an introduction to l evy processes with applications in finance

8/13/2019 AN INTRODUCTION TO L EVY PROCESSES WITH APPLICATIONS IN FINANCE

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AN INTRODUCTION TO LEVY PROCESSES

WITH APPLICATIONS IN FINANCE

ANTONIS PAPAPANTOLEON

Abstract. These lectures notes aim at introducing Levy processes inan informal and intuitive way, accessible to non-specialists in the field.In the first part, we focus on the theory of Levy processes. We analyzea toy example of a Levy process, viz. a Levy jump-diffusion, which yet

offers significant insight into the distributional and path structure of aLevy process. Then, we present several important results about Levyprocesses, such as infinite divisibility and the Levy-Khintchine formula,the Levy-Ito decomposition, the Ito formula for Levy processes and Gir-sanovs transformation. Some (sketches of) proofs are presented, stillthe majority of proofs is omitted and the reader is referred to textbooksinstead. In the second part, we turn our attention to the applicationsof Levy processes in financial mo deling and option pricing. We discusshow the price process of an asset can be modeled using Levy processesand give a brief account of market incompleteness. Popular models inthe literature are presented and revisited from the point of view of Levyprocesses, and we also discuss three methods for pricing financial deriva-tives. Finally, some indicative evidence from applications to market datais presented.

Contents

Part 1. Theory 21. Introduction 22. Definition 53. Toy example: a Levy jump-diffusion 64. Infinitely divisible distributions and the Levy-Khintchine formula 85. Analysis of jumps and Poisson random measures 116. The Levy-Ito decomposition 127. The Levy measure, path and moment properties 14

8. Some classes of particular interest 178.1. Subordinator 178.2. Jumps of finite variation 178.3. Spectrally one-sided 188.4. Finite first moment 18

2000 Mathematics Subject Classification. 60G51,60E07,60G44,91B28.Key words and phrases. Levy processes, jump-diffusion, infinitely divisible laws, Levy

measure, Girsanovs theorem, asset price modeling, option pricing.These lecture notes were prepared for mini-courses taught at the University of Piraeus

in April 2005 and March 2008, at the University of Leipzig in November 2005 and atthe Technical University of Athens in September 2006 and March 2008. I am grateful forthe opportunity of lecturing on these topics to George Skiadopoulos, Thorsten Schmidt,

Nikolaos Stavrakakis and Gerassimos Athanassoulis.1


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2 ANTONIS PAPAPANTOLEON

9. Elements from semimartingale theory 1810. Martingales and Levy processes 21

11. Itos formula 2212. Girsanovs theorem 2313. Construction of Levy processes 2814. Simulation of Levy processes 2814.1. Finite activity 2914.2. Infinite activity 29

Part 2. Applications in Finance 3015. Asset price model 3015.1. Real-world measure 3015.2. Risk-neutral measure 31

15.3. On market incompleteness 3216. Popular models 3316.1. BlackScholes 3316.2. Merton 3416.3. Kou 3416.4. Generalized Hyperbolic 3516.5. Normal Inverse Gaussian 3616.6. CGMY 3716.7. Meixner 3717. Pricing European options 3817.1. Transform methods 3817.2. PIDE methods 40

17.3. Monte Carlo methods 4218. Empirical evidence 42Appendix A. Poisson random variables and processes 44Appendix B. Compound Poisson random variables 45Appendix C. Notation 46Appendix D. Datasets 46Appendix E. Paul Levy 46Acknowledgments 47References 47

Part 1. Theory

1. Introduction

Levy processes play a central role in several fields of science, such asphysics, in the study of turbulence, laser cooling and in quantum field theory;inengineering, for the study of networks, queues and dams; ineconomics, forcontinuous time-series models; in the actuarial science, for the calculationof insurance and re-insurance risk; and, of course, in mathematical finance.A comprehensive overview of several applications of Levy processes can befound inPrabhu (1998),inBarndorff-Nielsen, Mikosch, and Resnick (2001),

inKyprianou, Schoutens, and Wilmott (2005)and inKyprianou (2006).


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INTRODUCTION TO LEVY PROCESSES 3

100

105

110

115

120

125

130

135

140

145

150

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

USD/JPY

Figure 1.1. USD/JPY exchange rate, Oct. 1997Oct. 2004.

