Pricing Options in Jump Diffusion Models Using Mellin Transforms *

This paper is concerned with the valuation of options in jump diffusion models. The partial integro-differential equation (PIDE) inherent in the pricing problem is solved by using the Mellin integral transform. The solution is a single integral expression independent of the distribution of the jump size. We also derive analytical expressions for the Greeks. The results are implemented and compared to other approaches.


Introduction
Pure diffusion models are in most cases not flexible enough to fit the empirical observations concerning the movements of stock prices.Jump diffusions are a natural extension of pure diffusions since they are able to account for large and sudden changes.This is accomplished by adding a second source of uncertainty into the diffusion dynamics.The second source is discontinuous and models the jumps in the underlying asset.Also, pricing of options in jump diffusions is consistent with the volatility smile often observed in financial markets.
In his seminal paper from 1976 Merton [1] introduced a first jump diffusion process for modeling the behavior of stock prices.Since his work jump diffusions have become a very popular tool in modeling equity, foreign exchange and commodity markets.Under the assumption of log-normally distributed jumps, Merton solved the European option pricing problem explicitly in closedform in terms of an infinite series of Black-Scholes prices.Other popular pure and non-pure jump diffusion processes are those proposed in [2][3][4][5][6] among others.Kou [7] proposed a double exponential jump-diffusion model where jump sizes are double exponentially distributed.The model allows for an analytical pricing of some path-dependent options, such as continuously monitored barrier and lookback, and perpetual American options [8].In [9], pricing formulas in Kou's model for double (single) barrier and touch options with time-de-pendent rebates are derived applying Laplace transforms.
In many cases, however, an explicit closed-form valuation of options in jump diffusions is not possible and one is restricted to numerical procedures.These procedures rely on the fact that prices of derivatives in a jump diffusion environment satisfy partial integro-differential equations (PIDEs).The methods discretize the asset-time domain and use either binomial trees, finite difference or finites element methods to solve the PIDEs.[10], for instance, uses an explicit multinomial tree based approach for the valuation.[11] develop another pricing procedure based on a lattice framework.Further recent developments concerning the numerical evaluation include the articles [12][13][14][15].
The application of Mellin transforms for the purpose of option pricing was firstly introduced in [16] in the geometric Brownian motion economy.It was extended to the stochastic volatility model of Heston by Frontczak (2011) [17].This paper extends the Mellin integral transform framework further to the valuation of options in jump-diffusion models.It is organized as follows: Chapter 2 reviews the problem formulation.Chapter 3 provides a solution for European derivatives as a single integral.Also simple formulas for the Greeks are derived.Although our solution is general in the sense that it does not depend on the distribution of the jump size, we choose the log-normal jump-diffusion model as an explicit example and show that our solutions for plain vanilla and digital puts may be rewritten as infinite series of Black-Scholes-Merton prices.Some numerical results are presented in Chapter 4. Chapter 5 concludes the paper and indicates possible further research projects.

Problem Statement
Let be the price of a dividend paying stock at time .The dividend is paid continuously at a rate and is proportional to the price of the stock.The evolution of the stock is affected by two sources of uncertainty: a continuous part modeled as a standard Brownian motion and a discontinuous part modeled as jumps with a Poisson arrival process .Therefore, under the physical probability the dynamics of the stock are given by

and
is the proportional change in the stock price.We also assume that the processes , sion process in (2.1) becomes a geometric Brownian motion as assumed in the Black-Scholes and Merton model.In order to guarantee that the discounted process is a martingale, we have where   f Y


is the probability density function of Y and  E  is the expectation operator.It is well known that markets with jump diffusions are not complete.Hence, an equivalent martingale measure is not unique.Nevertheless, if we assume that the jump risk is diversifiable as has been done in [1] we can consider the risk-neutral dynamics of , (2.3)   where  is replaced by the riskless interest rate r .
The solution to the process above equals and where subscripts denote partial derivatives.For a derivation see [1,18].The terminal value of the derivative equals , where

 
g  is the payoff function of the contract under consideration.For a plain vanilla European put with price we have where X is the strike price of the put and   S T denotes the terminal stock price 1 .For the jump size distribution different distributions are commonly used in the financial literature.Prominent candidates are among others:  Log-normally distributed jumps ( [1]): The probability density function of Y is given by where , and > 0 Y  and  are constants.
 Double exponentially distributed jumps ( [7]): The probability density function of Y is asymmetric and equals where , > 0 where , , > 0 Y   and     denotes the gamma function. One-sided exponentially distributed jumps2 : The probability density function   f Y may be seen as a special case of the probability density function of gamma , where is a prespecified level.

X
distributed jumps with 1 The particular choice of the distribution of is crucial in order to solve the problem analytically in the -domain.The only two models where Y is assumed to be continuous and that have been solved explicitly in terms of an infinite series are to the best of our knowledge the Merton (1976) model and the model of Kou (2002).

