Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets

Подождите немного. Документ загружается.

6.3 Cox-Ingersoll-Ross Processes 357

6.3.1 CIR Processes and BESQ

From Theorem 1.5.5.1 on the solutions to SDE, the equation

= k(θ − r

) dt + σ



|dW

= x (6.3.1)

admits a unique solution which is strong. For θ =0andx = 0, the solution

is r

= 0, and from the comparison Theorem 1.5.5.9, we deduce that, in the

case kθ > 0, r

≥ 0forx ≥ 0. In that case, we omit the absolute value and

consider the positive solution of

= k(θ − r

) dt + σ

√

= x. (6.3.2)

This solution is called a Cox-Ingersoll-Ross (CIR) process or a square-root

process (See Feller [342]). For σ = 2, this process is the square of the norm

of a δ-dimensional OU process, with dimension δ = kθ (see Subsection 2.6.5

and the previous Subsection 6.2.6), but this equation also makes sense even

if δ is not an integer.

We shall denote by

kθ,σ

the law of the CIR process solution of the

equation (6.3.1). In the case σ =2,wesimplywrite

kθ,2

kθ

.Now,

the elementary change of time A(t)=4t/σ

reduces the study of the solution

of (6.3.2)tothecaseσ = 2: indeed, if Z

= r(4t/σ

), then

= k



(θ − Z

) dt +2



with k



=4k/σ

and B a Brownian motion.

Proposition 6.3.1.1 The CIR process (6.3.2)isaBESQprocesstrans-

formed by the following space-time changes:

= e

−kt



− 1)



where (ρ(s),s≥ 0) is a BESQ

process, with dimension δ =

4kθ

Proof: The proof is left as an exercise for the reader. A more general case

will be presented in the following Theorem 6.3.5.1. 

It follows that for 2kθ ≥ σ

, a CIR process starting from a positive initial

point stays strictly positive. For 0 ≤ 2kθ < σ

, a CIR process starting from

a positive initial point hits 0 with probability p ∈]0, 1[ in the case k<0

(P(T

< ∞)=p) and almost surely if k ≥ 0(P(T

< ∞)=1).Inthecase

0 < 2kθ, the boundary 0 is instantaneously reﬂecting, whereas in the case

2kθ < 0, the process r starting from a positive initial point reaches 0 almost

surely. Let T

=inf{t : r

=0} and set Z

= −r

. Then,

=(−δ + λZ

)dt + σ



|dB

where B is a BM. We know that Z

≥ 0, thus r

takes values in R

−

358 6 A Special Family of Diﬀusions: Bessel Processes

Absolute Continuity Relationship

A routine application of Girsanov’s theorem (See Example 1.7.3.5 or Propo-

sition 6.2.7.2) leads to (for kθ > 0)

kθ

=exp



[x + kθt − ρ

] −





kθ

. (6.3.3)

Comments 6.3.1.2 (a) From an elementary point of view, if the process r

reaches 0 at time t, the formal equality between dr

and kθdt explains that the

increment of r

is positive if kθ > 0. Again formally, for k>0, if at time t,the

inequality r

>θholds (resp. r

<θ), then the drift k(θ−r

) is negative (resp.

positive) and, at least in mean, r is decreasing (resp. increasing). Note also

that E(r

) → θ when t goes to inﬁnity. This is the mean-reverting property.

(b) Here, we have used the notation r for the CIR process. As shown above,

this process is close to a BESQ ρ (and not to a BES R).

r.v. r

and for the process (I



ds, t ≥ 0). Dassios and Nagaradjasarma

[214] present an explicit computation of the joint moments of r

and I

, and,

in the case θ = 0, the joint density of the pair (r

6.3.2 Transition Probabilities for a CIR Process

From the expression of a CIR process as a time-changed squared Bessel process

given in Proposition 6.3.1.1, using the transition density of the squared Bessel

process given in (6.2.2), we obtain the transition density of the CIR process.

