Financial modeling with levy processes

Maybe the most important result of the theory of L´evy processes is thatthe jump measure of a general L´evy process is also a Poisson random measure.definition to show that X + Y is also

Peter Tankov Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires

Universit´ e Paris-Diderot (Paris 7) Email: peter.tankov@polytechnique.org web: www.math.jussieu.fr/∼tankov Notes of lectures given by the author at the Institute of Mathematics of the Polish Academy of Sciences

in October 2010 This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

Contents

7 The Esscher transform and absence of arbitrage in

11.1 Large/small strikes 60

mar-tingale measure 74

his-torical measure 80

by allowing the stock prices to jump while preserving the independence andstationarity of returns There are ample reasons for introducing jumps infinancial modeling First of all, asset prices do jump, and some risks sim-ply cannot be handled within continuous-path models Second, the well-documented phenomenon of implied volatility smile in option markets shows

that the risk-neutral returns are non-gaussian and leptokurtic While thesmile itself can be explained within a model with continuous paths, the factthat it becomes much more pronounced for short maturities is a clear indica-tion of the presence of jumps In continuous-path models, the law of returnsfor shorter maturities becomes closer to the Gaussian law, whereas in real-ity and in models with jumps returns actually become less Gaussian as thehorizon becomes shorter Finally, jump processes correspond to genuinelyincomplete markets, whereas all continuous-path models are either complete

or ’completable’ with a small number of additional assets This fundamentalincompleteness makes it possible to carry out a rigorous analysis of the hedg-ing error and find ways to improve the hedging performance using additionalinstruments such as liquid European options

tractability, which makes it possible to perform many computations itly and to present deep results of modern mathematical finance in a simplemanner This has led to an explosion of the literature on option pricing

literature which now contains hundreds of research papers and several graphs However, some fundamental aspects such as asymptotic behavior ofimplied volatility or the computation of hedge ratios have only recently beengiven a rigorous treatment

textbooks such as [20, 60] for a more financial perspective or [3, 44] for amore mathematical perspective

L´evy processes are a class of stochastic processes with discontinuous paths,which is at the same time simple enough to study and rich enough for appli-cations, or at least to be used as building blocks of more realistic models.Definition 1 A stochastic process X is a L´evy process if it is c`adl`ag, satisfies

• Independent increments;

• Stationary increments;

From these properties it follows that

• X is continuous in probability: ∀ε, lims→0P [|Xs+t− Xt| > ε] = 0.

• At any fixed time, the probability of having a jump is zero: ∀t, P [Xt− =

Xt] = 1

shown that any L´evy process which has a.s.continuous trajectories is a nian motion with drift

and a symmetric positive definite matrix A such that

Xt= γt + Wt,where W is a Brownian motion with covariance matrix A

Proof This result is a consequence of the Feller-L´evy central limit theorem,

here a short proof (for the one-dimensional case)

the stationarity and independence of increments

n| > ε] Since

P [sup

k

|ξnk| > ε] = 1 − (1 − P [|ξ1n| > ε])n,

we get that lim

n (1 − an)n = 1, from which it follows that lim

n n log(1 − an) = 0.But n log(1 − an) ≤ −nan≤ 0 Therefore,

Step 3 The equations (1) and (2) allow to prove that for every function fsuch that f (x) = o(|x|2) in a neighborhood of 0, limnnE[f (X1

n)] = 0 whichimplies that ε > 0

log E[eiuX1] = n log E[eiuX1n1X1

Definition 2 Let (τi)i≥1be a sequence of exponential random variables with

i=1τi Then the process

1 For all t ≥ 0, the sum in (8) is finite a.s

2 The trajectories of N are piecewise constant with jumps of size 1 only

3 The trajectories are c`adl`ag

6 The characteristic function of the Poisson process is

E[eiuNt] = exp{λt(eiu− 1)}

7 The Poisson process is a L´evy processThe Poisson process counts the events with exponential interarrival times

In a more general setting, one speaks of a counting process

process

n≥1

1t≥Tn

is called a counting process

In other words, a counting process is an increasing piecewise constantprocess with jumps of size 1 only and almost surely finite

The first step towards the characterization of L´evy processes is to acterize L´evy processes which are counting processes

(Nt) is a Poisson process

Proof The proof uses the characterization of the exponential distribution byits memoryless property: if a random variable T satisfies

P [T > t + s|T > t] = P [T > s]

for all t, s > 0 then T has exponential distribution

stationarity of increments give us:

P [T1 > t + s|T1 > t] = P [Nt+s = 0|Nt= 0]

= P [Nt+s− Nt = 0|Nt= 0] = P [Ns = 0] = P [T1 > s],

which means that the first jump time T1 has exponential distribution.Now, it suffices to show that the process (XT 1 +t− XT 1)t≥0 is independentfrom T1 and has the same law as (Xt)t≥0 Let f (t) := E[eiuX t] Then usingonce again the independence and stationarity of increments we get that f (t +s) = f (t)f (s) and Mt := ef (t)iuXt is a martingale Let Tn