In mathematical finance, Levy processes are becoming extremely fashion-able because they can describe the observed reality of financial markets ina more accurate way than models based on Brownian motion. In the realworld, we observe that asset price processes have jumps or spikes, and riskmanagers have to take them into consideration; in Figure1.1we can observesome big price changes (jumps) even on the very liquid USD/JPY exchangerate. Moreover, the empirical distribution of asset returns exhibits fat tails

and skewness, behavior that deviates from normality; see Figure 1.2 for acharacteristic picture. Hence, models that accurately fit return distributionsare essential for the estimation of profit and loss (P&L) distributions. Simi-larly, in the risk-neutral world, we observe that implied volatilities are con-stant neither across strike nor across maturities as stipulated by the Blackand Scholes (1973) (actually, Samuelson 1965) model; Figure1.3depicts atypical volatility surface. Therefore, traders need models that can capturethe behavior of the implied volatility smiles more accurately, in order tohandle the risk of trades. Levy processes provide us with the appropriatetools to adequately and consistently describe all these observations, both inthe real and in the risk-neutral world.

The main aim of these lecture notes is to provide an accessible overviewof the field of Levy processes and their applications in mathematical financeto the non-specialist reader. To serve that purpose, we have avoided mostof the proofs and only sketch a number of proofs, especially when they offersome important insight to the reader. Moreover, we have put emphasis onthe intuitive understanding of the material, through several pictures andsimulations.

We begin with the definition of a Levy process and some known exam-ples. Using these as the reference point, we construct and study a Levy

jump-diffusion; despite its simple nature, it offers significant insights and anintuitive understanding of general Levy processes. We then discuss infinitelydivisible distributions and present the celebrated LevyKhintchine formula,

which links processes to distributions. The opposite way, from distributions


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0.02 0.01 0.0 0.01 0.02

0

20

40

60

80

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

to processes, is the subject of the Levy-Ito decomposition of a Levy pro-cess. The Levy measure, which is responsible for the richness of the class ofLevy processes, is studied in some detail and we use it to draw some conclu-sions about the path and moment properties of a Levy process. In the nextsection, we look into several subclasses that have attracted special atten-tion and then present some important results from semimartingale theory.A study of martingale properties of Levy processes and the Ito formula for

Levy processes follows. The change of probability measure and Girsanovstheorem are studied is some detail and we also give a complete proof in thecase of the Esscher transform. Next, we outline three ways for constructingnew Levy processes and the first part closes with an account on simulationmethods for some Levy processes.

The second part of the notes is devoted to the applications of Levy pro-cesses in mathematical finance. We describe the possible approaches in mod-eling the price process of a financial asset using Levy processes under thereal and the risk-neutral world, and give a brief account of market incom-pleteness which links the two worlds. Then, we present a primer of popularLevy models in the mathematical finance literature, listing some of theirkey properties, such as the characteristic function, moments and densities(if known). In the next section, we give an overview of three methods for pric-ing options in Levy-driven models, viz. transform, partial integro-differentialequation (PIDE) and Monte Carlo methods. Finally, we present some em-pirical results from the application of Levy processes to real market financialdata. The appendices collect some results about Poisson random variablesand processes, explain some notation and provide information and links re-garding the data sets used.

Naturally, there is a number of sources that the interested reader shouldconsult in order to deepen his knowledge and understanding of Levy pro-cesses. We mention here the books ofBertoin (1996), Sato (1999), Apple-baum (2004), Kyprianou (2006)on various aspects of Levy processes. Cont

and Tankov (2003) andSchoutens (2003)focus on the applications of Levy


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1020

3040

5060

7080

90 1 2

3 4

5 6

7 8

9 10

10

10.5

11

11.5

12

12.5

13

13.5

14

maturitydelta (%) or strike

impliedvol(%)

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

processes in finance. The books ofJacod and Shiryaev (2003) and Prot-ter (2004) are essential readings for semimartingale theory, while Shiryaev(1999) blends semimartingale theory and applications to finance in an im-pressive manner. Other interesting and inspiring sources are the papers byEberlein (2001),Cont (2001), Barndorff-Nielsen and Prause (2001),Carr et

al. (2002), Eberlein and Ozkan(2003) andEberlein (2007).

2. Definition

Let (, F, F, P) be a filtered probability space, whereF =FT and thefiltrationF = (Ft)t[0,T] satisfies the usual conditions. Let T[0, ] denotethe time horizon which, in general, can be infinite.