Proposition 6.3.2.1 Let r be a CIR process following (6.3.2). The transition

density

kθ,σ

t+s

∈ dy|r

= x)=f

(x, y)dy is given by

(x, y)=

2c(t)





ν/2

exp



−

x + ye

2c(t)





c(t)



xye



{y≥0}

where c(t)=

− 1) and ν =

2kθ

− 1.

Proof: From the relation r

= e

−kt

c(t)

,whereρ is a BESQ

(ν)

, we obtain

kθ,σ

t+s

∈ dy|r

= x)=e

(ν)

c(t)

(x, ye

)dy .

Denoting by (r

(x); t ≥ 0) the CIR process with initial value r

= x,the

random variable Y

= r

(x)e

[c(t)]

−1

has density

P(Y

∈ dy)/dy = c(t)e

−kt

(x, yc(t)e

−kt

{y>0}

−α/2

2α

ν/2

−y/2

ν/2

(

√

yα)1

{y≥0}

where α = x/c(t). 

6.3 Cox-Ingersoll-Ross Processes 359

Remark 6.3.2.2 This density is that of a noncentral chi-square χ

(δ, α) with

δ =2(ν + 1) degrees of freedom, and non-centrality parameter α.Usingthe

notation of Exercise 1.1.12.5, we obtain

kθ,σ

<y)=χ



4kθ

c(t)

;

c(t)



where the function χ

(δ, α; ·), deﬁned in Exercise 1.1.12.5, is the cumulative

distribution function associated with the density

f(x; δ, α)=2

−δ/2

exp



−

(α + x)



−1

∞



n=0





n!Γ (n + δ/2)

{x>0}

−α/2

2α

ν/2

−x/2

ν/2

(

√

xα)1

{x>0}

6.3.3 CIR Processes as Spot Rate Models

The Cox-Ingersoll-Ross model for the short interest rate has been the object

of many studies since the seminal paper of Cox et al. [206] where the authors

assume that the riskless rate r follows a square root process under the

historical probability given by

θ − r

) dt + σ

√

Here,

θ−r) deﬁnes a mean reverting drift pulling the interest rate towards its

long-term value

θ with a speed of adjustment equal to

k. In the risk-adjusted

economy, the dynamics are supposed to be given by:

θ − r

) − λr

)dt + σ

√

where (W





√

ds, t ≥ 0) is a Brownian motion under the risk-

adjusted probability Q where λ denotes the market price of risk.

Setting k =

k + λ, θ =

θ/k), the Q -dynamics of r are

= k(θ −r

)dt + σ

√

Therefore, we shall establish formulae under general dynamics of this form,

already given in (6.3.2).

Even though no closed-form expression as a functional of W can be written

for r

, it is remarkable that the Laplace transform of the process, i.e.,

kθ,σ



exp



−



du φ(u)r



is known (see Exercise 6.2.5.8).

360 6 A Special Family of Diﬀusions: Bessel Processes

Theorem 6.3.3.1 Let r be a CIR process, the solution of

= k(θ − r

)dt + σ

√

The conditional expectation and the conditional variance of the r.v. r

are

given by, for s<t,

kθ,σ

)=r

−k(t−s)

+ θ(1 − e

−k(t−s)

Var(r

)=r

−k(t−s)

− e

−2k(t−s)

)

θσ

(1 − e

−k(t−s)

)

Proof: From the deﬁnition, for s ≤ t, one has

= r

+ k



(θ − r

)du + σ



√

Itˆo’s formula leads to

= r

+2k



(θ − r

du +2σ



)

3/2

+ σ



= r

+(2kθ + σ

)



du − 2k



du +2σ



)

3/2

It can be checked that the stochastic integrals involved in both formulae are

martingales: indeed, from Proposition 6.3.1.1, r admits moments of any order.

