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 number of jumps per unit time. The jump size may follow any distribution, but a common choice is a log-normal distribution
is the standard normal distribution,
is the average jump size, and
is the volatility of jump size. The three parameters
characterize the jump diffusion model.
For European call and put options, closed-form solutions for the price can be found within the jump diffusion model in terms of Black-Scholes prices. If we write
as the Black-Scholes price of a call or put option with spot
, interest rate
(assumed constant for simplicity), and time to expiry
, then the corresponding price within the jump diffusion model can be written as:
term in this series corresponds to the scenario where
jumps occur during the life of the option.
It can be shown that for all derivatives with convex
payoff (which includes regular call and put options) the price always increases when jumps are present (i.e., when
)—regardless of the average jump direction. Thus, holding other parameters constant, the option price is a minimum for
(i.e., the Black-Scholes case) and increases both for
. This increase in price can be interpreted as compensation for the extra risk taken by the option writer due to the presence of jumps, since this risk cannot be eliminated by delta hedging (see Joshi 2003, Section 15.5).
R. Merton, Continuous-Time Finance
, Oxford: Blackwell, 1998.
M. Joshi, The Concepts and Practice of Mathematical Finance
, Cambridge: Cambridge University Press, 2003.