to execute the FFT algorithm is the calculation of the characteristic function of ln S T, ln P T, T p under the forward measure: Proposition 4.1. The characteristic function of ln S T, ln P T, T p under F T is given by: where φ F T T u 1, u = E F T exp iu 1 ln S T + iu ln P T, T p F = exp iu 1 µ F T S T + iu µ F T P T,T + 1 p Σu 1, u E F T N T exp iu 1 ln V i,4. µ F T σ S T = ln S Sr + σ S + E Q V 1 η N T σ Sr σ r Bs, T ds ln P, T σ r i=1 Bs, T ds, 4.3 µ F T P T,T p = ln P, T p + σ r ln P, T σ r Bs, T p Bs, T ds σ r Bs, T ds, Bs, T p ds 4.4 Σu 1, u = u 1 + u σr Bs, T ds u σr Bs, T p ds u 1σSrT T +u 1 + u u σr Bs, T Bs, T p ds u 1 σ r σ Sr Bs, T ds E F T +u 1 u σ r σ Sr Bs, T p.ds u 1σST, 4.5 i=1 N T η 1 η exp iu 1 ln V i = exp η N T p + q η 1 iu 1 η + iu 1 The proof of this proposition and the detail of integrals in µ F T S T, µ F T P T,T p, Σu 1, u depending on Bs, T, Bs, T p, are provided in appendix B. The calculation of the expected real bonus, eq.., doesn't require any change of measure. The only information necessary to run the FFT algorithm is the characteristic function of ln S T, ln P T, T p under the measure P : 5

6 Proposition 4.. The characteristic function of ln S T, ln P T, T p under the real measure P is as follows: where φ P T u 1, u = E P exp iu 1 ln S T + iu ln P T, T p F = exp iu 1 µ P S T + iu µ P P T,T + 1 p Σu N T 1, u E P exp iu 1 ln V i, 4.7 σ µ P S T = ln S + σ Sr λ r + σ S λ S Sr + σ S +σ Sr σ r Bs, T ds + σ r + E Q V 1 η N i=1 T ln P, T Bs, T ds, 4.8 µ P P T,T p = ln P, T p λ r σ r ln P, T + σ Sr σ r Bs, T p ds σ r Bs, T ds + σ r Bs, T p ds Bs, T ds, 4.9 Σu 1, u = u 1 + u σr Bs, T ds u σr Bs, T p ds u 1σSrT T +u 1 + u u σr Bs, T Bs, T p ds u 1 σ r σ Sr Bs, T ds +u 1 u σ r σ Sr Bs, T p ds u 1σST, i=1 N T E exp P η 1 η iu 1 ln V i = exp η N T p + q η 1 iu 1 η + iu 1 The proof of this proposition is provided in appendix C and the detail of integrals in µ P S T, µ P P T,T p, Σu 1, u depending on Bs, T, Bs, T p, are developed in appendix B. 5 FFT pricing. The characteristic functions of ln S T, ln P T, T p being determined under P and F T, we now develop the FFT method in order to compute the expected payo: V M = E M [ P T, Tp + S T Ke rg.t ] + F, 5.1 where M is either the forward measure M = F T adopt the following notations for logprices: or either the real measure M = P. We s = ln S T, p = ln P T, T p, and the joint density of s, p under M is denoted qt M s, p. The expectation 5.1 may then be rewritten as follows: V M = e p + e s Ke rg.t qt M s, pdpds, 5. Ω 6

7 where Ω is the domain on which the payo is positive: Ω = { s, p R e s + e p Ke rg.t }, and its frontier is displayed on gure 5.1 dotted line. Figure 5.1: Domain In order to approach the integral 5., the domain Ω is sliced in rectangular strips. Let N be a constant usually a power of. One builds next a N N equally spaced grid Λ s Λ p : { Λ s = {k 1,j } = j 1 } Nλ 1 R j N 1, Λ p = {k,j } = { j 1 } Nλ R j N 1. Furthermore, we dene an index function k j, for j =,..., N 1: { k j = min k,q Λ p e k,q + e k1,j+1 Ke rgt }, q that allows us to dene a rectangular strip Ω j : Ω j := [k 1,j ; k 1,j+1 [k j, + such that the domain Ω is approximate by Ω, the union of Ω j : Ω = j= An illustration of this discrete domain is provided on gure 5.1. Since Ω Ω and that the payo of 5.1 is positive over Ω, we have the following relation: Ω j 7