= E[eiuXt]E[eivT1n]

The proof is finished with an application of the dominated convergence orem

to model asset prices because the condition that the jump size is always equal

to 1 is too restrictive, but it can be used as building block to construct richermodels

Definition 4 (Compound Poisson process) The compound Poisson processwith jump intensity λ and jump size distribution µ is a stochastic process(Xt)t≥0 defined by

N is a Poisson process with intensity λ independent from {Yi}i≥1

In other words, a compound Poisson process is a piecewise constant cess which jumps at jump times of a standard Poisson process and whosejump sizes are i.i.d random variables with a given law

a compound Poisson process with jump intensity λ and jump size distribution

µ Then X is a piecewise constant L´evy process and its characteristic function

is given by

E[eiuXt] = exp

tλZ

R

(eiux− 1)µ(dx)



Example 1 (Merton’s model) The Merton (1976) model is one of the firstapplications of jump processes in financial modeling In this model, to takeinto account price discontinuities, one adds Gaussian jumps to the log-price.

processes: we shall use it in the next section to give a full characterization

of their path structure

Definition 5 (Random measure) Let (Ω, P, F ) be a probability space and(E, E ) a measurable space Then M : Ω × E → R is a random measure if

• For every ω ∈ Ω, M (ω, ·) is a measure on E

• For every A ∈ E, M (·, A) is measurable

Definition 6 (Poisson random measure) Let (Ω, P, F ) be a probabilityspace, (E, E ) a measurable space and µ a measure on (E, E ) Then M :

Ω × E → R is a Poisson random measure with intensity µ if

• For all A ∈ E with µ(A) < ∞, M (A) follows the Poisson law withparameter E[M (A)] = µ(A)

• For any disjoint sets A1, An, M (A1), , M (An) are independent

In particular, the Poisson random measure is a positive integer-valuedrandom measure It can be constructed as the counting measure of randomlyscattered points, as shown by the following proposition

Proposition 5 Let µ be a σ-finite measure on a measurable subset E of Rd.Then there exists a Poisson random measure on E with intensity µ.

inde-pendent random variables such that P [Xi ∈ A] = µ(A)µ(E), ∀i and ∀A ∈ B(E),and let M (E) be a Poisson random variable with intensity µ(E) independent

is a Poisson random measure on E with intensity µ

Assume now that µ(E) = ∞, and choose a sequence of disjoint able sets {Ei}i≥1 such that µ(Ei) < ∞, ∀i and S

Corollary 1 (Exponential formula) Let M be a Poisson random measure

on (E, E ) with intensity µ, B ∈ E and let f be a measurable function withR

JX(A) = #{t : ∆Xt 6= 0 and (t, ∆Xt) ∈ A}

The jump measure of a set of the form [s, t] × A counts the number ofjumps of X between s and t such that their sizes fall into A For a countingprocess, since the jump size is always equal to 1, the jump measure can beseen as a random measure on [0, ∞)

Poisson random measure on [0, ∞) with intensity λ × dt

Maybe the most important result of the theory of L´evy processes is thatthe jump measure of a general L´evy process is also a Poisson random measure.

definition to show that X + Y is also a L´evy process

Exercise 2 Show that the memoryless property characterizes the tial distribution: if a random variable T satisfies

exponen-∀t, s > 0, P [T > t + s|T > t] = P [T > s]

then either T ≡ 0 or T has exponential law

parameters λ and λ0 then N + N0 is a Poisson process with parameter λ + λ0.Exercise 5 Let X be a compound Poisson process with jump size distribu-tion µ Establish that

• E[|Xt|] < ∞ if and only if R

R|x|f (dx) and in this caseE[Xt] = λt

Z

R

xf (dx)

• E[|Xt|2] < ∞ if and only if R

Rx2f (dx) and in this caseVar[Xt] = λt

Z

R

x2f (dx)

• E[eX t] < ∞ if and only if R

Rexf (dx) and in this caseE[eXt] = exp

λtZ

R

(ex− 1)f (dx)



Exercise 6 The goal is to show that to construct a Poisson random measure

on R, one needs to take two Poisson processes and make the first one runtowards +∞ and the second one towards −∞

random measure defined by

M (A) = #{t > 0 : t ∈ A, ∆Nt= 1} + #{t > 0 : −t ∈ A, ∆Nt0 = 1}.Show that M is a Poisson random measure with intensity λ

3 Path structure of a L´ evy process

measure ν defined by

is called the L´evy measure of X

Theorem 1 (L´evy-Itˆo-decomposition) Let X be an Rd-valued L´evy process

with intensity dt × ν

2 The L´evy measure ν satisfies R

Rd(kxk2∧ 1)ν(dx) < ∞

co-variance matrix A such that

The triple (A, ν, γ) is called the characteristic triple of X

The proof is based on the following lemma

Proof In view of the independence and stationarity of increments, it isenough to show that X1 and Y1 are independent Let Mt = E[eeiuXtiuXt] and

