Jump-diffusion models driven by Lévyprocesses

José E. Figueroa-López∗

Purdue UniversityDepartment of Statistics

West Lafayette, IN 47907-2066figueroa@stat.purdue.edu

Abstract: During the past and this decade, a new generation of continuous-time fi-nancial models has been intensively investigated in a quest to incorporate the so-calledstylized empirical features of asset prices like fat-tails, high kurtosis, volatility clus-tering, and leverage. Modeling driven by “memoryless homogeneous” jump processes(Lévy processes) constitutes one of the most plausible directions in this enterprise.The basic principle is to replace the underlying Brownian motion of the Black-Scholesmodel with a type of jump-diffusion process. In this chapter, the basic results and toolsbehind jump-diffusion models driven by Lévy processes are covered, providing an ac-cessible overview, coupled with their financial applications and relevance. The materialis drawn upon recent monographs (c.f. [18], [47], [46]) and papers in the field.

Keywords and phrases: Lévy processes, Poisson random measures, jump-diffusionfinancial models, exponential Lévy models.

1. An overview of financial models with jumps

The seminal Black-Scholes model [10] provides a framework to price options based on thefundamental concepts of hedging and absence of arbitrage. One of the key assumptions of theBlack-Scholes model is that the stock price process t→ St is given by a geometric Brownianmotion (GBM), originally proposed by Samuelson [45]. Concretely, the time-t price of thestock is postulated to be given by

St = S0eσWt+µt, (1)

where {Wt}t≥0 is a standard Brownian motion. This model is plausible since Brownian motionis the model of choice to describe the evolution of a random measurement whose value is theresult of a large-number of small shocks occurring through time with high-frequency. Thisis indeed the situation with the log return process Xt = log(St/S0), whose value at a given

∗ Research partially supported by a grant from the US National Science Foundation (DMS-0906919).1

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 2

time t (not “very” small) is the superposition of a high number of small movements drivenby a large number of agents posting bid and ask prices almost at all times.

However, the limitations of the GBM were well-recognized almost from its inception.For instance, the time series of log returns, say log{S∆/S0}, . . . , log{Sk∆/S(k−1)∆}, exhibitleptokurtic1 distributions which are inconsistent with the Gaussian distribution postulatedby the GBM. As expected the discrepancy from the Gaussian distribution is more markedwhen ∆ is small (say a day or smaller). Also, the volatility, as measured for instance by thesquare root of the realized variance of log returns, exhibits clustering and leverage effects,which contradicting the random-walk property of a GBM. Specifically, when plotting thetime series of log returns against time, there are periods of high variability followed by lowvariability periods suggesting that high volatility events “cluster” in time. Leverage refersto a tendency towards a volatility growth after a sharp drop in prices, suggesting thatvolatility is negatively correlated with returns. These and other stylized statistical featuresof asset returns are widely known in the financial community (see e.g. [17] and [9] for moreinformation). In the risk-neutral world, it is also well known that the Black-Scholes impliedvolatilities of call and put options are not flat neither with respect to the strike nor to thematurity, as it should be under the Black-Scholes model. Rather implied volatilities exhibita smile or smirk curve shape.

In a quest to incorporate the stylized properties of asset prices, many models have beenproposed during the last and this decade, most of them derived from natural variations ofthe Black-Scholes model. The basic idea is to replace the Brownian motion W in (1), withanother related process such as a Lévy process, a Wiener integral

∫ t0 σsdWs, or a combination

of both, leading to a jump-diffusion model or a semimartingale model. The simplest jump-diffusion model is of the form

St := S0eσWt+µt+Zt , (2)

where Z := {Zt}t≥0 is a “pure-jump” Lévy process. Equivalently, (2) can be written as

St := S0eXt , (3)

where Xt is a general Lévy process. Even this simple extension of the GBM, called geometricLévy model or exponential Lévy model, is able to incorporate several stylized features ofasset prices such as heavy tails, high-kurtosis, and asymmetry of log returns. There are otherreasons in support of jumps in the dynamics of the stock prices. On one hand, often event-driven information produces “sudden” and “sharp” price changes at discrete unpredictabletimes. Second, in fact stock prices are made up of discrete trades occurring through time ata very high frequency. Hence, processes exhibiting infinitely many jumps in any finite timehorizon [0, T ] are arguably better approximations to such high-activity stochastic processes.

1Fat tails with high kurtosis.

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 3

Merton [35], following Press [40], proposed one of the earliest models of the form (2), takinga compound Poisson process Z with normally distributed jumps (see Section 2.1). However,earlier Mandelbrot [34] had already proposed a pure-jump model driven by a stable Lévyprocess Z. Merton’s model is considered to exhibit light tails as all exponential momentsof the densities of log(St/S0) are finite, while Mandelbrot’s model exhibit very heavy tailswith not even finite second moments. It was during the last decade that models exhibitingappropriate tail behavior were proposed. Among the better known models are the varianceGamma model of [13], the CGMY model of [11], and the generalized hyperbolic motion of[6, 7] and [19, 20]. We refer to Kyprianou et. al. [32, Chapter 1] and Cont and Tankov [18,Chapter 4] for more extensive reviews of different types of geometric Lévy models in finance.