Definition 2.1. A cadlag, adapted, real valued stochastic process L =(Lt)0tT with L0 = 0 a.s. is called a Levy process if the following con-ditions are satisfied:

(L1): Lhas independent increments, i.e. Lt Ls is independent ofFsfor any 0s < tT.

(L2): L has stationary increments, i.e. for any 0s, tT the distri-bution ofLt+s Lt does not depend on t.

(L3): Lis stochastically continuous, i.e. for every 0tT and >0:limst P(|Lt Ls|> ) = 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 linear drift, a Brow-nian motion and a compound Poisson process is again a Levy process; itis often called a jump-diffusion process. We shall call it a Levy jump-diffusion process, since there exist jump-diffusion processes which are not

Levy processes.


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0.0 0.2 0.4 0.6 0.8 1.0

0.

0

0.

01

0.

02

0.

03

0.

04

0.

05

0.0 0.2 0.4 0.6 0.8 1.0

0.

1

0.

0

0.1

0.2

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

0.0 0.2 0.4 0.6 0.8 1.0

0.4

0.2

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8 1.0

0.1

0.0

0.1

0.2

0.3

0.4

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

3. Toy example: a Levy jump-diffusion

Assume that the process L = (Lt)0tT is a Levy jump-diffusion, i.e. aBrownian motion plus a compensated compound Poisson process. The pathsof this process can be described by

Lt = bt + Wt+ Ntk=1

Jk t(3.1)where b R, R0, W = (Wt)0tT is a standard Brownian motion,N = (Nt)0tT is a Poisson process with parameter (i.e. IE[Nt] = t)and J = (Jk)k1 is an i.i.d. sequence of random variables with probabilitydistribution F and IE[J] =


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The characteristic function ofLt is

IE

eiuLt

= IE

exp

iu

bt + Wt+Ntk=1

Jk t= exp

iubt

IE

exp

iuWt

exp

iu Ntk=1

Jk t

;

since all the sources of randomness are independent, we get

= exp

iubt

IE

exp

iuWt

IE

exp

iu

Nt

k=1Jk iut

;

taking into account that

IE[eiuWt ] = e122u2t, WtNormal(0, t)

IE[eiuPNt

k=1 Jk ] = et(IE[eiuJ1]), NtPoisson(t)

(cf. also AppendixB) we get

= exp

iubt

exp 1

2u22t

exp

t

IE[eiuJ 1] iuIE[J]= exp iubt exp

1

2

u22t exp tIE[eiuJ 1 iuJ];and because the distribution ofJ is Fwe have

= exp

iubt

exp 1

2u22t

exp

t

R

eiux 1 iuxF(dx).

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

IE

eiuLt

= exp

t

iub u22

2 +

R

(eiux 1 iux)F(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 and thatequation (3.2) is a special case of the celebrated Levy-Khintchine formula.

Remark 3.1. Note that time factorizes out, and the drift, diffusion andjumps parts are separated; moreover, the jump part factorizes to expectednumber of jumps () and distribution of jump size (F). It is only naturalto ask if these features are preserved for all Levy processes. The answer isyes for the first two questions, but jumps cannot be always separated into a

product of the form F.


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4. Infinitely divisible distributions and the Levy-Khintchineformula

There is a 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.

LetXbe a real valued random variable, denote its characteristic functionbyXand its law by PX, hence X(u) =

R

eiuxPX(dx). Let denotethe convolution of the measures and , i.e. ( )(A) =

R(A x)(dx).

Definition 4.1. The law PXof 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 suchthat

X d=X

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

(1/n)n .(4.1)

Equivalently, the lawPXof a random variable X is infinitely divisibleif forall n N there exists another law PX(1/n) of a random variable X(1/n) suchthat

PX=PX(1/n) . . . PX(1/n) ntimes

.(4.2)

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

Characterization 4.2. The law of a random variable X is infinitely divis-ible, if for all n N, there exists a random variable X(1/n), such that

X(u) = X(1/n)(u)n.(4.3)Example 4.3 (Normal distribution). Using the characterization above, wecan easily deduce that the Normal distribution is infinitely divisible. LetXNormal(, 2), then we have

X(u) = exp

iu 12

u22

= exp

n(iu

n1

2u2

2

n)

=

exp

iu

n1

2u2

2

n

n= X(1/n)(u)

n,

whereX(1/n) Normal(n , 2

n).

Example 4.4(Poisson distribution). Following the same procedure, we caneasily conclude that the Poisson distribution is also infinitely divisible. LetXPoisson(), then we have

X(u) = exp

(eiu 1)

=

exp

n

(eiu 1)n

=

X(1/n)(u)n

,

whereX(1/n)

Poisson(

n).


