# Option Prices in the Kou Jump Diffusion Model

Option Prices in the Kou Jump Diffusion Model

The well-known weaknesses of diffusion-based models of option prices have led to a variety of models involving (in addition to a "diffusive component" driven by an (exponential) Brownian motion) randomly occurring discontinuous jumps. These models are known as "jump diffusions" and form a subset of the set of models driven by an (exponential) Lévy process. The first such model was due to Robert Merton. In Merton's model the jump sizes are normally distributed. Kou introduced a similar model, in which the distribution of jump sizes is an asymmetric exponential. Both models possess certain features that they share with observed market prices and which are not present in the popular Black–Scholes model, such as the leptokurtic feature, but Kou's model has several advantages over Merton's. One of these is the fact that, thanks to the "memoryless property" of the exponential distribution, it is possible to obtain "explicit formulas" for many important types of options. This Demonstration uses such a formula for the price of a European call and put option, shown on the graph by a solid brown curve, which is compared with that of the corresponding option in the Black–Scholes model shown by a dashed green curve. Note than increasing the jump intensity (average jump frequency) parameter will make the jump diffusion price of options increasingly higher than the Black–Scholes one, reflecting the increased jump risk.