The geometric Lévy model (2) cannot incorporate volatility clustering and leverage effectsdue to the fact that log returns will be independent identically distributed. To cope with thisshortcoming, two general classes of models driven by Lévy processes have been proposed. Thefirst approach, due to Barndorff-Nielsen and Shephard (see e.g. [7] and references therein),proposes a stochastic volatility model of the form

St := S0e∫ t0budu+

∫ t0σudWu , (4)

where σ is a stationary non-Gaussian Ornstein-Uhlenbeck process

σ2t = σ20 +

∫ t0ασ2sds+ Zαt,

driven by a subordinator Z (i.e. a non-decreasing Lévy process) (see Shephard [48] andAndersen and Benzoni [3] for two recent surveys on these and other related models). Thesecond approach, proposed by Carr et. al. [12, 14], introduces stochastic volatility via arandom clock as follows:

St = S0eZτ(t) , with τ(t) :=

∫ t0r(u)du. (5)

The process τ plays the role of a “business” clock which could reflect non-synchronous tradingeffects or a “cumulative measure of economic activity”. Roughly speaking, the rate processr controls the volatility of the process; for instance, in time periods where r is “high”, the“business time” τ runs faster resulting in more frequent jump times. Hence, positive mean-revering diffusion processes {r(t)}t≥0 are plausible choices to incorporate volatility clustering.

To account for the leverage phenomenon, different combinations of the previous modelshave been considered leading to semimartingale models driven by Wiener and Poisson ran-dom measures. A very general model in this direction assumes that the log return process

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 4

Xt := log (St/S0) is given as follows (c.f. [27], [49]):

Xt = X0 +∫ t

0bsds+

∫ t0σsdWs +

∫ t0

∫|x|≤1

δ(s, x)M̄(ds, dx) +∫ t

0

∫|x|>1

δ(s, x)M(ds, dx)

σt = σ0 +∫ t

0b̃sds+

∫ t0σ̃sdWs +

∫ t0

∫|x|≤1

δ̃(s, x)M̄(ds, dx) +∫ t

0

∫|x|>1

δ̃(s, x)M(ds, dx),

where W is a d−dimensional Wiener process, M is the jump measure of an independentLévy process Z, defined by

M(B) := #{(t,∆Zt) ∈ B : t > 0 such that ∆Zt 6= 0},

and M̄(dt, dx) := M(dt, dx)−ν(dx)dt is the compensate Poisson random measure of Z, whereν is the Lévy measure of Z. The integrands (b, σ, etc.) are random processes themselves,which could even depend on X and σ leading to a system of stochastic differential equations.One of the most active research fields in this very general setting is that of statistical inferencemethods based on high-frequency (intraday) financial data. Some of the studied problemsinclude the prediction of the integrated volatility process

∫ t0 σ

2sds or of the Poisson integrals∫ t

0

∫R\{0} g(x)M(dx, ds) based on realized variations of the process (see e.g. [27, 28], [33],

[51, 52], [37], [8]), testing for jumps ([8], [37], [1]), and the estimation in the presence of“microstructure” noise ([2], [38, 39]).

In this work, the basic methods and tools behind jump-diffusion models driven by Lévyprocesses are covered. The chapter will provide an accessible overview of the probabilis-tic concepts and results related to Lévy processes, coupled whenever is possible with theirfinancial application and relevance. Some of the topics include: construction and characteri-zation of Lévy processes and Poisson random measures, statistical estimation based on high-and low-frequency observations, density transformation and risk-neutral change of measures,arbitrage-free option pricing and integro-partial differential equations. The material is drawnupon recent monographs (c.f. [18], [47], [46]) and recent papers in the field.

2. Distributional properties and statistical estimation of Lévy processes

2.1. Definition and fundamental examples

A Lévy process is a probabilistic model for an unpredictable measurement Xt that evolves intime t, in such a way that the change of the measurement in disjoint time intervals of equalduration, say Xs+∆ −Xs and Xt+∆ −Xt with s+ ∆ ≤ t, are independent from one anotherbut with identical distribution. For instance, if St represents the time-t price of an asset andXt is the log return during [0, t], defined by

Xt = log (St/S0) ,

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 5

then the previous property will imply that daily or weekly log returns will be independentfrom one another with common distribution. Formally, a Lévy process is defined as follows:

Definition 1. A Lévy process X = {Xt}t≥1 is a Rd-valued stochastic process (collectionof random vectors in Rd indexed by time) defined on a probability space (Ω,F ,P) such that

(i) X0 = 0;(ii) X has independent increments: Xt1−Xt0 , . . . , Xtn−Xtn−1 are independent for any

0 ≤ t0 < · · · < tn;(iii) X has stationary increments: the distribution of Xt+∆−Xt is the same as X∆, for

all t,∆ ≥ 0;(iv) its paths are right-continuous with left-limits (rcll);(v) it has no fixed jump-times; that is, P(∆Xt 6= 0) = 0, for any time t.The last property can be replaced by asking that X is continuous in probability, namely,

XsP−→ Xt, as s→ t, for any t. Also, if X satisfies all the other properties except (iv), then

there exists a rcll version of the process (see e.g. Sato [46]).There are three fundamental examples of Lévy processes that deserve some attention:

Brownian motion, Poisson process, and compound Poisson process.

Definition 2. A (standard) Brownian motion W is a real-valued process such that (i) W0 =0, (ii) it has independent increments, (iii) Wt−Ws has normal distribution with mean 0 andvariance t− s, for any s < t, and (iv) it has continuous paths.

It turns out that the only real Lévy processes with continuous paths are of the formXt = σWt + bt, for constants σ > 0 and b.

A Poisson process is another fundamental type of Lévy process that is often used asbuilding blocks of other processes.

