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1. INTRODUCTION AND SUMMARY

The benchmark model for security prices is geometric Brownian motion. A relatively simple yetpowerful generalization is the class of all continuous-time process where all non-overlappingincrements log( / )t t t t X X S S =2 1 2 1 are independent random variables with stationarydistributions. This set of processes tXare the Lvy processes or processes with stationaryindependent increments. [see the monographs by Sato (1996) and Bertoin (1999)]. They consistof a combination of a linear drift, a Brownian motion, and an independent jump process Here weprovide a solution to the associated European-style option valuation problem.

These models help explain some, but not all, of the well-documented deviations from thebenchmark model. Lvy process models can be good fits to daily stock return distributions which

are characterized by wide tails and excess kurtosis. [For example, see Eberlein, Keller, andPrause (1998)]. The so-called jump-diffusions (which are members of the class where the jumpsare compound Poisson processes) offer a compelling explanation for the relatively steep smilesobserved in expiring index options like the S&P500.1 It should be noted that the independentincrement assumption is counter-factual in some respects.2 Nevertheless, because of theirsuccesses, flexibility, and analytic tractability, continued financial applications are likely. For arecent survey of applications, in finance and elsewhere, see Barndorff-Nielsen, Mikosch, andResnick (2001).

Jump-diffusion models are distinguished by their jump amplitude distributions. Two examplesare Merton (1976) who solved for option prices with log-normally distributed jumps, and Kou(2000), who did the same for a double exponential distribution. To obtain an option formula, theauthors relied upon particular properties of those distributions. Mertons solution relies upon aproduct of lognormal variates being lognormally distributed. Kous derivation stresses theimportance of the memoryless property of the exponential distribution. Our results make clearthat, in fact, no special properties are needed: we obtain an option formula for any jumpdistribution.

In addition, many of the original results for these models are very complicated. Special functionsand complicated expressions are required when option formulas are given in S-space or stockprice space.3 However, more recently, it has been recognized that option values for both the Lvy

process problem and related proportional returns problems are much simpler in Fourier space4.For example, Carr and Madan (1999) have derived relatively simple formulas for call options onLvy processes, working in Fourier space. Bakshi and Madan (2000), although not workingdirectly on Lvy processes, derive an applicable Black-Scholes-style formula for call optionsusing characteristic functions and some more complicated formulas for general claims. Raible

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(2000) obtained an option formula which is very similar to our general result (see below).However, he presents it as a mixture of Fourier and two-sided Laplace transforms. Because we

use the generalized Fourier transform consistently, our strip condition is more transparent. InLewis (2000), we obtained related inversion formulas for options under stochastic volatility, aproportional returns problem. Here we generate the value of the general claim under a Lvyprocesses as an integral of Fourier transforms. Once you have our main result, the residuecalculus provides a standard approach to variations. For example, we show that the Black-Scholes style formula is simply obtained by moving integration contours.

The formula can be easily explained with a little background. Assume that under a pricingmeasure a stock price evolves as exp ( )T TS S r q T X = +0 , where r q is the cost of carry,Tis the expiration time for some option, and TX is some Lvy process satisfying

exp( )TX= 1E

. For Lvy processes, the important role of the characteristic functions( ) exp( )t tu iuX = E , u R is well-known. Like all characteristic functions, they are Fouriertransforms of a density and typically have an analytic extension (a generalizedFouriertransform) u z C , regular in some strip XS parallel to the real z-axis.

Somewhat less appreciated, yet key to our approach, is the recognition that option payofffunctions also have simple generalized Fourier transforms. Using the variable logTx S= , thesetransforms are ( ) exp( ) ( )w z izx w x dx= , where ( )w x is the payoff function

5. For example, ifKis a strike price, the call option payoff is ( ) ( )xw x e K += and so, by a simple integration,

( ) /( ) izw z K z iz += 2 , Im z> 1 . Note that ifzwere real, this regular Fourier transformswould not exist. As shown in Lewis (2000), payoff transforms ( )w zfor typical claims exist andare regular in their own strips wS in the complex z-plane, just like characteristic functions. Then,the initial option value ( )V S0 is given by simply integrating the (conjugate) product of these twotransforms times a phase factor. To do the integration legally, one has to keep zwithin theintersection of the two strips of regularity wS and XS ( XS is the reflection ofXS across thereal z-axis). With that synopsis, the formula is

(1.1) Option values: ( ) ( ) ( )rT i

izYT

i

eV S e z w z dz

+

= 0 2 ,

where ln ( )Y S r q T = + 0 , z u i= + , V w Xz = S S S.

Although very compact, (1.1) contains (as special cases), the Black-Scholes model, Mertonsjump-diffusion model, Kous jump-diffusion model, and all of the pure jump models that havebeen introduced. Indeed, it applies to the entire class of exponential Lvy processes which have

exp( )TX < E . The proof of it and our use of the residue calculus to obtain variations is ourprimary contribution in this paper.

