Mixture models of Stochastic Volatility and Local Volatility

Question

As far as I can see on this website the stochastic volatilty models seem to be preferred to local volatility models, mainly due to the fact that stochastic volatility is 2D diffusive process whilst local volatility models are a 1D diffusive process.

Why do you see stochastic local volatility mixture models come up in academic papers? What are their advantages and disadvantages versus either a stochastic volatility model or a local volatility model.

Emphasis on exotic options modelling is also helpful.

Answer 1

Stochastic local volatility model means $dS_t/S_t=...dt+\sigma_t L(S_t,t)dW_t$ with $\sigma_t$ the stochastic part (modeled for instance as in the Heston model, or any other dynamics deemed appropriate) and $L(S_t,t)$ the local part.

The local part $L(S_t,t)$ is computed from "Dupire’s unified theory of volatility" which states that $$ σ_{\text{local}}(S,t)^2 =E[(σ_tL(S_t,t))^2|S_t=S] = E[σ_t^2|S_t=S]L(S,t)^2 $$ so that $$ \boxed{L(S,t)^2 = \frac{σ_{\text{local}}(S,t)^2}{E[σ_t^2|S_t=S]}} $$ $σ_{\text{local}}(S,t)$ is the local volatility computed from the implied volatility surface using the Dupire formula, and $E[σ_t^2|S_t=S]$ can be efficiently computed using for instance a 2D ADI finite difference scheme for the Fokker Planck equation (a.k.a. forward Kolmogorov equation) associated with the model.

With a a stochastic local volatility model you can therefore have a realistic stochastic dynamics for the instantaneous volatility while at the same time have a perfect fit of the current implied volatility surface, which means the model consistently prices vanilla options and, unlike pure local volatility models, does a decent job at pricing exotics that depend on the dynamics of future volatility.