Definition 3. A Poisson process N is an integer-valued process such that (i) N0 = 0, (ii) ithas independent increments, (iii) Nt −Ns has Poisson distribution with parameter λ(t− s),for any s < t, and (iv) its paths are rcll. The parameter λ is called the intensity of theprocess.

The Poisson process is frequently used as a model to counts events of certain type (say,car accidents) occurring randomly through time. Concretely, suppose that T1 < T2 < . . .represent random occurrence times of a certain event and let Nt be the number of eventsoccurring by time t:

Nt =∞∑i=1

1{Ti≤t}. (6)

Then, if the events occur independently from one another, homogeneously in time, and withan intensity of λ events per unit time, {Nt}t≥0 given by (6) will be approximately a Poisson

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 6

process with intensity λ. This fact is a consequence of the Binomial approximation to thePoisson distribution (see, e.g., Feller [21] for this heuristic construction of a Poisson process).It turns out that any Poisson process can be written in the form (6) with {Ti}i≥1 (calledarrival times) such that the waiting times

τi := Ti − Ti−1,

are independent exponential r.v.’s with common mean 1/λ (so, the bigger the λ, the smallerthe expected waiting time between arrivals and the higher the intensity of arrivals).

To introduce the last fundamental example, the compound Poisson process, we recallthe concept of probability distribution. Given a random vector J in Rd defined on someprobability space (Ω,P), the distribution of J is the mapping ρ defined on sets A ⊂ Rd asfollows:

ρ(A) := P(J ∈ A).Thus, ρ(A) measures the probability that the random vector J belongs to the set A. Acompound Poisson process with jump distribution ρ and jump intensity λ is a process of theform

Zt :=Nt∑i=1

Ji,

where {Ji}i≥1 are independent with common distribution ρ and N is a Poisson process withintensity λ that is independent of {Ji}i. When d = 1, one can say that the compound Poissonprocess Z := {Zt}t≥0 is like a Poisson process with random jump sizes independent from oneanother. A compound Poisson process is the only Lévy process that has piece-wise constantpaths with finitely-many jumps in any time interval [0, T ]. Note that the distribution of thecompound Poisson process Z is characterized by the finite measure:

ν(A) := λρ(A), A ⊂ Rd,

called the Lévy measure of Z. Furthermore, for any finite measure ν, one can associate acompound Poisson process Z with Lévy measure ν (namely, the compound Poisson processwith intensity of jumps λ := ν(Rd) and jump distribution ρ(dx) := ν(dx)/ν(Rd)).

For future reference, it is useful to note that the characteristic function of Zt is given by

Eei〈u,Zt〉 = exp{t∫

Rd

(ei〈u,x〉 − 1

)ν(dx)

}(7)

Also, if E|Ji| =∫|x|ρ(dx) < ∞, then EZt = t

∫xρ(dx) and the so-called compensated

compound Poisson process Z̄t := Zt − EZt has characteristic function

Eei〈u,Z̄t〉 = exp{t∫

Rd

(ei〈u,x〉 − 1− i 〈u, x〉

)ν(dx)

}. (8)

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 7

One of the most fundamental results establishes that any Lévy process can be approxi-mated arbitrarily close by the superposition of a Brownian motion with drift, σWt + bt, andan independent compound Poisson process Z. The reminder Rt := Xt− (σWt + bt+Zt) is apure-jump Lévy process with jump sizes smaller than say an ε > 0, which can be taken arbi-trarily small. The previous fundamental fact is a consequence of the Lévy-Itô decompositionthat we review in Section 3.2.

2.2. Infinitely divisible distributions and the Lévy-Khinchin formula

The marginal distributions of a Lévy process X are infinitely-divisible. A random variable ξis said to be infinitely divisible if for each n ≥ 2, one can construct n i.i.d. r.v’s . ξn,1, . . . , ξn,nsuch that

ξD= ξn,1 + · · ·+ ξn,n.

That Xt is infinitely divisible is clear since

Xt =n−1∑k=0

(X(k+1)t/n −Xkt/n

),

and {X(k+1)t/n − Xkt/n}n−1k=0 are i.i.d. The class of infinitely divisible distributions is closelyrelated to limits in distribution of an array of row-wise i.i.d. r.v.’s:

Theorem 1 (Kallenberg [30]). ξ is infinitely divisible iff for each n there exists i.i.d. randomvariables {ξn,k}knk=1 such that

kn∑k=1

ξn,kD−→ ξ, as n→∞.

In term of the characteristic function ϕξ(u) := Eei〈u,ξ〉, ξ is infinitely divisible if and onlyif ϕξ(u) 6= 0, for all u, and its distinguished nth-root ϕξ(u)1/n is the characteristic function ofsome other variable for each n (see Lemma 7.6 in [46]). This property of the characteristicfunction turns out to be sufficient to determine its form in terms of three “parameters”(A, b, ν), called the Lévy triplet of ξ, as defined below.

Theorem 2 (Lévy-Khinchin formula). ξ is infinitely divisible iff

Eei〈u,ξ〉 = exp{i 〈b, u〉 − 1

2〈u,Au〉+

∫ (ei〈u,x〉 − 1− i 〈u, x〉1|x|≤1

)ν(dx)

}, (9)

for some symmetric nonnegative-definite matrix A, a vector b ∈ Rd, and a measure ν (calledthe Lévy measure) on Rd0 := Rd\{0} such that∫

Rd0(|x|2 ∧ 1)ν(dx)

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 8

Moreover, all triplets (A, b, ν) with the stated properties may occur.

