an introduction to levy processes´ with applications...

AN INTRODUCTION TO LEVY PROCESSESWITH APPLICATIONS IN FINANCE

ANTONIS PAPAPANTOLEON

Abstract. These notes aim at introducing Levy processes in an infor-mal and intuitive way. Several important results about Levy processes,such as the Levy-Khintchine formula, the Levy-Ito decomposition andGirsanov’s transformations, are discussed. Applications of Levy pro-cesses in financial modeling are presented and some popular models infinance are revisited from the point of view of Levy processes.

1. Introduction – Motivation

Levy processes play a fundamental role in Mathematical Finance, as wellas in other fields of science, such as Physics (turbulence, laser cooling),Engineering (telecommunications, queues, dams) and the Actuarial science(insurance risk). A comprehensive overview of some applications of Levyprocesses can be found in Prabhu (1998), in Barndorff-Nielsen, Mikosch,and Resnick (2001) and in Kyprianou (2005).

Levy processes have become increasingly popular in Mathematical Fi-nance because they can describe the observed reality of financial marketsin a more accurate way than models based on Brownian motion. In the“real” world, we observe that asset price processes have jumps or spikes, seeFigure 1.1, and risk-managers have to take them into account. Moreover,the empirical distribution of asset returns exhibits fat tails and skewness,behavior that deviates from normality, see Figure 1.2. Hence, models thataccurately fit return distributions are essential for the estimation of profitand loss (P&L) distributions. In the “risk-neutral” world, we observe thatimplied volatilities are constant neither across strike nor across maturitiesas stipulated by the Black and Scholes (1973) (actually, Samuelson 1965)model, see Figure 1.3. Therefore, traders need models that can capture thebehavior of the implied volatility smiles more accurately, in order to handlethe risk of trades. Levy processes provide us with the appropriate frame-work to adequately describe all these observations, both in the “real” andin the “risk-neutral” world.

Paul Levy. Processes with independent and stationary increments arenamed Levy processes after the French mathematician Paul Levy (1886-1971), who made the connection with infinitely divisible laws, characterizedtheir distributions (Levy-Khintchine formula) and described their structure

These lecture notes were prepared for mini-courses taught at the University of Piraeusin April 2005 and the University of Leipzig in November 2005. I am grateful for theopportunity of lecturing on these topics to George Skiadopoulos and Thorsten Schmidt.

1

2 ANTONIS PAPAPANTOLEON

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Oct 1997 Oct 1998 Oct 1999 Oct 2000 Oct 2001 Oct 2002 Oct 2003 Oct 2004

USD/JPY

Figure 1.1. USD/JPY foreign exchange rate, October1997-October 2004.

−0.02 −0.01 0.0 0.01 0.02

020

4060

80

Figure 1.2. Empirical distribution of daily log-returns forthe GBP/USD exchange rate and fitted Normal distribution

(Levy-Ito decomposition). Paul Levy is one of the founding fathers of thetheory of stochastic processes and made major contributions to the fieldof probability theory. Among others, Paul Levy contributed to the studyof Gaussian variables and processes, the law of large numbers, the centrallimit theorem, stable laws, infinitely divisible laws and pioneered the studyof processes with independent and stationary increments.

More information about Paul Levy and his scientific work, can be foundat the websites

http://www.cmap.polytechnique.fr/~rama/levy.html

and

http://www.annales.org/archives/x/paullevy.html

(in French).

http://www.cmap.polytechnique.fr/~rama/levy.html

http://www.annales.org/archives/x/paullevy.html

INTRODUCTION TO LEVY PROCESSES 3

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90 1 2 3 4 5 6 7 8 9 10

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10.5

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maturitydelta (%) or strike

impl

ied

vol (

%)

Figure 1.3. Implied volatilities of vanilla options on theEUR/USD exchange rate on November 5, 2001.

2. Definition

Let (Ω,F ,F, P ) be a filtered probability space, where F = FT and the fil-tration F = (Ft)t∈[0,T ] satisfies the usual conditions (cf. Jacod and Shiryaev2003, I.1.3). Note that, by abuse of the standard notation, whenever wewrite t ≥ 0 that means t ∈ [0, T ].

Definition 2.1. A cadlag, adapted, real valued stochastic process L =(Lt)t≥0 with L0 = 0 a.s. is called a Levy process if the following conditionsare satisfied:

(L1): L has independent increments, i.e. Lt−Ls is independent of Fsfor any 0 ≤ s < t ≤ T .

(L2): L has stationary increments, i.e. for any s, t ≥ 0 the distributionof Lt+s − Lt does not depend on t.

(L3): L is stochastically continuous, i.e for every t ≥ 0 and ε > 0:lims→t P (|Lt − Ls| > ε) = 0.

The simplest Levy process is the linear drift, a deterministic process.Brownian motion is the only (non-deterministic) Levy process with contin-uous sample paths. Other examples of Levy processes are the Poisson andcompound Poisson processes. Notice that the sum of a Brownian motionand a compound Poisson process is again a Levy process; it is often calleda “jump-diffusion” process; we shall call it a “Levy jump-diffusion” process,because there exist jump-diffusion processes which are not Levy processes.

3. First example: a Levy jump-diffusion

Assume that the process L = (Lt)t≥0 is a Levy jump-diffusion, i.e. aBrownian motion plus a compensated compound Poisson process. It can be


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0.02

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0.04

0.05

0.0 0.2 0.4 0.6 0.8 1.0

−0.

10.

00.

10.

2

Figure 2.4. Examples of Levy processes: linear drift (left)and Brownian motion

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4−

0.2

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0.6

0.8

0.0 0.2 0.4 0.6 0.8 1.0

−0.

10.

00.

10.

20.

30.

4

Figure 2.5. Examples of Levy processes: compound Pois-son process (left) and Levy jump-diffusion

described by

Lt = bt+ σWt +( Nt∑k=1

Jk − tλκ)

(3.1)

where b ∈ R, σ ∈ R+, W = (Wt)t≥0 is a standard Brownian motion,N = (Nt)t≥0 is a Poisson process with parameter λ (i.e. IE[Nt] = λt)and J = (Jk)k≥1 is an i.i.d. sequence of random variables with probabilitydistribution F and IE[J ] = κ < ∞; F describes the distribution of jumpsize. All sources of randomness are mutually independent.

It is well known that Brownian motion is a martingale; moreover, thecompensated compound Poisson process is a martingale. Therefore, L =(Lt)t≥0 is a martingale if and only if b = 0.

The characteristic function of Lt is

IE[eiuLt

]= IE

[exp

(iu(bt+ σWt +

Nt∑k=1

Jk − tλκ))]

= exp[iubt

]IE[exp

(iuσWt

)exp

(iu( Nt∑k=1

Jk − tλκ))]

;


since all the sources of randomness are independent, we get

= exp[iubt

]IE[exp

(iuσWt

)]IE[exp

(iu

Nt∑k=1

Jk − iutλκ)]

;

taking into account that

IE[eiuWt ] = e−12σ2u2t, Wt ∼ Normal(0, σ2t)

IE[eiuPNt

k=1 Jk ] = eλt(IE[eiuJ−1]), Nt ∼ Poisson(λt)

(cf. also Appendix B) we get

= exp[iubt

]exp

[− 1

2u2σ2t

]exp

[λt(IE[eiuJ − 1]− iuIE[J ]

)]= exp

[iubt

]exp

[− 1

2u2σ2t

]exp

[λt(IE[eiuJ − 1− iuJ ]

)];

and because the distribution of J is F we have

= exp[iubt

]exp

[− 1

2u2σ2t

]exp

[λt

∫R

(eiux − 1− iux

)F (dx)

].

