# Mixture models of Stochastic Volatility and Local Volatility

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.

• stochastic local volatility mixture = realistic stochastic dynamics of the volatility (stoch vol component) + perfect fit to the current implied vol surface (local vol component) – Antoine Conze Mar 25 '18 at 11:31
• i wouldn't say that people on this site prefer stochastic volatility to local vol - perhaps a stochastic local vol model like this though. – will Mar 25 '18 at 13:19
• when would you prefer local vol over stochastic vol then? – Permian Mar 25 '18 at 15:32
• Can anyone show how @AntoineConze's comment is true? I cant see this works mathematically – Permian Mar 25 '18 at 15:51
• Stochastic local volatility model means $dS_t/S_t = ... dt + \sigma_t L(S_t,t) dW_t$ with $\sigma_t$ your favorite diffusive process (Heston, ...) for the stochastic part of the volatility. Then the local part $L(S,t)$ is computed from "Dupire’s unified theory of volatility" $\sigma_{\text{local}}(S,t)^2=E[ (\sigma_t L(S_t,t))^2 | S_t=S]=E[ \sigma_t^2 | S_t=S] L(S_t,t)^2$. $E[ \sigma_t^2 | S_t=S]$ can be efficiently computed using for instance a 2D finite difference scheme for the Fokker Planck equation. – Antoine Conze Mar 25 '18 at 17:42

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.