Parameter sensitivity of CIR process

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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 Prob- ability, Institute of Mathematical Statistics (IMS), 2013, 18 (34), pp.1-6. �10.1214/ECP.v18-2035�.

�hal-00690856v2�

Parameter sensitivity of CIR process

S. M. OULD ALY

Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées 5, boulevard Descartes, 77454 Marne-la-Vallée cedex 2, France

email: [email protected]

May 27, 2013

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 ofV.

1 Introduction

The CIR process is defined as the unique solution of the following stochastic differential equation:

dVt = (a−bVt)dt+σp

VtdWt, V0 =v, (1.1)

where a, σ, v ≥ 0and 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

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 0 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 (2a > σ²). 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 Lp 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.

2 Differentiability

For technical reasons, we will rather consider the square root ofV^v, denotedX^v. Through- out this paper, we assume that

2a ≥σ² (2.1)

Under this assumption, we have for any T, v > 0, P(∀t∈[0, T] : V_t^v >0) = 1. The process X^v is the unique solution of the following stochastic differential equation

dX_t^v = a

2 − σ² 8

1 X_t^v − b

2X_t^v

dt+σ

2dW_t, X₀^v =√

v. (2.2)

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 7−→ X^v(a), i.e the existence of a process X˙_a so that

lim→0

sup

s≤t

X_s^v(a+)−X_s^v(a)

−X˙_a(s) p

= 0 (2.3)

We have the following result

Proposition 2.1. Let b∈ R and σ, x≥0. For every a ∈]σ²,+∞[, let X_a be the unique

solution of the SDE :

dX_t = a

2− σ² 8

1 X_t − b

2X_t

dt+ σ

2dW_t, X(0) =x

and let a₀ > σ². Then the function a 7−→X_a is L_p-differentiable at a₀, for any 1 ≤p ≤

2a0

σ² −1 and its derivative (X˙_a) is given by X˙_a(t) =

Z t 0

1 2X_sexp

−b

2(t−u)−(a 2 − σ²

8 ) Z t

s

du X_u²

ds. (2.4)

Proof: LetX be the unique solution of the stochastic differential equation dX_t =

a+ 2 − σ²

8 1

X_t − b 2X_t

dt+σ

2dWt, X₀ =√ v.

For >0, define R₀(t) := X_t−X_t. We can easily see that R₀ is given by R₀(t) =U_t

Z t 0

(U_s)⁻¹ 1 2X_sds,

where

U = exp

− Z t

0

α_sds

, with α_t =

a+ 2 − σ²

8 1

X_sX_s + b 2.

We have, using the fact that for any s≤t, e^−Rstαudu ≤e^−bt/2∨1 a.s,

|R₀(t)|

≤ t(e^−bt/2∨1)

2 sup

s≤t

1 X_s^v.

On the other hand, we have, using Lemma 2.3.2 of [4] ,

∀p <2 2a

σ² −1

, E

sup

s≤t

1 Xs^p

<+∞. (2.5)

In particular, we have for any p∈

1,2 ^2a_σ2 −1 , kR₀k_p ≤C.

Let’s now set

X˙_a(t) := lim

→0

R₀

(t) = U_t⁰ Z t

0

(U_s⁰)⁻¹ 1 2X_sds.

We have

X˙_a p

≤C. Furthermore, X˙_a is solution of the stochastic differential equation:

dX˙_a(t) = − a

2 − σ² 8

1 X_t² + b

2

X˙_a(t)dt+ 1 2Xt

dt.

Let R₁(t) =X_t−Xt−X˙a(t). The process R₁ is a solution of the stochastic differential equation

dR₁(t) =

−α_tR₁(t)−X˙_a(t)

α_t−

(a 2 − σ²

8 ) 1 X_t² + b

2

dt.

On the other hand, we have α_t−

(a

2− σ² 8 ) 1

X_t² + b 2

=− α_t

X_t − b 2X_t

R₀(t) + 2X_t².

