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

Подождите немного. Документ загружается.

630 11 L´evy Processes

Hence,

κ(α + q, β)=κ(q,0) exp





∞



∞

−1

(1 − e

−αt−βx

−qt

P(X

∈ dx)



In particular, for q =1,

κ(α +1,β)=κ(1, 0) exp





∞



∞

−1

−t

− e

−(α+1)t−βx

)P(X

∈ dx)





Note that, for α =0andβ = −iu, one obtains from equality (11.4.3)

E(e

iuΣ

κ(q, 0)

κ(q, −iu)

From Proposition 11.2.1.5, setting m

=inf

s≤t

,wehave

law

= X

− Σ

Let



X := −X be the dual process of X. The Laplace exponent of the dual

ladder process is, for some constant



κ(α, β)=



k exp





∞



∞

−1

−t

− e

−αt−βx

)P(−X

∈ dx)





k exp





∞



−∞

−1

−t

− e

−αt−βx

)P(X

∈ dx)



From the Wiener-Hopf factorization and duality, one deduces

E(e

ium

κ(q, 0)

κ(q, iu)

and

E(e

iu(X

−Σ

)

κ(q, 0)

κ(q, iu)

= E(e

ium

) .

Note that, by deﬁnition

κ(q, 0) = k exp





∞

dt t

−1

−t

− e

−qt

)P(X

≥ 0)



κ(q, 0) =



k exp





∞

dt t

−1

−t

− e

−qt

)P(X

≤ 0)



hence,

κ(q, 0)κ(q, 0) = k



k exp





∞

dt t

−1

−t

− e

−qt

)



= k



kq ,

11.4 Fluctuation Theory 631

where the last equality follows from Frullani’s integral, and we deduce the

following relationship between the ladder exponents

κ(q, −iu) κ(q, −iu)=k



k (q + Φ(u)) . (11.4.4)

Example 11.4.1.4 InthecaseoftheL´evy process X

= σW

+ μt, one has

Φ(u)=−iμu +

, hence the quantity

q+Φ(u)

q − iμu +

a + iu

b − iu

with

a =

μ +



+2σ

,b=

−μ +



+2σ

The ﬁrst factor of the Wiener-Hopf factorization E(e

iuΣ

) is the characteristic

function of an exponential distribution with parameter a, and the second

factor E(e

iu(X

−Σ

)

) is the characteristic function of an exponential on the

negative half axis with parameter b.

Comment 11.4.1.5 Since the publication of the paper of Bingham [100],

many results have been obtained about ﬂuctuation theory. See, e.g., Doney

[260], Greenwood and Pitman [407], Kyprianou [553] and Nguyen-Ngoc and

Yor [670].

11.4.2 Pecherskii-Rogozin Identity

For x>0, denote by T

the ﬁrst passage time above x deﬁned as

=inf{t>0:X

>x}

and by O

= X

− x the overshoot which can be written

= Σ

− x = H

− x

where η

=inf{t : H

>x} and where H is deﬁned in (11.4.2). Indeed,

= τ(η

Proposition 11.4.2.1 (Pecherskii-Rogozin Identity.) For every triple of

positive numbers (α, β, q),



∞

−qx

E(e

−αT

−βO

)dx =

κ(α, q) − κ(α, β)

(q − β)κ(α, q)

. (11.4.5)

Proof: See Pecherskii and Rogozin [702], Bertoin [78] or Nguyen-Ngoc and

Yor [670] for diﬀerent proofs of the Pecherskii-Rogozin identity. 

Comment 11.4.2.2 Roynette et al. [745] present a general study of over-

shoot and asymptotic behavior. See Hilberink and Rogers [435] for application

to endogeneous default, Kl¨uppelberg et al. [526] for applications to insurance.

632 11 L´evy Processes

11.5 Spectrally Negative L´evy Processes

A spectrally negative L´evy process X is a real-valued L´evy process with

no positive jumps, equivalently its L´evy measure is supported by (−∞, 0).

Then, X admits positive exponential moments

E(exp(λX

)) = exp(tΨ (λ)) < ∞, ∀λ>0

where

Ψ(λ)=λm +



−∞

λx

− 1 − λx1

{−1<x<0}

)ν(dx) .

We recall that the process e

λX

−tΨ(λ)

is a martingale. Introduce the inverse

of Ψ,



(q)=inf{λ ≥ 0:Ψ(λ) ≥ q}.

Then, the process

(exp(Ψ



(q)X

− tq),t≥ 0)

is a martingale.

11.5.1 Two-sided Exit Times

We present here some results on two-sided exit times. We look for formulae

for two-sided exit, i.e., for 0 <x<a,

−

=inf{t ≥ 0:X

≤ 0},

=inf{t ≥ 0:X

≥ a}.

Assume now that X is a spectrally negative F-L´evy process with X

= x.