Main Results
The following theorem characterizes the solution to (2.4).The functional form of the solution is simple and is expressed as a single integral.
Theorem 3.1 The solution of the PIDE (2.4) is given by In particular, for the European put   Proof: For a locally Lebesgue integrable function , the Mellin transform  , denote the Mellin transform of a financial derivative with price  ,  F S t .Since the Mellin transform is a linear integral operator it follows that the PIDE from (2.4) can be transformed into a homogeneous ordinary differential equation (ODE) of the form where The solution to this ODE equals H  follows immediately.The final expression for the price follows from the inversion theorem for Mellin transforms.  The previous theorem gives the analytic solution to (2.4) in a general manner, i.e. without specifying the distribution of Y .The solution has a simple functional form and prices are obtained very quickly using standard quadrature routines.It is also flexible to account for different jump size distributions and types of derivatives by specifying it follows that  Double exponentially distributed jumps ( [7]): We have 3 For a digital put we have    .In each of the above cases, the price   , F S t can be determined easily by a single integration.Another advantage of the approach is that hedging parameters, commonly known as Greek letters, are easily determined analytically.Popular Greeks widely used for risk management purposes are the delta, the gamma, and the vega of a contract 4 .The results for Greeks are summarized in the next proposition.
Proposition 3.2 Independently of the distribution of the jump size Y , the analytical expressions for the delta, gamma, and vega can be stated as follows: , where and > 0 c   H  is given in Equation (3.2).In particular if

 
, F S t is a European plain vanilla put option the Greeks are determined as Proof: The expressions follow directly for Theorem 3.1 since we can differentiate under the integral sign.For the Greeks of a plain vanilla put one uses the Equation for     from (3.4) and simplifies. Similar to the derivative's price the Greeks are char-acterized as a single integral which may be evaluated easily.Furthermore, hedge parameters for a range of different types of derivatives may be obtained by specifying the functional form for     .
N x is the cumulative standard normal distribution function and the adjusted parameters are given by , , , e , !
and the assertion follows. Now we are going to prove Proposition 3.3.
, e e e e e 1

T t h T t T t h T t T t g t
where , e e e e e 1 , , .
The transformation e z where is defined in (3.26).From [20] p. 337, a   Using this result and simplifying yields after some algebra One observes that the expression for the summands in the Equation for ) We arrive at the more compact form Merton's solution follows now from the identity where and > 0 c     is given in Equation (3.4).Since we have been able to derive an analytical expression for the inverse Mellin transform of  ,  g t   , convolution allows to write for the European put price The integrals above can be evaluated straightforwardly.Standard transformations convert both integrals into normal integrals.We get with the temporary quantities , , , , , , ,  (3.36)   and , , , , .
Finally, observing that where the modified parameters

Numerical Results
This section presents the numerical results from implementing the analytical solutions derived in this paper.We start with options in the log-normal model of Merton (1976).The input parameters can be found in Table 1.
The parameters were found in [12] and are based on a calibration using European options on the S&P 500 stock index.Prices are determined for three different stock price levels (in-, out-, and at-the-money options).The solution based on Mellin transforms is given in Theorem 3.1.The integral is evaluated numerically using Gauss-Lobatto quadrature.To get some information about the convergence of the results we truncate the limits of integration gradually at and choose i200  c 2 c  .We also compute the "exact" solution using the infinite series representation with .Tables 2 and 3 show the results for the log-normal model.

 n
The next table summarizes the results for a digital put.Our first and most important finding is that the framework proposed in this article is able to produce highly accurate prices for both types of derivatives.Even if the integration limits are truncated at the results are correct up to the third/fourth position after decimal point i50     in each of the stock price scenarios.For the prices coincide up to the eighth decimal place and there are no noteworthy differences between the two frameworks.100 y  We conclude the numerical analysis considering put options in Kou's double exponential model.The "exact" solution in form of an infinite series can by found in the original article of Kou.The input parameters are the same as above except for the jump component: Similar to the log-normal model, the numerical approximation of the integral solution produces satisfactory results.The convergence rate is comparable to the previous model.In summary, the numerical experiments carried out demonstrate that the new framework besides its theoretical insights produces prices with a convincing degree of precision.The Mellin transform approach must be regarded as a capable alternative to existing methods.

Conclusions
The purpose of this paper is to explore the valuation of options where the underlying follows a jump-diffusion.The Mellin integral transform approach was applied to solve the homogeneous PIDE analytically.The proposed solution has a very compact form as one single integral which may be evaluated easily and quickly by quadrature.The final pricing formula offers a powerful and flexible tool for handling a variety of jump size distributions and payoff structures.For log-normally distributed jumps, it was shown explicitly that in case plain vanilla puts our integral expression equivalent to the infinite series solution of BSM-prices.
The numerical results are convincing and logically consistent.They come along with the theoretical findings.
Future research projects are multifaceted.An extension of the proposed approach to American options seems to be possible.Also multi-asset options in jumpdiffusions could be valued within this framework.
  G  and     , respectively, for the contract under consideration.To give explicit examples, we state the expressions for   G  in the four special cases for discussed above: Y  Log-normally distributed jumps ([1]): Since for each real or complex s we have

n
and the integral can be viewed as the  -th moment of a log-normally distributed random variable with mean n 3.1)with the special choices of