It follows that R₁ can be written as R₁(t) =U_t

Z t 0

(U_s)⁻¹

X˙_a(t)

α_t

X_t − b 2X_t

R₀(t)− ² 2X_t²

ds,

Using (2.5) and the fact that for anys≤t, we havee^−Rstαudu ≤1∨e^−bt/2andRt

0α_se^−Rstαududs = 1−e^−R0tαudu, we get

∀1≤p < 2a

σ² −1, kR₁k_p ≤C².

The differentiability with respect to b is obtained in the same. The proof of the next Proposition is almost identical to Proposition 2.1.

Proposition 2.2. Let x, a, σ≥0 so that 4a >3σ². For everyb∈R, letX_b be the unique solution of the SDE :dX_t=

a

2 −^σ₈²

1

Xt −₂^bX_t

dt+^σ₂dW_t, X₀ =xand let b₀ ∈R. The function b7−→ X_b is L_p-differentiable at b₀, for any 1≤ p <2(_σ^2a2 −1) and its derivative X˙_b is given by

X˙_b(t) =− Z t

0

X_s 2 exp

−b

2(t−u)−(a 2 −σ²

8 ) Z t

s

du X_u²

ds (2.6)

We now consider the differentiability of X with respect to the parameter σ. We propose an extension of the result of Benhamou et al (cf. [2]) who show that σ 7−→X is C² in neighborhood of 0. We will show that this function isC¹ in [0,√

a[ and C^∞ around 0.

Proposition 2.3. For any σ ∈[0,√

a[, the functionσ 7−→X is C¹ at σ in L_p-sense, for every p∈[1,^2a_σ2 −1[ and its derivative is the unique solution of the SDE :

dX˙_σ(t) = − σ 4Xt

− a

2 −σ² 8

X˙_σ(t) Xt

− b 2

X˙_σ(t)

!

dt+1

2dW_t. (2.7)

Proof: LetX be the unique solution of the SDE : dX_t =

a

2 − (σ+)² 8

1 X_t − b

2X_t

dt+σ+

2 dW_t, X₀ =√ v.

Let set R₀(t) =X_t−X_t. In particular,R₀ solves the stochastic differential equation:

dR₀(t) =

a

2 − (σ+)² 8

1 X_t − b

2X_t− a

2 −σ² 8

1 X_t + b

2X_t

dt+ 2dW_t

=

− a

2 −(σ+)² 8

1 X_sX_s + b

2

R₀(t)− 2σ+² 8X_t

dt+

2dW_t.

It follows that R₀ can be written as R₀(t) =U_t

Z t 0

(U_s)⁻¹

−2σ+²

8X_s ds+ 2dW_s

,

where U is given by

U_t = exp

− Z t

0

α_sds

(2.8) and

α_s = a

2− (σ+)² 8

1 X_sX_s + b

2. (2.9)

Applying the Itô formula to the product (U_t)⁻¹W_t, we have R₀(t) = −2σ+²

8 U_t Z t

0

(U_s)⁻¹ds X_s +

2W_t+U_t Z t

0

W_sd(U)⁻¹_s .

On the other hand, using the fact that α_t ≥b/2, a.s, we know that for anys≤t, we have 0≤U_t(U_s)⁻¹ ≤1∨e^−bt/2, a.s. It follows that

|R₀(t)| ≤ c(t) Z t

0

ds X_s +

2

sup

s≤t

W_s+ sup

s≤t

W_s(1−U_t)U_t

≤ c(t) sup

s≤t

1

X_s +sup

s≤t

W_s(1 +U_t)U_t.

Using (2.5), we have, for any 1≤p < 2 ^2a_σ2 −1 ,

kR₀k_p ≤C. (2.10)

Let’s now set

X˙_σ(t) := U_t⁰ Z t

0

(U_s⁰)⁻¹

− σ

4X_sds+ 1 2dW_s

. We have

X˙_σ p

≤C. Furthermore, we can easily see that X˙_σ is solution to the stochastic differential equation:

dX˙_σ(t) =− a

2 − σ² 8

1 X_t² + b

2

X˙_σ(t)dt− σ

4X_tdt+1 2dW_t.

Set R₁(t) =X_t−X_t−X˙_σ(t). The processR₁ solves the stochastic differential equation:

dR₁(t) =

−α_tR₁(t)−X˙_σ(t)

α_t−

(a 2 −σ²

8 ) 1 X_t² + b

2

− ² 8X_t

dt.

