Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space

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1 Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle Avenue Michel Crépeau 174 La Rochelle Cedex Abstract We present a Malliavin calculus approach to sensitivity analysis of European options in a jump-diffusion model. The lack of differentiability due to the presence of a jump component is tackled using partial differentials with respect to the absolutely continuous Gaussian part. The method appears to be particularly efficient to compute sensitivities with respect to the volatility parameter of the jump component. Key words: Sensitivity analysis, Greeks, Malliavin calculus, jump-diffusion models, markets with jumps, Wiener space. Classification: 9A9, 9A1, 9A6, 6H7, 6J75. 1 Introduction Consider an option whose value is defined as the average discounted gain on an underlying asset S ξ t depending on a parameter ξ: C ξ = e R T rtdt E [fs ξt ]. where f is called the payoff function, e.g. fx = x K + for European options and f = 1 [K,+ [ for a binary option. Let Delta, Vega, Gamma, Rho and Theta be the Greek parameters measuring the sensitivity of option prices and defined by Delta = C x, Gamma = C x, C Rho = r, C Vega = σ, C Theta = T. Fast Monte-Carlo methods for the numerical computation of sensitivities in continuous markets have been developed in [6], [5], and also [], using integration by parts formulas 1

2 on probability spaces and differential tools of the Malliavin calculus on the Wiener space. In [7], a Malliavin type gradient operator has been used for Asian options in a market with jumps driven by a standard Poisson process N t t R+. This gradient operator acts on smooth functionals F = ft 1,..., T n, of the Poisson process by differentiation with respect to the jump times T k k 1 of the Poisson component N t t R+, as k=n D w F = wt k k ft 1,..., T n, where w is a functional parameter, cf. [9]. Functionals of the form k=1 F t, N t dt 1.1 do belong to the domain of D w due to the smoothing effect of the integral. In particular it turned out in [7] that D w can be applied to differentiate the value of an Asian option. However the L domain of D w does not contain the value N T at time T of the Poisson process, excluding in particular European claims of the form fn T from this analysis. Ssee [1] for a different way to compute Greeks in jump models. In this paper we consider a jump diffusion model driven by the sum of a Brownian motion and a jump process. Precisely, we consider an asset S t t R+ risk neutral probability are given by whose dynamics under the ds t S t = rtdt + σ 1 tdw t + σ tdx t 1. where W t t R+ is a Brownian motion, X t t R+ is a jump process, rt represents the interest rate, and σ 1 t, σ t are volatility parameters. In order to compute these sensitivities we will use the integration by parts formula of Malliavin calculus, cf. Proposition 1 below. We will compute the Greeks corresponding to the sensitivity of European option prices with respect to parameters such as spot price x, interest rate r, or volatility σ. We use both the Malliavin calculus and finite differences, and compare the results obtained by each method. The formulas obtained are similar to the ones obtained for continuous markets, except in the case of derivation with respect to the volatility parameter of the jump component.

3 We proceed as follows. In Sections and 3 we recall some basic notions of stochastic calculus and Malliavin calculus. The simulation graphs obtained show that the Malliavin method yields better numerical results in terms of accuracy and computation speed than the finite difference method. This paper is based on [4], see also [3] for another approach to the computation of Greeks in jump models with respect to the Gaussian part of the process. Stochastic calculus with jumps In this section we recall some notions on stochastic calculus with respect to semimartingales with jumps, according to [1]. Given a complete filtered probability space Ω, F, F t t R+, P, recall that a mapping τ : Ω R + is a stopping time if {τ t} F t, t R +, and that F τ = {A F : A {τ t} F t, t R + }. Given X t t R+ a stochastic process, we define the linear operator I X : S L by n I X H = H X + H i XTi+1 X Ti, i=1 for H t S a simple predictable process, i.e. n H t = H 1 {} t + H i 1 Ti,T i+1 ]t.1 where = T 1 T n+1 < is a sequence of stopping times and H i L Ω, F Ti, i n. Note that this definition is independent of the choice of the representation of the simple process H. The set S of simple predictable processes is endowed with the topology of uniform convergence in t, ω denoted by S u. We denote by L the space of random variables endowed with the topology of convergence in probability. Definition 1 [1] A process X t is called a total semimartingale if X t is càdlàg, adapted and if I X : S u L is continuous, in the sense that if a sequence H n n N of simple predictable processes converges uniformly to H then I X H n n N converges in probability to I X H. A process X s s R+ is called a semimartingale if, for all t, X s t s R+ is a total semimartingale. i=1 3