8 V M = e p + e s Ke rgt qt M s, pdpds Ω e p + e s Ke rgt qt M s, pdpds Ω = Π M 1 Ke rgt Π M 5.3 Where Π M 1 = Π M = e p + e s qt M s, pdpds Ω qt M s, pdpds Ω Before any further developments, we draw the attention of the reader on the fact that the inequality 5.3 means that the expected terminal payo is bounded from below by the numerical estimation. It entails that the bonus option the call option of eq..1 is then slightly underestimated whereas the price of the guarantee the put option of eq..1 obtained by the put call parity eq. 4.1 is therefore slightly overestimated. The discretization infers hence a small safety margin in the calculation of the fair prot sharing rate γ. The next steps consist to calculate Π 1 and Π which may be split as follows: Π M 1 = = = j= j= j= e p + e s qt M s, pdpds Ω j e p + e s qt M s, pdpds e p + e s qt M s, pdpds k j k 1,j+1 k j k 1,j Π M 1 k 1,j, k j Π M 1 k 1,j+1, k j 5.4 Π M = = j= j= k 1,j k j qt M s, pdpds qt M s, pdpds k 1,j+1 k j Π M k 1,j, k j Π M k 1,j+1, k j 5.5 Where Π M 1 k 1, k = Π M k 1, k = k 1 k k 1 k e p + e s q M T s, pdpds q M T s, pdpds The functions Π M 1 k 1, k and Π M k 1, k don't have a nite Fourier Transform because they are not square integrable. E.g. Π M 1 k 1, k tends to E M P T, T p + S T when k 1 and k tend to. To obtain a square integrable function, having a well dened Fourier transform, we multiply Π M 1,k 1, k by an exponentially decaying term: π M 1 k 1, k = expα 1 k 1 + α k Π M 1 k 1, k, π M k 1, k = expα 1 k 1 + α k Π M k 1, k. 8

9 For a range of positives values α 1, α we expect that π 1, k 1, k are well square integrable. The Fourier transform of Π M 1 is denoted χ M 1 v 1, v and may be expressed in term of the characteristic function of of ln S T, ln P T, T p : χ M 1 v 1, v = = = = e iv1k1+vk π M 1 k 1, k dk dk 1 e α1+iv1k1+α+ivk p e p + e s qt M s, p k 1 s k e p + e s q M T s, pdpdsdk dk 1 e α1+iv1k1+α+ivk dk dk 1 dpds e p + e s qt M s, p exp α 1 + iv 1 s + α + iv p dpds α 1 + iv 1 α + iv = φm T v 1 α 1 i, v α + 1 i + φ M T v 1 α i, v α i α 1 + iv 1 α + iv. Similarly, the Fourier transform of π M denoted χ M v 1, v is reformulated as follows: χ M v 1, v = φm T v 1 α 1 i, v α i α 1 + iv 1 α + iv The Fourier transforms of χ M 1,v 1, v are then easily computable since they depend on φ M T.,. whose analytical expressions are provided by propositions 4.1 and 4.. On all N N vertices of the grid Λ 1 Λ, Π M 1,k 1,j, k,l are obtained by an inverse Fourier transform : Π M 1 k 1,j, k,l = e α1k1,j αk,l π Π M k 1,j, k,l = e α1k1,j αk,l π e iv1k1,j+vk,l χ M 1 v 1, v dv dv 1, 5.6 e iv1k1,j+vk,l χ M v 1, v dv dv A naive approach to value Π M 1,k 1,j, k,l would be to discretize the double integrals present in eq. 5.6 and 5.7 and to compute them by the trapezoid rule. This way of doing is particularly inecient and require On 4 operations. A better method is to use a two dimensional FFT that computes for any input array {INv 1,m, v,n : m =,..., N 1, n =,..., N 1}, the following output array: OUT k 1,j, k,l = m= n= e.π.i N v1,m.k1,j+v,n.k,l INv 1,m, v,n j =,..., N 1, l =,..., N in ON. log N. In order to discretize the integrals of eq. 5.6 and 5.7, we introduce the integration steps 1 and. So as to rewrite the discrete integrals as the right term of eq. 5.8, we impose furthermore that: If we dene the following mesh points, λ 1 1 = λ = π N v 1,m := m N 1 v,n := n N 9