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Any path-independent European-style payoff, plain vanilla or exotic, may be valued. The

expression (1.1) is obviously just a single integral. Its real-valued and readily evaluated6

.Typically the integrand is a short expression, often with only elementary functions. See Tables2.1 and 3.1 for examples of ( )Tz and ( )w zrespectively.

Consider the call option again. Since ( ) /( ) zw z K z iz += , the integrand in (1.1) is aregular function ofzin the largerstrip XS, except for simple poles at z= 0 and z i= . Usingthe residue calculus, you can move the integration contour around in XS. Of course, you pick upresidue contributions if you move contours across (or along!) Im ,z= 0 1 . We do this and obtaina number of variations on (1.1) including the Black-Scholes style formula that we mentionedearlier. Finally, it turns out that exp( ) ( )TizY z is the (conjugate of the) Fourier transform of

the transition density for log S0 to reach log TSafter the elapse ofT. This allows us to interpret(1.1) simply as a Parseval identity. In fact, the proof of that clarifies what space of payofffunctions are handled by our theory and which types are excluded.In the next section, we review some fundamental aspects of Lvy processes, their applications tofinance, and their analytic characteristic functions. This material is almost entirely standard andexperts in those topics could well skim for our notation and then jump to the proof of (1.1) inSection 3.

2. BACKGROUND

2.1 The Framework

We consider a marketplace in which a stock price or security price tS 0 follows anexponential Lvy process tX(defined below) on a continuous-time probability space ( , , ) FQ .We stress that Qis a fixed martingale pricing measure. The pricing measure has the same nullsets as an objective or statistical measure P, and the two measures are related by anunspecified Girsanov change-of-measure transformation7. However, Pplays no direct role in ourdiscussion all expectations and stochastic processes are defined relative to Q.

A stock buyer receives a continuous dividend yield q; she could finance her purchase at theriskless rate of interest r. The net financing cost (the cost of carry) is r q . It is convenient toexplicitly remove this constant we then investigate option valuation where

[ ]exp ( )t tS S r q t X = +0 underQ, where tXis a Lvy process. Following Sato (1999), weadopt the following definition:

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Definition (Lvy Process) An adapted real-valued process tX , with X=0 0 , is called a Lvy

process if:

(i) it has independent increments; that is, for any choice ofn 1 and nt t t < < 0 , { }Pr | |s t sX X + > 0 ast 0 .(iv) as a function of t, it is right-continuous with left limits .

Processes that satisfy (i) and (ii) are called processes with stationary (or time-homogeneous)independent increments (PIIS). Some authors (e.g. Bertoin) simply define a Lvy process to be a

PIIS processwith X=

00

. Such processes can be thought of as analogs of random walks incontinuous time.

To prevent an arbitrage opportunity, the stock price (net of the cost of carry) must be a localmartingale underQ. In fact, we maintain throughout the stronger assumption: tS, net of thecarry, is a Q-martingale. That is, [ ] exp[( ) ]tS S r q t = 0E or [exp ]tX= 1E . For those Lvyprocesses with [exp ]tX< E , this normalization can be achieved by a drift adjustment.

Types of Lvy processes. In general, Lvy processes are a combination of a linear drift, aBrownian motion, and a jump process. When jumps occur, tXjumps by \ { }tX x = 0R ,(the notation means that we exclude zero as a possible jump amplitude x). Now consider anyclosed interval A R that does not contain the origin. Then, the cumulative number of jumps inthe time interval [0,t] with a size that belongs to A , call it tN, is a random variable which isalso a measure. This integer-valued Q-random measure is usually written ([ , ], )tN t A= 0 .With A fixed, then tN has a Poisson distribution with a mean value ( )At x dx . Here we haveintroduced the Lvy measure ( )x dx , which measures the relative occurrence of different jumpamplitudes [see Sato, 2001, Theorem 1.4].

Two types of Lvy processes with a jump component can be distinguished. In type I (the Poissoncase), we have ( )x dx < R . Then, we can write ( ) ( )x f x = , where ( )x dx = R is thePoisson intensity (the mean jump arrival rate), and ( ) ( )f x dx dF x= , where ( )F x is a

cumulative probability distribution8. We will also call this case the jump-diffusion case. Anexample of this type is Mertons (1976) jump-diffusion model, where x is normally distributed:

( ) ( ) exp[ ( ) / ] /( )x f x x = = 1 22 2 22 2 .

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In the alternative type II case, ( )x dx = R and no overall Poisson intensity can be defined. Asimple example is the Lvy stable process: ( ) / | |x c x += , < 0 and x < 0 . Carr and Wu (2000) have proposed a special case of thismodel for stocks. Notice that in the type II example, the source of the divergence is the failure of( )x to be integrable at the origin: there are too many small jumps. The divergence at the origin

is always the source of the integrability failure of ( )x in type II models the Lvy-Khintchinerepresentation (see below) guarantees that ( )x is always integrable at large | |x .

