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

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640 11 L´evy Processes

= G(t; ν

−1

,ν

) − G(t; ν

−1

,ν

) ,

where ν

,i =1, 2 are given below. Indeed, the characteristic function can be

factorized:

E(exp(iuX

)) =



1 −



−t/ν





−t/ν

with

−1



θν +



+2νσ



> 0 ,

−1



−θν +



+2νσ



> 0 .

The L´evy density of the process X is

|x|

exp(−ν

|x|), for x<0 ,

exp(−ν

x), for x>0 .

The Variance-Gamma process has inﬁnite activity. The density of the r.v. X

θx/σ

1/ν

√

2πσΓ(1/2)



+2σ

/ν



2ν

−





(θ

+2σ

/ν)



where K

is the modiﬁed Bessel function.

In the risk-neutral world, stock prices driven by a Variance-Gamma process

have dynamics

= S

exp



rt + X



1 − θν −



Indeed, from

E(e

) = exp



−



1 − θν −



we get that the process (S

−rt

,t≥ 0) is a martingale. The parameters ν and

θ give control on skewness and kurtosis.

The tempered stable process is a L´evy process without Gaussian

component and with L´evy density

1+Y

−xM

{x>0}

−

|x|

1+Y

−

{x<0}

where C

> 0,M

≥ 0, and Y

< 2. It was introduced by Koponen [537]and

used by Bouchaud and Potters [112] for ﬁnancial modelling.

11.9 Valuation of Contingent Claims 641

The CGMY model, introduced by Carr et al. [150] is a particular case

of the tempered stable process (i.e., the case where C

= C

−

= C and

= Y

−

= Y ). The L´evy density is

Y +1

−Mx

{x>0}

|x|

Y +1

{x<0}

with C>0,M ≥ 0,G≥ 0andY<2.

If Y<0, there is a ﬁnite number of jumps in any ﬁnite interval; if not, the

process has inﬁnite activity. If Y ∈ [1, 2[, the process has inﬁnite variation.

This process is also called KoBol (see Schoutens [766]).

11.9 Valuation of Contingent Claims

11.9.1 Perpetual American Options

In this section, we present the model derived in Chesney and Jeanblanc [174].

Suppose that the dynamics of the risky asset (a currency or a paying dividend

asset) are

= S

(11.9.1)

under the chosen risk-neutral probability Q where X is a L´evy process with

parameters (m, σ

,ν). We assume that E(e

λX

) < ∞ for every λ ∈ R. As seen

in Subsection 11.7.3, one has

− (r −δ)+m +



− 1 − x1

|x|≤1

)ν(dx)=0. (11.9.2)

By deﬁnition, the value of a perpetual American call with strike K is

)=sup

τ∈T

E((S

− K)e

−rτ

)whereT is the set of stopping times.

Mordecki [659] has proved that one can reduce attention to ﬁrst hitting times,

i.e., to stopping times τ of the form

T (L)=inf{t ≥ 0:S

≥ L},

where L is a ﬁxed boundary, with L ≥ S

. Therefore,

)=sup



T (L)

− K)e

−rT(L)



Let us deﬁne f as

f(x, L)=E



−rT(L)

(x exp(X

T (L)

) − K)



Then, the value of the American call is

(x)=sup

L≥x

f(x, L) .

642 11 L´evy Processes

Let  =ln(L/x), T

()=inf{t ≥ 0,X

≥ } the hitting time of  for the

process X,andO



the overshoot deﬁned in terms of X by X

T ()

=  + O



(Obviously, T

()=T

(L):=T



.) We introduce the function

g(x, )=f(x, L)=f (x, xe



)=xE(e

−rT



) − KE(e

−rT



)

= xe



E(e

−rT



) − KE(e

−rT



) .

Then, C

(x)=sup

≥0

g(x, ) .

General Formula

On the one hand, we deﬁne

ϕ(q, x)=xα(q,r) − Kβ(q,r)

with

α(q, r)=



+∞

−q

E(e

−rT



)d , (11.9.3)

β(q, r)=



+∞

−q

E(e

−rT



)d . (11.9.4)

It is then not diﬃcult to check that

ϕ(q, x)=



∞

−q

¯g(x, )d

where

¯g(x, )=g(xe

−

,)=xE(e

−rT



) − KE(e

−rT



)=f(xe

−

,x) .

Thus

ϕ(q, x)=



∞

−q

¯g(x, )d =



−q ln(x/y)

¯g(x, ln(x/y))



f(y, x)e

−q ln(x/y)

dy . (11.9.5)

On the other hand, as in Gerber and Landry [386], by deﬁnition of the

perpetual call exercise boundary b

for x<b

,f(x, b

)=sup

L≥x

f(x, L)

hence, assuming that f is diﬀerentiable,

∂f

∂L

(x, b

)=0,x<b

. (11.9.6)

11.9 Valuation of Contingent Claims 643

Therefore, by diﬀerentiation of (11.9.5) with respect to x at point b

∂ϕ

∂x

(q, b

f(b

)

−

ϕ(q, b

)

hence

α(q, r)=

− K

−

α(q, r) − Kβ(q,r)) .