Now, since t is a common factor, we re-write the above equation as

IE[eiuLt

]= exp

[t(iub− u2σ2

2+∫R

(eiux − 1− iux)λF (dx))].(3.2)

Since the characteristic function of a random variable determines its dis-tribution, we have a “characterization” of the distribution of the randomvariables underlying the Levy jump-diffusion. We will soon see that this dis-tribution belongs to the class of infinitely divisible distributions. We will alsosee that equation (3.2) is a special case of the celebrated Levy-Khintchineformula.

4. Infinitely divisible distributions and the Levy-Khintchineformula

There is strong interplay between Levy processes and infinitely divisibledistributions. We first define infinitely divisible distributions and give someexamples and then describe their relationship to Levy processes.

Let X be a real valued random variable, denote its characteristic functionby ϕX and its law by PX , hence ϕX(u) =

∫R e

iuxPX(dx).

Definition 4.1. The law PX of a random variable X is infinitely divisible,if for all n ∈ N there exist i.i.d. random variables X(1/n)

1 , . . . , X(1/n)n such

that

Xd= X

(1/n)1 + . . .+X(1/n)

n .(4.1)

Equivalently, the law PX of a random variable X is infinitely divisible if forall n ∈ N there exists another law PX(1/n) of a random variable X(1/n) such


that

PX = PX(1/n) ∗ . . . ∗ PX(1/n)︸︷︷︸n times

.(4.2)

Alternatively, we can characterize an infinitely divisible random variableusing its characteristic function.

Definition 4.2. The law of a random variable X is infinitely divisible, if forall n ∈ N, there exists a random variable X(1/n), such that

ϕX(u) =(ϕX(1/n)(u)

)n.(4.3)

Example 4.3 (Normal distribution). Using the second definition, we caneasily see that the Normal distribution is infinitely divisible. Let X ∼Normal(µ, σ2), then we have

ϕX(u) = exp[iuµ− 1

2u2σ2

]= exp

[iun

µ

n− 1

2u2n

σ2

n

]= exp

[n(iu

µ

n− 1

2u2σ

2

n)]

=

(exp

[iuµ

n− 1

2u2σ

2

n

])n=(ϕX(1/n)(u)

)nwhere X(1/n) ∼ Normal(µn ,

σ2

n ).

Example 4.4 (Poisson distribution). Following the same procedure, wecan easily deduce that the Poisson distribution is infinitely divisible. LetX ∼ Poisson(λ), then we have

ϕX(u) = exp[λ(eiu − 1)

]= exp

[nλ

n(eiu − 1)

]=

(exp

[λn

(eiu − 1)])n

=(ϕX(1/n)(u)

)nwhere X(1/n) ∼ Poisson(λn).

Remark 4.5. Other examples of infinitely divisible distributions are thecompound Poisson distribution, the exponential, the Γ-distribution, the geo-metric, the negative binomial, the Cauchy distribution and the strictly stabledistribution. Counter-examples are the uniform and binomial distributions.

The next theorem provides a complete characterization of random vari-ables with infinitely divisible distributions via their characteristic functions;this is the celebrated Levy-Khintchine formula.


Theorem 4.6. The law PX of a random variable X is infinitely divisible ifand only if there exists a triplet (b, c, ν), with b ∈ R, c ∈ R+ and a measuresatisfying ν(0) = 0 and

∫R(1 ∧ |x|2)ν(dx) <∞, such that

IE[eiuX ] = exp[ibu− u2c

2+∫R

(eiux − 1− iux1l|x|<1)ν(dx)].(4.4)

Proof. See Theorem 8.1 in Sato (1999).

The triplet (b, c, ν) is called the Levy or characteristic triplet and theexponent in (4.4)

ψ(u) = iub− u2c

2+∫R

(eiux − 1− iux1l|x|<1)ν(dx)(4.5)

is called the Levy or characteristic exponent. Moreover, b ∈ R is called thedrift term, c ∈ R+ the Gaussian or diffusion coefficient and ν the Levymeasure.

Now, consider a Levy process L = (Lt)t≥0; using the fact that, for anyn ∈ N and any t > 0,

Lt = Lt/n + (L2t/n − Lt/n) + . . .+ (Lt − L(n−1)t/n)(4.6)

together with the stationarity and independence of the increments, we con-clude that the random variable Lt is infinitely divisible.

Moreover, for all u ∈ R and all t ≥ 0, define

ψt(u) = log IE[eiuLt ];(4.7)

making use of (4.6) and the stationarity and independence of the increments,we have for any m ∈ N

mψ1(u) = ψm(u)(4.8)

and hence, for any rational t > 0

tψ1(u) = ψt(u).(4.9)

For an irrational t, we can choose a sequence of rationals that decreases tot and use right continuity of L to prove that (4.9) holds for all t ≥ 0.

Therefore, we have that for every Levy process, the following propertyholds

IE[eiuLt ] = etψ(u)(4.10)

= exp[t(ibu− u2c

2+∫R

(eiux − 1− iux1l|x|<1)ν(dx))]

where ψ(u) := ψ1(u) is the characteristic exponent of L1 := X (say), arandom variable with an infinitely divisible distribution.

We have seen so far, that every Levy process can be associated with thelaw of an infinitely divisible distribution. The opposite, i.e. that given anyrandom variable X, whose law is infinitely divisible, we can construct a Levyprocess L = (Lt)t≥0 such that L1 := X, is also true. This will be the subjectof the next section.


5. The Levy-Ito decomposition

Theorem 5.1. Consider a triplet (b, c, ν) where b ∈ R, c ∈ R+ and νis a measure satisfying ν(0) = 0 and

∫R(1 ∧ |x|2)ν(dx) < ∞. Then,

there exists a probability space (Ω,F , P ) on which four independent Levyprocesses exist, L(1), L(2), L(3) and L(4), where L(1) is a constant drift, L(2)

is a Brownian motion, L(3) is a compound Poisson process and L(4) is asquare integrable (pure jump) martingale with an a.s. countable number ofjumps on each finite time interval of magnitude less that 1. Taking L =L(1) + L(2) + L(3) + L(4), we have that there exists a probability space onwhich a Levy process L = (Lt)t≥0 with characteristic exponent

ψ(u) = iub− u2c

2+∫R

(eiux − 1− iux1l|x|<1)ν(dx)(5.1)

for all u ∈ R, is defined.

Proof. See chapter 4 in Sato (1999) or chapter 2 in Kyprianou (2005).

The Levy-Ito decomposition is a hard mathematical result to prove; here,we go through some steps of the proof because it reveals much about thestructure of the paths of a Levy process. We split the Levy exponent (5.1)into four parts

ψ = ψ(1) + ψ(2) + ψ(3) + ψ(4)

where

ψ(1)(u) = iub, ψ(2)(u) =u2c

2,

ψ(3)(u) =∫

|x|≥1

(eiux − 1)ν(dx),

ψ(4)(u) =∫

|x|<1

(eiux − 1− iux)ν(dx).

The first part corresponds to a deterministic linear process (drift) with pa-rameter b, the second one to a Brownian motion with coefficient c andthe third part to the characteristic function of a compound Poisson pro-cess with arrival rate λ := ν(R \ (−1, 1)) and jump magnitude F (dx) :=

ν(dx)ν(R\(−1,1))1l|x|≥1.

The last part is the most difficult to handle; let ∆L(4) denote the jumpsof the Levy process L(4) and µL

(4)denote the random measure counting

the jumps of L(4). Next, one constructs a compensated compound Poissonprocess

L(4,ε)t =

∑0≤s≤t

∆L(4)s 1l1>|∆L(4)

s |>ε − t( ∫

1>|x|>ε

xν(dx))

=

t∫0

∫1>|x|>ε

xµL(4)

(dx, ds)− t( ∫

1>|x|>ε

xν(dx))


and shows that the jumps of L(4) form a Poisson point process; using resultsfor Poisson point processes, we get that the characteristic function of L(4,ε)

is

ψ(4,ε)(u) =∫

ε<|x|<1

(eiux − 1− iux)ν(dx).

