Details

In the jump diffusion model, the stock price follows the random process . The first two terms are familiar from the Black–Scholes model: drift rate , volatility , and random walk (Wiener process) . The last term represents the jumps: is the jump size as a multiple of stock price, while is the number of jump events that have occurred up to time . is assumed to follow the Poisson process , where is the average frequency with which jumps occur. The jump size follows a log-normal distribution , where is the standard normal distribution, is the average jump size, and is the volatility of jump size.

Snapshot 1: when there are no jumps, the jump diffusion model reduces to the Black–Scholes model, in which returns follow a normal distribution

Snapshot 2: the effect of jumps can be observed clearly by "turning down" the volatility of the diffusive component to zero

Snapshot 3: one limitation of the jump diffusion model is that is the volatility in the diffusive component is constant, which results in unrealistic-looking behavior when the jumps are large: in reality, volatility tends to increase dramatically around the time that large jumps occur.

References:

R. Merton, Continuous-Time Finance, Cambridge, MA: B. Blackwell, 1998.

M. Joshi, The Concepts and Practice of Mathematical Finance, New York: Cambridge University Press, 2003.