Calibrating and pricing with a stochastic-local volatility model

The constant volatility plain vanilla Black–Scholes model is clearly inadequate to reproduce even plain vanilla option prices observed in the market. Efforts to build a pricing model with modified dynamics that allow a better fit have mostly proceeded in one of two directions. One variant is the local volatility (LV) framework, which posits a volatility process that varies over time but is nonstochastic given the stock price. The other variant introduces a fully stochastic volatility (SV) process in which volatility has its own stochastic driving factor that is only imperfectly correlated with the returns process. Both types of models can be calibrated to market prices of at the money options. But the LV model’s ability to match today’s implied volatility surface exactly entails dynamics that can lead to large inaccuracies for exotic options, while SV models generally feature mean reversion in the volatility and tend to have trouble with far in- or out-of-the-money options. In this article, the authors propose a combined “stochastic-local volatility” model. The main structure comes from the Heston SV model, but in the returns equation, the volatility from the variance equation is multiplied by a “leverage factor” that allows the model to fit the volatility surface better. Fitting the model to the data presents some econometric challenges, but the authors show that the result works well.