A general integral representation. One can take a differential of ([ , ], )tN t A= 0 , writing( , )tdN dt dx= and use these differential random measures in an integration theory [see Jacod

and Shiryaev (1987)]. With that theory, one can decompose any Lvy process tXinto theform9:

(2.1) ( )\{ }

( , ) ( ) ( )tt h tX t B x ds dx h x x ds dx = + + 0 0R ,

where tB is a Q-Brownian motion, h and are constants, and ( )h x is a truncation function, tobe explained. The Brownian motion and the jump process are independent. This representation isunique in the sense that, once the truncation function is fixed, then there is only one set ofcharacteristics { , , ( )}h x for a given { : }tX t 0 . If the truncation function is changed, thedrift h changes but the pair { , ( )}x is invariant. The purpose of the truncation function is tomake the integral in (2.1) exist near the origin, where the integrand must be taken as a whole.Such a function is only necessary in some Type II models where ( )x diverges as ( / | | )O x +21 ,

<

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Definition (Characteristic Function). For z C (z a complex number) and Ima z b< < , wecall [ ]( ) exp( )t tz iz X = E the characteristic function of the process tX .

Remark. Let ( )tp x be the transition probability density for a Lvy process to reach tX x= afterthe elapse of time t. For Ima z b< < , the characteristic function of the process is identical tothe characteristic function of this transition density, which is also the generalized Fouriertransform of the transition density10.

[ ]( ) ( ) exp( ) ( ) ,t t tz p x izx p x dx = R

F Ima z b< < .Definition (Infinitely Divisible Characteristic Functions) . A characteristic function ( )tz is saidto be infinitely divisible, if for every positive integer n, it is the nth power of some characteristic

function.

The characteristic function of Lvy processes are infinitely divisible; this is a simpleconsequence of the PIIS properties.11

THEOREM(Lvy-Khintchine Representation). If ( )Tz , Ima z b< < , is an infinitely divisiblecharacteristic function, then it has the representation

(2.2) { }\{ }( ) exp ( ) ( )izxT hz iz T z T T e izh x x dx = + 2 212 0 1R ,

where min( , ) ( )x x dx < 2

1R .

PROOF: For a proof (when zis real), we refer to Sato (1999, Theorem 8.1). For an extension tocomplex z, we refer to Lukacs Theorem below. However, just proceeding formally, its easy tosee how the representation (2.2) follows from the representation (2.1). We will only take thesimplest case where the deterministic part of the integral in (2.1) exists on its own. Using thatassumption, and the independence of the Brownian motion and the jump process, we haveimmediately from (2.1):

{ }\{ }[exp( )] exp ( ) ( )T hiz X iz T izT h x x ds = 0RE

[ ] ( )\{ }exp( ) exp ( , )T

Tiz B iz x dt dx 0 0RE E

Now it is well-known that [ ] ( )exp( ) expTiz B z T = 2 212E . It is also well-known12 that

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( ) ( )\{ } \{ }{ }exp ( , ) exp ( ) ( )t t tT

izx izx

t TN Niz x dt dx e T e x dx

<

= =

0 0 0

0

1 1R R

E E .

Combining these results yields (2.2).

The Lvy-Khintchine representation has the form ( ) exp ( )Tz T z = , where ( )z is calledthe characteristic exponent13. The normalization ( )Tz = =0 1 and the martingale identity

( )Tz i = = 1 imply that ( ) ( )i = =0 014. Since, for a sensible stock market model,

( )tz must exist at both z= 0 and z i= , it would be helpful if it existed for all zinbetween. Indeed, all the examples in Table 2.1 exist forzwithin a horizontal strip

{ : Im }z a z b= < . Analyticity in stripsis typical, based on this theorem:

THEOREM(Lukacs, 1970, Theorem 7.1.1): If a characteristic function ( )z is regular15 in theneighborhood of z= 0 , then it is also regular in a horizontal strip and can be represented in

this strip by a Fourier integral. This strip is either the whole z-plane, or it has one or two

horizontal boundary lines. The purely imaginary points on the boundary of the strip of regularity

(if this strip is not the whole plane) are singular points of( )z .

Remarks. Because of the representation ( ) exp ( )tz t z = , singularities of ( )tz aresingularities of ( )z . Hence, an immediate corollary of Lukacs theorem is that the strips ofregularity for the analytic characteristic functions of Lvy processes are time-independent16.Clearly, a goodLvy process (good for the purpose of building a stock price model), has an

analytic characteristic function regular within a strip: Ima z b , where a 1 , b 0 . If aparticular Lvy process is a good one, say at t= 1 , then it is a good one for all t. Our optionvaluation formula only applies to good Lvy processes.

Examples. Example of characteristic functions for Lvy processes that have been proposed forthe stock market are shown in Table 2.1. [Three entries are adapted from Raible (2000)]. Foreach process, there is a constant drift parameter , determined by solving ( )t i = 1 . We havealready mentioned the first two models in the table, both of which have Brownian motioncomponents and are type I or Poisson types.