Then, the process (exp λ(X

− x) − tψ(λ)) is a (P

, F) martingale and, for

a>x, the optional stopping theorem (with a careful check of integrability

conditions) implies

−qT

<∞}

)=e

−Ψ



(q)(a−x)

We can express the Laplace transforms of the two-sided exit times in terms

of a family of two scale functions z

(q)

and w

(q)

, q>0

exp(−qT

−

)

= z

(q)

(x) −



(q)

(x) ,

exp(−qT

−

}

= w

(q)

(x)/w

(q)

(a) ,

exp(−qT

−

}

= z

(q)

(x) − z

(q)

(a)w

(q)

(x)/w

(q)

(a) .

The functions z

(q)

and w

(q)

are characterized via their Laplace transforms

11.5 Spectrally Negative L´evy Processes 633



∞

−βx

(q)

(x)dx =

Ψ(β) − q

, for β>Ψ



(q) , (11.5.1)

(q)

(x)=1+q



(q)

(y)dy .

As a consequence, one obtains



∞

−βx

(q)

(x)dx =

Ψ(β)

β(Ψ(β) − q)

,β>Ψ



(q) .

11.5.2 Laplace Exponent of the Ladder Process

Proposition 11.5.2.1 Let X be a spectrally negative L´evy process and Θ be

an exponential variable with parameter q, independent of X.Then,Σ

has

an exponential law with parameter Ψ



(q).

Proof: For any x>0,

P(Σ

≥ x)=P(T

≤ Θ)=E





∞

−qt



= E(e

−qT

)=e

−xΨ



(q)

where we have used the fact that, since X is spectrally negative, X

= x. 

Proposition 11.5.2.2 The Laplace exponent of the ladder process is

κ(α, β)=Ψ



(α)+β.

The Laplace exponent of the dual ladder process is

κ(α, β)=k



α − Ψ(β)



(α) − β

Proof: The absence of positive jumps ensures that H

= Σ

= t,and

κ(α, β)=Ψ



(α)+β, κ(α, β)=k



α − Ψ(β)



(α) − β

(See formula (11.4.4)). 

11.5.3 D. Kendall’s Identity

Kendall’s identity states that for a L´evy process with no positive jumps, if

x>0andT

=inf{t : X

≥ x},then

t P(T

∈ dt) dx = x P(X

∈ dx) dt

634 11 L´evy Processes

In particular, if X

admits a density p

(t, x), then T

has a density p

(t, x)

and

(t, x)=xp

(t, x) .

(See Borovkov and Burq [110] or Dozzi and Vallois [265] for a proof.) Note

that, in the case of Brownian motion, this equality was obtained in 3.1.4.1.

Example 11.5.3.1 Let X

= λt − γ

where γ is a Gamma process. Then

E(f(λt − γ

)) =

Γ (t)



∞

dyf(λt − y) y

t−1

−y

Γ (t)



λt

−∞

dxf(x)(λt − x)

t−1

−(λt−x)

hence,

(t, x)=

Γ (1 + t)

(λt − x)

t−1

−(λt−x)

{x<λt}

11.6 Subordinators

11.6.1 Deﬁnition and Examples

Recall (see Deﬁnition 11.2.1.3)thataL´evy process Z which takes values in

[0, ∞[ is called a subordinator. The Laplace transform E(e

−θZ

)ofZ

exists

for θ ≥ 0andis

E(e

−θZ

) = exp



−t



mθ +



]0,∞[

(1 − e

−θx

)ν(dx)



The constant m is called the drift of Z.IfZ has no drift and f is a positive

function, the exponential formula leads to



exp −



∞

f(s)dZ



=exp



−



∞



(1 − e

−xf(s)

)ν(dx)



The term subordinator originates from the following operation:

Deﬁnition 11.6.1.1 Let Z be a subordinator and X an independent L´evy pro-

cess. The process



= X

is a L´evy process, called the subordinated

L´evy process, i.e., it is the X process subordinated by Z.

Example 11.6.1.2 We present some examples of subordinators (S) and

subordinated processes (SP), in various degrees of generality:

11.6 Subordinators 635

• Compound Poisson Process (S). A ν-compound Poisson process X

where the support of ν is included in ]0, ∞[, is a subordinator. If X is a

ν-compound Poisson process, its quadratic variation

[X]



s≤t

(ΔX

)



k=1

is a compound Poisson process and a subordinator.

• (S) Let B be a BM, and

=inf{t ≥ 0:B

≥ r}.

The process (τ

,r ≥ 0) is a 1/2-stable subordinator, whose L´evy measure

√

2πx

3/2

{x>0}

dx (see Example 11.2.3.5).

• (SP) Let X be a Brownian motion and Z an independent subordinator.