Now, due to the Pecherski-Rogozin identity (see Subsection 11.4.2), the

functions α and β are known in terms of the ladder exponent κ:

α(q, r)=

κ(r, q) − κ(r, −1)

(q +1)κ(r, q)

,β(q, r)=

κ(r, q) − κ(r, 0)

qκ(r, q)

(11.9.7)

and an easy computation leads to

κ(r, 0)

κ(r, −1)

K. (11.9.8)

Proposition 11.9.1.1 The boundary of a perpetual American call is given

κ(r, 0)

κ(r, −1)

where κ is the ladder exponent of X.

Comment 11.9.1.2 If E(e

) < ∞, using the Wiener-Hopf factorization,

Mordecki [659] proves that the boundaries for perpetual American options

are given by

= KE(e

),b

= KE(e

)

where m

=inf

s≤t

and Θ is an exponential r.v. independent of X with

parameter r, hence b

1 − ln E(e

)

. This last equality follows from

E(e

ium

)E(e

iuΣ

κ(r, 0)κ(r, 0)

κ(r, −iu)κ(r, iu)

r + Φ(u)

with Φ(u)=−ln E(e

iuX

). The equality between E(e

)and

κ(r,0)

κ(r,−1)

obtained in Nguyen-Ngoc and Yor [671] using Wiener-Hopf decomposition.

A Particular Case: X without Positive Jumps

If the process X has no positive jumps, then, we obtain the well-known result

derived e.g. in Zhang [874] (see Subsection 10.7.2). Indeed, in that case, if Ψ

is the Laplace exponent of X,

644 11 L´evy Processes

κ(α, β)=Ψ



(α)+β

where Ψ



(z)isthepositivenumbery such that Ψ (y)=z. Hence

= K



(r)



(r) − 1

Note that

Ψ(λ)=λm + λ



λx

− 1 − λx1

|x|≤1

)ν(dx) . (11.9.9)

The function Ψ is convex, Ψ(0) = 0 and Ψ(1) = r − δ. (This last property is

a consequence of the martingale criterion (11.9.2).)

A Second Particular Case: X without Negative Jumps

We now assume that the jumps of X are positive hence the dual process



X = −X has no positive jumps. Then

κ(u, k)=

u −



Ψ(k)





(u) − k

(11.9.10)

where



Ψ is the L´evy exponent of −X, given by



Ψ(k)=Ψ (−k)and





(z)is

the positive root of



Ψ(y)=z. Relying on equations (11.9.8)and(11.9.10), the

exercise boundary is given by:

r −



Ψ(0)





(r)





(r)+1

r −



Ψ(−1)

Using that



Ψ(0) = 0,



Ψ(−1) = Ψ(1) = r − δ,and





(r)=−Ψ

,n

(r), where

,n

(r) is the negative root of Ψ(k)=r, we obtain the following proposition:

Proposition 11.9.1.3 The exercise boundary for a perpetual call written on

the exponential of a L´evy process without negative jumps is

,n

(r) − 1

,n

(r)

K. (11.9.11)

Comment 11.9.1.4 It is straightforward to check that in the case ν =0,

the formula coincides with the usual formula (3.11.12). Indeed, in this case,

,n

(r) − 1

,n

(r)



(r)



(r) − 1

(11.9.12)

where Ψ

is the L´evy exponent in the case φ =0,sothatΨ



(r)isthepositive

root of bk +

k(k −1) = r and Ψ

,n

(r) is the negative root. Usual relations

between the sum and the product of roots and the coeﬃcients for a second

order polynomial, lead to the result, b

−1

K with γ

−ν+

√

+2r

(see

(3.11.10)).

11.9 Valuation of Contingent Claims 645

The Perpetual American Currency Put

The put case can be solved by using the symmetrical relationship (see Fajardo

and Mordecki [339]) between the American call and put boundaries:

(K, r, δ, Ψ)b

(K, δ, r,



Ψ)=K

where



Ψ(u)=Ψ(1 − u) − Ψ (1) . (11.9.13)

And by relying on (11.9.11), the exercise boundary b

of the perpetual put in

a jump-diﬀusion setting with negative jumps can be obtained:



,n

(δ)



,n

(δ) − 1

. (11.9.14)

ThecasewherethesizeofthejumpsofX is a strictly positive constant

ϕ corresponds to ν(dx)=λδ

(dx), with λ>0. Set φ = e

− 1. In that case,

the Laplace exponent of the L´evy process X is

Ψ(u)=uμ + u

+ λ(e

uϕ

− 1)

