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
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7rTzj4fWz2snnVHil0CX2i-0sta2uEvIi0m8QOri2HOqBBMxW6N-9MxSMy_xL1eR2p9HLjj2agw1n2xWpA7OyvE9ih17uM9iRTP9uTZB0X55BIKi6By5DKr9lHLJYYKChdttS2_zIlTM/s1600/Screen+Shot+2014-04-04+at+19.19.44+.png)
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.