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Remark 4.5. Other examples of infinitely divisible distributions are thecompound Poisson distribution, the exponential, the -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. We will use the followingpreparatory result (cf.Sato 1999,Lemma 7.8).

Lemma 4.6. If(Pk)k0 is a sequence of infinitely divisible laws andPkP, thenP is also infinitely divisible.

Theorem 4.7. The lawPXof a random variableX is infinitely divisible ifand only if there exists a triplet(b,c,), withb R, c R0 and a measuresatisfying({0}) = 0 and R(1 |x|2)(dx)1}

|eiux 1|(dx)

0 as u0.


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

(u) =iub u2c

2 +

R

(eiux 1 iux1{|x|


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5. Analysis of jumps and Poisson random measures

The jump process L= (Lt)0tT associated to the Levy process L isdefined, for each 0tT, viaLt = Lt Lt,

where Lt = limst Ls. The condition of stochastic continuity of a Levyprocess yields immediately that for any Levy process L and anyfixed t >0,then Lt= 0 a.s.; hence, a Levy process has nofixed times of discontinuity.

In general, the sum of the jumps of a Levy process does not converge, inother words it is possible that

st

|Ls|= a.s.

but we always have that st

|Ls|2


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Definition 5.2. The measure defined by

(A) = IE[

L

(1, A)] = IEs1 1A(Ls())is the Levy measureof the Levy process L.

Now, using that L(t, A) is a counting measure we can define an in-tegral with respect to the Poisson random measure L. Consider a setA B(R\{0}) such that 0 / A and a function f : R R, Borel measur-able and finite on A. Then, the integral with respect to a Poisson randommeasure is defined as follows:

A

f(x)L(; t, dx) =st

f(Ls)1A(Ls()).(5.2)

Note that each A f(x)L(t, dx) is a real-valued random variable and gen-erates a cadlag stochastic process. We will denote the stochastic process by0

A f(x)

L(ds, dx) = (t

0

A f(x)

L(ds, dx))0tT.

Theorem 5.3. Consider a set A B(R\{0}) with 0 / A and a functionf : R R, Borel measurable and finite onA.A. The process(

t0

A f(x)

L(ds, dx))0tT is a compound Poisson processwith characteristic function

IE

exp

iu

t0

A

f(x)L(ds, dx)

= exp

t

A

(eiuf(x) 1)(dx)

.(5.3)

B. IffL1(A), then

IE t0

A

f(x)L(ds, dx)= t A

f(x)(dx).(5.4)

C. IffL2(A), then

Var t

0

A

f(x)L(ds, dx)= t

A|f(x)|2(dx).(5.5)

Sketch of Proof. The structure of the proof is to start with simple functionsand pass to positive measurable functions, then take limits and use domi-nated convergence; cf. Theorem 2.3.8 in Applebaum (2004).

6. The Levy-Ito decomposition

Theorem 6.1. Consider a triplet(b,c,) wherebR, cR0 and is ameasure satisfying({0}) = 0 and

R(1|x|2)(dx)


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for allu R, is defined.

Proof. See chapter 4 in Sato (1999)or chapter 2 in Kyprianou (2006).

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 (6.1)into four parts

= (1) + (2) + (3) + (4)

where

(1)(u) =iub, (2)(u) =u2c

2 ,

(3)(u) = |x|1 (eiux

1)(dx),

(4)(u) =

|x||L(4)s |>} t

1>|x|>

x(dx)

=

t0

1>|x|>

xL(4)

(dx, ds) t

1>|x|>

x(dx)

and shows that the jumps of L(4) form a Poisson process; using Theorem5.3we get that the characteristic exponent ofL(4,) is

(4,)(u) =


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1

2

3

4

5

1

2

3

4

5

Figure 7.6. 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 7.7. The density of the Levy measure of an NIG(left) and an-stable process.

whereL(ds, dx) =(dx)ds. Here L(1) is a constant drift, L(2) a Brownianmotion, L(3) a compound Poisson process and L(4) a pure jump martingale.This result is the celebrated Levy-It o decompositionof a Levy process.

7. The Levy measure, path and moment properties

The Levy measure is a measure on R that satisfies

({0}) = 0 andR

(1 |x|2)(dx)


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0.2

0.4

0.6

0.8

1

Figure 7.8. 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).