= u



r − δ −



+ u

+ λ((1 + φ)

− 1 − uφ)

with μ = r − δ − λφ −

. Recall that in the case of jumps of constant size

(K, r, δ, λ, ϕ)b

(K, δ, r, λ(1 + ϕ), −

1+ϕ

)=K

The price of a perpetual American call option can be decomposed as

(x)=δx

+∞



n=0



+∞

−(δ+λ(1+φ))s

(λ(1 + φ)s)

N(d

,n; s))ds

− rK

+∞



n=0



+∞

−(r+λ)s

(λs)

N(d

,n; s))ds (11.9.15)

with

(z,n; s)=

ln(x/z)+(r − δ − λφ + σ

/2)s + n ln(1 + φ)

√

(z,n; s)=d

(z,n; s) − σ

√

Comment 11.9.1.5 The perpetual exercise boundary for the put can also be

obtained by relying on the procedure used for the call. It can also be checked

that, when there is no dividend and when the L´evy measure corresponds to a

646 11 L´evy Processes

compound Poisson process, this formula is the same as in Gerber and Landry

[386], i.e.,

= rK



m + σ



(xe

− x1

|x|≤1

)ν(dx)



−1

See Chesney and Jeanblanc [174] for details.

Comment 11.9.1.6 The problem of American options is studied in Alili

and Kyprianou [8], Asmussen et al. [24], Boyarchenko and Levendorskii [117],

Chan [160, 159], Kyprianou and Pistorius [555] and Mordecki [658]. Russian

options are presented in Avram et al. [32], Asmussen et al. [24]andMordecki

and Moreira [660]. Barrier and lookback options are studied in Avram et al.

[31] and Nguyen-Ngoc and Yor [670, 671].

Il faut imaginer Sisyphe heureux.

A. Camus, Le Mythe de Sisyphe

List of Special Features, Probability Laws,

and Functions

A.1 Main Formulae

For the reader’s convenience, we give here a list of some main formulae and

the page numbers where they appeared. They are presented, for each topic,

in alphabetical order. We have not given details of notation, which should be

clear from the context.

A.1.1 Absolute Continuity Relationships

Cameron-Martin’s Formula. (Page 73)

(ν)

[F (X

,t≤ T )] = W

(0)

νX

−ν

T/2

F (X

,t≤ T )]

Girsanov’s Theorem. (Page 73)

(f)

[F (X

,t≤ T )]

= W

(0)



exp





f(X

)dX

−



)ds



F (X

,t≤ T )



Squared Radial Ornstein-Uhlenbeck Processes and Squared Bessel

Processes. (Page 356)

=exp



−

− x − at) −





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

Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4,

 Springer-Verlag London Limited 2009

647

648 A List of Special Features, Probability Laws, and Functions

Poisson Processes of Intensities λ and (1 + β)λ. (Page 466)

(1+β)λ



(1 + β)

−λβt



A.1.2 Bessel Processes

The CIR Process r is a space-time changed BESQ process ρ. (Page 357)

= e

−kt



− 1)



Laplace Transform for the BESQ. (Page 343)

[exp(−λρ

)] =

(1 + 2λt)

δ/2

exp



−

λx

1+2λt



Scale Functions.

For a squared Bessel process (Page 336)

−x

1−(δ/2)

for δ>2; ln x for δ =2; x

1−(δ/2)

for δ<2

Transition Densities. (Page 343)

For a squared Bessel process of index ν

(ν)

(x, y)=





ν/2

exp



−

x + y





√



ForaBesselprocessofindexν

(ν)

(x, y)=





exp



−

+ y







A.1 Main Formulae 649

A.1.3 Brownian Motion

Let W be a Brownian motion, and M

=sup

s≤t

Brownian Bridges.

The following processes are Brownian bridges on [0,T] (Page 237)

=(T −t)



T − s

;0≤ t ≤ T

=(T −t) W



T − t



, 0 ≤ t ≤ T

Hitting Times.

Law of the hitting time of y>0foradriftedBM:X

= W

+ νt, ν>0

(Page 148)

P(T

(X) ∈ dt)=

√

2πt

exp



−

(y − νt)



{t≥0}

Joint law of W

and ﬁrst hitting time of 0. (Page 142)

∈ dx, T

≥ t)=

{x≥0}

√

2πt



exp



−

(z − x)



− exp



−

(z + x)



Joint Law of B and its Local Time. (Page 222)

P(|B

|∈dx, L

∈ d)=1

{x≥0}

{≥0}

2(x + )

√

2πt

exp



−

(x + )



dx d

L´evy’s Equivalence Theorem. (Page 218)

(|W

|,L

; t ≥ 0)

law

=(M

− W

; t ≥ 0)

Occupation Time Formula. (Page 212)



f(W

) ds =



+∞

−∞

f(x) dx