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.

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.