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 can be read from the Levy measure: for example, Figures7.6and7.7reveal that the compound Poisson process has a finite numberof jumps on every time interval, while the NIG and -stable processes havean infinite one; we then speak of an infinite activityLevy process.

Proposition 7.1. LetL be a Levy process with triplet(b,c,).

(1) If (R) 0 is ascaling parameter. The last parameter,

R affects the heaviness of thetails and allows us to navigate through different subclasses. For example, for

= 1 we get the hyperbolic distribution and for =12 we get the normalinverse Gaussian (NIG).

The characteristic function of the GH distribution is

GH(u) = eiu 2 2

2 (+ iu)2

2K

2 (+ iu)2K

2 2 ,while the first and second moments are

E[L1] = + 2

K+1()

K()

and

Var[L1] =2

K+1()

K() +

24

2

K+2()K()

K2+1()

K2()

,

where=

2 2.The canonical decomposition of a Levy process driven by a generalized

hyperbolic distribution (i.e. L1GH) is

Lt= tE[L1] +

t

0 Rx(L GH)(ds, dx)


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and the Levy triplet is (E[L1], 0, GH). The Levy measure of the GH distri-

bution has the following form

GH(dx) =ex

|x|

0

exp(

2y+ 2 |x|)2y(J2||(

2y ) + Y2||(

2y ))

dy+ e|x|1{0}

;hereJ and Y denote the Bessel functions of the first and second kind withindex . We refer toRaible (2000, section 2.4.1) for a fine analysis of thisLevy measure.

The GH distribution contains as special or limiting cases several knowndistributions, including the normal, exponential, gamma, variance gamma,hyperbolic and normal inverse Gaussian distributions; we refer to Eberleinand v. Hammerstein (2004) for an exhaustive survey.

16.5. Normal Inverse Gaussian. The normal inverse Gaussian distribu-tion is a special case of the GH for =12 ; it was introduced to finance inBarndorff-Nielsen (1997). The density is

fNIG(x) =

exp

2 2 + (x )K1

1 + (x )2

1 + ( x )

2,

while the characteristic function has the simplified form

NIG(u) = eiu exp(2 2)exp(

2 (+ iu)2) .

The first and second moments of the NIG distribution are

E[L1] = +

2 2 and Var[L1] = 2 2 +

2

(

2 2)3 ,

and similarly to the GH, the canonical decomposition is

Lt = tE[L1] +

t0

R

x(L NIG)(ds, dx),

where now the Levy measure has the simplified form

NIG(dx) = ex

|x|K1(|x|)dx.

The NIG is the only subclass of the GH that is closed under convolution,i.e. ifXNIG(,, 1, 1) andYNIG(,, 2, 2) andXis independentofY, then

X+ YNIG(,, 1+ 2, 1+ 2).Therefore, if we estimate the returns distribution at some time scale, then

we know it in closed form for all time scales.


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16.6. CGMY. The CGMY Levy process was introduced by Carr, Geman,Madan, and Yor (2002); another name for this process is (generalized) tem-

pered stable process (see e.g. Cont and Tankov 2003). The characteristicfunction ofLt, t[0, T] is

Lt(u) = exp

tC(Y)(M iu)Y + (G + iu)Y MY GY.The Levy measure of this process admits the representation

CGMY(dx) =CeMx

x1+Y 1{x>0}dx + C

eGx

|x|1+Y1{x0, G >0, M >0, andY 0, < < , > 0,then the density is

fMeixner(x) = 2cos 22

2(2) expx

+ ix 2 .

The characteristic function Lt, t[0, T] is

Lt(u) =

cos 2

cosh ui2

2t,

and the Levy measure of the Meixner process admits the representation

Meixner(dx) =exp

xx sinh(x) .


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The Meixner process is a pure jump Levy process with canonical decompo-sition

Lt = tE[L1] +

t0

R

x(L Meixner)(ds, dx),

and Levy triplet (E[L1], 0, Meixner).

17. Pricing European options

The aim of this section is to review the three predominant methodsfor pricing European options on assets driven by general Levy processes.Namely, we review transform methods, partial integro-differential equation(PIDE) methods and Monte Carlo methods. Of course, all these methods

can be used under certain modifications when considering more generaldriving processes as well.

The setting is as follows: we consider an asset S= (St)0tT modeled asan exponential Levy process, i.e.