The remaining (pure jump) models are all type II. The third table entry is the Variance Gammaprocess. The option value was obtained by Madan, Carr, and Chang (1998). The process is builtup by sampling Brownian motion with drift at random times, time increments which themselvesare described by another Lvy process. Clearly, the sampled process is a pure jump process.

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The next table entry is the Normal Inverse Gaussian (NIG) process, another pure jump processapplied to stock returns by Barndorff-Nielsen (1998). There are 4 real parameters: ( , , , )

where, roughly speaking, and are shape parameters (steepness, tail decay), and and are drift and scale parameters, respectively. [Also see Lillestl, (1998)].

Following is the Generalized Hyperbolic process. This pure jump Lvy process incorporates theVG and NIG process as special cases, as well as another special case just called the hyperbolicprocess. There are 5 real parameters: ( , , , , ) where the first 4 have a similar interpretationas before and is an additional shape parameter. [For surveys, see Bibby and Srensen (2001)and the dissertation of Prause (1999)]

The last entry is the (maximally) skewed stable process of Carr and Wu (2000). The general

stable Lvy process has four parameters ( , , , ) , where ( , ] 0 2

is the index,[ , ] 1 1 is a skewness, R is a drift, and 0 is a scale parameter. The transitionprobability distribution ( )t tp X x= typically has a power-law decay as x when < 2and so [exp( )]tXE does not exist. However, Carr and Wu have noted that for= 1 , whichthey term maximum skewness, then the decay is more rapid, [exp( )]tX < E , and thecharacteristic function is given by the entry in the table. Note that (unless , = 1 2 ) thecharacteristic function has a branch point singularity at z= 0 and we need a branch cut. [Theauthors estimated . . = 1 61 01 , based upon daily option quotes for S&P500 index option overa year ending in May, 2000. They generally suggest that ( , ) 1 2 for stock prices]. For a stockprice model, we want the characteristic function to be regular for at least Im z <

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(2.3) [ ]( ) ( ) ( , ) ( )xt tt

dSr q dt dB e dt dx x dt dx

S

= + + 1

R

,

where ( )x integrates ( )xe 1 . Under (2.3), tS(net of the carry) is clearly a local martingale.For every type I model, ( ) ( )x f x = . For example, in Kous double exponential model,

( )f x = exp | | / /( )x 2 , <

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The upper limit x = in (3.2) does not exist unless Im z > 1 . Applying this restriction, then(3.2) is well-defined, in fact, regular in that strip:

(3.3) Call option: ( ) izKw zz iz

+

=

1

2 , Im z> 1

This result is typical: option payoffs have Fourier transforms (w.r.t. log TS) as long as we admita complex-valued transform variable. Then, ( )w zexists forIm zrestricted to an interval; i.e.,

wz S, just like a characteristic function. We can go through the same exercise for variousstandard payoff functions and see what restrictions are necessary for their Fourier transforms toexist. The results are given in Table 3.1 below.

So by introducing generalized transforms, we are able to handle standard payoffs which areunbounded at x = (the call), or constant at x = (the put) and which have no regular

Fourier transform. All of the typical payoff functions that one might encounter in practice arehandled. But, not every mathematically possible payoff function has a generalized Fouriertransform.

Definition. We say a function ( )f x is Fourier integrable in a strip if there exists a pair of

real numbers a and b, where a b < , such that the generalized Fourier transform[ ] ( ) ( )f z f x= F exists and is regular for z u i v= + , u R , and ( , )a b .

Our valuation theorem below will apply to payoff functions which are (i) Fourier integrable in a

strip, and (ii) bounded for| |x 0 . Similarly, as x , we have[ ]( )exp ( )w x b x 0 for every > 0 . For example, for a call option with strikeK, the

requirement is that ( )x xe K e + 0 , as x + . This condition again implies that( , ) 1 , as we found before. Similarly, for the put option, the requirement is that

( )x xK e e + 0 , as x . This condition implies that ( , ) 0 . Note thatexponential growth in | |x means power law behavior in TSas ,TS 0 .

Property (ii) arises from a technical requirement (see below) that ( ) exp( )w x x be bounded.We already know this term is bounded at (since its zero). To keep ( )exp( )w x xbounded everywhere, we also need ( )w x bounded for | |x < . This is a very mild restriction,in the sense that it only excludes payoff functions that would be unlikely to be offered in the

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marketplace. For example, consider a payoff that behaves like | | | log( / ) |x x S K =2 20 asS K , where K< < 0 . While this payoff is integrable with respect to x nearx x= 0 , its

not bounded and so excluded. In practice, such a payoff would be a pretty dangerous offeringfor the seller.