Finally, there exists a Levy process L(4) which is a square integrable mar-tingale and L(4,ε) → L(4) uniformly on [0, T ] as ε→ 0+ with Levy exponentψ(4).

Therefore, we can decompose any Levy process into four independentLevy processes L = L(1) + L(2) + L(3) + L(4), i.e.

Lt = bt+√cWt +

t∫0

∫|x|≥1

xµL(ds,dx)(5.2)

+( t∫

0

∫|x|<1

xµL(ds,dx)− t

∫|x|<1

xν(dx))

where L(1) is a constant drift, L(2) is a Brownian motion, L(3) a compoundPoisson process and L(4) is a pure jump martingale. This result is thecelebrated Levy-Ito decomposition of a Levy process.

6. The Levy measure and path properties

The Levy measure ν is a measure on R that satisfies

ν(0) = 0 and∫R

(1 ∧ |x|2)ν(dx) <∞.(6.1)

That means, that a Levy measure has no mass at the origin, but singularities(i.e. infinitely many jumps) can occur around the origin (i.e. small jumps).Intuitively speaking, the Levy measure describes the expected number ofjumps of a certain height in a time interval of length 1.

For example, the Levy measure of the Levy jump-diffusion is ν(dx) =λ · F (dx); from that we can deduce that the expected number of jumps, ina time interval of length 1, is λ and the jump size is distributed accordingto F .

More generally, if ν is a finite measure, i.e. ν(R) =∫

R ν(dx) = λ < ∞,then F (dx) := ν(dx)

λ , which is a probability measure. Thus, λ is the expectednumber of jumps and F (dx) the distribution of the jump size x. If ν(R) = ∞,then an infinite number of (small) jumps is expected.

The Levy measure is responsible for the richness of the class of Levyprocesses and carries useful information about the structure of the process.Path properties of a Levy process can be read from the Levy measure: forexample, Figures 6.6 and 6.7 reveal that the compound Poisson process hasa finite number of jumps on every time interval, while the NIG and α-stableprocesses have an infinite one; we then speak of an infinite activity Levyprocess.


1

2

3

4

5

1

2

3

4

5

Figure 6.6. The distribution function of the Levy measureof the standard Poisson process (left) and the density of theLevy measure of a compound Poisson process with double-exponentially distributed jumps.

1

2

3

4

5

1

2

3

4

5

Figure 6.7. The density of the Levy measure of an NIG(left) and an α-stable process.

Proposition 6.1. Let L be a Levy process with triplet (b, c, ν).(1) If ν(R) <∞ then almost all paths of L have a finite number of jumps

on every compact interval. In that case, the Levy process has finiteactivity.

(2) If ν(R) = ∞ then almost all paths of L have an infinite number ofjumps on every compact interval. In that case, the Levy process hasinfinite activity.


Whether a Levy process has finite variation or not also depends on theLevy measure (and on the presence or absence of a Brownian part).

Proposition 6.2. Let L be a Levy process with triplet (b, c, ν).(1) If c = 0 and

∫|x|≤1 |x|ν(dx) < ∞ then almost all paths of L have

finite variation.(2) If c 6= 0 or

∫|x|≤1 |x|ν(dx) = ∞ then almost all paths of L have

infinite variation.


The different functions a Levy measure has to integrate in order to havefinite activity or variation, are graphically exhibited in Figure 6.8. The


compound Poisson process has finite measure, hence it has finite variationas well; on the contrary, the NIG has an infinite measure and has infinitevariation.

0.2

0.4

0.6

0.8

1

Figure 6.8. The Levy measure must integrate |x|2 ∧ 1 (redline); it has finite variation if it integrates |x| ∧ 1 (blue line);it is finite if it integrates 1 (orange line).

Moreover, the Levy measure carries information about the finiteness ofthe moments of a Levy process; this is extremely useful information in math-ematical finance, for the existence of a martingale measure. The finitenessof the moments of a Levy process is related to the finiteness of an integralover the Levy measure (more precisely, the restriction of the Levy measureto jumps larger than 1 in absolute value, i.e. big jumps).

Proposition 6.3. Let L be a Levy process with triplet (b, c, ν). Then

(1) Lt has finite p-th moment for p ∈ R+ (IE|Lt|p < ∞) if and only if∫|x|≥1 |x|

pν(dx) <∞.(2) Lt has finite p-th exponential moment for p ∈ R (IE[epLt ] < ∞) if

and only if∫|x|≥1 e

pxν(dx) <∞.

Proof. The proof of these results can be found in Theorem 25.3 in Sato(1999). Actually, the conclusion of this theorem holds for the general class ofsubmultiplicative functions (cf. Definition 25.1 in Sato 1999), which containsexp(px) and |x|p ∨ 1 as special cases.

In order to gain some understanding of this result and because it blendsbeautifully with the Levy-Ito decomposition, we will give a rough proof ofthe sufficiency for the second part (inspired by Kyprianou 2005).

Recall from the Levy-Ito decomposition, that the characteristic exponentof a Levy process was split into four independent parts, the third of whichis a compound Poisson process with arrival rate λ := ν(R \ (−1, 1)) andjump magnitude F (dx) := ν(dx)

ν(R\(−1,1))1l|x|≥1. Finiteness of IE[epLt ] implies


finiteness of IE[epL(3)t ], where

IE[epL(3)t ] = e−λt

∑k≥0

(λt)k

k!

∫R

epxF (dx)

k

= e−λt∑k≥0

tk

k!

∫R

epx1l|x|≥1ν(dx)

k

.

Since all the summands are finite, the one corresponding to k = 1 must alsobe finite, therefore

e−λt∫R

epx1l|x|≥1ν(dx) <∞ =⇒∫

|x|≥1

epxν(dx) <∞.

1

2

3

4

Figure 6.9. A Levy process has first moment if the Levymeasure integrates |x| for |x| ≥ 1 (blue line) and secondmoment if it integrates x2 for |x| ≥ 1 (orange line).

The graphical representation of the functions the Levy measure mustintegrate so that a Levy process has finite moments is given in Figure 6.9.The NIG process possesses moments of all order, while the α-stable doesnot; one can already see from Figure 6.7 that the tails of the Levy measureof the α-stable are heavier than that of the NIG.

7. Some cases of particular interest

We already know that a Brownian motion, a (compound) Poisson processand a Levy jump-diffusion are Levy processes, their Levy-Ito decompositionand their characteristic functions. Here, we present some other special casesof Levy processes that are of special interest.

7.1. Subordinator. A subordinator is an a.s. increasing (in t) Levy pro-cess. Equivalently, for L to be a subordinator, the triplet must satisfyν(−∞, 0) = 0, c = 0,

∫(0,1) xν(dx) <∞ and γ = b+

∫(0,1) xν(dx) > 0.


The Levy-Ito decomposition of a subordinator is

(7.1) Lt = γt+

t∫0

∫(0,∞)

xµL(ds,dx)

and the Levy-Khintchine formula takes the form

(7.2) IE[eiuLt ] = exp[t(iuγ +

∫(0,∞)

(eiux − 1)ν(dx))].

Two examples of subordinators are the Poisson and the inverse Gaussianprocess, cf. Figures 7.10 and A.13.

7.2. Jumps of finite variation. A Levy process has jumps of finite varia-tion if and only if

∫|x|≤1 |x|ν(dx) <∞. In this case, the Levy-Ito decompo-

sition of L resumes the form

(7.3) Lt = γt+ cWt +

t∫0

∫R

xµL(ds,dx)


(7.4) IE[eiuLt ] = exp[t(iuγ − u2c

2+∫R

(eiux − 1)ν(dx))],

where γ is defined as above.Moreover, if ν([−1, 1]) < ∞, which means that ν(R) < ∞ then L is a

compound Poisson process.