Then, if



= X

,andν denotes the L´evy measure of



X,wehave

E(e

iθ

)=E



exp −



=exp



−t



ν(dz)(1 − e

−θ

z/2

)



=exp



−t



ν(dx)(1 − e

iθx

)



Using

1 − e

−θ

z/2

√

2πz



∞

−∞

dx e

−

(1 − e

iθx

) ,

we obtain

ν(dx)=dx



ν(dz)

√

2πz

−

a formula which goes back at least to Huﬀ [450].

• Cauchy Process (SP). We present the most well-known example of the

subordination operation. Let B be a Brownian motion and, for t ≥ 0

deﬁne τ

=inf{s ≥ 0:B

≥ t}.LetW be a BM, independent of B.

The process



= W

is a Cauchy process (i.e., W

has the Cauchy law

with parameter t)whoseL´evy measure is dx/(πx

). The proof relies on

the equality

E(e

)=E



exp −



= e

−|u|t

This result will be vastly generalized in the following Proposition 11.6.2.1.

• Gamma Process (S). The Gamma process (G(t; a, ν),t ≥ 0) is a

L´evy process with G(1; a, ν)havingaGammaΓ (a, ν) distribution (See

636 11 L´evy Processes

Example 11.1.1.9 or Appendix.) The Gamma process is an increasing L´evy

process, hence a subordinator, with L´evy measure

exp(−xν)1

{x>0}

dx .

11.6.2 L´evy Characteristics of a Subordinated Process

Proposition 11.6.2.1 (Changes of L´evy Characteristics Under Sub-

ordination.) Let X be a (m

,ν

) L´evy process and Z a subordinator

with drift β and L´evy measure ν

, independent of X.The process



= X

is a L´evy process with characteristic exponent

Φ(u)=i m u +

u 



Au −



iu  x

− 1 − iu x1

{|x|≤1}

)ν(dx)

with

m = βm



(ds)1

{|x|≤1}

x P(X

∈ dx) ,



A = βA

ν(dx)=βν

(dx)+



∞

(ds)P(X

∈ dx) .

Proof: The proof may be performed in two parts. First, one establishes that



X is a L´evy process, and second, its L´evy characteristics may be computed.

WerefertoSato[761], Theorem 30.1 for the details. 

Comment 11.6.2.2 A precise study of subordinators with many examples

may be found in Bertoin [79, 80].

Exercise 11.6.2.3 Let B be a Brownian motion, X

= μt + σB

and N a

Poisson process with intensity λ independent of B. Prove that the process

= X

is a compound Poisson process. Give the law of Z

. (See also

Exercise 8.6.3.7.) 

11.7 Exponential L´evy Processes as Stock Price

Processes

We now present, in the following sections, some applications to ﬁnance.

11.7.1 Option Pricing with Esscher Transform

Let S

= S

rt+X

where under the historical probability P the process X is

a real-valued L´evy process with characteristic triple (m, σ

,ν). The process S

is called an exponential L´evy process.

11.7 Exponential L´evy Processes as Stock Price Processes 637

Let us assume that E(e

αX

) < ∞,forα ∈ [−, ]. In terms of the

L´evy measure ν, this condition can be written



{|x|≥1}

αx

ν(dx) < ∞.

This implies that X has ﬁnite moments of all orders.

Proposition 11.7.1.1 We assume that E(e

αX

) < ∞ for any α in an open

interval ]a, b[ with b − a>1 and that there exists a real number θ such that

Ψ(θ)=Ψ (θ +1). Then, the process e

−rt

= S

is a martingale under the

Esscher transform P

(θ)

deﬁned as P

(θ)

= Z

where Z

θX

E(e

θX

)

Proof: In order to prove that (e

,t ≥ 0) is a martingale, since X

is a L´evy process under the Esscher transform Z, one has to check that

(θ)

) = 1, which follows from the choice of θ and

(θ)

θX

)

(θ+1)X

)=e

t(Ψ(θ+1)−Ψ (θ))



We assume that the e.m.m. chosen by the market is the probability P

(θ)

deﬁned in Proposition 11.7.1.1. The value of a contingent claim h(S

)is

= e

−r(T −t)

(θ)

(h(S

)|F

)

= e

−r(T −t)

θX

)

(h(ye

r(T −t)+X

T −t

θX

T −t

)

y=S

11.7.2 A Diﬀerential Equation for Option Pricing

Let S

= S

where X is a (m, σ

,ν)-L´evy process under the risk-neutral

probability Q. We assume that E

) < ∞.

By deﬁnition of a risk-neutral probability, the discounted price process

= e

−rt

,t ≥ 0) is a Q-strictly positive martingale with initial value

equal to 1. We know that e

−tΨ(1)

is a martingale, hence Z is a martingale if

Ψ(1) = r. In other terms, we assume that the Q-characteristic triple (m, σ

,ν)

of X is such that

m = r − σ

/2 −



− 1 − y1

{|y|≤1}

)ν(dy) .