Generalized Fourier transforms are inverted by integrating along a straight line in the complex z-plane, parallel to the real axis, with zwithin the strip of regularity.20 For example, fixing

Im z= , then the payoff functions are given by:

( ) ( )i

izx

iw x e w z dz

+

= 12 , wz S.

If ( )f x is a complex-valued function of a real variable x, then by f L , wemean that| ( ) |f x dx

< , where | |f is the modulus off . With that notation, we need the following

Parseval-style identity adapted from Titchmarsh (1975), and using our ( )2 normalization:

THEOREM 3.1 (Titchmarsh, Theorem 39): Let both ( ) , ( )f x g x L 1 and one of them is bounded.Assume that *( ) ( )f x y g x dx is continuous aty = 0 . In addition, assume that, with u R ,the Fourier transforms [ ] ( ) ( )f u f x= F and [ ]( ) ( )g u g x= F exist. Then,(3.4) * *( ) ( ) ( ) ( )f x g x dx f u g u du

= 12 .

We need a notation for a reflected strip. If ( , )a b and u is any real number, then Sconsists

of all z u i= + and*

Sconsists of all z u i= .

THEOREM 3.2.(Option Valuation). Let ( )V S0 be the current price of a European-style optionwith a payoff function ( )w x 0 , where log Tx S= . Assume that( )w x is Fourier integrable ina strip and bounded for | |x < , with transform ( )w z , wz S. Let

[ ]exp ( )t tS S r q t X = +0 , where tX is a Lvy process and [ ]exp tX is a martingale. Assumethat TX has the analytic characteristic function ( )Tz , regular in the strip

{ : ( , )}X z u i a b = = + S , where a < 1 and b > 0 . Then, if V w XS S S is notempty, the option value is given by

(3.5) ( ) ( ) ( )rT i

izYTieV S e z w z dz

+

= 0 2 , where Im z= , *V w Xz = S S S,and ln ( )Y S r q T = + 0 . Moreover, VSis not empty when the payoff is a call or put option.

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PROOF: From martingale pricing, since the payoff function is (ln )Tw S , then

(3.6) [ ]( ) (ln )rT TV S e w S =0 E ( )rT i izTi

e S w z dz

+

= 2 E , Im z= , wz S

[ ]{ }exp ln ( ) exp( ) ( )rT i

Ti

e iz S r q T izX w z dz

+

= + 02 E .

In the first line we just inserted the transform representation for (ln )Tw S . In the second line, weinserted [ ]exp ( )T TS S r q T X = +0 . Next, we bring the expectation inside the integral, whichrequires an exchange of integration order. This exchange is just made formally here; we validateit by Theorem (3.4) below, an alternative proof of (3.5). Certainly a necessary condition for theexchange to be valid is that [ ]exp( ) ( )TizX z = E exists. Now ( )T z exists if *Xz S, but

zis already restricted to wz S. So the whole integrand exists and is regular if*

V w Xz = S S S. Substituting ln ( )Y S r q T = + 0 , we have (3.5). Now* { : ( , )}X u i = + S , where < 0 and > 1 . So *XS intersects both Im z< 0 and

Im z> 1 , which shows that VSis not empty for puts or calls

Next, an alternative proof will show that (3.5) is a consequence of Theorem 3.1. To accomplishthat, first let lns S=0 0 and lnT Ts S= and consider the distribution function for the log-stockprice to reach a terminal value after the elapse ofT. Define both the distribution function and itsdensity with

( , ) Pr{ | } ( , )

x

T T TQ x s s x s q s d = < = 0 0 0 .

LEMMA3.3. Equation (3.5) is true if the following two integrals are equal:

(3.7) ( , ) ( ) ( , ) ( ) T T T T T q x s w x dx q u i s w u i du

= + 0 012 ,

*w Xu i+ S S.

PROOF: Recall from the definition that ( ) exp( ) ( )T Tz izx p x dx = R , where ( )Tp x is thetransition density for the Lvy process. Introduce a distribution function for that:

( ) Pr{ } ( )x

T T TP x X x p d

= < = .

Now Pr{ | } Pr{ ( ) | }T T Ts x s s r q T X x s X x Y < = + + < = < 0 0 0

recalling that ln ( )Y S r q T = + 0 . In other words, ( , ) ( )T TQ x s P x Y = 0 and so bydifferentiating w.r.t. x, we have ( , ) ( )T Tq x s p x Y = 0 . By taking the Fourier transform of bothsides of this last identity, we have ( , ) ( )exp( ) T Tq z s p z izY =0 forzin some strip, call it qS. But,

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of course, ( ) ( )T Tp z z , so ( )exp( ) ( , )T Tz izY q z s = 0 and the strip is q X=S S. Takingz z , we have ( ) exp( ) ( , )T Tz izY q z s = 0 , where Xz S. Now the first expression on

the right-hand-side of (3.6) may also be written as the (discounted) integral of the transitiondensity and the payoff function. This shows that (3.5) is equivalent to the statement:

( ) ( , ) ( ) ( , ) ( ) rT i

rTT T T T T

i

eV S e q x s w x dx q z s w z dz

+

= = 0 0 02 ,

*w Xz S S.

