Brownian motion and normal distribution have been widely used, for example, in Black-Scholes option pricing framework, to study the return of assets. However, two puzzles that surfaced from many empirical investigations, have received much attention recently, namely the leptokurtic feature that the return distribution of assets may have a higher peak and two asymmetric heavier tails than those of the normal distribution and an empirical abnormality called "volatility smile" in option pricing. To incorporate both the leptokurtic feature and "volatility smile", a jump diffusion model is proposed in which the price of the underlying asset is modeled by two parts - a continuous part driven by Brownian motion, and a jump part with the jump size having a double exponential distribution.

In addition to the leptokurtic feature and "volatility smile", the model is simple enough to produce analytical solutions for a variety of option pricing problems, including options, future options, and such interest rates derivatives as caps and floors in terms of the $Hh$ function. Although there are many models that can incorporate some of the three properties (leptokurtic feature, "volatility smile" and analytical tractability) the current jump diffusion model can incorporate all three together under a unified framework.

Steve Kou is an Assistant Professor in the Department of Industrial Engineering and Operations Research at Columbia University, where he teaches Financial Engineering.  He is a specialist in mathematical finance and is well-known internationally for his research on numerical pricing of discrete exotic options, such as discrete barrier and lookback options; option pricing in imperfect markets; market LIBOR models with jump risk; pricing of electricity options; and jump diffusion models and their closed form solutions for both equity and interest rate derivatives. Some of his results have been widely used in Wall Street and have been incorporated into such standard MBA textbooks.


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