4 Next, we recall the construction of the stochastic exponential. Theorem 1 [1] Let X t t R+ exists a unique semimartingale Z t t R+ Moreover, Z t t R+ is given by Z t = exp X t 1 [X, X] t denote a semimartingale starting from X =. Then there Z t = 1 + s t which satisfies the equation t Z s dx s. 1 + X s exp X s + 1 X s, t R +. The next result Girsanov theorem, often used to construct risk neutral probabilities, has been used for the computation of option sensitivities with respect to the interest rate. Theorem [1] Let W t t R+ be a Brownian motion under historical probability P, let F t t R+ be the filtration generated by W t t R+, let θ t t R+ be an F t -adapted process, and let t W Q t = θ s ds + W t and t Mt = exp θ s dw s 1 t θ sds, t R +, and define the probability measure Q by QF = MT dp, F F.. F Then the process W Q t is a Brownian motion under Q and more generally we have, E Q [F W Q ] = E P [F W ], F L 1 Ω, P. 3 Malliavin calculus on the Wiener space In this section we recall the basics of Malliavin calculus, cf. e.g. [8], [11], in view of applications to sensitivity analysis. Let W t t R+ denote a d-dimensional Brownian motion, and let C denote the space of random variables F of the form F = f h 1 tdw t,..., h n tdw t, f SR n, 4

5 h 1,..., h n L R +, where SR n is the space of rapidly decreasing C functions on R n. Given F C, the gradient of F is the process D t F t R+ in L Ω R + defined by D t F = n f h 1 tdw t,..., x i i=1 h n tdw t h i t, t R +, a.s. We also define the norm F 1, = E[F ] [ 1/ 1/ + E D t F dt], F C, and denote by D 1, the completion of C with respect to the norm 1,. The gradient operator D is a closed linear mapping defined on D 1, and taking its values in L Ω R +. The gradient D has the derivation property, i.e. if φ : R n R is continuously differentiable with bounded partial derivatives, and F = F 1,..., F n a random vector whose components belong to D 1,, then φf D 1, and: D t φf = n i=1 φ x i F D t F i, t R +, a.s. Next we recall other properties of the gradient D, cf. [8], [6]. Property 1 Let X t t R+ be the solution to the stochastic differential equation dx t = bx t dt + σx t dw t, where b and σ are continuously differentiable functions. Let Y t t R+ denote the first variation process defined by the stochastic differential equation: dy t = b X t Y t dt + σ X t Y t dw t, Y = 1, Then X t D 1,, t R +, and its gradient is given by: hence if f C b 1 Rn we have D s X t = Y t Y 1 s σx s 1 {s t}, s R +, a.s., D s fx T = fx T Y T Y 1 s σx s 1 {s t}, s R +, a.s. The gradient operator D admits an adjoint δ called divergence operator or Skorohod integral, which satisfies the following integration by parts formula. 5

6 Property Let u L Ω R +. Then u Dom δ if and only if for all F D 1, we have [ ] E[ DF, u H ] = E D t F utdt Ku F 1,, where Ku is constant independent of F D 1,. If u Dom δ, δu is defined by the relation E[F δu] = E[ DF, u H ], F D 1,. An important property of the divergence operator δ is that its domain Dom δ contains the adapted processes in L Ω R +. Moreover, for such processes the action of δ coincides with that of Itô s stochastic integral. Property 3 For all adapted stochastic process u L Ω R + we have: We have the following property. δu = utdw t. Property 4 Let F D 1,. For all u Dom δ such that F δu D tf utdt L Ω we have In the sequel we use the notation D w F = δf u = F δu D t F utdt. D t F wtdt, F D 1,, w L Ω R +. The next proposition contains the main result used for the computation of sensitivities. Proposition 1 Let F ξ ξ be a family of random variables, continuously differentiable in Dom D with respect to ξ. Let w t t [,T ] a process verifying D w F ξ, a.s. on { ξ F ξ }, ξ a, b. We have ξ E [ f F ξ] [ = E f F ξ δ w ] ξf ξ D w F ξ for all function f such that f F ξ L Ω, ξ a, b. Proof. If f Cb R, we have ξ E [ ff ξ ] = E [ f F ξ ξ F ξ] [ ] Dw ff ξ = E D w F ξ ξ F ξ [ = E f F ξ δ w ] ξf ξ. D w F ξ 3.1 The general case is obtained by approximation of f by functions in C b R. 6