Letting z u i= + in this last equation shows that (3.5) is true if (3.7) is true.

THEOREM 3.4 Under the assumptions of Theorem 3.2, equation (3.7) is true; hence, so is (3.5)

PROOF: Let u be real. Also, let 0 be any fixed real number, such that*

w Xu i+ 0 S S. Also

define ( ) ( , ) exp( )Tf x q x s x= 0 0 and ( ) ( )exp( )g x w x x= 0 , wheres0 is fixed and need notbe displayed. Note that ifz u iv= + 0 , then ( , ) exp( ) ( , ) *( )T Tq z s iux v x q x s dx f u = + =0 00and ( ) exp( ) ( ) ( ) w z iux v x w x dx g u= =0 . Thus, proving (3.7) valid is equivalent to provingthat, regardless of the choice for0 :

(3.8) *( ) ( ) ( ) ( )f x g x dx f u g u du

= 12 ,

*w Xu i+ 0 S S

We showed above that ( , )Tq x s0 is Fourier integrable in XS. That is, ( , )exp( )Tq x s izx dx

0( , )exp( )Tq x s iux x dx

= 0 exists for all Xz u i= + S. In particular, chooseu = 0 and = 0 . Obviously any real part is a valid choice. And = 0 is a valid choice because if0

is the imaginary part of some number in XS (which it is), then 0 is the imaginary part ofsome number in XS. With that choice, we have shown that ( , )exp( )Tq x s x dx

00

( )f x dx= exists. Since ( )f x is a non-negative real, then ( ) | ( )|f x dx f x dx =

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Contour variations. The general formula (3.5) , with wS taken from Table 3.1, provides astarting point formula that has many variations. The variations are obtained by the use of the

residue calculus. We stress that we maintain the assumptions of the theorem, which means that{ : Im }X z z = S , where < 0 and> 1 . We illustrate with the call option. Then,using the Table 3.1 entries, we see that call option price ( , , )C S K Tis given by

(3.9) ( , , ) ( )rT i

izkT

i

Ke dz C S K T e z z iz

+

=

1

122

, ( , ) 1 1

using ( )log ( )Sk r q T K

= + .

Since > 1 , the open interval ( , )1 is not empty and the integral exists. The phase factor usesthe dimensionless moneyness k, which is also a natural moneyness measure for the implied

volatility smile. With that variable, at-the-money means that the forward stock price (the pricefor delivery at expiration) is equal to the strike.

The integrand in (3.9) is regular throughout *XS, except for simple poles at z= 0 and z i= .The pole at z= 0 has a residue /( )rTKe i 2 and the pole at z i= has a residue /( )qTSe i 2 .Lets move the integration contour to ( , )2 0 1 ; then by the residue theorem, the call optionvalue must also equal the integral along Im z = 2 minus i2 times the residue at z i= . Thatgives us a first alternative formula

(3.10) ( , , ) rT i

qT izk T

i

Ke dz C S K T Se e z z iz

+

=

2

2 22

, 2 0 1

This is actually our preferred integration formula for the call option21. For example, with/=2 1 2 , which is symmetrically located between the two poles, this last formula becomes

(3.11) ( )( ) /( , , ) ReqT r q T iuk iT duC S K T Se SKe e uu

+ = + 22

120

4

1 .

Next, we move the contour from (3.9) to ( , ) 3 0 . Then you pick up both poles. Moreover,the integral along Im z = 3 is also the put option formula ( , , )P S K T, since Im z< 0 (seeTable 3.1). The net result is simply the put/call parity relation, qT rT C P Se Ke = + , which

is our second alternative formula.

Finally, we can move the contours to exactly Im z= 1 and Im z= 0 . Then the integrals becomeprincipal value22 integrals and you pick up one-half of the residues. For example, start with theintegral in (3.9) written in the form

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( )( )

rT iizk

T

i

Ke i ie z dz I I

z z i

+

= +

1

1

1 2

2

Now move I1 to exactly Im z= 0 ; the residue theorem combined with letting z u= yields

( )( ) ( )rT rT

iukT

Ke i Ke iI i e u duu

= + 11 2 2 2 2 P ,

where Pdenotes the Cauchy principle value and we have taken u z= . Similarly, moveI2 toexactly Im z= 1 ; the residue theorem yields

( )( ) ( )qT rT i

izkT

i

Se i Ke iI i e z dz z i

+

= +

12 2

22 2

P .

Now let z u i= + , and we have ( )( ) ( )

qT qT iuk

TSe i Se iI i e u i du

u

= 12 2 2 2 2 P .

The integrals are real and may be simplified by taking the real part. Putting the results togetheryields the call option price in the Black-Scholes form:

(3.12) ( , , ) qT rT C S K T Se Ke = 1 2 ,

where( )

Reiuk

Te u i duiu

= +

10

1 1

2,

( )Re

iukTe u du

iu

= +

20

1 1

2.