St= S0exp Lt, 0tT ,(17.1)where L = (Lt)0tT has the Levy triplet (b,c,). We assume that theasset is modeled directly under a martingale measure, cf. section15.2,hencethe martingale restriction on the drift term b is in force. For simplicity, weassume thatr >0 and = 0 throughout this section.

We aim to derive the price of a European option on the asset S with

payoff function g maturing at time T, i.e. the payoff of the option is g(ST).

17.1. Transform methods. The simpler, faster and most common methodfor pricing European options on assets driven by Levy processes is to de-rive an integral representation for the option price using Fourier or Laplacetransforms. This blends perfectly with Levy processes, since the represen-tation involves the characteristic function of the random variables, which isexplicitly provided by the Levy-Khintchine formula. The resulting integralcan be computed numerically very easily and fast. The main drawback ofthis method is that exotic derivatives cannot be handled so easily.

Several authors have derived valuation formulae using Fourier or Laplacetransforms, see e.g.Carr and Madan (1999),Borovkov and Novikov (2002)

andEberlein, Glau, and Papapantoleon (2008). Here, we review the methoddeveloped by S. Raible (cf.Raible 2000, Chapter 3).

Assume that the following conditions regarding the driving process of theasset and the payoff function are in force.

(T1): Assume thatLT(z), the extended characteristic function ofLT,exists for all z C withzI1[0, 1].

(T2): Assume thatPLT, the distribution ofLT, is absolutely continu-ous w.r.t. the Lebesgue measure \ with density .

(T3): Consider an integrable, European-style, payoff function g(ST).(T4): Assume that x eRx|g(ex)| is bounded and integrable for

all R

I2

R.

(T5): Assume that I1 I2=.


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Furthermore, let Lh(z) denote the bilateral Laplace transform of a func-tion h at z

C, i.e. let

Lh(z) :=R

ezxh(x)dx.

According to arbitrage pricing, the value of an option is equal to its dis-counted expected payoff under the risk-neutral measure P. Hence, we get

CT(S, K) = erTIE[g(ST)] = e

rT

g(ST)dP

= erTR

g(S0ex)dPLT(x) = e

rT

R

g(S0ex)(x)dx

because PLT is absolutely continuous with respect to the Lebesgue measure.Define the function (x) =g(ex) and let = log S0, then

CT(S, K) = erT

R

( x)(x)dx= erT( )() =:C(17.2)

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

The idea now is to apply a Laplace transform on both sides of ( 17.2) andtake advantage of the fact thatthe 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 the

Laplace transforms to recover the option value.Applying Laplace transforms on both sides of (17.2) for C z = R+iu,RI1 I2, u R, we get that

LC(z) = erT

R

ezx( )(x)dx

= erTR

ezx(x)dx

R

ezx(x)dx

= erTL(z)L(z).

Now, inverting this Laplace transform yields the option value, i.e.

CT(S, K) = 1

2i

R+iRi

ezLC(z)dz

= 1

2

R

e(R+iu)LC(R+ iu)du

= eR

2

R

eiuerTL(R+ iu)L(R+ iu)du

= erT+R

2 ReiuL(R+ iu)LT(iR u)du.


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Here, L is the Laplace transform of the modified payoff function (x) =g(ex) and LT is provided directly from the Levy-Khintchine formula.

Below, we describe two important examples of payoff functions and theirLaplace transforms.

Example 17.1 (Call and put option). A European call option pays offg(ST) = (ST K)+, for some strike price K. The Laplace transform of itsmodified payoff function is

L(z) = K1+z

z(z+ 1)(17.3)

for z C withz= RI2= (, 1).Similarly, for a European put option that pays offg(ST) = (K ST)+,

the Laplace transform of its modified payoff function is given by (17.3) for

z C withz = RI2= (0, ).Example 17.2 (Digital option). A European digital call option pays offg(ST) = 1{ST>K}. The Laplace transform of its modified payoff function is

L(z) =Kz

z(17.4)

for z C withz= RI2= (, 0).Similarly, for a European digital put option that pays offg(ST) = 1{ST


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Now, for notational but also computational convenience, we work withthe driving process L and not the asset price process S, hence we derive a

PIDE involvingf(Lt, t) :=G(St, t), or in other words

f(Lt, t) = er(Tt)IE[g(S0e

LT)] =Vt, 0tT .(17.8)Let us denote by if the derivative offwith respect to the i-th argument,2i fthe second derivative offwith respect to thei-th argument, and so on.