7 4 Sensitivity analysis 4.1 Market model We consider a jump diffusion model in which the dynamics of the underlying asset price is given by ds t = rtdt + σ 1 tdw t + σ tdx t, S = x, 4.1 S t where rt denotes the interest rate, X t t R+ is a jump semimartingale, W t t R+ is an independent Brownian motion, and σ 1 t, σ t are volatility parameters, respectively relative to the continuous and jump components. The solution of 4.1 is given by: t t t S t = x exp rsds + σ 1 sdw s + σ sdx s 1 t σ1 sds 1 t σ sd[x, X] s t 1 + σ s X s exp σ s X s + 1 σ s X s s= with µt = rt σ1t/. Note that if X t t R+ X s <, s t is a.s. of finite variation, i.e. a.s., then t S t = x exp σ sdx s 1 t exp rsds + t = exp σ sdxs c + t t t σ 1 sds σ 1 sdw s rsds + s t t s t σ s X s 1 + σ s X s σ 1 sdw s 1 where X c s s R + denotes the continuous part of X s s R+. 4. Delta t σ1sds s t 1 + σ s X s, Delta represents the variation of the option price with respect to the initial price x of the underlying asset: Delta = e R T rtdt x E [fsx T ], where S x T = xs1 T = xs T. From Proposition 1, Delta = e R T rtdt E [ fs x T δ w ] xst x, D w ST x 7

8 and from Property 1, Hence D w S T = = Delta = e R T rtdt E = 1 x D s S T wsds S T σ 1 s1 {s T } wsds = S T σ 1 swsds. [ e R T rtdt fs T δ σ 1twtdt E w x σ 1twtdt [ fs T ] wtdw t ] Vega Vega measures the sensitivity of the option price with respect to the volatility parameter Vega 1 In case of a derivation with respect to the volatility parameter σ 1 we consider the perturbed process S ε t t T given by: ds ε t = rtsε t dt + σ 1t + ε σ 1 ts ε t dw t + σ ts ε t dx t, 4.3 where ε > and σ 1 : [, T ] R is a bounded perturbation function. We have ε ST ε = Sε T σ 1 tdw t Letting C ε = E[fST ε ], from Proposition 1 we have Vega 1 = C ε ε = ε= e R T rtdt σ 1twtdt E σ 1 tσ 1 t + ε σ 1 tdt. [ ] fs T δ w σ 1 tdw t σ 1 σ 1 tdt. As for the Delta, the obtained formula coincides with that of [6]. 8

9 4.3. Vega We choose here to differentiate with respect to the volatility σ related to the jump process, in this case the formula we obtain is different from the one obtained for Vega 1. Consider the process S ε t t T defined by: ds ε t = rts ε t dt + σ 1 ts ε t dw t + σ ts ε t + ε σ tdx t 4.4 where ε is a small real parameter and σ : [, T ] R is a bounded perturbation function. Given the option price C ε = e R T rt E [fs ε T ], Sε = x, we need to compute C ε ε ε=. By Proposition 1 we have C ε ε = R T e [ rt E fst ε δ w ] εst ε, D w ST ε with t St ε = A t exp σ s + ε σ s dxs c 1 + σ s + ε σ s X s s t and Hence We then obtain Vega = ε S ε T = S ε T e R T rt 4.4 Gamma t t A t = x exp µsds + σ 1 sdw s. Xs s T σ 1twtdt E fs T σ s X t s σ s + ε σ s X s wtdw t Xs s T σ sdxs c. σ s X t s σ s X s σ sdxs c. Delta is more sensitive to variations when the exercise price of an option is close to the spot price x. Gamma is used to evaluate the sensitivity of Delta with respect to the variations of x. We have Gamma = e R T rtdt x E [fs T ] 9

