Parameter sensitivity of CIR process

ISSN:1083-589X in PROBABILITY

Parameter sensitivity of CIR process

S. M. Ould Aly

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.

Keywords:CIR process ;sensitivity.

AMS MSC 2010:65C30 ; 62P05.

Submitted to ECP on May 18, 2012, final version accepted on March 13, 2013.

1 Introduction

The CIR process is defined as the unique solution of the following stochastic differ- ential equation:

dV_t= (a−bV_t)dt+σp

V_tdW_t, V₀=v, (1.1)

wherea, 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 equa- tion 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 parametersa, b andσ inLp 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.

∗Université Paris-Est Marne-la-Vallée, France. E-mail:[email protected]

2 Differentiability

For technical reasons, we will rather consider the square root ofV^v, denoted X^v. Throughout 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 processX^vis 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

2dWt, X₀^v=√

v. (2.2)

We start by studying the differentiability ofX with respect to the parametera. We consider here theL_p-differentiability of the functiona7−→X^v(a), i.e the existence of a processX˙aso that

→0lim

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. Letb∈R^andσ, x≥0. For everya∈]σ²,+∞[, letX_a be the unique solution of the SDE :

dX_t= a

2 −σ² 8

1 Xt

− b 2X_t

dt+σ

2dW_t, X(0) =x

and leta0 > σ². Then the functiona7−→Xa isLp-differentiable ata0, for any1 ≤p≤

2a₀

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

X˙_a(t) = Z t

0

1 2Xs

exp

−b

2(t−u)−(a 2−σ²

8 ) Z t

s

du X_u²

ds. (2.4)

Proof. LetXbe 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, defineR₀(t) :=X_t−Xt. We can easily see thatR₀is given by R₀(t) =U_t

Z t 0

(U_s)⁻¹ 1 2Xs

ds,

where

U= exp

− Z t

0

α_sds

, with α_t= a+

2 −σ² 8

1 X_sXs

+b 2.

We have, using the fact that for anys≤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 anyp∈

1,2 _σ^2a₂ −1 , kR₀k_p≤C.

Let’s now set

X˙a(t) := lim

→0

R₀

(t) =U_t⁰ Z t

0

(U_s⁰)⁻¹ 1 2Xs

ds.

We have

X˙a

_p≤C. Furthermore,X˙ais 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.

LetR₁(t) =X_t−X_t−X˙_a(t). The processR₁is 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 thatR₁can 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 tobis obtained in the same. The proof of the next Proposition is almost identical to Proposition 2.1.

Proposition 2.2. Letx, a, σ≥0so that4a >3σ². For everyb∈R^{, let}Xbbe the unique solution of the SDE :dX_t=

a 2−^σ₈²

1

Xt −^b₂X_t

dt+^σ₂dW_t, X₀=xand letb₀∈R^{. The} functionb7−→X_b isL_p-differentiable atb₀, for any1≤p <2(^2a_σ2 −1)and its derivative X˙bis given by

X˙b(t) =− Z t

0

Xs

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 thatσ7−→X is C²in neighborhood of 0. We will show that this function isC¹in[0,√

a[andC^∞ around 0.

Proposition 2.3. For anyσ∈ [0,√

a[, the functionσ 7−→X isC¹at σinLp-sense, for everyp∈[1,_σ^2a2 −1[and its derivative is the unique solution of the SDE :

dX˙σ(t) = − σ 4X_t −

a 2 −σ²

8

X˙σ(t) X_t − b

2 X˙σ(t)

! dt+1

2dWt. (2.7)

Proof. LetXbe the unique solution of the SDE : dX_t=

a

2 −(σ+)² 8

1 X_t −b

2X_t

dt+σ+

2 dWt, X₀=√ v.

Let setR₀(t) =X_t−X_t. In particular,R₀solves 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

2Xt

dt+

2dWt

=

− a

2 −(σ+)² 8

1 X_sXs

+b 2

R₀(t)−2σ+² 8Xt

dt+

2dW_t.