The following remarks are important:

Remark 1. The previous result implies that the time-t marginal distribution of a Lévyprocess {Xt}t≥0 is identified with a Lévy triplet (At, bt, νt). Given that X has stationary andindependent increments, it follows that Eei〈u,Xt〉 =

{Eei〈u,X1〉

}t, for any rational t and by

the right-continuity of X, for any real t. Thus, if (A, b, ν) is the Lévy triplet of X1, then(At, bt, νt) = t(A, b, ν) and

ϕXt(u) := Eei〈u,Xt〉 = etψ(u), where (11)

ψ(u) := i 〈b, u〉 − 12〈u,Au〉+

∫ (ei〈u,x〉 − 1− i 〈u, x〉1|x|≤1

)ν(dx). (12)

The triple (A, b, ν) is called the Lévy or characteristic triplet of the Lévy process X.

Remark 2. The exponent (12) is called the Lévy exponent of the Lévy process {Xt}t≥0. Wecan see that its first term is the Lévy exponent of the Lévy process bt. The second term isthe Lévy exponent of the Lévy process ΣWt, where W = (W

1, . . . ,W d)T are d−independentWiener processes and Σ is a d × d lower triangular matrix in the Cholesky decompositionA = ΣΣT . The last term in the Lévy exponent can be decomposed into two terms:

ψcp(u) =∫|x|>1

(ei〈u,x〉 − 1

)ν(dx), ψlccp(u) =

∫|x|≤1

(ei〈u,x〉 − 1− i 〈u, x〉

)ν(dx).

The first term above is the Lévy exponent of a compound Poisson process Xcp with Lévymeasure ν1(dx) := 1|x|>1ν(dx) (see (8)). The exponent ψ

lccp corresponds to the limit indistribution of compensated compound Poisson processes. Concretely, suppose that X(ε) isa compound Poisson process with Lévy measure νε(dx) := 1ε

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 9

2.3. Short-term distributional behavior

The characteristic function (11) of X determines uniquely the Lévy triple (A, b, ν). Forinstance, the uniqueness of the matrix A is a consequence of the following result:

limh→0

h · logϕXt(h−1/2u

)= − t

2〈u,Au〉 ; (14)

see pp. 40 in [46]. In term of the process X, (14) implies that{1√hXht

}t≥0

D−→ {ΣWt}t≥0 , as h→ 0. (15)

where W = (W 1, . . . ,W d)T are d−independent Wiener processes and Σ is a lower triangularmatrix such that A = ΣΣT .

From a statistical point of view, (15) means that, when Σ 6= 0, the short-term increments{X(k+1)h −Xkh}nk=1, properly scaled, behave like the increments of a Wiener process. In thecontext of the exponential Lévy model (34), the result (15) will imply that the log returnsof the stock, properly scaled, are normally distributed when the Brownian component of theLévy process X is non-zero. This property is not consistent with the empirical heavy tails ofhigh-frequency financial returns. Recently, Rosiński [44] proposes a pure-jump class of Lévyprocesses, called tempered stable (TS) Lévy processes , such that{

1

h1/αXht

}t≥0

D−→ {Zt}t≥0 , as h→ 0, (16)

where Z is a stable process with index α < 2.

2.4. Moments and short-term moment asymptotics

Let g : Rd → R+ be a nonnegative locally bounded function and X be a Lévy process withLévy triplet (A, b, ν). The g-moment of a random variable ξ is defined by Eg(ξ). In the caseof a compound Poisson process, it is clear that

Eg(Xt) 0 if and only if∫|x|>1 g(x)ν(dx) 0such that g(x+ y) ≤ Kg(x)g(y) (resp. g(x+ y) ≤ K(g(x) + g(y))), for all x, y. Examples ofthis kind of functions are g(x1, . . . , xd) = |xj|p, for p ≥ 1, and g(x1, . . . , xd) = exp{|xj|β}, forβ ∈ (0, 1].

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 10

Hence, X(t) := (X1(t), . . . , Xd(t)) := Xt has finite mean if and only if∫{|x|>1} |x|ν(dx) <

∞. In that case, by differentiation of the characteristic function, it follows that

EXj(t) = t(∫{|x|>1}

xjν(dx) + bj

),

Similarly, E|X(t)|2 1} |x|2ν(dx)

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 11

Woerner [51] and also Figueroa-López [22] used the previous short-term ergodic property toshow the consistency of the statistics

β̂π(ϕ) :=1

tn

n∑k=1

ϕ(Xtk −Xtk−1

), (19)

towards the integral parameter β(ϕ) :=∫ϕ(x)ν(dx), when tn → ∞ and max{tk − tk−1} →

0, for test functions ϕ as in (a). When ν(dx) = s(x)dx, Figueroa-López [22] applied theestimators (19) to analyze the asymptotic properties of nonparametric sieve-type estimatorsŝ for s. The problem of model selection was analyzed further in [24, 25], where it was provedthat sieve estimators s̃

Tcan match the rate of convergence of the minimax risk of estimators

ŝ. Concretely, it turns out that

lim supT→∞

E‖s− s̃T‖2

inf ŝ sups∈Θ E‖s− ŝ‖2

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 12

The above method can be applied to devise non-parametric estimation of the Lévy measureby replacing the Fourier transform ϕX1 by its empirical version:

ϕ̂X1

(u) :=1

n

n∑k=1

exp {i 〈u,Xk −Xk−1〉} .