10 with δ w xs T D w S T From this we deduce Gamma = e R T e R T rtdt x 4.5 Rho rtdt [ ] = e R T rtdt x E w fs T δ x σ 1twtdt [ ] = e R T rtdt x E δw fs T x σ 1twtdt e R T [ rtdt = x σ 1twtdt E fs T δ w ] xs T δw D w S T 1 x δw x σ 1twtdt e R T rtdt σ 1twtdt E [fs T δw], wδw = δ x σ 1twtdt = δwδw D tδwwtdt x T σ 1twtdt = δw 1 x. T σ 1twtdt x E [fs T ] = 4.5 E fs T wtdw t σ wtdw t wtdw t 1twtdt T T σ 1twtdt σ. 1twtdt Rho represents the sensitivity, i.e. the first derivative, of the option price with respect to the interest rate parameter. Consider the process S ε t t T defined by: ds ε t = rt + ε rtsε t dt + σ 1tS ε t dw t + σ ts ε t dx t 4.6 where ε > is small and r is a bounded perturbation function from [, T ] to R. Given the option price C ε = e R T rt+ε rtdt E [fs ε T ], let Z T ε ε = exp rtσ1 1 tdw t ε + ε rtσ1 1 t dt, 1

11 and define the probability measure Q ε, equivalent to P, by dq ε = Z T ε dp. Then 4.6 reads with dw ε t W ε t t T ds ε t = rts ε t dt + σ 1 ts ε t dw ε t + σ ts ε t dx t, 4.7 = dw t + ε rtσ 1 1 tdt. From Theorem Girsanov Theorem, the process is a standard Brownian motion under the probability Q ε. Hence W ε t T has same law as W, and S ε t T has same law as S t t T, and we have C ε = e R T rt+ε rtdt E P [fst ε ] = e R T rt+ε rtdt E Q ε [ ] fs 1 Z ε T ε T ] [ = e R T 1 rt+ε rtdt E P fs ZT ε T with ZT ε = exp ε rtσ 1 1 tdw t + ε with respect to ε we get, after evaluation in ε = : [ Rho = e R T T rtdt E fs T 5 Numerical simulations rtσ 1 1 t dt, 4.8. By differentiation of 4.8 ] rt σ 1 t dw t e R T T rtdt rtdte [fs T ]. 4.9 In this section we present numerical simulations for the Greek parameters Delta, Vega, Gamma and Rho, for a European binary option with exercise price K. Expectations are computed numerically via the Monte Carlo method, i.e. using the approximation E[X] 1 N N Xi i=1 due to the law of large numbers, given a large number N of independent samples Xi i=1,...,n. We consider a simplified model where the parameters σ 1, σ, r are independent of time and where the jump process is taken to be a compensated Poisson process with intensity λ. Under these conditions the price of the underlying asset is given by: S t = x1 + σ Nt exp α σ 1 t + σ 1 W t, with α = r σ λ, t [, T ], and the value of the binary option is: ] X ξ = e rt E [1 [K, [ S ξt, ξ = x, r, σ 1, σ. 11

12 For the Malliavin calculus approach we choose w t to be the constant function wt = 1/T, t T, which yields D w S T = S T σ 1 wsds = σ 1 S T. The intensity of the Poisson component is fixed equal to λ = 1/1. The following graphs allow to compare the efficiency of both methods. 5.1 Simulation of Delta From 4., Delta is given by Delta = e rt xσ 1 T E [fs T W T ]. To approximate Delta by finite differences we use Delta = C x1+ε C x1 ε. xε Delta e+7 1e+8 1.5e+7 e+8 Numbers of samples Figure 1: Delta with N = 1 8, x = 1, r =., σ 1 =.1, σ =.1, T = 1, K = 1 and e =.1. 1