It follows thatR₀can be written as R₀(t) =U_t

Z t 0

(U_s)⁻¹

−2σ+² 8X_s ds+

2dWs

,

whereUis 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 Xs

+

2Wt+U_t Z t

0

Wsd(U)⁻¹_s .

On the other hand, using the fact thatα_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 Xs

+ 2

sup

s≤t

Ws+ sup

s≤t

Ws(1−U_t)U_t

≤ c(t) sup

s≤t

1 Xs

+sup

s≤t

Ws(1 +U_t)U_t.

Using (2.5), we have, for any1≤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 4Xs

ds+1 2dW_s

.

We have

X˙σ

_p≤C. Furthermore, we can easily see thatX˙σis solution to the stochas- tic differential equation:

dX˙_σ(t) =− a

2 −σ² 8

1 X_t² +b

2

X˙_σ(t)dt− σ 4Xt

dt+1 2dW_t.

SetR₁(t) =X_t−X_t−X˙_σ(t). The processR₁solves 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

Xt

− b 2Xt

R₀(t)−2σ+² 8X_t² .

It follows thatR₁can be written as R₁(t) =U_t

Z t 0

(U_s)⁻¹

− ² 8Xs

ds+X˙σ(s) α_s

Xs

− b 2Xs

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}

8Xs

ds+|X˙_σ(t)|

b 2Xs

|R₀(s)|+2σ+² 8X_s²

ds+

Z t

0

U_t(U_s)⁻¹α_t Xs

|X˙σ(s)||R₀(s)|ds

≤ c(t) Z t

0

² 8Xs

ds+|X˙_σ(t)|

b 2Xs

|R₀(s)|+2σ+² 8X_s²

ds

+c2(t) sup

s≤t

|X˙σ(s)||R₀(s)|

Xs

! .

Finally, using (2.5), we have, for any1≤p < _σ^2a2 −1 , kR₁k_p≤C².

Proposition 2.4. Under the assumptions of Propositions 2.1, 2.2, 2.3, the solution of the SDE(1.1)is differentiable with respect to the parametersa,bandσ. Its derivatives, denoted byV˙_a,V˙_bandV˙_σrespectively, are given as

V˙_a(t) = √ V_t

Z t 0

√1 Vs

exp

−b

2(t−u)−(a 2 −σ²

8 ) Z t

s

du Vu

ds, V˙b(t) = −√

Vt

Z t 0

√ Vsexp

−b

2(t−u)−(a 2 −σ²

8 ) Z t

s

du V_u

ds,

V˙σ(t) = 2 σVt− 2

σ pVt





√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

Vu

du



(2.11). Proof. AsV_t=X_t²,V is differentiable with respect to the parametersa,bandσunder the assumptions of Propositions 2.3, 2.1, 2.2. The derivativesV˙σis 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 processZt:= ˙Vσ(t)−_σ²Vtis solution of the SDE : dZt=

−2a σ −bZt

dt+σ Zt

2√ Vt

dW_t², Z0=−2 σx.

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

σV_t^α−b(1 +α)ZtV_t^α+ (αa+α²

2 σ²)ZtV^α−1)dt+ (α+1

2)ZV^α−12dW_t².

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

dY_t=

−2a

σV_t⁻¹² −b

2Y_t−(a 2 −σ²

8 )Yt

V_t

dt , Y₀=−2 η

√v.

We can easily solve this equation, we get Yt:= Vσ(t)−_σ²Vt

√Vt

=−2 σ

√ve^−γt−2a σ

Z t 0

e^{−(γt−γu)}

√Vu

du, a.s,

where

γ_t:= b 2t+ (a

2−σ² 8 )

Z t 0

dr Vr

. (2.12)

Thus V˙σ(t) = 2

σVt− 2 σ

pVt





√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

Vu

du



, a.s.

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Acknowledgments.I would like to thank Professor Damien Lamberton for many useful discussions and anonymous referee who read the first version and helped me improve the presentation.