Parameter sensitivity of CIR process

Transcription

1 Parameter sensitivity of CIR process Sidi Mohamed Ould Aly To cite this version: Sidi Mohamed Ould Aly. Parameter sensitivity of CIR process. Electronic Communications in Probability, Institute of Mathematical Statistics IMS, 13, 18 34, pp.1-6. <1.114/ECP.v18-35>. <hal-69856v> HAL Id: hal Submitted on 7 May 13 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Parameter sensitivity of CIR process S. M. OULD ALY Université Paris-Est, Laboratoire d Analyse et de Mathématiques Appliquées 5, boulevard Descartes, Marne-la-Vallée cedex, France sidi-mohamed.ouldali@univ-paris-est.fr May 7, 13 Abstract We study the differentiability of the CIR process with respect to its parameters. We give a stochastic representation for these derivatives in terms of the paths of V. 1 Introduction The CIR process is defined as the unique solution of the following stochastic differential equation: dv t = a bv t dt + σ V t dw t, V = v, 1.1 where a, σ, v and b R see [8] for the existence and uniqueness of the solution of the SDE. This process is widely used in finance to model short term interest rate see [3] but also used to model stochastic volatility in the Heston stochastic volatility model. The option prices in these models depend in the values of the parameters of CIR process. On the other hand, these parameters are often calibrated to market prices of derivatives, so they tend to change their values regularly. The knowledge of the derivatives of the CIR process with respect to its parameters is therefore crucial for the study the sensitivities of prices in these models. The most common approach to study the sensitivity of stochastic differential equation with respect to its parameters is to use the Malliavin calculus, especially for the sensitivity 1

3 with respect to the initial value. The Malliavin derivative gives a stochastic representation of the sensitivity of process with respect to its initial value. We note that the coefficients of 1.1 are neither differentiable in nor globally Lipschitz, so the standard results see e.g [9],[5] cannot be used here. Nevertheless, for the special case of CIR process, Alòs and Ewald [1] show the existence of Malliavin derivative of the CIR process under assumption a > σ. In mathematical finance, the sensitivities of option prices with respect to not only the initial point, but also other parameters, need to be studied very carefully. In this article, we study the differentiability of the solution of 1.1 with respect to the parameters a, b and σ in L p sense see next section. We show that, under some assumptions, this process is differentiable with respect to these parameters and give a stochastic representation of its derivatives. Differentiability For technical reasons, we will rather consider the square root of V v, denoted X v. Throughout this paper, we assume that a σ.1 Under this assumption, we have for any T, v >, P t [, T ] : V v t > = 1. The process X v is the unique solution of the following stochastic differential equation dx v t = a σ 1 8 Xt v b Xv t dt + σ dw t, X v = v.. We start by studying the differentiability of X with respect to the parameter a. We consider here the L p -differentiability of the function a X v a, i.e the existence of a process Ẋa so that We have the following result lim sup Xs v a + Xs v a s t Ẋas p =.3 Proposition.1. Let b R and σ, x. For every a ]σ, + [, let X a be the unique

4 solution of the SDE : dx t = a σ 1 b 8 X t X t dt + σ dw t, X = x and let a > σ. Then the function a X a is L p -differentiable at a, for any 1 p a σ 1 and its derivative Ẋa is given by Ẋ a t = 1 b t σ exp t u a X s 8 s du X u ds..4 Proof: Let X be the unique solution of the stochastic differential equation a + dxt = σ 1 8 Xt b X t dt + σ dw t, X = v. For >, define R t := X t X t. We can easily see that R is given by R t = U t U s 1 1 X s ds, where U = exp a + αsds, with αt = σ 1 + b 8 XsX s. We have, using the fact that for any s t, e s α u du e bt/ 1 a.s, R t tebt/ 1 sup s t 1 X v s. On the other hand, we have, using Lemma.3. of [4], [ a p < σ 1, E sup s t In particular, we have for any p [ 1, a σ 1 [, R p C. 1 X p s ] <

5 Let s now set R Ẋ a t := lim t = U t U s 1 1 X s ds. We have Ẋa C. Furthermore, Ẋ a is solution of the stochastic differential equation: p a dẋat = σ 1 8 Xt + b Ẋ a tdt + 1 X t dt. Let R 1t = X t X t Ẋat. The process R 1 is a solution of the stochastic differential equation dr 1t = αtr 1t Ẋat αt a σ [ 8 1 Xt + b ] dt. On the other hand, we have α t a σ 8 1 Xt + b α = t X t b X t R t + X t. It follows that R 1 can be written as R 1t = U t α Us Ẋ 1 a t t X t b X t R t X t ds, Using.5 and the fact that for any s t, we have e s αudu 1 e bt/ and α s e s αudu ds = 1 e α u du, we get 1 p < a σ 1, R 1 p C. The differentiability with respect to b is obtained in the same. The proof of the next Proposition is almost identical to Proposition.1. Proposition.. Let x, a, σ so that 4a > 3σ. For every b R, let X b be the unique a solution of the SDE : dx t = σ 1 8 X t b X t dt+ σdw t, X = x and let b R. The function b X b is L p -differentiable at b, for any 1 p < a 1 and its derivative σ Ẋ b is given by X s Ẋ b t = exp b t σ t u a 8 s 4 du X u ds.6

