Jump-diffusion: where Geometric Brownian Motion meets jumps

Jumps observed in mountains

In 1976, Robert C. Merton introduced the jump-diffusion model to the world of financial mathematics. Jump diffusion is a mixture model, it incorporates a jump process and a diffusion process. In this previous article (where options are priced using Monte Carlo method and Black-Scholes dynamics), it can be seen that the model lacks a certain “disaster factor”: the price paths that are generated do not include jumps.

Example of jumps in the path of DM/USD exchange rate with 5-minutely data, source: lpsm.paris

The figure above shows, the investor can locate “jumps” in the movement of DM/USD. Diffusion-type processes with continuous paths, in particular, a Geometric Brownian Motion, cannot capture these extreme movements that are present in the market.

Different causes of jumps and their sizes, US Jumps by year, “What Triggers Stock Market Jumps?” — Scott R. Baker, Nicholas Bloom, Steve Davis, Marco Sammon

Summary

To price option contracts with a jump-diffusion model, the investor first needs to understand how jumps are characterised; in this approach, the Poisson process is utilized to represent extreme events. Furthermore, the Poisson process will be combined with the Brownian Motion, and finally with…