Assume that H is a function which is regular enough so that

V (t, S)=e

−r(T −t)

(H(S

)|S

= S)

belongs to C

1,2

. Then, since e

−rt

V (t, S

)isaQ-martingale, Itˆo’s formula yields

638 11 L´evy Processes

rV =

∂

V + ∂

V + rS∂



(V (t, Se

) − V (t, S) − S(e

− 1)∂

V (t, S)) ν(dy) .

Introducing the function u(t, y)=e

r(T −t)

V (t, S

)wherey =ln(S/S

), we

see that u satisﬁes

∂

u +



r −



∂

u +

∂



(u(t, y + z) − u(t, z) − y(e

− 1)∂

u(τ,x)) ν(dz)=0.

See Cont and Tankov [192] Chapter 12, for regularity conditions.

11.7.3 Put-call Symmetry

Let us study a ﬁnancial market with a riskless asset with constant interest rate

r, and a price process (a currency) S

= S

where X is a Q-L´evy process

such that E(e

) < ∞. The choice of Q as an e.m.m. implies that the process

= e

−(r−δ)t

,t ≥ 0) is a Q-strictly positive martingale with initial

value equal to 1. We know that e

−tΨ(1)

is a martingale, hence Z is a

martingale if Ψ (1) = r−δ. In other words, we assume that the Q-characteristic

triple (m, σ

,ν)ofX is such that

m = r − δ − σ

/2 −



− 1 − y1

{|y|≤1}

)ν(dy) .

Then,

−rT

− K)

)=E

−δT

− KS

)

= E

−δT

− KS

)

with



= Z

. The process X is a



Q-L´evy process, with characteristic

exponent Ψ(λ+1)−Ψ(1) (see Proposition 11.3.1.1). The process S

= e

−X

is the exponential of the L´evy process, Y = −X which is the dual of the

L´evy process X, and the characteristic exponent of Y is Ψ(1 − λ) − Ψ(1).

Hence, the following symmetry between call and put prices holds:

,K,r,δ,T,Ψ)=P

(K, S

,δ,r,T,



Ψ) ,

where



Ψ(λ)=Ψ(1 − λ) − Ψ (1).

Comment 11.7.3.1 This result was proved by Fajardo and Mordecki [339]

and generalized by Eberlein and Papapantoleon [293].

11.8 Variance-Gamma Model 639

11.7.4 Arbitrage and Completeness

We give here some results related to arbitrage opportunities. The reader can

refer to Cherny and Shiryaev [170] and Selivanov [778]forproofs. LetX be

a(m, σ

,ν)L´evy process, not identically equal to 0 and S

= e

for t<T.

The set

M = {Q ∼ P : S is an (F, Q)-martingale}

is empty if and only if X is increasing or if X is decreasing. If M is not

empty, then M is a singleton if and only if one of the following two conditions

is satisﬁed: σ =0,ν= λδ

, or σ =0,ν=0. Hence, there is no arbitrage

in a L´evy model (except if the L´evy process is increasing or decreasing) and

the market is incomplete, except in the basic cases of a Brownian motion and

of a Poisson process.

The same kind of result holds for a time-changed exponential model: if

= e

X(τ

)

where τ is an increasing process, independent of X, such that

P(τ

>τ

) > 0, then the set M = {Q ∼ P : S is an (F

, Q)-martingale} is

empty if and only if X is increasing or decreasing. If M is not empty, then

M is a singleton if and only if τ is deterministic and continuous and one of

the following two conditions is satisﬁed: σ =0,ν= λδ

, or σ =0,ν=0.

Comment 11.7.4.1 Esscher transforms appear while looking for speciﬁc

changes of measures, which minimize some criteria, such as variance minimal

martingale measure, f

-minimal martingale measure or minimal entropy

martingale measure. See Fujiwara and Miyahara [369] and Kl¨oppel et al.[525].

11.8 Variance-Gamma Model

In a series of papers, Madan and several co-authors [155, 609, 610, 612]

introduce and exploit the Variance-Gamma Model (see also Seneta [779]).

The Variance-Gamma process is a L´evy process where X

has a Variance-

Gamma law (see  Subsection A.4.6)VG(σ, ν, θ). Its characteristic function

E(exp(iuX

)) =



1 − iuθν +

νu



−t/ν

The Variance-Gamma process may be characterized as a time-changed BM

with drift as follows: let W be a BM, and γ(t)aG(t;1/ν, 1/ν) process

independent of W .Then

= θγ(t)+σW

γ(t)

is a VG(σ, ν, θ) process. The Variance-Gamma process is a ﬁnite variation

process and is the diﬀerence of two increasing L´evy processes. More precisely,

it is the diﬀerence of two independent Gamma processes