Nt = E[eeiuYtiuYt] Then M and N are martingales on [0, 1] From the

Z,

E[eiuZt] = E[eiuZ1]t and E[eiuZ1] 6= 0, ∀u

This means that M is bounded By Proposition 3, the number of jumps of Y

on [0, 1] is a Poisson random variable Therefore, N has integrable variation

on this interval By the martingale property and dominated convergence wefinally get

Poisson process, which means that JX([t1, t2] × A) follows the Poisson lawwith parameter (t2− t1)ν(A) and that JX([t1, t2] × A) is independent from

JX([s1, s2] × A) if t2 ≤ s1 Let us now take two disjoint sets A and B ByLemma 1, NA and NB are independent, which proves that JX([s1, s2] × A)and JX([t1, t2] × B) are also independent, for all s1, s2, t1, t2

It remains to show that

Z

kxk≤δ

kxk2ν(dx) < ∞for some δ > 0 Let

and Rε

t Since (Xε

t) is a L´evy process, Lemma 1 implies that

Xtε and Rεt are independent In addition, |E[eiuXt]| > 0 for all t, u Thismeans that

E[eiuXt] = E[eiuRεt]E[eiuXtε]

Therefore, |E[eiuX ε

not depend on ε By the exponential formula, this is equivalent to

exp

tZ

|x|≥ε

(eiux− 1)ν(dx)

 ≥ C > 0,

which gives R|x|≥ε(1 − cos(ux))ν(dx) ≤ ˜C < ∞ Since this result is true forall u, the proof of part 2 is completed

com-pensation of small jumps and the fact that the L´evy measure integrates kxk2

near zero: introducing the process

Mtε =Z

and so, since the space L2 is complete, for every t, Mtε converges in L2 as

ε → 0 Using Doob’s inequality we show that the convergence is uniform in

process independent from N and M in view of Lemma 1 We conclude withProposition 1

Corollary 2 (L´evy-Khintchine representation) Let X be a L´evy process withcharacteristic triple (A, ν, γ) Its characteristic function is given by

E[eiuXt] = exp

t

Example 2 (The variance gamma process) One of the simplest examples

process with stationary independent increments and such that for all t, the

pt(x) = λ

ct

Γ(ct)x

func-tion has a very simple form:

E[eiuXt] = (1 − iu/λ)−ct

Yt= µXt+ σBXt

saying that the price follows a geometric Brownian motion on a stochastictime scale given by the gamma process [34] The variance gamma processprovides another example of a L´evy processes with infinite intensity of jumps,and its characteristic function is given by

Compute the probability that X will have at least one negative jump of sizebigger than ε > 0 on the interval [0, T ]

Exercise 8 Let X be a L´evy process with L´evy measure ν(dx) = λν0(dx),where ν0 has no atom and satisfies ν0(R) = ∞ For all n ∈ N, let kn > 0

L´evy measure ν which satisfies R

decom-position, show that the trajectories of X have a.s finite variation (a functionhas finite variation if it can be represented as the difference of two increasingfunctions)

equation (11) Show that the variance gamma process can be represented as

a difference of two independent gamma process, and use this result to deduce

The main application of the stochastic integral in finance is the representation

of self-financing portfolios: in the absence of interest rates, when the price

of the risky asset is a continuous process S, and the quantity of the asset isdenoted by φ, the portfolio value is

S must be right-continuous, since the price jumps arrive inexpectedly On

portfolio manager up to date t; it must therefore be left-continuous Thefollowing example illustrates this: suppose that the asset price is given by

be the time of the first jump of N If one could use the (c`adl`ag) strategy

clearly be an arbitrage opportunity, since

The simplest (and the only one which can be realized in practice) form

of a portfolio strategy is such where the portfolio is only rebalanced a finitenumber of times We define a simple predictable process by

where T0 = 0, (Ti)i≥0 is a sequence of stopping times, and for each i, φi is

in probability (ucp)

The sequence of processes (Xn) is said to converge ucp to the process X iffor every t, (Xn−X)∗

t converges to 0 in probability, where Zt∗ := sup0≤s≤t|Zs|

continuous processes, with the same topology It is then possible to show thatthe space Sucpis dense in Lucp, and to associate the ucp topology with a metric

on Ducp, for which this space will be complete To extend the stochastic

depends on the integrator S, and we shall limit ourselves to the integratorsfor which this property holds

Definition 9 The process S ∈ D is a semimartingale if the stochastic

Ducp.Every adapted c`adl`ag process of finite variation on compacts is a semi-martingale This follows from

sup

0≤t≤T

...

continuous processes, with the same topology It is then possible to show thatthe space Sucpis dense in Lucp, and to associate the ucp topology with a metric