13 .3.5 Numbers of samples= e Delta Strike K Figure : Delta as a function of K with r =.5, σ 1 =.15, σ = Numbers of samples = e Delta Maturity T Figure 3: Delta as a function of T with r =.5, σ 1 =.15, σ =.1. 13

14 5. Simulations of Vega 5..1 Vega 1 Vega 1 is defined by We have C σ1 σ 1 [ = e rt E fs T δ w ] σ 1 S T D w S T σ1 S T = S T σ 1 T + W T and hence δ w σ 1 S T w = δ σ 1 T + W T = T + W T WT D w S T σ 1 σ 1 T 1, σ 1 [ W Vega 1 = e rt E fs T T W T 1 ]. T σ 1 σ 1 By the finite difference method, Vega 1 is given by Vega 1 = C σ 1 1+ε C σ1 1 ε σ 1 ε Vega e+8 1e+9 1.5e+8 e+9 Numbers of samples Figure 4: Vega 1 with x = 1, r =.5, σ 1 =.15, σ =.1, T = 1, K = 1 and e =.1. 14

15 1.5 1 Numbers of samples= e+4.5 Vega Strike K Figure 5: Vega 1 as a function of K r =.5, σ 1 =.15, σ = Numbers of samples = e Vega Maturity T Figure 6: Vega 1 as a function of T r =.5, σ 1 =.15, σ =.1. 15

16 5.. Vega Vega is defined by We have and hence C σ σ [ = e rt E fs T δ w ] σ S T D w S T NT σ S T = S T λt 1 + σ δ w σ S T w NT = δ λt = W T D w S T σ σ σ 1 T [ Vega = e rt E fs T W T σ 1 T By the finite difference method, Vega is given by Vega = C σ 1+ε C σ 1 ε σ ε ] NT λt. 1 + σ NT λt, 1 + σ Note that for the numerical computation of Vega 1, the finite difference method performs similarly to the Malliavin method Vega e+8 1e+9 1.5e+8 e+9 Numbers of samples Figure 7: Vega with x = 1, r =.1, σ 1 =., σ =.1, T = 1, K = 1 and e =.1. 16

17 3.5 3 Numbers of samples= e+4.5 Vega Strike K Figure 8: Vega as a function of K with r =.5, σ 1 =.15, σ =.1. 3 Numbers of samples = e+4.5 Vega Maturity T Figure 9: Vega as a function of T with r =.5, σ 1 =.15, σ =.1, x = 1. 17

18 T K Figure 1: Vega as a function of K and T : x = 1, r =.5, σ 1 =.15, σ =.1, and e =.1 Malliavin method, 1 samples T K Figure 11: Vega as a function of K and T : x = 1, r =.5, σ 1 =.15, σ =.1, and e =.1 finite differences, 1 samples. 18

19 5.3 Simulations of Gamma From 4.5 we have [ Gamma = e rt 1 E fs T x σ 1 T By the finite difference method, Gamma is given by W T σ 1 T 1 ] σ 1 T W T Gamma = C x1+ε C x + C x1 ε x ε Gamma e+8 1e+9 1.5e+8 e+9 Numbers of samples Figure 1: Gamma with x = 1, r =.1, σ 1 =., σ =.1, T = 1, K = 1 and e = Simulations of Rho From the equation 4.9 and assuming that the perturbation function r is constant and equal to 1 we have Rho = e rt σ 1 = e rt E [ E fs T [ fs T WT By the finite difference method, Rho is given by σ 1 Rho = C r1+ε C r1 ε εr ] dw t T e rt E [fs T ] ] T 19

20 3.5 Numbers of samples= e Rho Strike K Figure 13: Rho as a function of K with r =.5, σ 1 =.15, σ = Numbers of samples = e+4.5 Rho References Maturity T Figure 14: Rho as a function of T with r =.5, σ 1 =.15, σ =.1. [1] M.-P. Bavouzet-Morel and M. Messaoud. Computation of Greeks using Malliavin s calculus in jump type market models. Preprint, 4. [] E. Benhamou. Smart Monte Carlo: various tricks using Malliavin calculus. Quant. Finance,

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