10 the discretized version of Π M 1 k 1,j, k,l is provided by: Π M 1 k 1,j, k,l e α1k1,j αk,l π m= n= = 1j+l e α1k1,j αk,l π 1 e iv1,mk1,j+v,nk,l χ M 1 v 1,m, v,n 1 m= n= In the same way, one gets the discrete version of Π M k 1,j, k,l : e πi N mj+nl 1 m+n χ M 1 v 1,m, v,n. Π M k 1,j, k,l 1 j+l e α1k1,j αk,l π 1 m= n= e πi N mj+nl 1 m+n χ M v 1,m, v,n Once that the elements Π M 1 k 1,j, k,l and Π M k 1,j, k,l are computed for all j =... N 1, l =,..., N 1 by the FFT algorithm, the values of Π M 1, Π M are inferred from relations 5.4, 5.5 and V is worth: V M = E M [ P T, Tp + S T Ke rgt ] F + Π M 1 Ke rg.t Π M For a given prot sharing rate γ, the price of the bonus is then: γe Q e rsds [ P T, T p + S T Ke rgt ] F + γp, T Π Q 1 KergT Π Q. The put option in eq..1, valuing the cost the guarantee, is obtained by the put-call parity: E Q e rsds [ Ke rgt P T, T p + S T ] F + P, T Π Q 1 Kerg.T Π Q + P, T Ke rgt P, T p S Finally, the expected real bonus under P is estimated by: γe P [ P T, Tp + S T Ke rgt ] F + = γ Π P 1 Ke rgt Π P 6 Applications. The Hull & White model has been tted to the european swap rates curve on the 31/1/8. The other market parameters are in table 6.1. The insurer's time horizon, T, is set to one year and the maturity T p of zero coupon bonds hold in portfolio is xed to three years. The discretization parameters used to run the FFT method are provided in table 6.. Table 6.1: Market parameters. Parameters Values Parameters Values a.17 η N 1 σ r.175 η 1 1 σ Sr -.15 η 5 σ S. p 3% λ S.3494 q 7% λ r -.36 T 1 year T p 3 years 1

11 Table 6.: FFT parameters. Parameters Values N 56 α 1, α.75 λ 1, λ.5 1,.π N.λ 1 Figure 6.1 shows some market prices of European call and put options, of maturity 1 year, written on a basket of stocks and bonds. Those prices are displayed for dierent strike prices, Ke rgt, and investment strategies, ρ. As we could have expected, the prices of calls and puts increase proportionally to the fraction of stocks hold in portfolio. The option prices are indeed positively correlated with the volatility of the underlying total asset, which is mainly determined by the amount of stocks. Figure 6.1: Call and put options by strike price. Once that the call and put options involved in eq..1 are computed, we can easily determine the fair participating rate γ of surplus that should be distributed by the insurer as bonus. For this purpose, one have assumed that the premium paid in by the insured is worth K = 1. The graph 6. illustrates the link between the fair participating rate γ, the interest rate guarantee and the structure of asset. 11

12 Figure 6.: Fair prot sharing rates by guarantee and stock ratio. One observes that the higher is the part of stocks hold in portfolio, the lower is the participating rate. E.g. for a guarantee of.%, the fair participating rate, γ, falls from 94%, for 1% of stocks, to 6%, for 9% of stocks. The prot sharing rate is also inversely proportional to the guarantee. If the portfolio counts 1% of stocks, γ goes from 98%, for r g =.%, to 55%, for r g =.5%. Figure 6.3: Fair prot sharing rates by guarantee and duration. 1

13 Figure 6.4: Expected returns by guarantee and stocks ratio. Figure 6.3 presents the relation between the fair PS rate γ, the interest rate guarantee and the duration of zero coupon bonds, for a portfolio composed of % of stocks. Increasing the duration of bonds reduces the fair participating rate, whatsoever the guarantee. The graph 6.4 presents the expected total return granted to policyholders. return is the sum of the guarantee and of the expected bonus under P : r g + γe P [ P T, Tp + S T Ke rgt ] F + This expected The gure reveals that the insured, who accepts a lower guarantee, will be rewarded on average more than a costumer choosing a higher protection. The average total return is also proportional to the amount of stocks purchased by the insurer. 7 Conclusions. In this paper, we have developed a method to value a fair prot sharing rate for participating insurance contracts. The most novel feature of our work is to consider this issue when the insurer's asset is made up of a basket of stocks, driven by a jump diusion model, and zero coupon bonds. Such modelling allows us to emphasize the dependence between the investment strategy, the guarantee and the optimal level of bonus. Furthermore, the dynamics of stocks includes a jump process that allows us to take into account the sudden upward and downward movements of the markets. The main drawback of our approach is that it doesn't provide any analytical expressions of embedded options prices, given that the total asset is not lognormally distributed. However, the characteristic function of logprices being analytically calculable, the FFT of options values may be eciently computed by the method of Dempster and Hong. The numerical applications reveal that the cost of the guarantee is proportional to the amount of stocks and to the duration of zero coupon bonds, hold in portfolio. One also observes that the higher is the guarantee or the quantity of stocks, the lower is the fair participating rate. Finally, if the insurer proposes a fair prot sharing rate, we have shown that an insured who accepts a lower guarantee will be rewarded on average more than a costumer choosing a higher protection. A future research could be to study the same issue in a multi-period setting. 13