7.3. Spectrally one-sided. A Levy processes is called spectrally negativeif ν(0,∞) = 0. Similarly, a Levy processes is called spectrally positive if −Lis spectrally negative.

7.4. Finite first moment. As we have seen already, a Levy process hasa finite first moment if and only if

∫|x|≥1 |x|ν(dx) < ∞. Therefore, the

Levy-Ito decomposition of L resumes the form

(7.5) Lt = b′t+ cWt +( t∫

0

∫R

xµL(ds,dx)− t

∫R

xν(dx))


(7.6) IE[eiuLt ] = exp[t(iub′ − u2c

2+∫R

(eiux − 1− iux)ν(dx))],

where b′ = b+∫|x|≥1 xν(dx).

Remark 7.1 (Assumption (M)). For the remaining parts we will work onlywith Levy process that have a finite first moment. We will refer to them asLevy processes that satisfy assumption (M). For the sake of simplicity, wesuppress the notation b′ and write b instead.


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−0.

50.

00.

5

0.0 0.2 0.4 0.6 0.8 1.0

010

2030

40

Figure 7.10. Simulated path of a normal inverse Gaussian(left) and an inverse Gaussian process

8. Elements from semimartingale theory

A semimartingale is a stochastic process X which admits the decomposi-tion

X = X0 +M +A

where X0 is finite and F0-measurable, M is a local martingale with M0 = 0and A is a finite variation process with A0 = 0. X is a special semimartingaleif A is predictable.

Any special semimartingale X has the following, so-called, canonical de-composition

Xt = X0 +Bt +Xct +

t∫0

∫R

x(µX − νX)(ds,dx)

whereXc is the continuous martingale part ofX and∫ t0

∫R x(µ

X−νX)(ds,dx)is the purely discontinuous martingale part of X. µX is called the randommeasure of jumps of X; it counts the number of jumps of specific size thatoccur in a time interval of specific length. νX is called the compensator ofµX (for more details, see chapter II in Jacod and Shiryaev 2003).

Returning to the Levy-Ito decomposition (5.2), we can easily see that aLevy process with triplet (b, c, ν) which satisfies assumption (M), has thefollowing canonical decomposition

Lt = bt+√cWt +

t∫0

∫R

x(µL − νL)(ds,dx),

wheret∫

0

∫R

xµL(ds,dx) =∑

0≤s≤t∆Ls

and

IE[ t∫

0

∫R

xµL(ds,dx)]

=

t∫0

∫R

xνL(ds,dx) = t

∫R

xν(dx);


one also writes νL(ds,dx) = ν(dx)ds.We denote the continuous martingale part of L by Lc and the purely

discontinuous martingale part of L by Ld, i.e.

Lct =√cWt and Ldt =

t∫0

∫R

x(µL − νL)(ds,dx).(8.1)

Remark 8.1. Every Levy process is also a semimartingale. Every Levy pro-cess with finite first moment (i.e. satisfies assumption (M)) is also a specialsemimartingale; conversely, every Levy process that is a special semimartin-gale, has a finite first moment.

9. Girsanov’s theorem

We will describe a special case of Girsanov’s theorem for semimartingales,where a Levy process remains a process with independent increments (PII)under the new measure.

Let P and P be probability measures defined on the filtered space (Ω,F ,F).Two probability measures P and P are equivalent measures, if P (A) = 0 ⇔P (A) = 0 and one writes P ∼ P .

Whenever P ∼ P , there exists a unique positive P -martingale η = (ηt)t≥0

such that IE[d ePdP

∣∣Ft] = ηt, ∀t ≥ 0 (the Radon-Nikodym derivative); η is calledthe density process of P with respect to P .

Conversely, given a measure P and a positive P -martingale η = (ηt)t≥0,one can define a measure P on (Ω,F), where P ∼ P , via the Radon-Nikodymderivative IE[d eP

dP

∣∣FT ] = ηT .

Theorem 9.1. Let L be a Levy process with triplet (b, c, ν) under P , thatsatisfies (M). It has the canonical decomposition

Lt = bt+√cWt +

t∫0

∫R

x(µL − νL)(ds,dx).(9.1)

(A1): Assume that P ∼ P . Then, there exists a deterministic process βand a measurable non-negative deterministic process Y , satisfying

t∫0

∫R

|x(Y (s, x)− 1

)|ν(dx)ds <∞,

and

t∫0

(c · β2

s

)ds <∞


such that the density process η = (ηt)t≥0 has the form

ηt = IE

[dPdP

∣∣∣Ft] = exp[ t∫

0

βs√cdWs −

12

t∫0

β2scds(9.2)

+

t∫0

∫R

(Y (s, x)− 1)(µL − νL)(ds,dx)

−t∫

0

∫R

(Y (s, x)− 1− ln(Y (s, x)))µL(ds,dx)].

(A2): Conversely, if η is a positive martingale of the form (9.2), then itdefines a probability measure P on (Ω,F), such that P ∼ P .

(A3): In both cases, we have that Wt = Wt −∫ t0 βsds is a P -Brownian

motion, νL(ds,dx) = Y (s, x)νL(ds,dx) is the P -compensator of µL

and L has the following canonical decomposition under P

Lt = bt+√cWt +

t∫0

∫R

x(µL − νL)(ds,dx),(9.3)

where

bt = bt+

t∫0

cβsds+

t∫0

∫R

x(Y (s, x)− 1

)νL(ds,dx).(9.4)

Proof. Theorems III.3.24, III.5.19 and III.5.35 in Jacod and Shiryaev (2003)yield the result.

Remark 9.2. In the statement of Theorem 9.1, we have implicitly assumedthat L has finite first moment under P as well, i.e.

IE|Lt| <∞, for all t ≥ 0.

Remark 9.3. The process L is not necessarily a Levy process under thenew measure P ; it depends on the tuple (β, Y ). We have the following cases

(G1): if (β, Y ) are deterministic and independent of time, then L re-mains a Levy process under P ; its triplet is (b, c, Y · ν).

(G2): if (β, Y ) are deterministic but depend on time, then L becomesa process with independent (but not stationary) increments underP , often called an additive process.

(G3): if (β, Y ) are neither deterministic nor independent of time, thenL is a semimartingale under P .

Remark 9.4. Notice that c, the diffusion coefficient, and µL, the randommeasure of jumps of L, did not change under the change of measure from P

to P . That happens because c and µL are path properties of the process anddo not change under an equivalent change of measure. Intuitively speaking,the paths do not change, the probability of certain paths occurring changes.


Example 9.5. Assume that L is a Levy process with canonical decompo-sition (9.1) under P . Assume that P ∼ P and the density process is

ηt = exp[β√cWt +

t∫0

∫R

αx(µL − νL)(ds,dx)(9.5)

−(cβ2

2+∫R

(eαx − 1− αx)ν(dx))t],

where β ∈ R+ and α ∈ R are constants.Then, comparing (9.5) with (9.2), we have that the tuple of functions

that characterize the change of measure is (β, Y ) = (β, f), where f(x) =eαx. Because (β, Y ) are deterministic and independent of time, L remainsa Levy process under P , its Levy triplet is (b, c, eαxν) and its canonicaldecomposition is given by equations (9.3) and (9.4).

Remark 9.6. Actually, the change of measure of example 9.5 correspondsto the so-called Esscher transformation or exponential tilting. In chapter 3 ofKyprianou (2005), one can find a significantly easier proof of Girsanov’s the-orem for Levy processes for the special case of the Esscher transform. Here,we reformulate the result of example 9.5 and give a short proof (inspired byEberlein and Papapantoleon 2005).