6 We now consider the differentiability of X with respect to the parameter σ. propose an extension of the result of Benhamou et al cf. [] who show that σ X is C in neighborhood of. We will show that this function is C 1 in [, a[ and C around. Proposition.3. For any σ [, a[, the function σ X is C 1 at σ in L p -sense, for every p [1, a σ 1[ and its derivative is the unique solution of the SDE : dẋσt = σ 4X t We a σ Ẋσ t b 8 X t Ẋσt dt + 1 dw t..7 Proof: Let X be the unique solution of the SDE : dx t = a σ Xt b X t dt + σ + dw t, X = v. Let set R t = X t X t. In particular, R solves the stochastic differential equation: dr t = = a σ + 1 b a 8 Xt X t σ 1 + b 8 X t X t dt + dw t [ a σ b ] R t σ + dt + 8 XsX s 8X t dw t. It follows that R can be written as where U is given by and R t = U t α s = Us 1 σ + ds + 8X s dw s, Ut = exp αsds Applying the Itô formula to the product U t 1 W t, we have R t = σ + Ut 8.8 a σ b 8 XsX s..9 Us 1 ds + X s W t + Ut 5 W s du 1 s.

7 On the other hand, using the fact that α t b/, a.s, we know that for any s t, we have U t U s 1 1 e bt/, a.s. It follows that Rt ct ct sup s t ds + X s 1 + sup X s s t sup s t Using.5, we have, for any 1 p < a σ 1, W s + sup W s 1 Ut Ut s t W s 1 + U t U t. R p C..1 Let s now set Ẋ σ t := Ut Us 1 σ ds + 1 4X s dw s. We have Ẋσ C. Furthermore, we can easily see that Ẋσ is solution to the stochastic p differential equation: a dẋσt = σ 1 8 Xt + b Ẋ σ tdt σ dt + 1 4X t dw t. Set R 1t = X t X t Ẋσt. The process R 1 solves the stochastic differential equation: dr 1t = αtr 1t Ẋσt αt a σ [ 8 1 Xt On the other hand, we can easily see that α t a σ 8 1 Xt + b It follows that R 1 can be written as R 1t = U t α = t X t b X t Us 1 α ds + Ẋσs s 8X s X s + b R t b X s ] R s + 8X t σ +. 8Xt dt. σ + ds. 8Xs 6

8 We have R1t U t U s 1 U t U s 1 α ds + Ẋσs t 8X s 8X s ds + Ẋσt + b R σ + X s X s + s 8Xs σ + ds + 8Xs b X s R s + Ut Us 1 α t Ẋσs R X s ds s b ct ds + Ẋσt R 8X s X s + s Ẋσs R +c t sup s. s t X s σ + ds 8Xs ds Finally, using.5, we have, for any 1 p < a σ 1, R 1 p C. Proposition.4. Under the assumptions of Propositions.3,.1,., the solution of the SDE 1.1 is differentiable with respect to the parameters a, b and σ. Its derivatives, denoted by V a, Vb and V σ respectively, are given as V a t = V t V b t = V t V σ t = σ V t σ 1 b σ exp t u a V s 8 b σ V s exp t u a 8 Vt ve b t a σ 8 dr Vr + a du s V u s du V u ds, ds, e b tu a σ 8 dr u Vr Vu du..11 Proof: As V t = X t, V is differentiable with respect to the parameters a, b and σ under the assumptions of Propositions.3,.1,.. The derivatives V σ is given as solution of the SDE : d V σ t = b V σ tdt + V t dw t + σ V σ t V t dw t, Vσ =. 7

9 One can see that the process Z t := V σ t σ V t is solution of the SDE : dz t = a σ bz t dt + σ Z t dwt, Z = V t σ x. On the other hand, applying Itô formula to the process ZV α, for α R, we have dzv α t = a σ V t α b1 + αz t Vt α + αa + α σ Z t V α1 dt + α α ZV dw t. It follows that, for α = 1, the process Y = ZV 1, Y has finite variation and is given as solution of dy t = a σ V t We can easily solve this equation, we get 1 b Y t a σ 8 Y t dt, Y = v. V t η Y t := V σt σ V t Vt = σ ve γ t a σ e γtγu Vu du, a.s, where Thus V σ t = σ V t σ γ t := b t + a σ 8 dr..1 V r Vt ve b t a σ 8 dr Vr + a e b tu a σ 8 dr u Vr Vu du, a.s. References [1] Alòs, E. and Ewald, C.-O.: Malliavin differentiability of the Heston volatility and applications to option pricing. Advances in Applied Probability, 4 1. pp , 8. [] Benhamou, E., Gobet, E. and Miri, M.: Times dependent Heston model, SIAM Journal on Financial Mathematics, Vol.1, pp.89-35, 1 [3] Cox, J, Ingersoll, J, and Ross, J. A.: Theory of the Term structure of Interest Rates. Econometrica, 53:385 47,

10 [4] De Marco, S.: On Probability Distributions of Diffusions and Financial Models with non-globally smooth coefficients, PhD dissertation, Université Paris Est et Scuola Normale Superiore di Pisa, 1. [5] Detemple, J., Garcia, R. and Rindisbacher, M.: Representation formulas for Malliavin derivatives of diffusion processes, Finance and Stochastics 9 3, 5. [6] Dufresne, D.: The integrated square-root process. Research Paper no. 9, Centre for Actuarial Studies, University of Melbourne, 1. [7] Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 1993 [8] Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland/Kodansha, [9] Nualart, D.: The Malliavin Calculus and Related Topics, Probability and Its Applications Springer, New York