14 Appendix A. The Hull and White model belongs to the category of non arbitrage model. Furthermore, the price of a zero coupon bond is an ane function of interest rates: Where P t, T = exp At, T Bt, T r t 7.1 Bt, T = 1 a 1 e a.t t At, T = log P, T P, t + Bt, T f, t σ r 4.a 3 1 e a.t t 1 e at For further details on ane models, we refer to Due 1. Appendix B. This appendix proves the proposition 4.1. Under the forward measure F T, the following processes: dw S,F t = dw S,Q t dw r,f t = dw r,q t + σ r Bt, T dt are Brownian motions. Under F T, the dynamics of assets are such that: dp t, T p P t, T p ds t = r t σ Sr σ r Bt, T E Q V 1 η N dt + σsr dw r,f t S t = r t + σ rbt, T p Bt, T dt σ r Bt, T p dw r,f t + σ S dw S,F t And by the Itô's lemma, we infer the following dynamics of logprices: d ln S t = r t σ Sr σ r Bt, T E Q V 1 η N σ Sr σ S dt d ln P t, T p = +σ Sr dw r,f t + σ S dw S,F t + ln V dn t r t + σrbt, T p Bt, T σ r Bt, T p By integration, we get that logprices at time T are given by: ln S T = ln S + dt σ r Bt, T p dw r,f t r s σ Sr σ r Bs, T E Q V 1 η N σ Sr σ S + V 1dN t ds N T + σ Sr dws r,f + σ S dws S,F + ln V i 7. i=1 ln P T, T p = ln P, T p + r s + σrbs, T p Bs, T σ r Bs, T p ds σ r Bs, T p dw r,f s 7.3 Logprices depend on the integral of r t, which may also be written as a stochastic integral with respect to Ws r,f. According to proposition 7.1 proved in appendix D, we can establish the following 14

15 decompositions of logprices: Where µ F T S T ln S T = µ F T S T + σ r Bs, T + σ Sr dws r,f + σ S dw S,F ln P T, T p = µ F T P T,T p + σ r Bs, T Bs, T p dw r,f s N T s + ln V i and µ F T P T,T p are respectively dened by expressions 4.3 and 4.4. The logprices being normal random variables, the characteristic function of the random vector ln S T, ln P T, T p is given by: Where φ F T T u 1, u = E F T exp iu 1 ln S T + iu ln P T, T p F = exp iu 1 µ F T S T + iu µ F T P T,T + 1 p Σu 1, u E F T i=1 N T exp iu 1 ln V i i=1 Σu 1, u = iu 1 + iu σ r Bs, T iu σ r Bs, T p + iu 1 σ Sr + iu 1 σ S ds The integrals depending on Bs, T, Bs, T p, present in eq and 4.5 are given by: Bs, T ds = 1 T B, T a Bs, T ds = 1 a T B, T 1 ab, T Bs, T p ds = 1 a T 1 a e atp T e atp Bs, T p ds = 1 a T a 3 e atp T e atp + 1.a 3 e atp T e atp Bs, T Bs, T p ds = 1 a T 1 a 3 e atp T e a.tp 1 1 e a.t a a 3 e atp T atp+t e Furthermore, according to Schreve 4 p 468, the characteristic function of a compound Poisson process is given by the following expression: N T E F T exp iu 1 ln V i = exp η N T φ Y u 1 1 i=1 Where φ Y u 1 is the characteristic function of Y = ln V. If ξ + and ξ respectively points out exponential random variables of intensities η 1 and η, one can infer that φ Y u 1 has the following expression: φ Y u 1 = E expiu 1 Y = pe expiu 1 ξ + + qe exp iu 1 ξ η 1 η = p + q η 1 iu 1 η + iu 1 15