Proposition 9.7. Let L = (Lt)t≥0 be a Levy process with canonical decom-position (9.1) under P and assume that IE[euLt ] < ∞ for all u ∈ [−p, p],p > 0. Assume that P ∼ P with Radon-Nikodym derivative ηT , where

ηt =eβL

cteαL

dt

IE[eβLct ]IE[eαLd

t ](9.6)

= exp(β√cWt +

t∫0

∫R

αx(µL − νL)(ds,dx)

−[cβ2

2+∫R

(eαx − 1− αx)ν(dx)]t

),

for β ∈ R, |α| < p. Then, L remains a Levy process under P , its Levy tripletis (b, c, fν), where f(x) = eαx and its canonical decomposition is given byequations

Lt = bt+√cWt +

t∫0

∫R

x(µL − νL)(ds,dx),(9.7)

b = b+ βc+∫R

x(eαx − 1

)ν(dx).(9.8)


Proof. Applying Theorem 25.17 in Sato (1999), the moment generating func-tion MLt of Lt exists for u ∈ C with <u ∈ [−p, p]. We get

IE[ezLt ] = IE[ezLtηt]

= IE[ezLteβLcteαL

dt IE[eβL

ct ]−1IE[eαL

dt ]−1]

= IE[ezbte(z+β)Lcte(z+α)Ld

t ]IE[eβLct ]−1IE[eαL

dt ]−1]

= exp

t[zb+(z + β)2c

2+∫R

x(e(z+α)x − 1− (z + α)x

)ν(dx)

−β2c

2−∫R

x(eαx − 1− αx

)ν(dx)

]= exp

t[zb+z2c

2+ zβc+

∫R

eαx(ezx − 1− zx

)ν(dx)

+∫R

zx(eαx − 1

)ν(dx)

]= exp

t[z(b+ βc+∫R

x(eαx − 1

)ν(dx)

)+z2c

2

+∫R

(ezx − 1− zx

)eαxν(dx)

] .

The statement follows by proving that eαxν(dx) is a Levy measure, i.e.∫R(1 ∧ x2)eαxν(dx) <∞. It suffices to note that

(9.9)∫

|x|≤1

x2eαxν(dx) ≤ C

∫|x|≤1

x2ν(dx) <∞,

because ν is a Levy measure, where C is a constant, while the other partfollows from the assumptions.

10. Martingales and Levy processes

We give a condition for a Levy process to be a martingale and discusswhen the exponential of a Levy process is a martingale.

Proposition 10.1. Let L = (Lt)t≥0 be a Levy process with Levy triplet(b, c, ν) and assume that IE|Lt| <∞. L is a martingale if and only if b = 0.

Proof. Follows from Theorem 5.2.1 in Applebaum (2004).

Proposition 10.2. Let L = (Lt)t≥0 be a Levy process with Levy exponentψ and assume that IE[euLt ] <∞, u ∈ R. The process M = (Mt)t≥0, definedas

Mt =euLt

etψ(u)


is a martingale.

Proof. We have that IE[euLt ] = etψ(u) <∞, for all t ≥ 0.For 0 ≤ s ≤ t, we can re-write M as

Mt =euLs

esψ(u)

eu(Lt−Ls)

e(t−s)ψ(u)= Ms

eu(Lt−Ls)

e(t−s)ψ(u).

Using the fact that a Levy process has stationary and independent incre-ments, we get

IE[Mt

∣∣∣Fs] = MsIE[eu(Lt−Ls)

e(t−s)ψ(u)

∣∣∣Fs] = Mse(t−s)ψ(u)e−(t−s)ψ(u)

= Ms

The stochastic exponential E(L) of a Levy process L = (Lt)t≥0 is thesolution Z of the stochastic differential equation

dZt = Zt−dLt, Z0 = 1

defined as

E(L)t = exp(Lt −

ct

2

) ∏0≤s≤t

(1 + ∆Ls

)e−∆Ls .(10.1)

The stochastic exponential of a Levy process that is a martingale is a localmartingale (cf. Jacod and Shiryaev 2003, Theorem I.4.61) and indeed amartingale when working of a finite time horizon (cf. Kallsen 2000, Lemma4.4).

11. Construction of Levy processes

Three popular methods to construct a Levy process are described below.(C1): Specifying a Levy triplet ; more specifically, whether there ex-

ists a Brownian component or not and what is the Levy measure.Examples of Levy process constructed this way include the stan-dard Brownian motion, which has Levy triplet (0, 1, 0) and the Levyjump-diffusion, which has Levy triplet (b, σ2, λF ).

(C2): Specifying an infinitely divisible random variable as the densityof the increments at time scale 1 (i.e. L1). Examples of Levy processconstructed this way include the standard Brownian motion, whereL1 ∼ Normal(0, 1) and the normal inverse Gaussian process, whereL1 ∼ NIG(α, β, δ, µ).

(C3): Time-changing Brownian motion with an independent increas-ing Levy process. Let W denote the standard Brownian motion; wecan construct a Levy process by “replacing” the (calendar) time tby an independent increasing Levy process τ , therefore Lt := Wτ(t),t ≥ 0. The process τ has the useful –in Finance– interpretationas “business time”. Models constructed this way include the stan-dard Brownian motion, the normal inverse Gaussian process, whereBrownian motion is time-changed with the inverse Gaussian process(hence its name) and the variance gamma process, where Brownianmotion is time-changed with the gamma process.


Naturally, some processes can be constructed using more than one meth-ods. Nevertheless, each method has some distinctive advantages which arevery useful in applications. The advantages of specifying a triplet (C1) arethat the characteristic function and the pathwise properties are known andallows the construction of a rich variety of models; the drawbacks are thatparameter estimation and simulation (in the infinite activity case) can bequite involved. The second method (C2) allows the easy estimation andsimulation of the process; on the contrary the structure of the paths mightbe unknown. The method of time-changes (C3) allows for easy simulation,yet estimation might be quite difficult.

12. Simulation of Levy processes

We shall briefly describe simulation methods for Levy processes. We con-centrate on finite activity Levy processes (i.e. Levy jump-diffusions) andsome special cases of infinite activity Levy processes, namely the normalinverse Gaussian and the variance gamma process. Here we do not discussmethods for simulation of random variables with known density; varioussimulation algorithms can be found in Devroye (1986) (also available onlineat http://jeff.cs.mcgill.ca/~luc/rnbookindex.html). Moreover, sev-eral speed-up methods for the Monte Carlo simulation of Levy processes arediscussed in Webber (2005).

12.1. Finite activity. Assume we want to simulate the Levy jump-diffusion

Lt = bt+ σWt +Nt∑k=1

Jk

where Nt ∼ Poisson(λt) and J ∼ F (dx).We can simulate a discretized trajectory of the Levy jump-diffusion L at

fixed time points t1, . . . , tn as follows:• simulate a standard normal variate• transform it into a normal variate with variance σ∆t, where ∆t =ti − ti−1 (denoted Gi)

• simulate a Poisson random variate with parameter λ∆t• simulate the law of jump sizes J , i.e. simulate F (dx)• if the Poisson variate is larger than zero, add the value of the jump.

The discretized trajectory is

Lti = bti +i∑

j=1

Gi +Nti∑k=1

Jk.

12.2. Infinite activity. We will describe simulation methods for two verypopular models in mathematical finance, the variance gamma and the nor-mal inverse Gaussian process. These two processes can be easily simu-lated because they are time-changed Brownian motions; we follow Contand Tankov (2003) closely. A general treatment of simulation methods forinfinite activity Levy processes can be found in Cont and Tankov (2003) andSchoutens (2003).

http://jeff.cs.mcgill.ca/~luc/rnbookindex.html


Assume we want to simulate a normal inverse Gaussian (NIG) processwith parameters σ, θ, κ; we can simulate a discretized trajectory at fixedtime points t1, . . . , tn as follows:

• simulate n independent inverse Gaussian variables Ii with parame-ters λi = (∆t)2

κ and µi = ∆t, where ∆t = ti − ti−1, i = 1, . . . , n• simulate n standard normal variables Gi• set ∆Li = θIi + σ

√IiGi


Lti =i∑

j=1

∆Lj .

