Bates Stochastic Volatility Jump Model

Black-Scholes assumes constant volatility and continuous price paths, which fails to match observed option prices: real implied volatility surfaces show a smile (out-of-the-money options priced richer than the model implies) that a constant-volatility model can't produce. The Heston model fixes part of this by letting volatility itself wander randomly over time, generating a smile — but it still assumes prices move continuously, so it struggles to explain short-dated smiles or price sudden crash risk properly. The Bates model adds the missing piece: on top of Heston's randomly evolving volatility, it lets the price itself jump discontinuously at random moments, with jump sizes and frequency chosen to reflect crash risk.

$$dS_t = \mu S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^S + S_t\,dJ_t, \qquad dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t^v ,$$

where $J_t$ is a jump process (a Poisson-timed sudden move) layered on top of Heston's ordinary variance dynamics $v_t$. In plain English: volatility drifts around slowly and continuously (as in Heston), while separately, at random rare moments, the price itself can gap — capturing both the gradual "vol goes up and down" behavior and the fast "something happened overnight" behavior in one model.

Worked example

A short-dated out-of-the-money put on an index is priced richly in the market relative to what Heston alone predicts, because a small chance of an overnight crash matters disproportionately for a short-dated option — there isn't enough time for gradual volatility diffusion to explain the premium. Adding a jump component with, say, a 2% annual probability of a 15% overnight drop lets Bates match that short-dated smile far better than Heston's continuous-path dynamics can, while Heston's own diffusion term still handles the longer-dated smile shape reasonably well on its own.