The first term /q

e C S

= 1 is also the delta. The second term Pr( )TS K = >2 . Theseintegrands are integrable as u 0 because ( )u is an analytic characteristic function in aneighborhood ofu = 0 and u i= . For example, nearu = 0 , ( ) ( )u u +1 0 , where| ( ) | < 0 because of analyticity23. Hence, the integrand in 2 tends toRe / ( ) Im ( )i u i k k + = +0 0 as u 0 , which is finite. Because of the martingaleidentity ( )i = 1 , the 1 integrand is also finite as u 0 . For numerical work (3.11) is moreefficient: in the Black-Scholes form (3.12) not only are there two integrations, but the integrandfalls off more slowly then (3.11) by a factor ofu.

As we have shown, the residue calculus provides a general mechanism for obtaining a number of

variation on the basic formula, most of which have been obtained before. For example, (3.9) wasderived by Carr and Madan (1999) in a specialized attack on the call option. The Black-Scholesform (3.12) was obtained in a more general setting by Bakshi and Madan (2000). A formula ofthe same style as (3.10) was obtained by Lewis (2000) in a stochastic volatility setting.

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4. CONCLUSIONS

The generalization of the Black-Scholes theory to the martingale pricing theory began acompetition in mathematical finance between the PDE approach and the probabilistic approachto solving certain problems. For the simple ones, its a tie because both methods usually work ina few steps. But for some complicated problems, one method seems to win out24. For optionvaluation under general Lvy processes, the probabilistic approach, in my opinion, is the clearvictor. One can introduce PDEs, but to get to the same results is a long trek around I know thisbecause, in fact, I originally obtained a version of (3.5) from a PDE.

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Table 2.1Characteristic Functions for Lvy Processes in Stock Price Models

Lvy Process Characteristic Function: [ ]T Ti XE=(z) exp( z )Strip of

Regularity XS

Lognormal Jump-diffusion ( ){ }

/exp iz ziz T z T T e + 2 2

2 2 21

21

Entire z-plane

DoubleExponential

Jump-diffusionexp iziz T z T T e

z

+ +

22 21

2 22

11

1

Im z

<

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Table 3.1Generalized Fourier Transforms for Various Financial Claims

Financial

Claim(Option)

Payoff

Function:

w x( )

Payoff

Transform:

[ ]w z w x ( ) ( )= F

Strip

of

regularity wS

Call option ( )xe K+

izK

z iz

+

1

2

Im z> 1

Put option

( )x

K e

+

izK

z iz

+

1

2

Im z< 0

Covered call or

cash-secured put( )min ,xe K izK

z iz

+

1

2

Im z<

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References

Aase, Knut K.(1988): Contingent Claims Valuation when the Security Price is a Combination of

an Ito Process and a Random Point Process, Stochastic Processes and their Applications, 28,185-220.

Bakshi, Gurdip and Dilip Madan (2000): Spanning and Derivative-security Valuation, J. ofFinancial Economics, 55, 205-238.

Barndorff-Nielsen, O.E., T Mikosch, and S. I. Resnick, eds (2001): Levy Processes: Theory andApplications, Birkhuser, Boston.

Barndorff-Nielsen, O.E (1998): Processes of Normal Inverse Gaussian Type, Finance andStochastics, 2, 41-68.

Bates, David S. (1988): Pricing options on jump-diffusion processes. Working paper 37-88,Univ. of Pennsylvania, Rodney L. White Center.

Bates, David S. (1991): The Crash of 87: Was it Expected? The Evidence from Options Markets ,J. Finance 46, July, 1009-1044.

Bertoin, Jean (1996): Lvy Processes, Cambridge University Press.

Bibby, Bo Martin and Michael Srensen (2001): Hyperbolic Processes in Finance, manuscript,Institute of Mathematical Sciences, Univ. of Copenhagen.

Breiman, Leo (1992), Probability, SIAM, Philadelphia.

Bremaud, Pierre (1974): The Martingale Theory of Point Processes over the Real Half LineAdmitting an Intensity. Control Theory, Numerical Methods and Computer System Modelling.Series: Lecture Notes in Economics and Mathematical Systems, Vol. 107, Springer-Verlag,Berlin, 519-542.

Carr, Peter and Dilip B. Madan (1999): Option Valuation using the Fast Fourier Transform, J.Computational Finance, 2, No.4, Summer, 61-73.

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Carr, Peter and Liuren Wu (2000): The Finite Moment Logstable Process and Option Pricing,manuscript, Feb. 21, 2000.

Colwell, David B. and Robert J. Elliott (1993): Discontinuous Asset Price and Non-AttainableContingent Claims, Math. Finance, 3, (July), 295-308.

Eberlein, Ernst, Ulrich Keller, and Karsten Prause (1998): New Insights into Smile, Mispricing,and Value at Risk: the Hyperbolic Model, J. Business, 71, 371-405.

