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
, 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
is the average frequency with which jumps occur. The jump size follows a log-normal distribution
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.