Jump diffusion models add a jump process to the Brownian motion responsible for the ”normal” propagation of the underlying. In such models, jumps repre- sent rare events like crashes. The Black Scholes model is extened through the addition of a compound Poisson jump process to capture the skewness and tail behaviour observed in real world data. For the jump process a compound Poisson process with an intensity and a jump size distribution f is used. With drift and Brownian motion
we get the following process
for the spot price. Depending on the jump size distribution f we can distinguish between different jump models:
- Merton Jump Diffusion Model: The Merton Jump Diffusion model uses a normal distribution for f. This extension adds three additional parameters to the Black Scholes model — the intensity of the compound Poisson process, the jump mean and the standard deviation of the jump size.
- Kou Model: Kou modelled the jump distribution f with an asymmetric double exponential distribution to distinguish between up- and down-moves. In the Kou model four additional parameters arise, namely the intensity of the compound Poisson process, the intensity of the exponential distribution for down jumps , the intensity of the exponential distribution for up jumps and the probability for the up jump.