Assume we want to simulate a variance gamma (VG) process with pa-rameters σ, θ, κ; we can simulate a discretized trajectory at fixed time pointst1, . . . , tn as follows:

• simulate n independent gamma variables Γi with parameter ∆tκ where

∆t = ti − ti−1, i = 1, . . . , n• set Γi = κΓi• simulate n standard normal variables Gi• set ∆Li = θΓi + σ

√ΓiGi


Lti =i∑

j=1

∆Lj .

13. Asset price model

We describe an asset price model driven by a Levy process, both underthe “real” and under the “risk-neutral” measure.

13.1. Real-world measure. Under the real-world measure, we model theasset price process as the exponential of a Levy process, that is

St = S0 expLt;(13.1)

here, L is the Levy process whose infinitely divisible distribution has beenestimated from the data set available for the particular asset. Therefore,the log-returns of the model have independent and stationary increments,which are distributed –along time intervals of specific length, for example1– according to an infinitely divisible distribution X, i.e. L1

d= X.Naturally, the path properties of the process L carry over to S; if, for

example, L is a pure-jump Levy process, then S is also a pure-jump process.This fact allows us to capture, up to a certain extend, the microstructure ofprice fluctuations, even on an intraday time scale.

The fact that the price process is driven by a Levy process, makes themarket, in general, incomplete; the only exceptions are the markets drivenby the Normal (Black-Scholes model) and Poisson distributions. Therefore,there exists a large set of equivalent martingale measures, i.e. candidatemeasures for risk-neutral valuation.


Eberlein and Jacod (1997) provide a thorough analysis and characteriza-tion of the set of equivalent martingale measures; moreover, they prove thatthe range of, e.g. call, option prices under all possible equivalent martingalemeasures spans the whole no-arbitrage interval (i.e [(S0−Ke−rT )+, S0] for aEuropean call option with strike K). In addition, Selivanov (2005) discussesthe existence and uniqueness of martingale measures for exponential Levymodels in finite and infinite time horizon and for various specifications ofthe no-arbitrage condition.

The Levy market can be completed using particular assets, such as mo-ment derivatives (e.g. variance swaps), and then there exists a unique equiv-alent martingale measure; see Corcuera, Nualart, and Schoutens (2005a,2005b). For example, if an asset is driven by a Levy jump-diffusion

Lt = bt+ σWt +Nt∑k=1

Jk(13.2)

where F (dx) ≡ 1, then the market can be completed using only varianceswaps on this asset.

The process (St)t≥0 is the solution of the stochastic differential equation

dSt = St−(dLt +12ct+ e∆Lt − 1−∆Lt),

where ∆Lt = Lt − Lt− is the jump of L at time t. We could also specify Sas the solution to the following SDE

dSt = St−dLt,

whose solution is the stochastic exponential St = S0E(Lt). The secondapproach is unfavorable for financial applications, because (a) the asset pricecan take negative values, unless jumps are restricted to be larger than −1(i.e. supp(ν) ⊂ [−1,∞)) and (b) the distribution of log-returns is not known.In the special case of the Black-Scholes model, the two approaches coincide.

Remark 13.1. For the connection between the natural and stochastic ex-ponential for Levy processes, we refer to Lemma A.8 in Goll and Kallsen(2000).

13.2. Risk-neutral measure. Under the risk neutral measure, denoted byP , we will model the asset price process as an exponential Levy process

St = S0 expLt(13.3)

where the Levy process L must satisfy Assumptions (M) and (EM).

Assumption (EM). We assume that the Levy process L has a finite firstexponential moment, i.e.

IE[eLt ] <∞.

There are various ways to choose the martingale measure such that itis equivalent to the real-world measure. We refer to Goll and Ruschendorf(2001) for a unified exposition – in terms of f -divergences – of the differentmethods for selecting an equivalent martingale measure (EMM). Note that,some of the proposed methods to choose an EMM preserve the Levy property


of log-returns; examples are the Esscher transformation and the minimalentropy martingale measure (cf. Esche and Schweizer 2005).

The market practice is to consider the choice of the martingale measureas the result of a calibration to market data of vanilla options. Hakalaand Wystup (2002) describe the calibration procedure in detail; moreover,Cont and Tankov (2004, 2005) and Belomestny and Reiß (2005) presentnumerically stable calibration methods for Levy driven models.

Because we have assumed that P is a risk neutral measure, the asset pricehas mean rate of return µ , r−δ and the discounted and re-invested process(e(r−δ)tSt)t≥0, is a martingale under P . Here r ≥ 0 is the (domestic) risk-free interest rate, δ ≥ 0 the continuous dividend yield (or foreign interestrate) of the asset.

The process L has the canonical decomposition

Lt = bt+ σWt +

t∫0

∫R

x(µL − νL)(ds,dx)(13.4)

and hence, the drift term b equals the expectation of L1 and can be writtenas

b = r − δ − σ2

2−∫R

(ex − 1− x)ν(dx).(13.5)

where σ ≥ 0 the diffusion coefficient and W a standard Brownian mo-tion under P . µL is the random measure of jumps of the process L andνL(dt,dx) = ν(dx)dt is the compensator of the jump measure µL, where νis the Levy measure of L1.

14. Popular models

14.1. Black-Scholes. The most famous asset price model based on a Levyprocess is that of Samuelson (1965), Black and Scholes (1973) and Merton(1973). The log-returns are normally distributed with mean µ and varianceσ2, i.e. L1 ∼ Normal(µ, σ2) and the density is

fL1(x) =1

σ√

2πexp

[− (x− µ)2

2σ2

].

The characteristic function is

ϕL1(u) = exp[iµu− σ2u2

2

],

the first and second moments are

E[L1] = µ, Var[L1] = σ2,

while the skewness and kurtosis are

skew[L1] = 0, kurt[L1] = 3.

The canonical decomposition of L is

Lt = µt+ σWt

and the Levy triplet is (µ, σ2, 0).


14.2. Merton. Merton (1976) was the one of the first to use a discontinuousprice process to model asset returns. The canonical decomposition of thedriving process is

Lt = µt+ σWt +Nt∑k=1

Jk

where Jk ∼ Normal(µJ , σ2J), k = 1, ..., hence the distribution of the jump

size has the density

fJ(x) =1

σJ√

2πexp

[− (x− µJ)2

2σ2J

].

The characteristic function of L1 is


2+ λ(eiµJu−σ2

Ju2/2 − 1

)],

and the Levy triplet is (µ, σ2, λ · fJ).The density of L1 is not known in closed form, while the first two moments

are

E[L1] = µ+ λµJ and Var[L1] = σ2 + λµ2J + λσ2

J

14.3. Kou. Kou (2002) proposed a jump-diffusion model similar to Mer-ton’s, where the jump size is double-exponentially distributed. Therefore,the canonical decomposition of the driving process is

Lt = µt+ σWt +Nt∑k=1

Jk

where Jk ∼ DbExpo(p, θ1, θ2), k = 1, ..., hence the distribution of the jumpsize has the density

fJ(x) = pθ1e−θ1x1lx≤0 + (1− p)θ2eθ2x1lx>0.

The characteristic function of L1 is


2+ λ( pθ1θ1 − iu

− (1− p)θ2θ2 + iu

− 1)],

and the Levy triplet is (µ, σ2, λ · fJ).The density of L1 is not known in closed form, while the first two moments

are

E[L1] = µ+λp

θ1− λ(1− p)

θ2and Var[L1] = σ2 +

λp

θ21

+λ(1− p)

θ22

.

