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 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.

R. Merton,

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