Therefore, the expectation of r

is given by

E(r

kθ,σ

)=r

+ k



θt −



E(r

)du



We now introduce Φ(t)=E(r

). The integral equation

Φ(t)=r

+ k(θt −



Φ(u)du)

can be written in diﬀerential form Φ



(t)=k(θ − Φ(t)) where Φ satisﬁes the

initial condition Φ(0) = r

. Hence

E[r

]=θ +(r

− θ)e

−kt

Inthesameway,from

E(r

)=r

+(2kθ + σ

)



E(r

)du − 2k



E(r

)du ,

setting Ψ(t)=E(r

)leadstoΨ



(t)=(2kθ + σ

)Φ(t) − 2kΨ(t), hence

6.3 Cox-Ingersoll-Ross Processes 361

Var [r

(1 − e

−kt

)[r

−kt

(1 − e

−kt

)] .

Thanks to the Markovian character of r, the conditional expectation can also

be computed:

E(r

)=θ +(r

− θ)e

−k(t−s)

= r

−k(t−s)

+ θ(1 − e

−k(t−s)

Var(r

)=r

−k(t−s)

− e

−2k(t−s)

)

θσ

(1 − e

−k(t−s)

)



Note that, if k>0, E(r

) → θ as t goes to inﬁnity, this is the reason why

the process is said to be mean reverting.

Comment 6.3.3.2 Using an induction procedure, or with computations

done for squared Bessel processes, all the moments of r

can be computed.

See Dufresne [279].

Exercise 6.3.3.3 If r is a CIR process and Z = r

,provethat



αZ

1−1/α

(kθ +(α − 1)σ

/2) − Z

αk



dt + αZ

1−1/(2α)

σdW

In particular, for α = −1, dZ

= Z

(k − Z

(kθ − σ

))dt − Z

3/2

σdW

is the

so-called 3/2 model (see Section 6.4 on CEV processes and the book of Lewis

[587]). 

6.3.4 Zero-coupon Bond

We now address the problem of the valuation of a zero-coupon bond, i.e., we

assume that the dynamics of the interest rate are given by a CIR process under

the risk-neutral probability and we compute E



exp



−





Proposition 6.3.4.1 Let r be a CIR process deﬁned as in (6.3.2) by

= k(θ −r

) dt + σ

√

and let

kθ,σ

be its law. Then, for any pair (λ, μ) of positive numbers

kθ,σ



exp



−λr

− μ





=exp[−A

λ,μ

(T ) − xG

λ,μ

(T )]

with

λ,μ

(s)=

λ(γ + k + e

γs

(γ − k)) + 2μ(e

γs

− 1)

λ(e

γs

− 1) + γ(e

γs

+1)+k(e

γs

− 1)

λ,μ

(s)=−

2kθ



2γe

(γ+k)s/2

λ(e

γs

− 1) + γ(e

γs

+1)+k(e

γs

− 1)



where γ =



+2σ

μ.

362 6 A Special Family of Diﬀusions: Bessel Processes

Proof: We seek ϕ : R × [|0,T] → R

such that the process

ϕ(r

,t)exp



−μ





is a martingale. Using Itˆo’s formula, and assuming that ϕ is regular, this

necessitates that ϕ satisﬁes the equation

−

∂ϕ

∂t

= −xμϕ + k(θ − x)

∂ϕ

∂x

∂

∂x

. (6.3.4)

Furthermore, if ϕ satisﬁes the boundary condition ϕ(x, T )=e

−λx

, we obtain

kθ,σ



exp



−λr

− μ





= ϕ(x, 0)

It remains to prove that there exist two functions A and G such that

ϕ(x, t)=exp(−A(T −t) −xG(T −t)) is a solution of the PDE (6.3.4), where

A and G satisfy A(0) = 0,G(0) = λ. Some involved calculation leads to the

proposition. 