16 Appendix C. The proof of proposition 4. is based upon the results of proposition 4.1. Given that the following processes: dw S,P t = dw S,F t λ r σ Sr dt λ S σ S dt dw r,p t = dw r,f t λ r σ r dt σ r Bt, T dt are Brownian motion under P and according relations 7. and 7.3, we obtain the following decompositions of logprices: ln S T = µ P S T + σ r Bs, T + σ Sr dws r,p + σ S dw S,P ln P T, T p = µ P P T,T p + σ r Bs, T Bs, T p dw r,p s N T s + ln V i Where µ P S T and µ P P T,T p are respectively dened by expressions 4.8 and 4.9. The logprices being normal random variables, the result of proposition 4. follows. i=1 Appendix D. In this appendix, we establish the expressions of r sds under the real and forward measures. Proposition 7.1. The integral of short term interest rates is a Gaussian random variable under P and F T : r s ds = log P, T σ r + σ r Bs, T dws r,f Bs, T ds r s ds = log P, T + σ r +λ r σ r Bs, T ds + Bs, T ds σ r Bs, T dw r,p s Proof. As mentioned in section 3, the dynamics of risk free rate under the risk neutral measure is given by: Consider a process Z s dened by: Taking into account 7.4, the dierential of Z s is so that: dr s = a bs r s ds + σ r dw r,q s 7.4 Z s = e as bs r s 7.5 dz s = ae as bs r s ds + e as b sds e as dr s = e as b sds e as σ r dw r,q s And then the process Z s may be rewritten as the following sum of integrals: Z s = Z + s e au b udu 16 s e au σ r dw r,q u 7.6

17 From relation 7.5, we know that r s = bs e as Z s Z = b r 7.7 It suces therefore to combine to get that r s = e as r + s ae as u budu + The short term rate r s is hence Gaussian under Q and s ae as u budu = [ ] u=s e as u f, u + u= Integrating expression 7.8 and taking into account that lead to the desired result. s s dwu r,q = dwu r,p + λ r du dwu r,q = dwu r,f σ r Bu, T du r = f, e as u σ r dw r,q u 7.8 σ r a e as u 1 e au du References [1] Briys E., de Varenne F a. On the risk of life insurance liabilities: debunking some common pitfalls. Journal of Risk and Insurance [] Briys E., de Varenne F b. Valuing risky xed rate debt: an extension. Journal of Financial Quantitative Analysis. 3, [3] Miltersen K., Perrson S. 3. Guaranteed investment contracts: distributed and undistributed excess return. Scandinavian Actuarial Journal. 4, [4] Bacinello A. 1. Fair pricing of life insurance policies with a minimum interest rate guaranteed. ASTIN Bulletin 31, [5] Bernard C., Le Courtois O., Quittard-Pinon F. 5. Market value of life insurance contracts under stochastic interest rates and default risk. Insurance: Mathematics and Economics. 36, [6] Grosen A., Jørgensen P.L.. Fair valuation of life insurance liabilities: the impact of interest rate guarantees, surrender options and bonus policies. Insurance: Mathematics and Economics. 6, [7] Grosen A., Jørgensen P.L.. Life insurance liabilities at market value: an analysis of insolvency risk, bonus policy and regulatory intervention rules in a barrier option framework. Journal of Risk and Insurance. 69 1, [8] Jørgensen P.L. 1. Life insurance contracts with embedded options. Journal of Risk and Finance 3 1, [9] Jensen B., Jørgensen P.L., Grosen A. 1. A nite dierence approach to the valuation of path dependent life insurance liabilities. The Geneva Papers on Risk and Insurance Theory. 6, [1] Carr P. Madan D Option valuation using the Fast Fourier Transform. The Journal of Computational Finance. 4,

18 [11] Dempster M.A.H., Hong S.S.G.. Spread option valuation and the Fast Fourier Transform. Working paper 6/, Judge Institute of Management Studies. University of Cambridge. [1] Cerny A. 4. Introduction to Fast Fourier Transform in nance. Tanaka Business school discussion papers. Imperial College London. [13] Hull J.C Options, futures and other derivatives. Third edition. Prentice Hall international. [14] Due D., 1. Dynamic asset pricing theory. Third edition. Princeton University Press. [15] Kou S.G.. A jump diusion model for option pricing. Management science. 48 8, [16] Kou S.G. Wang H. 4. Option pricing under a double diusion model. Management science. 59, [17] Shreve S.E. 4. Stochastic calculus for nance II: Continuous time models. Springer. 18