On the other hand, we can easily see that α_t−

(a

2 − σ² 8 ) 1

X_t² + b 2

=− α_t

X_t − b 2X_t

R₀(t)− 2σ+² 8X_t² .

It follows that R₁ can be written as R₁(t) = U_t

Z t 0

(U_s)⁻¹

− ²

8X_sds+X˙_σ(s) α_s

X_s − b 2X_s

R₀(s) + 2σ+² 8X_s²

ds.

We have

|R₁(t)| ≤ Z t

0

U_t(U_s)^{−1 2}

8X_sds+|X˙_σ(s)|

α_t X_s + b

2X_s

|R₀(s)|+2σ+² 8X_s²

ds

≤ Z t

0

U_t(U_s)^{−1 2}

8X_sds+|X˙_σ(t)|

b

2X_s|R₀(s)|+ 2σ+² 8X_s²

ds+

Z t

0

U_t(U_s)⁻¹α_t

X_s|X˙_σ(s)||R₀(s)|ds

≤ c(t) Z t

0

²

8X_sds+|X˙σ(t)|

b

2X_s|R₀(s)|+2σ+² 8X_s²

ds

+c₂(t) sup

s≤t

|X˙_σ(s)||R₀(s)|

X_s

! .

Finally, using (2.5), we have, for any 1≤p < ^2a_σ2 −1 , kR₁k_p ≤C².

Proposition 2.4. Under the assumptions of Propositions 2.3, 2.1, 2.2, the solution of the SDE (1.1) is differentiable with respect to the parameters a, b and σ. Its derivatives, denoted by V˙a, V˙b and V˙σ respectively, are given as

V˙_a(t) = √ V_t

Z t 0

√1

V_sexp

−b

2(t−u)−(a 2 − σ²

8 ) Z t

s

du V_u

ds, V˙_b(t) = −√

V_t Z t

0

√

V_sexp

−b

2(t−u)−(a 2− σ²

8 ) Z t

s

du Vu

ds,

V˙_σ(t) = 2

σV_t− 2 σ

pV_t





√ve^{−2bt−(a2−σ}

2 8 )Rt

0 dr Vr +a

Z t 0

e^{−b2(t−u)−(a2−σ}

2 8 )Rt

u dr

√ Vr

V_u du



(2.11).

Proof: As V_t=X_t², V is differentiable with respect to the parameters a, b and σ under the assumptions of Propositions 2.3, 2.1, 2.2. The derivatives V˙_σ is given as solution of the SDE :

dV˙_σ(t) =−bV˙_σ(t)dt+p

V_tdW_t²+σV˙_σ(t) 2√

Vt

dW_t, V˙_σ(0) = 0.

One can see that the process Zt := ˙Vσ(t)− ²_σVt is solution of the SDE : dZt=

−2a σ −bZt

dt+σ Z_t 2√

V_tdW_t², Z0 =−2 σx.

On the other hand, applying Itô formula to the process ZV^α, for α∈R^∗, we have d(ZV^α)(t) = (−2a

σ V_t^α−b(1 +α)Z_tV_t^α+ (αa+α²

2 σ²)Z_tV^α−1)dt+ (α+1

2)ZV^α−12dW_t².

It follows that, for α=−¹₂, the processY =ZV⁻¹²,Y has finite variation and is given as solution of

dY_t=

−2a σ V⁻

1 2

t − b

2Y_t−(a 2− σ²

8 )Y_t V_t

dt , Y₀ =−2 η

√v.

We can easily solve this equation, we get Y_t:= V_σ(t)− _σ²V_t

√V_t =−2 σ

√ve^−γt − 2a σ

Z t 0

e^{−(γt−γu)}

√V_u du, a.s,

where

γ_t := b 2t+ (a

2 − σ² 8 )

Z t 0

dr

V_r. (2.12)

Thus

V˙_σ(t) = 2

σV_t− 2 σ

pV_t





√ve^{−b2t−(a2−σ}

2 8 )Rt

0 dr Vr +a

Z t 0

e^{−b2(t−u)−(a2−σ}

2 8 )Rt

u dr

√ Vr

V_u du



, a.s.

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