given discrete observations X1, . . . , Xn of the process. Recently, similar nonparametric meth-ods have been proposed in the literature to estimate the Lévy density s(x) = ν(dx)/dxof a real Lévy process X (c.f. [36], [16], and [26]). For instance, based on the incrementsX1−X0, . . . , Xn−X(n−1), Neumann and Reiss [36] consider a nonparametric estimator for sthat minimizes the distance between the “population” characteristic function ϕ

X1(·; s) and

the empirical characteristic function ϕ̂X1

(·). By appropriately defining the distance metric,Neumann and Reiss (2008) showed the consistency of the proposed estimators. Another ap-proach, followed for instance by Watteel and Kulperger [50] and Comte and Genon-Catalot[16], relies on an “explicit” formula for the Lévy density s in terms of the derivatives of thecharacteristic function ϕ

X1. For instance, under certain regularity conditions,

F (xs(x)) (·) = −iϕ′X1

(·)ϕX1

(·),

where F(f)(u) =∫eiuxf(x)dx denotes the Fourier transform of a function f . Hence, an

estimator for s can be built by replacing ψ by a smooth version of the empirical estimateϕ̂X1

and applying inverse Fourier transform F−1.

3. Path decomposition of Lévy processes

In this part, we show that the construction in (13) holds true a.s. (not only in distribution)and draw some important consequences. The fundamental tool for this result is a probabilisticcharacterization of the random points {(t,∆Xt) : t s.t. ∆Xt 6= 0} as a Poisson point processon the semi-plane R+ × R\{0}. Due to this fact, we first review the properties of Poissonrandom measures, which are also important building blocks of financial models.

3.1. Poisson random measures and point processes

Definition 4. Let S be a Borel subset of Rd, let S be the set of Borel subsets of S, and letm be a σ−finite measure on S. A collection {M(B) : B ∈ S} of Z̄+-valued random variablesdefined on a probability space (Ω,F ,P) is called a Poisson random measure (PRM) (orprocess) on S with mean measure m if

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 13

(1) for every B ∈ S, M(B) is a Poisson random variable with mean m(B);(2) if B1, . . . , Bn ∈ S are disjoint, then M(B1), . . . ,M(B1) are independent;(3) for every sample outcome ω ∈ Ω, M(·;ω) is a measure on S.Above, we used some basic terminology of real analysis. For all practical purposes, Borel

sets of Rd are those subsets that can be constructed from basic operations (complements,countable unions, and intersections) of elementary sets of the form (a1, b1] × . . . (ad, bd]. Ameasure m is a mapping from S to [0,∞] such that

m(∅) = 0, and m( ∞⋃i=1

Bi

)=∞∑i=1

m(Bi),

for any mutually disjoints Borel sets {Bi}i≥1. A measure is said to be σ-finite if there existsmutually disjoints {Bi}i≥1 such that Rd =

⋃∞i=1Bi and m(Bi)

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 14

are assigned independently from one another. The distribution of the score can actuallydepend on the point xi. Concretely, let σ(x, dx

′) be a probability measure on S ′ ⊂ Rd′ ,for each x ∈ S (hence, σ(x, S ′) = 1). For each i, generate a r.v. x′i according σ(xi, dx′)(independently from any other variable). Consider the so-called marked Poisson process

M ′(·) =∞∑i=1

δ(xi,x′i)(·).

Proposition 2. M ′ is a Poisson random measure on S × S ′ with mean measure

m′(dx, dx′) = σ(x, dx′)m(dx).

As an example, consider the following experiment. We classify the points of the Poissonprocess M into k different types. The probability that the point xi is of type j is pj(xi)(necessarily pj(·) ∈ [0, 1]), independently from any other classification. Let {yji} be thepoints of {xi} of type j and let M j be the counting measure associated with {yji}:

M j :=∑

δ{yji }

We say that the process M1 is constructed from M by thinning.

Proposition 3. M1, . . . ,Mk are independent Poisson random measures with respective meanmeasures m1(dx) := p1(x)m(dx), . . . ,mk(dx) := pk(x)m(dx).

Example 1. Suppose that we want to simulate a Poisson point process on the unit circleS := {(x, y) : x2 + y2 ≤ 1} with mean measure:

m′(B) =∫∫

B∩S

√x2 + y2dxdy.

A method to do this is based on the previous thinning method. Suppose that we generate a“homogeneous” Poisson point process M on the square R := {(x, y) : |x| ≤ 1, |y| ≤ 1} withan intensity of λ = 8 points per unit area. That is, the mean measure of M is

m(B) =1

4

∫∫Bλdxdy.

Let {(xi, yi)}i denote the atoms of the Poisson random measure M . Now, consider the fol-lowing thinning process. We classify the point (xi, yi) of type 1 with probability p(xi, yi) :=12

√x2i + y

2i and of type 2 with probability 1− p(xi, yi). Suppose that {(x1i , y1i )}i are the point

of type 1. Then, this process is a Poisson point process with mean measure m′.

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 15

3.1.2. Integration with respect to a Poisson random measure

Let M be a Poisson random measure as Definition 4. Since M(·;ω) is an atomic randommeasure for each ω, say M(·;ω) = ∑∞i=1 δxi(ω)(·), one can define the integral

M(f) :=∫Sf(x)M(dx) =

∞∑i=1

f(xi),

for any measurable nonnegative deterministic function f . This is a R̄+ = R ∪ {∞}-valuedr.v. such that

E[e−∫f(x)M(dx)

]= exp

{−∫ (

1− e−f(x))m(dx)

}, E

[∫f(x)M(dx)

]=∫f(x)m(dx);

see [30, Lemma 10.2]. Also, if B ∈ S is such that m(B)

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 16

3.2. The Lévy-Itô decomposition

The following result, called the Lévy-Itô decomposition, is fundamental for the theory ofLévy processes. It says that any Lévy process X is the superposition of a constant drift bt, aBrownian component ΣWt, a compound Poisson process X

cpt , and the limit of compensated

Poisson processes. As stated below, it characterizes not only Lévy processes but also processeswith independent increments (called additive processes).