Kou, S.G. (2000): A Jump Diffusion Model for Option Pricing with Three Properties:Leptokurtic Feature, Volatility Smile, and Analytical Tractability, preprint, ColumbiaUniversity, Feb. 2000.

Lewis, Alan L. (1998): Applications of Eigenfunction Expansions in Continuous-Time Finance,Math. Finance, 8, No.4, October 1998, 349-383

Lewis, Alan L. (2000): Option Valuation under Stochastic Volatility, Finance Press, NewportBeach, California.

Lillestl, Jostein (1998). Fat and Skew: Can NIG Cure? On the Prospects of Using the NormalInverse Gaussian Distribution in Finance, June 15, 1998 manuscript, The Norwegian School ofEconomics and Business Administration.

Madan, Dilip B., Peter Carr, and Eric C. Chang (1998): The Variance Gamma Process andOption Pricing, manuscript, University of Maryland, June 1998 (forthcoming, European FinanceReview).

Merton, R.C.(1976): Option Pricing When Underlying Stock Returns are Discontinuous, J. ofFinancial Economics, 3, Jan-Mar, 125-144. Reprinted as Ch. 9 in Continuous-Time Finance.Basil Blackwell, Cambridge Mass (1990).

Naik, Vasanttilak and Moon Lee (1990): General Equilibrium Pricing of Options on the MarketPortfolio with Discontinuous Returns, The Review of Financial Studies, 3, number 4, 493-521.

Prause, Karsten (1999): The Generalized Hyperbolic Model: Estimation, Financial Derivatives,and Risk Measures, Dissertation, Faculty of Mathematics, Univ. of Freiburg, Germany

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Raible, Sebastian (2000): Lvy Processes in Finance: Theory, Numerics, and Empirical Facts,Dissertation, Faculty of Mathematics, Univ. of Freiburg, Germany.

Sato, Ken-Iti (1999): Lvy Processes and Infinitely Divisible Distributions. CambridgeUniversity Press.

Titchmarsh, E.C. (1975): Introduction to the Theory of Fourier Integrals, Oxford UniversityPress. Reprint of the 1948 second edition.

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Notes

1

First, when there is the possibility of a significantly negative jump (a crash), then expiring out-of-the-money putsbecome worthless like T . (see Mertons 1976 series solution). Here is the jump arrival rate under themartingale pricing distribution, and Tis the time to option expiration. This is a much slower decay in Tthan the

Black-Scholes formula, and if you try to fit the Black-Scholes formula to it (with an implied volatility), the only way

it works is for the implied volatility to grow very large as T 0 . Second, the jump rate is very sensitive topreferences. Statistically, you might expect a market crash once every 10 years for example ( . ) = 0 10 . But since

you are risk-averse, your effective rate for , which you use to price options is once every 4 years for another

example ( . ) = 0 25 . These two effects both work in the same direction to substantially boost the implied volatility

above the statistical volatility of the index. Very steep smiles are easy to fit because the model generates arbitrarily

large implied volatilities as T 0 . For preferences effects in Mertons model, see Bates (1991,1998) and Naikand Lee (1990). For general measure changes in jump-diffusions, see Aase (1998) and Colwell and Elliott (1993).

2 The independent increment property is too idealized for the real world. For example, while actual security returnshave very low autocorrelations, squared or absolute returns often have significant positive autocorrelations, and

volatility clustering effects are well-documented. These effects are missing from a pure Lvy process model.

3 See, for example, the S-space call option formula obtained by Madan, Carr, and Chang (1998).

4 Proportional returns problems, which may have many stochastic factors, have /t tdS Sindependent oftS.

5 Equivalently, this transform is a complex Mellin transform using TSas the integration variable.

6 We tested our results in the Mathematica system, where (1.1) is generally set up in just a few lines of code andtakes typically a couple seconds (on an old desktop system) to evaluate any of the half-dozen Lvy examples fromthe literature. Since Mathematica is an interpretive system, this is a worst case scenario for run-times. When the

highest numerical efficiency is important, consider applying the Fast Fourier Transform method of Carr and Madan(1999) to (1.1) or any of the variants given in Sec. 3.

7 For example, the change of measure induced by an equilibrium model with a power utility function ( )u S S = isvery popular, often applied in an ad-hoc manner as an Esscher transform. This particular measure change causes asimple translation z z i + in the Fourier transform of the Lvy measure ( )z , defined below. Some, but not allof the models in Table 2.1 can maintain their parameterized form under this particular change of measure.

8 A cumulative probability distribution ( )F x is any non-decreasing function with ( )F = 0 and ( )F = 1 .

9 This is a rearrangement of terms in Jacod and Shiryaev (1987) Theorem II.2.34.

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10 Sometimes the term complex Fourier transform is used. A comprehensive reference is Titchmarsh (1975). The

reason for the limitation to the strip Ima z b