Corollary 6.3.4.2 In particular, taking λ =0,

kθ,σ



exp(−μ



ds)



= e

θt/σ



cosh

γt

sinh

γt



−2kθ/σ

exp



−2μx

k + γ coth

γt



where γ

= k

+2μσ

These formulae may be considered as extensions of L´evy’s area formula for

planar Brownian motion. See Pitman and Yor [716].

Corollary 6.3.4.3 Let r be a CIR process deﬁned as in (6.3.2) under the risk-

neutral probability. Then, the price at time t of a zero-coupon bond maturing

at T is

kθ,σ



exp



−









=exp[−A(T −t)−r

G(T −t)] = B(r

,T−t)

with

B(r, s)=exp(−A(s) − rG(s))

and

6.3 Cox-Ingersoll-Ross Processes 363

G(s)=

2(e

γs

− 1)

(γ + k)(e

γs

− 1) + 2γ

k + γ coth(γs/2)

A(s)=−

2kθ



2γe

(γ+k)s/2

(γ + k)(e

γs

− 1) + 2γ



= −

2kθ



+ln



cosh

γs

sinh

γs



−1



where γ =

√

+2σ

The dynamics of the zero-coupon bond P (t, T )=B(r

,T − t) are, under

the risk-neutral probability

P (t, T )=P (t, T )(r

dt + σ(T −t, r

)dW

)

with σ(s, r)=−σG(s)

√

Proof: The expression of the price of a zero-coupon bond follows from the

Markov property and the previous proposition with A = A

0,1

,G = G

0,1

.Use

Itˆo’s formula and recall that the drift term in the dynamics of the zero-coupon

bond price is of the form P(t, T )r

. 

Corollary 6.3.4.4 The Laplace transform of the r.v. r

kθ,σ

−λr



2λ˜c +1



2kθ/σ

exp



−

λ˜c˜x

2λ˜c +1



with ˜c = c(T )e

−kT

and ˜x = x/c(T ), c(T )=

− 1).

Proof: The corollary follows from Proposition 6.3.4.1 with μ =0.Itcan

also be obtained using the expression of r

as a time-changed BESQ (see

Proposition 6.3.1.1). 

One can also use that the Laplace transform of a χ

(δ, α) distributed

random variable is



2λ +1



δ/2

exp



−

λα

2λ +1



and that, setting c(t)=σ

− 1)/(4k), the random variable r

/c(t)is

(δ, α) distributed, where α = x/c(t).

Comment 6.3.4.5 One may note the “aﬃne structure” of the model: the

Laplace transform of the value of the process at time T is the exponential

of an aﬃne function of its initial value E

−λr

)=e

−A(T )−xG(T)

.Fora

complete characterization and description of aﬃne term structure models, see

Duﬃe et al. [272].

364 6 A Special Family of Diﬀusions: Bessel Processes

Exercise 6.3.4.6 Prove that if

=(δ

− kr

)dt + σ



,i=1, 2

where W

are independent BMs, then the sum r

+ r

is a CIR process. 

6.3.5 Inhomogeneous CIR Process

Theorem 6.3.5.1 If r is the solution of

=(a − λ(t)r

)dt + σ

√

= x (6.3.5)

where λ is a continuous function and a>0, then

,t≥ 0)

law



(t)





(s)ds



,t≥ 0



where (t) = exp





λ(s)ds



and ρ is a squared Bessel process with dimension

4a/σ

Proof: Let us introduce Z

= r

exp(



λ(s)ds)=r

(t). From the integration

by parts formula,

= x + a



(s)ds + σ





(s)



Deﬁne the increasing function C(u)=



(s) ds and its inverse

A(t)=inf{u : C(u)=t}. Apply a change of time so that

A(t)

= x + a



A(t)

(s)ds + σ



A(t)



(s)



The process σ



A(t)



(s)

√

is a local martingale with increasing

process σ



A(t)

(s) Z

ds =4



A(u)

du, hence,

:=Z

A(t)

= x+

t+2





A(u)

= ρ

t+2



√

(6.3.6)

where B is a Brownian motion. 

