Over the last decade many banks have started various kinds of Levy processes to value options and other financial derivatives, especially within the Equity and FX asset classes. This has been necessitated due to the observance of fat tails and period jumps in the financial markets.

In our CFE course we cover two of the popular Levy processes and implement them on Excel spreadsheet to value derivatives.

A Levy process is a stochastic process used in the analysis of asset prices. Since the Normal distribution, which forms the basis of a Weiner process (Brownian motion), cannot explain the fat tails observed in the markets, many practitioners have resorted to using Levy processes for modeling asset prices. In fact, Levy process is a family of stochastic processes and there are many kinds of Levy processes.

In simple terms, a Levy process is a Weiner process (random walk) with a jump. Of course, the jump can be a complex process.

A stock (asset) price can modeled as a Levy process as shown below:

Where, is a stochastic Levy process and is the initial value of the asset. In fact, the exponential Brownian motion (geometric Brownian motion) that has been extensively used in valuation of financial derivatives in a Black-Scholes economy for the past 40 years is a special case of a Levy process.

An exponential Brownian motion, with a drift , and Weiner process, can be written as:

In the above equation, , the Levy process, contains no jump term. Now consider the Kou’s Double Exponential Jump Diffusion (stochastic) process, which is a type of Levy process:

In the above equation, , the Levy process contains, which represents the mean size of upward and downward jumps and a compound Poisson process given by .

There are four main kinds of Levy processes that are used in the field of Quantitative Finance. These are:

- Jump Diffusion processes (such as compound Poisson processes, etc.)
- Generalized tempered stable processes (Variance Gamma process, etc.)
- Generalized Hyperbolic processes
- Meixner processes

The Jump Diffusion process is a Brownian motion with jumps. It has a constant drift, a stochastic volatility (weiner process) and a jump component. Generally, the jump is modeled as a compound Poisson process and the simplest example is that of Merton’s Jump Diffusion process. A more complex jump diffusion process is the Kou’s Double Exponential Jump Diffusion process mentioned above andwhich, according to many experts and quants, handles the task of calibration in the vanilla and barrier FX options very well.

A Variance gamma process, part of the generalized tempered stable processes, is a Levy process where the time change is random and there is no diffusion component. Hence it is a pure jump process. The random time change in the VG process is modeled as a gamma process. According to many quants VG process handles the real life phenomenon of frequent small jumps and infrequent large jumps in the equity markets quite well.

**Reference: **
For a better and more detailed discussion on Levy Processes see Exotic Option Pricing and Advanced Levy Models by Paul Wilmott, Wim Shoutens and Andreas Kyprianou

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