14.4. Generalized Hyperbolic. The generalized hyperbolic model was in-troduced by Eberlein and Prause (2002) following the works on the hyper-bolic model by Eberlein and Keller (1995); the class of hyperbolic distribu-tions was invented by O E. Barndorff-Nielsen in relation the so-called “sandproject” (cf. Barndorff-Nielsen 1977). The increments of time length 1 fol-low a generalized hyperbolic distribution with parameters α, β, δ, µ, λ, i.e.


L1 ∼ GH(α, β, δ, µ, λ) and the density is

fGH(x) = c(λ, α, β, δ)(δ2 + (x− µ)2

)(λ− 12)/2

×Kλ− 12

(α√δ2 + (x− µ)2

)exp

(β(x− µ)

),

where

c(λ, α, β, δ) =(α2 − β2)λ/2

√2παλ−

12Kλ

(δ√α2 − β2

)and Kλ denotes the Bessel function of the third kind with index λ (cf.Abramowitz and Stegun 1968). Parameter α > 0 determines the shape,0 ≤ |β| < α determines the skewness, µ ∈ R the location and δ > 0 is ascaling parameter. The last parameter, λ ∈ R affects the heaviness of thetails and allows us to navigate through different subclasses. For example,for λ = 1 we get the hyperbolic distribution and for λ = −1

2 we get thenormal inverse Gaussian (NIG).

The characteristic function of the GH distribution is

ϕGH(u) = eiuµ( α2 − β2

α2 − (β + iu)2)λ

2 Kλ

(δ√α2 − (β + iu)2

)Kλ

(δ√α2 − β2

) ,

while the first and second moments are

E[L1] = µ+βδ2

ζ

Kλ+1(ζ)Kλ(ζ)

and

Var[L1] =δ2

ζ

Kλ+1(ζ)Kλ(ζ)

+β2δ4

ζ2

(Kλ+2(ζ)Kλ(ζ)

−K2λ+1(ζ)K2λ(ζ)

),

where ζ = δ√α2 − β2.

The canonical decomposition of a Levy process driven by a generalizedhyperbolic distribution (i.e. L1 ∼ GH) is

Lt = tE[GH] +

t∫0

∫R

x(µL − νGH)(ds,dx)

and the Levy triplet is (E[GH], 0, νGH); the Levy measure of the GH distri-bution is known, but has a quite complicated expression, see Raible (2000,section 2.4.1)

The GH distribution contains as special or limiting cases several knowndistributions, including the normal, exponential, gamma, variance gamma,hyperbolic and normal inverse Gaussian distributions; see Eberlein andv. Hammerstein (2004).

14.5. Normal Inverse Gaussian. The normal inverse Gaussian distribu-tion is a special case of the GH for λ = −1

2 ; it was introduced to finance inBarndorff-Nielsen (1997). The density is

fNIG(x) =α

πexp

(δ√α2 − β2 + β(x− µ)

)K1

(αδ√

1 + (x−µδ )2)

√1 + (x−µδ )2

,


while the characteristic function has the simplified form

ϕNIG(u) = eiuµexp(δ

√α2 − β2)

exp(δ√α2 − (β + iu)2)

.

The first and second moments of the NIG distribution are

E[L1] = µ+βδ√α2 − β2

and Var[L1] =δ√

α2 − β2+

β2δ

(√α2 − β2)3

,

and similarly to the GH, the canonical decomposition is

Lt = tE[NIG] +

t∫0

∫R

x(µL − νNIG)(ds,dx),

where now the Levy measure has the form

νNIG(dx) = eβxδα

π|x|K1(α|x|)dx.

The NIG is the only subclass of the GH that is closed under convolution,i.e. ifX ∼ NIG(α, β, δ1, µ1) and Y ∼ NIG(α, β, δ2, µ2) andX is independentof Y , then

X + Y ∼ NIG(α, β, δ1 + δ2, µ1 + µ2).

Therefore, if we estimate the returns distribution at some time scale, thenwe know it –in closed form– for all time scales.

15. Pricing European options

We review the idea of Sebastian Raible for the valuation of Europeanoptions using Laplace transforms; see chapter 3 and appendix B.1 in Raible(2000). We assume that the following conditions are in force.

15.1. Summary of assumptions. The necessary assumptions regardingthe return distribution of the asset are:

(D1): Assume that ϕLT(z), the extended characteristic function of LT ,

exists for all z ∈ C with =z ∈ I1 ⊃ [0, 1].(D2): Assume that PLT

, the distribution of LT , is absolutely contin-uous w.r.t. the Lebesgue measure λ\ with density ρ.

The necessary assumptions regarding the payoff function are:

(P1): Consider a European-style payoff function f(ST ) that is inte-grable.

(P2): Assume that x 7→ e−Rx|f(e−x)| is bounded and integrable forall R ∈ I2 ⊂ R.

Finally, we need the following assumption involving both the payoff func-tion and the return distribution of the asset:

(B1): Assume that I1 ∩ I2 6= ∅.


15.2. Raible’s method.

Definition 15.1. Let Lh(z) denote the bilateral Laplace transform of afunction h at z ∈ C, i.e. let

Lh(z) :=∫R

e−zxh(x)dx.

The next Theorem gives an explicit expression for the value of an optionwith payoff function f(·) and driving process L.

Theorem 15.2. Assume that the above mentioned conditions are in forceand let g(x) := f(e−x) denote the modified payoff function of an option withpayoff f(x) at time T . Choose an R ∈ I1∩ I2. Letting V (ζ) denote the priceof this option, as a function of ζ := − logS0, we have

V (ζ) =eζR−rT

2π

∫R

eiuζLg(R+ iu)ϕLT(iR− u)du,(15.1)

whenever the integral on the r.h.s. exists (at least as a Cauchy principalvalue).

Proof. According to arbitrage pricing, the value of an option is equal to itsdiscounted expected payoff under the risk-neutral measure P . We get

V = e−rT IE[f(ST )]

= e−rT∫Ω

f(ST )dP

= e−rT∫R

f(S0ex)dPLT(x)

= e−rT∫R

f(S0ex)ρ(x)dx

because PLTis absolutely continuous with respect to the Lebesgue measure

(by D2). Define the function g(x) = f(e−x) and let ζ = − logS0, then

V = e−rT∫R

g(ζ − x)ρ(x)dx = e−rT (g ∗ ρ)(ζ)(15.2)

which is a convolution of g with ρ at the point ζ, multiplied by the discountfactor.

The idea now is to apply a Laplace transform on both sides of (15.2) andtake advantage of the fact that the Laplace transform of a convolution equalsthe product of the Laplace transforms of the factors. The resulting Laplacetransforms are easier to calculate analytically. Finally, we can invert theLaplace transforms to recover the option value.

For the functions g and ρ we have that x 7→ e−Rx|g(x)| is bounded,∫R

e−Rx|g(x)|dx <∞


for R ∈ I2 (by P2) and∫R

e−Rxρ(x)dx = ϕLT(iR) <∞

for R ∈ I1 (by D1). Therefore, for R ∈ I1 ∩ I2 (use B1) the prerequisites ofTheorem B.2 in Raible (2000) are met and we get that, the convolution in(15.2) is continuous and absolutely convergent. Applying Theorem B.2, weget, for z = R+ iu, u ∈ R

LV (z) = e−rT∫R

e−zx(g ∗ ρ)(x)dx

= e−rT∫R

e−zxg(x)dx∫R

e−zxρ(x)dx

= e−rTLg(z)Lρ(z).

Moreover, from the same Theorem we have that ζ 7→ V (ζ) is continuous and∫R

e−Rζ |V (ζ)|dζ <∞.

Hence, the prerequisites of Theorem B.3 in Raible (2000) are satisfied andwe can invert this Laplace transform to recover the option value.

V (ζ) =1

2πi

R+i∞∫R−i∞

eζzLV (z)dz

=12π

∫R

eζ(R+iu)LV (R+ iu)du

=eζR

2π

∫R

eiζue−rTLg(R+ iu)Lρ(R+ iu)du

=e−rT+ζR

2π

∫R

eiζuϕLT(iR− u)Lρ(R+ iu)du.