Assume thatfC2,1(R[0, T]), i.e. it is twice continuously differentiablein the first argument and once continuously differentiable in the secondargument. An application of Itos formula yields:

d(ertVt) = d(ertf(Lt, t))

=rertf(Lt, t)dt + ert2f(Lt, t)dt+ ert1f(Lt, t)dLt+1

2ert21 f(Lt, t)dLct

+ ertR

f(Lt

+ z, t) f(Lt, t) 1f(Lt

, t)z

L(dz, dt)

= ert

rf(Lt, t)dt + 2f(Lt, t)dt + 1f(Lt, t)bdt

+ 1f(Lt, t)

cdWt+

R

1f(Lt, t)z(L L)(dz, dt)

+1

2

21 f(Lt, t)cdt

+R

f(Lt


, t)z

(L L)(dz, dt)

+

R

f(Lt


, t)z

(dz)dt

.

Now, the stochastic differential of the bounded variation part of the optionprice process is

ert

rf(Lt, t) + 2f(Lt, t) + 1f(Lt, t)b +12

21 f(Lt, t)c

+R

f(Lt


, t)z(dz),while the remaining parts constitute of the local martingale part.

As was already mentioned, the bounded variation part vanishes identi-cally. Hence, the price of the option satisfies the partial integro-differentialequation

0 =rf(x, t) + 2f(x, t) + 1f(x, t)b + c2

21 f(x, t)(17.9)

+

R f(x + z, t) f(x, t) 1f(x, t)z

(dz),


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for all (x, t) R (0, T), subject to the terminal conditionf(x, T) =g(ex).(17.10)

Remark 17.3.Using the martingale condition (15.8) to make the drift termexplicit, we derive an equivalent formulation of the PIDE:

0 =rf(x, t) + 2f(x, t) +

r c2

1f(x, t) +

c

221 f(x, t)

+

R

f(x + z, t) f(x, t) (ez 1)1f(x, t)

(dz),

for all (x, t) R (0, T), subject to the terminal conditionf(x, T) =g(ex).

Remark 17.4. Numerical methods for solving the above partial integro-differential equations can be found, for example, in Matache et al. (2004),inMatache et al. (2005) and inCont and Tankov (2003, Chapter 12).

17.3. Monte Carlo methods. Another method for pricing options is touse a Monte Carlo simulation. The main advantage of this method is thatcomplex and exotic derivatives can be treated easily which is very impor-tant in applications, since little is known about functionals of Levy processes.Moreover, options on several assets can also be handled easily using MonteCarlo simulations. The main drawback of Monte Carlo methods is the slowcomputational speed.

We briefly sketch the pricing of a European call option on a Levy drivenasset. The payoff of the call option with strike Kat the time of maturity Tisg(ST) = (ST K)+ and the price is provided by the discounted expectedpayoff under a risk-neutral measure, i.e.

CT(S, K) = erTIE[(ST K)+].

The crux of pricing European options with Monte Carlo methods is to sim-ulate the terminal value of asset price ST = S0exp LT see section 14forsimulation methods for Levy processes. Let STk for k = 1, . . . , N denotethe simulated values; then, the option price CT(S, K) is estimated by theaverage of the prices for the simulated asset values, that is

CT(S, K) = e

rTN

k=1(STk K)+,

and by the Law of Large Numbers we have thatCT(S, K)CT(S, K) as N .18. Empirical evidence

Levy processes provide a framework that can easily capture the empiri-cal observations both under the real world and under the risk-neutralmeasure. We provide here some indicative examples.

Under the real world measure, Levy processes are 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 ofgeneralized hyperbolic distributions (cf. section16.4). In


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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 18.11. Densities of hyperbolic (red), NIG (blue) andhyperboloid distributions (left). Comparison of the GH (red)and Normal distributions (with equal mean and variance).

0.02 0.0 0.02

0

20

40

60

0.02 0.01 0.0 0.01 0.02

0.0

2

0.0

1

0.0

0.0

1

0.0

2

empiricalquantiles

Figure 18.12. Empirical distribution and Q-Q plot ofEUR/USD daily log-returns with fitted GH (red).

Figure18.11,various densities of generalized hyperbolic distributions and acomparison of the generalized hyperbolic and normal density are plotted.