Theorem 3. [13.4, Kallenberg] Let {Xt}t≥0 be a rcll process in Rd with X(0) = 0. Then,X has independent increments without fixed jumps times if and only if, there exists Ω0 ∈ Fwith P(Ω0) = 1 such that for any ω ∈ Ω0,

Xt(ω) = bt(ω) +Gt(ω) +∫ t

0

∫{|x|>1}

x M(ω; ds, dx) +∫ t

0

∫{|x|≤1}

x (M −m)(ω; ds, dx), (22)

for any t ≥ 0, and for a continuous function b with b0 = 0, a continuous centered Gaussianprocess G with independent increments and G0 = 0, and an independent Poisson randommeasure M on [0,∞)× Rd0 with mean measure m satisfying∫ t

0

∫Rd0

(|x|2 ∧ 1) m(ds, dx) 0. (23)

The representation is almost surely unique, and all functions b, processes G, and measuresm with the stated properties may occur.

Note that the above theorem states that the jump random measure MX of X, defined by

MX((s, t]×B) :=∑

u∈(s,t]:∆Xu 6=01{∆Xu ∈ B},

is almost surely a Poisson process with mean measure m(dt, dx). In the case of a Lévy process(that is, we also assume that X has stationary increments), the previous theorem impliesthat MX is a Poisson random measure in R+×R0 with mean measure m(dt, dx) = ν(dx)dt,for a measure ν satisfying (10). In that case, the representation (22) takes the following form:

Xt = bt+ ΣWt +∫ t

0

∫{|x|>1}

x M(ds, dx) +∫ t

0

∫{|x|≤1}

x (M −m)(ds, dx), (24)

where W is a d-dimensional Wiener process. The third term is a compound Poisson processwith intensity of jumps ν(|x| > 1) and jump distribution 1{|x|>1}ν(dx)/ν(|x| > 1). Similarly,the last term can be understood as the limit of compensated Poisson processes as follows:∫ t

0

∫{|x|≤1}

x(M −m)(ds, dx) = limε↓0

∫ t0

∫{ε

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 17

3.3. Some sample path properties

One application of the Lévy-Itô decomposition (24) is to determine conditions for certainpath behavior of the process. The following are some cases of interest (see Section 19 in [46]for these and other path properties):

1. Path-continuity : The only continuous Lévy processes are of the form bt+ σWt.2. Finite-variation: A necessary and sufficient condition for X to have a.s. paths of

bounded variation is that σ = 0 and∫{|x|≤1}

|x|ν(dx) 0 x M(ds, dx),

and∫ t

0

∫x 0.

3. A non-decreasing Lévy process is called a subordinator . Necessary and sufficient con-ditions for X to be a subordinator are that b0 > 0, σ = 0, and ν((−∞, 0)) = 0.

4. Simulation of Lévy processes

4.1. Approximation by skeletons and compound Poisson processes

Accurate path simulation of a pure jump Lévy processes X = {X (t)}t∈[0,1], regardless ofthe relatively simple statistical structure of their increments, present some challenges whendealing with infinite jump activity (namely, processes with infinite Lévy measure). One of themost popular simulation schemes is based on the generation of discrete skeletons. Namely,the discrete skeleton of X based on equally spaced observations is defined by

X̃t =∞∑k=1

X k−1n

1[ k−1n ≤t<kn

]] =∞∑k=1

∆k1{t≥ kn},

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 18

where ∆k = Xk/n − X(k−1)/n are i.i.d. with common distribution L(X1/n). Popular classeswhere this method is applicable are Gamma, variance Gamma, Stable, and Normal inverseGaussian processes (see [18, Section 6.2]). Lamentably, the previous scheme has limitedapplications since in most cases a r.v. with distribution L(X1/n) is not easy to generate.

A second approach is to approximate the Lévy process by a finite-jump activity Lévyprocesses. That is, suppose that X is a pure-jump Lévy process, then, in light of the Lévy-Itô decomposition (24), the process

X0,εt ≡ t(b−

∫{|x|≥ε}

x ν(dx)

)+∑s≤t

∆Xs1{|∆Xs|≥ε} (26)

converges uniformly on any bounded interval to X a.s. (as usual ∆Xt ≡ Xt − Xt−). Theprocess

∑s≤t ∆Xs1{|∆Xs|≥ε} can be simulated using a compound Poisson process of the form∑Nεt

i=1 Jεi , where N

εt is a homogeneous Poisson process with intensity ν(|x| ≥ ε) and {Jεi }

∞i=1

are i.i.d with common distribution νε(dx) ≡ 1{|x|≥ε}ν(dx)/ν(|x| ≥ ε). Clearly, such a schemeis unsatisfactory because all jumps smaller than ε are totally ignored. An alternative methodof simulation approximates the small jumps with a Wiener motion.

