Jump diffusion models are used in the pricing of financial derivatives to account for events that cause asset value jumps (discontinuities) within an underlying asset price process. A jump diffusion model combines a Brownian motion process with a compound Poisson process that represents the jump component.
The Brownian motion component represents diffusion or regular price changes, while the Poisson component represents infrequent jumps. These jumps may be caused by events such as news releases, economic data announcements, or exogenous shocks to the market.
Jump diffusion models provide a better framework to capture extreme price movements, which are not well-captured by a purely continuous diffusion model. They are particularly useful in modeling prices of assets with kurtosis higher than that of the normal distribution, such as equity prices, commodity prices, and foreign exchange rates.
The presence of jumps creates additional risk, and hence, affects the value of options and other derivatives. Pricing of jump models can be done using Monte Carlo simulation or using Fourier transform methods.
One commonly used jump diffusion model is the Merton model, which assumes that the underlying asset price process follows a log-normal distribution, modified by a compound Poisson process. The compound Poisson process represents the jump component, and the model is parameterized by the intensity of the jump process and the volatility of the asset price. Other popular jump diffusion models include the Bates model and Kou model, which allow for stochastic volatility in addition to jumps in the asset price.
In summary, jump diffusion models play a crucial role in financial derivative pricing by providing a more accurate description of asset price movements and allowing for the inclusion of infrequent but disruptive jumps in the underlying asset price process.