Jumpdiffusion

Jump diffusion is a class of stochastic processes used to model quantities that evolve with both continuous fluctuations and sudden, discrete changes. It extends a pure diffusion model by incorporating jumps, allowing for abrupt moves in value that occur at random times.

A common formulation for a jump-diffusion process S_t is given by the stochastic differential equation dS_t = μ S_t dt + σ S_t dW_t + (J - 1) S_{t-} dN_t, where W_t is a standard Brownian motion, N_t is a Poisson process with intensity λ, and J is a positive random multiplier representing the relative jump size. The jumps occur at the times of N_t, and between jumps the process follows a geometric Brownian motion. The average jump impact is κ = E[J - 1], and the process can also be written as S_t = S_0 exp((μ - λ κ - 0.5 σ^2) t + σ W_t) ∏_{i=1}^{N_t} J_i, with J_i i.i.d. copies of J.

A common variant is the Merton jump-diffusion model, where jump sizes are multiplicative with J = e^Y and Y is typically distributed (for example, Y ~ N(m, s^2), making jumps lognormal). This framework enables closed-form or semi-closed-form pricing for certain derivative instruments by averaging over the jump distribution, often requiring numerical methods for general cases.

Applications of jump diffusion appear in options pricing, risk management, and credit or energy markets, where sudden events contribute to heavy tails and skewness in asset returns. The model provides a tractable way to capture both continuous volatility and discrete, abrupt movements, and it has been subject to extensive calibration and extension since its introduction in the 1970s.