Proposition 6.3.2.1 admits an immediate extension.

Proposition 6.3.5.2 The transition density of the inhomogeneous process

(6.3.5) is

6.4 Constant Elasticity of Variance Process 365

P(r

∈ dy|r

= x) (6.3.7)

(s, t)

2

∗

(s, t)

exp



−

x + y(s, t)

2

∗

(s, t)



y(s, t)



ν/2





xy(s, t)



∗

(s, t)



where ν =(2a)/σ

− 1, (s, t) = exp





λ(u)du



,

∗

(s, t)=



(s, u)du.

Comment 6.3.5.3 Maghsoodi [615], Rogers [736] and Shirakawa [790] study

the more general model

=(a(t) − b(t)r

)dt + σ(t)

√

under the “constant dimension condition”

a(t)

(t)

= constant .

As an example (see e.g. Szatzschneider [817]), let

=(δ + β(t)X

)dt +2



and choose r

= ϕ(t)X

where ϕ is a given positive C

function. Then



δϕ(t)+



β(t)+



(t)

ϕ(t)





dt +2



ϕ(t)r

satisﬁes this constant dimension condition.

6.4 Constant Elasticity of Variance Process

The Constant Elasticity of Variance (CEV) process has dynamics

= Z

(μdt + σZ

) . (6.4.1)

The CEV model reduces to the geometric Brownian motion for β =0andtoa

particular case of the square-root model for β = −1/2 (See equation (6.3.2)).

In what follows, we do not consider the case β =0.

Cox [205] studied the case β<0, Emanuel and MacBeth [320]thecase

β>0 and Delbaen and Shirakawa [238]thecase−1 <β<0. Note that the

choice of parametrization is

= S

(μdt + σS

θ/2

)

in Cox, and

= S

μdt + σS

in Delbaen and Shirakawa.

366 6 A Special Family of Diﬀusions: Bessel Processes

In [428], Heath and Platen study a model where the num´eraire portfolio

follows a CEV process.

The CEV process is intensively studied by Davydov and Linetsky [225, 227]

and Linetsky [592]; the main part of the following study is taken from their

work. See also Beckers [64], Forde [354] and Lo et al. [599, 601]. Occupation

time densities for CEV processes are presented in Leung and Kwok [582]in

the case β<0. Atlan and Leblanc [27] and Campi et al. [138, 139]present

a model where the default time is related with the ﬁrst time when a CEV

process with β<0 reaches 0.

One of the interesting properties of the CEV model is that (for β<0) a

stock price increase implies that the variance of the stock’s return decreases

(this is known as the leverage eﬀect). The SABR model introduced in Hagan

et al. [417, 418] to ﬁt the volatility surface, corresponds to the case

= α

,dα

= να

where W and B are correlated Brownian motions with correlation ρ.This

model was named the s

tochastic alpha-beta-rho model, hence the acronym

SABR.

6.4.1 Particular Case μ =0

Let S follow the dynamics

= σS

1+β

which is the particular case μ =0in(6.4.1). Let T

(S)=inf{t : S

=0}.

 Case β>0. We deﬁne X

σβ

−β

for t<T

(S)andX

= ∂,where∂

is a cemetery point for t ≥ T

(S). The process X satisﬁes

β +1

dt − dW



ν +



dt − dW

It is a Bessel process of index ν =1/(2β) (and dimension δ =2+

> 2). The

process X does not reach 0 and does not explode, hence the process S enjoys

the same properties. From S

=(σβX

)

−1/β

= kX

−2ν

, we deduce that the

process S is a strict local martingale (see Table 6.1, page 338). The density of

the r.v. S

is obtained from the density of a Bessel process (6.2.3):

∈ dy)=

βt

1/2

−2β−3/2

exp



−

−2β

+ y

−2β

2σ





−β



dy .