4.2. Approximation of the small jumps of a Lévy processes

Consider a Lévy process with Lévy triple (σ2, b, ν). Define the following processes:

X1,εt := bεt+ σWt +∫ t

0

∫|x|≥ε

xM(dx, ds),

where bε = b −∫ε

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 19

(a) {σ−1(ε)Xεt }t≥0 converges in distribution to a standard Brownian motion {B(t)}t≥0 ifand only if

limε→0

σ(ε)

ε=∞. (27)

(b) Under (27), it holds that

supx∈R

∣∣∣P(Xt ≤ x)− P(X2,εt ≤ x)∣∣∣ ≤ c∫|x|≤ε x

3ν(dx)

σ3(ε)≤ c ε

σ(ε).

The first part of the above theorem provides a way to approximate the small-jumps com-ponent of X properly scaled by a Wiener process. Condition (27) can be interpreted as anassumption requiring that the size of the jumps of σ−1(ε)Xε are asymptotically vanishing.Part (b) suggests that the distribution of certain Lévy processes (with infinite jump activ-ity) can be approximated closely by the combination of a Wiener process with drift and acompound Poisson process.

4.3. Simulations based on series representations

Throughout, X = {Xt}t∈[0,1] is a Lévy process on Rd with Lévy measure ν and withoutBrownian component (which can be simulated separately). Let M := MX be the jumpmeasure of the process X, which we assumed admits the following representation:

Condition 1. The following series representation holds:

M(·) =∞∑i=1

δ(Ui,H(Γi,Vi)) (·) , a.s. (28)

for a homogeneous Poisson process {Γi}∞i=1 on R+ with unit intensity rate, an independentrandom sample {Ui}∞i=1 uniformly distributed on (0, 1), and an independent random sample{Vi}∞i=1 with common distribution F on a measurable space S, and a measurable functionH : (0,∞)× S → Rd.

Remark 3. Representation (28) can be obtained (in law) if the Lévy measure has the de-composition

ν(B) =∫ ∞

0σ(u;B)du, (29)

where σ(u;B) = P [H(u,V) ∈ B]. It is not always easy to obtain (29). The following are typ-ical methods: the inverse Lévy measure method, Bondensson’s method, and Thinning method(see Rosinski [43] for more details).

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 20

DefineA(s) =

∫ s0

∫SH(r, v)I (‖H(r, v)‖ ≤ 1)F (dv)dr. (30)

Condition 2.A(Γn)− A(n)→ 0, a.s. (31)

Lemma 1. The limit in (31) holds true if any of the following conditions is satisfied:i. b ≡ lims→∞A(s) exists in Rd;ii. the mapping r → ‖H(r, v)‖ is nonincreasing for each v ∈ S.Proposition 1. If the conditions 1 and 2 are satisfies then, a.s.

Xt = bt+∞∑i=1

(H(Γi, Vi)I (Ui ≤ t)− tci) , (32)

for all t ∈ [0, 1], where ci ≡ A(i)− A(i− 1).Remark 4. The series (32) simplifies further when

∫|x|≤1 |x|ν(dx) < ∞, namely, when X

has paths of bounded variation. Concretely, a.s.

Xt = (b− a) t+∞∑i=1

JiI (Ui ≤ t) , (33)

where a =∫|x|≤1 xν(dx). The vector b0 ≡ b− a is the drift of the Lévy process.

5. Density Transformation of Lévy processes

The following two results describes Girsanov-type theorems for Lévy processes. Concretely,the first result provides conditions for the existence of an equivalent probability measureunder which X is still a Lévy process, while the second result provides the density process.These theorems have clear applications in mathematical finance as a devise to define risk-neutral probability measures. The proofs can be found in Section 33 of Sato [46]. Girsanov-type theorems for more general processes can be found in Jacod and Shiryaev [29] (see alsoApplebaum [4] for a more accessible presentation).

Theorem 5. Let {Xt}t≤T be a real Lévy process with Lévy triple (σ2, b, ν) under some prob-ability measure P. Then the following two statements are equivalent:

(a) There exists a probability measure Q ∼ P such that {Xt}t≤T is a Lévy process withtriplet (σ′2, b′, ν ′) under Q.

(b) All the following conditions hold.

(i) ν ′(dx) = k(x)ν(dx), for some function k : R→ (0,∞).

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 21

(ii) b′ = b+∫x1|x|ε

(k(x)− 1)ν(dx)))

,

with EP [ξ] ≡ 1. The convergence on the right-hand side of the formula above is uniform int on any bounded interval.

6. Exponential Lévy models

As it was explained in the introduction, the simplest extension of the GBM (1) is the Geo-metric or exponential Lévy model:

St = S0eXt , (34)

where X is a general Lévy process with Lévy triplet (σ2, b, ν) defined on a probability space(Ω,P). In this part, we will review the financial properties of this model. As in the Black-Scholes model for option pricing, we shall also assume the existence of a risk-free asset Bwith constant interest rate r. Concretely, B is given by any of the following two equivalentdefinitions:

dBt = rBtdtB0 = 1

, or Bt = ert. (35)

The following are relevant questions: (1) Is the market arbitrage-free?; (2) Is the market com-plete?; (3) Can the arbitrage-free prices of European simple claim X = Φ(ST ) be computedin terms of a Black-Scholes PDE?.

6.1. Stochastic integration and self-financing trading strategies

As in the classical Black-Scholes model, the key concept to define arbitrage opportunities isthat of a self-financing trading strategy. Formally, this concept requires the development of atheory of stochastic integration with respect to Lévy processes and related processes such as

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 22

(34). In other words, given a suitable process {βt}0≤t≤T , representing the number of sharesof the stock held at time t, we want to define the integral

Gt :=∫ t

0βudSu, (36)

representing the net gain/loss in the stock at time t. Two different treatments of the generaltheory of stochastic integration with respect to semimartingales can be found in Jacod andShiryaev [29], and Protter [41]. More accessible presentations of the topic are given in, e.g.,Applebaum [4], and Cont and Tankov [18]. Our goal in this part is only to recall the generalideas behind (36) and the concept of self-financibility.

