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How Stochastic Local Volatility Fits Vanillas and Models Exotic Risk

Article Quant Q&A · Author: Trajan
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Summary

The document explains why a stochastic local volatility model can combine features of local volatility and stochastic volatility. It writes the asset’s instantaneous volatility as a stochastic factor multiplied by a state- and time-dependent local factor. The stochastic component can follow a model such as Heston, while the local component is calibrated to the observed implied volatility surface.

The local multiplier is obtained by dividing Dupire local variance by the conditional expected stochastic variance given the asset price. That conditional expectation can be estimated by solving the model’s forward Kolmogorov equation, for example with a two-dimensional alternating-direction implicit finite-difference method. The stated benefit is a fit to the current vanilla surface alongside stochastic future volatility dynamics, which can improve the treatment of volatility-dependent exotic options relative to pure local volatility. The answer gives no numerical comparison, implementation details, or discussion of calibration stability and model-specific limitations.

Key ideas

  • The model combines a stochastic volatility factor with a local multiplier depending on asset price and time.
  • The local multiplier is calibrated using Dupire local volatility and conditional expected stochastic variance.
  • A forward Kolmogorov equation can be used to compute the conditional variance needed for calibration.
  • The construction aims to fit vanilla option prices while retaining stochastic future volatility dynamics.
  • The document claims this can improve exotic option modeling compared with pure local volatility, but gives no quantitative evidence.

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OptionsVolatilityDerivatives pricingStatistics
Full text
# Mixture models of Stochastic Volatility and Local Volatility


# 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.

## Answer by Antoine Conze (score 7, accepted)

https://quant.stackexchange.com/a/39716

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.