Why combine them

Heston's smile is produced by variance that takes time to move, so it is mild at short maturities and grows with the horizon. Merton's smile is produced by jumps that can strike at any instant, so it is steepest at the front and fades as diffusion takes over. Index markets show a steep short-dated skew that persists at long maturities: neither model alone fits both ends. Bates (1996) puts the two together.

The model

$$ dS_t = (r - q - \lambda k)\, S_t\, dt + \sqrt{v_t}\, S_t\, dW^{(1)}_t + (e^{J} - 1)\, S_{t^-}\, dN_t, \qquad dv_t = \kappa(\theta - v_t)\, dt + \sigma \sqrt{v_t}\, dW^{(2)}_t $$

with correlated Brownian motions as in Heston and a Poisson process \(N_t\) of intensity \(\lambda\) whose jumps \(J\) are normal with mean \(\mu_J\) and standard deviation \(\sigma_J\), as in Merton. Eight parameters in all.

Pricing

The diffusion and the jumps are independent, so the characteristic function of \(\ln S_T\) is a product: Heston's characteristic function times the compound-Poisson factor \(\exp\!\big(\lambda T\,(E[e^{iuJ}] - 1) - iu\,\lambda k T\big)\). Prices then follow from the same two integrals as in Heston. The calculator prices Heston and Bates through one shared routine that takes any characteristic function, which is how quants build model libraries: the pricing engine is written once.

Reading the smiles

On the calculator the Bates smiles are drawn solid and the Heston smiles for the same variance parameters dashed. At a tenth of a year the jumps put the Bates wings far above Heston's; at two years the two almost coincide. Turn the intensity to zero and they merge everywhere. Set the volatility of variance near zero and the Bates smile becomes Merton's, flattening with maturity.

In practice

Bates is the standard "one step up" from Heston on equity desks, and the model most often fitted to index surfaces when a single model must cover every maturity. Its cost is identifiability: jumps and the volatility of variance both fatten tails, and a calibration can trade one for the other. Desks fix or bound the jump parameters from short-dated quotes first, then fit the variance process to the rest.