We first note that the process β should not only be adapted to the information process{Ft}0≤t≤T generated by the stock price (i.e. βt depends only on the stock prices up to time t),but also should be predictable, which roughly speaking means that its value at any time t canbe determined from the information available right before t. As usual, (36) can be defined forsimple trading strategies in a natural manner and then, this definition can be extended to acertain class of processes β as the limits of stochastic integrals for simple trading strategies.Concretely, assuming a buy and hold trading strategy of the form: βt := 1{τ1

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 23

Once a trading strategy has been defined, one can easily define a self-financing strategyon the market (34-35), as a pair (α, β) of adapted processes such that the so-called valueprocess Vt := αtBt + βtSt, satisfies that dVt = αtBtrdt + βtdSt. Intuitively, the change ofthe portfolio value dVt during a small time interval [t, t + dt] is due only to the changes inthe value of the primary assets in the portfolio and not due to the infusion or withdrawal ofmoney.

6.2. Conditions for the absence of arbitrage

Let us recall that an arbitrage opportunity during a given time horizon [0, T ] is just a self-financing trading strategy (α, β) such that its value process {Vt}0≤t≤T satisfies the followingthree conditions:

(i) V0 = 0, (ii) VT ≥ 0, a.s. (iii) P(VT > 0) > 0.

According to the first fundamental theorem of finance, the market (34-35) is arbitrage-free ifthere exists an equivalent martingale measure (EMM) Q; that is, if there exists a probabilitymeasure Q such that the following two conditions hold:

(a) Q(B) = 0 if and only if P(B) = 0;(b) The discounted price process S∗t := B

−1t St, for 0 ≤ t ≤ T , is a martingale under Q.

In order to find conditions for the absence of arbitrage, let us recall that for any functionk satisfying (iv) in Theorem 5, and any real η ∈ R, it is possible to find a probabilitymeasure Q(η,k) equivalent to P such that, under Q(η,k), X is a Lévy process with Lévy triplet(σ2, b′, ν ′) given as in Theorem 5. Thus, under Q(η,k), the discounted stock price S∗ is alsoan exponential Lévy model

S∗t = S0eX∗t ,

with X∗ being a Lévy process with Lévy triplet (σ2, b′−r, ν ′). It is not hard to find conditionsfor an exponential Lévy model to be a martingale (see, e.g., Theorem 8.20 in [18]). Concretely,S∗ is a martingale under Q(η,k) if and only if

b+∫x1{|x|≤1}(k(x)− 1)ν(dx) + ση − r +

σ2

2+∫

R0(ex − 1− x1{|x|≤1})k(x)ν(dx) = 0. (38)

It is now clear that if ν ≡ 0, there will exist a unique EMM of the form Q(η,k), but if ν 6= 0,there will exist in general infinitely-many of such EMM. In particular, we conclude that theexponential Lévy market (34-35) is incomplete. One popular EMM for exponential Lévymodels is the so-called Esscher transform, where k(x) = eθx, and η is chosen to satisfy (38).

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 24

6.3. Option pricing and integro-partial differential equations

As seen in the previous part, the exponential Lévy market is in general incomplete, andhence, options are not superfluous assets whose payoff can be perfectly replicated in an idealfrictionless market. The option prices are themselves subject to modeling. It is natural toadopt an EMM that preserve the Lévy structure of the log return process Xt = log(St/S0)as in the previous section. From now on, we adopt exactly this option pricing model andassume that the time-t price of a European claim with maturity T and payoff X is given by

Πt = EQ{e−r(T−t)X

∣∣∣Su, u ≤ t} ,where Q is an EMM such that

St = S0ert+X∗t ,

with X∗ being a Lévy process under Q. Throughout, (σ2, b∗, ν∗) denotes the Lévy triplet ofX∗ under Q.

Note that in the case of a simple claim X = Φ(ST ), there exists a function C : R+×R+ →R+ such that

Πt := C(t, St(ω)).

Indeed, by the Markov property, one can easily see that

C(t, x) = e−r(T−t)EQ[Φ(xer(T−t)+X

∗T−t)]. (39)

The following theorem shows that C satisfies an integro-partial differential equation (IPDE).The IPDE equation below is well-known in the literature (see e.g. [15] and [42]) and its proofcan be found in [18, Proposition 12.1].

Proposition 5. Suppose the following conditions:

1.∫|x|≥1 e

2xν∗(dx) 0 or lim infε↘0 ε

−β ∫|x|≤ε |x|2ν∗(dx) 0.Then, the function C(t, x) in (39) is continuous on [0, T ] × [0,∞), C1,2 on (0, T ) × (0,∞)and verifies the integro-partial differential equation:

∂C(t, x)

∂t+ rx

∂C

∂x(t, x) +

1

2σ2x2

∂2C

∂x2(t, x)− rC(t, x)

+∫

R0

(C(t, xey)− C(t, x)− x(ey − 1)∂C

∂x(t, x)

)ν∗(dy) = 0,

on [0, T )× (0,∞) with terminal condition C(T, x) = Φ(x), for all x.

J.E. Figueroa-López/Jump-diffusion models driven by Lévy processes 25

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