A typical example of the behavior of asset returns can be seen in Figures1.2and 18.12. The fitted normal distribution has lower peak, fatter flanksand 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 nor-mal distribution. 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 dis-tributions allows the implied volatility smiles produced by a Levy modelto accurately capture the shape of the implied volatility smiles observed inthe market. A typical volatility surface can be seen in Figure 1.3. Figure18.13exhibits the volatility smile of market data (EUR/USD) and the cal-ibrated implied volatility smile produced by the NIG distribution; clearly,

the resulting smile fits the data particularly well.


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0 20 40 ATM 60 80 100

13.5

14.5

15.5

16.5

17.5

delta (%)

impliedvolatility(%)

market prices

model prices

Figure 18.13. Implied volatilities of EUR/USD options andcalibrated NIG smile.

Appendix A. Poisson random variables and processes

Definition A.1. LetXbe a Poisson distributed random variable with pa-rameter R0. Then, for n N the probability distribution is

P(X=n) = en

n!and the first two centered moments are

E[X] = and Var[X] =.

Definition A.2. A cadlag, adapted stochastic processN = (Nt)0tT withNt : R0 N {0} is called a Poisson process if

(1) N0= 0,(2) Nt Ns is independent ofFs for any 0s < t < T,(3) Nt Ns is Poisson distributed with parameter (t s) for any 0

s < t < T.

Then, 0 is called the intensityof the Poisson process.Definition A.3. Let N be a Poisson process with parameter . We shallcall the process N= (Nt)0tT withNt :

R0

R where

Nt:= Nt t(A.1)a compensated Poisson process.

A simulated path of a Poisson and a compensated Poisson process can beseen in FigureA.14.

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

Proof. We have that

(1) the process Nis adapted to the filtration because Nis adapted (bydefinition);

(2) IE[|Nt|]


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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.14. Plots of the Poisson (left) and compensatedPoisson process.

(3) finally, let 0s < t < T, thenIE[Nt|Fs] = IE[Nt t|Fs]

= IE[Ns (Nt Ns)|Fs] (s (t s))=Ns IE[(Nt Ns)|Fs] (s (t s))=Ns s=Ns.

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

IE[eiuNt ] = exp t(eiu 1)and

IE[eiuNt ] = exp

t(eiu 1 iu).Appendix B. Compound Poisson random variables

Let N be a Poisson distributed random variable with parameter 0and J = (Jk)k1 an i.i.d. sequence of random variables with law F. Then,by conditioning on the number of jumps and using independence, we havethat the characteristic function of a compound Poisson distributed randomvariable is

IEeiuPNk=1 Jk= n0

IEeiuPNk=1 JkN=nP(N=n)=n0

IE

eiuPn

k=1 Jk

en

n!

=n0

R

eiuxF(dx)

n enn!

= exp

R(eiux 1)F(dx)

.


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Appendix C. Notation

d

= equality in law,L(X) law of the random variable Xa b= min{a, b}, a b= max{a, b}C z = + i, with , R; thenz = andz= 1A denotes the indicator of the generic eventA, i.e.

1A(x) =

1, ifxA,0, ifx /A.

Classes:Mloc local martingalesAloc processes of locally bounded variationVprocesses of finite variationGloc() functions integrable wrt the compensated random measure

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

be downloaded from

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

Appendix E. Paul Levy

Processes with independent and stationary increments are named Levyprocessesafter the French mathematician Paul Levy (1886-1971), who madethe connection with infinitely divisible laws, characterized their distributions(Levy-Khintchine formula) and described their path structure (Levy-Ito de-composition). Paul Levy is one of the founding fathers of the theory of

stochastic processesand made major contributions to the field of probabil-ity theory. Among others, Paul Levy contributed to the study of Gaussianvariables and processes, the law of large numbers, the central limit theorem,stable laws, infinitely divisible laws and pioneered the study of processeswith 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).


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Acknowledgments

A large part of these notes was written while I was a Ph.D. student atthe University of Freiburg; I am grateful to Ernst Eberlein for various inter-esting and illuminating discussions, and for the opportunity to present thismaterial at several occasions. I am grateful to the various readers for theircomments, corrections and suggestions. Financial support from the DeutscheForschungsgemeinschaft (DFG, Eb 66/9-2) and the Austrian Science Fund(FWF grant Y328, START Prize) is gratefully acknowledged.

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Financial and Actuarial Mathematics, Vienna University of Technology,Wiedner Hauptstrasse 8/105, 1040 Vienna, Austria

E-mail address: [email protected]: http://www.fam.tuwien.ac.at/~papapan

an introduction to l evy processes with applications in finance

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