15.3. Examples of options.

Example 15.3 (Plain vanilla option). A European plain vanilla call optionpays off f(ST ) = (ST−K)+, for some strike price K. The Laplace transformof its modified payoff function g (normalized for strike K = 1) is

Lg(z) =1

z(z + 1)(15.3)

for z ∈ C with <z = R ∈ I2 = (−∞,−1).Similarly, for a European plain vanilla put option that pays off f(ST ) =

(K−ST )+, the Laplace transform of its modified payoff function g (normal-ized for strikeK = 1) is given by (15.3) for z ∈ C with <z = R ∈ I2 = (0,∞).


0.1

0.2

0.3

0.4

0.5

0.6

0.1

0.2

0.3

0.4

0.5

0.6

Figure 16.11. Densities of hyperbolic (red), NIG (blue) andhyperboloid (green) distributions (left). Comparison of theGH (red) and Normal (blue) distributions.

Example 15.4 (Digital option). A European digital call option pays offf(ST ) = 1lST>K. The Laplace transform of its modified payoff function gis

Lg(z) = −1z

(K

S0

)z(15.4)

for z ∈ C with <z = R ∈ I2 = (−∞, 0).Similarly, for a European digital put option that pays off f(ST ) = 1lST<K,

the Laplace transform of its modified payoff function g is

Lg(z) =1z

(K

S0

)z(15.5)

for z ∈ C with <z = R ∈ I2 = (0,∞).

15.4. Examples of distributions.

Example 15.5 (Normal). The Normal distribution has a (known) Lebesguedensity and its moment generating function MN (u) exists for all u ∈ R,therefore we have that I1 = R.

Example 15.6 (Generalized Hyperbolic). The Generalized Hyperbolic dis-tribution has a (known) Lebesgue density and its moment generating func-tion MGH(u) exists for all u ∈ (−α − β, α − β), therefore we have thatI1 = (−α− β, α− β).

16. Empirical evidence

Levy processes provide a framework that can easily capture the empiricalobservations both under the “real world” and the “risk-neutral” measure.We provide here some indicative examples.

Under the “real world” measure, Levy processes can be generated by dis-tributions that are flexible enough to capture the observed fat-tailed andskewed (leptokurtic) behavior of asset returns. One such class of distribu-tions is the class of generalized hyperbolic distributions (cf. section 14.4). InFigure 16.11, various densities of generalized hyperbolic distributions and acomparison of a generalized hyperbolic and normal density are plotted.

A typical example of the behavior of asset returns can be seen in Figures1.2 and 16.12. The fitted normal distribution has lower peak, fatter flanks


−0.02 0.0 0.02

020

4060

−0.02 −0.01 0.0 0.01 0.02

−0.

02−

0.01

0.0

0.01

0.02

quantiles of GH

empi

rical

qua

ntile

s

Figure 16.12. Empirical distribution of EUR/USD dailylog-returns with fitted GH (red) and the corresponding Q-Qplot.

and lighter tails than the empirical distribution; this means that, in real-ity, tiny and large price movements occur more frequently, and small andmedium size movements occur less frequently, than predicted by the normaldistribution. On the other hand, the generalized hyperbolic distributiongives a very good statistical fit of the empirical distribution; this is furtherverified by the corresponding Q-Q plot.

Under the “risk-neutral” measure, the flexibility of the generating distri-butions allows to accurately capture the shape of the implied volatility smileand to a lesser extend the shape of the whole volatility surface.

17. Related literature

The following articles are recommended for further reading.• Eberlein and Keller (1995), Cont (2001), Eberlein (2001), Barndorff-

Nielsen and Prause (2001), Carr et al. (2002), Eberlein and Ozkan(2003).

The following books are recommended for further reading.• Levy processes: Bertoin (1996), Sato (1999), Applebaum (2004),

Kyprianou (2005)• Levy processes with applications in finance: Cont and Tankov (2003),

Schoutens (2003)• Semimartingale theory: Protter (2004), Jacod and Shiryaev (2003)• Semimartingale theory with applications in finance: Shiryaev (1999).

Appendix A. Poisson random variables and processes

Definition A.1. Let X be a Poisson distributed random variable with pa-rameter λ ∈ R+. Then, for n ∈ N the probability distribution is

P (X = n) = e−λλn

n!and the first two centered moments are

E[X] = λ and Var[X] = λ.


Definition A.2. An adapted cadlag process Nt : Ω×R+ → N∪0 is calleda Poisson process if

(1) N0 = 0,(2) Nt −Ns is independent of Fs for any 0 ≤ s < t < T ,(3) Nt −Ns is Poisson distributed with parameter λ(t− s) for any 0 ≤

s < t < T .

Then, λ ≥ 0 is called the intensity of the Poisson process.

Definition A.3. Let N be a Poisson process with parameter λ. We shallcall the process N t : Ω× R+ → R where

N t := Nt − λt(A.1)

a compensated Poisson process.

A simulated path of a Poisson and a compensated Poisson process can beseen in Figure A.13.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

Figure A.13. Plots of the Poisson (left) and compensatedPoisson process.

Proposition A.4. The compensated Poisson process defined by (A.1) is amartingale.

Proof. We have

(1) The process N is adapted to the filtration because N is adapted (bydefinition),

(2) IE[|N t|] <∞ because IE[|Nt|] <∞,(3) Let 0 ≤ s < t < T , then

IE[N t|Fs] = IE[Nt − λt|Fs]= IE[Ns − (Nt −Ns)|Fs]− λ(s− (t− s))

= Ns − IE[(Nt −Ns)|Fs]− λ(s− (t− s))= Ns − λs

= N s.


Remark A.5. The characteristic functions of the Poisson and compensatedPoisson processes are respectively

IE[eiuNt ] = exp(λt(eiu − 1)

)and

IE[eiuNt ] = exp(λt(eiu − 1− iu)

).

Appendix B. Compound Poisson random variables

Let N be a Poisson distributed random variable with parameter λ > 0and J = (Jk)k≥1 an i.i.d. sequence of random variable with law F . Then, byconditioning on the number of jumps and and using independence, we havethat the characteristic function of a compound Poisson distributed randomvariable is

IE[eiuPN

k=1 Jk ] =∑n≥0

IE[eiu

PNk=1 Jk

∣∣N = n]P (N = n)

=∑n≥0

IE[eiu

Pnk=1 Jk

]e−λ

λn

n!

=∑n≥0

∫R

eiuxF (dx)

n

e−λλn

n!

= exp

λ ∫R

(eiux − 1)F (dx)

.

Appendix C. Notation

d= equality in lawa ∧ b = mina, ba ∨ b = maxa, bLet C 3 z = x+ iy, x, y ∈ R. Then <z = x and =z = y.

1lA(x) =

1, x ∈ A,0, x /∈ A.

Appendix D. Datasets

The EUR/USD implied volatility data are from 5 November 2001. Thespot price was 0.93, the domestic rate (USD) 5% and the foreign rate (EUR)4%. The data are available at

http://www.mathfinance.de/FF/sampleinputdata.txt.The USD/JPY, EUR/USD and GBP/USD foreign exchange time se-

ries correspond to noon buying rates (dates: 22/10/1997 − 22/10/2004,4/1/99−3/3/2005 and 1/5/2002−3/3/2005 respectively). The data can bedownloaded from

http://www.newyorkfed.org/markets/foreignex.html.

http://www.mathfinance.de/FF/sampleinputdata.txt

http://www.newyorkfed.org/markets/foreignex.html


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Department of Mathematical Stochastics, University of Freiburg, D–79104Freiburg, Germany

E-mail address: [email protected]

URL: http://www.stochastik.uni-freiburg.de/~papapan

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