A regularity structure for rough volatility
A regularity structure for rough volatility
This paper introduces a regularity structure framework for rough volatility models, which are non-Markovian and non-semimartingale when the Hurst parameter H<1/2. The authors develop a renormalized Wong-Zakai approximation theory, showing that naive approximations diverge but can be corrected by subtracting diverging terms. Explicit renormalization constants are provided for both mollifier and wavelet approximations, with a phase transition at H=1/4. The framework is applied to option pricing, reducing the problem to sampling renormalized integrals and total variance, and to large deviations, yielding rate functions for both simple and non-simple rough volatility. The results enable rigorous analysis of rough volatility models and provide tools for numerical simulation and smile asymptotics.
Highlights
- 1Develops a regularity structure framework for rough volatility models, enabling renormalized Wong-Zakai approximations.
- 2Provides explicit renormalization constants for mollifier and wavelet approximations of fractional Brownian motion.
- 3Establishes short-time large deviation principles for both simple and non-simple rough volatility models.
- 4Derives a skew formula for non-simple rough volatility, generalizing known results for Markovian and rough Heston models.
- 5Offers a pathwise approach to option pricing under rough volatility, avoiding direct simulation of the orthogonal Brownian motion.
Methods
- MHairer's regularity structures for renormalized approximations of rough volatility.
- MWong-Zakai type approximations with renormalization for fractional Brownian motion.
- MLarge deviation theory via contraction principle and continuity of the reconstruction map.
- MStochastic Volterra equations for non-simple rough volatility models.
Results
- RA Wong-Zakai theorem for simple rough volatility: convergence of renormalized approximations to the Itô price process.
- ROption prices can be expressed as limits of expectations involving renormalized integrals and total variance.
- RShort-time large deviation principle for log-price under simple rough volatility with rate function involving fractional integrals.
- RExtension of large deviations to non-simple rough volatility (stochastic Volterra dynamics).
- RPhase transition at H=1/4 for replacing time-dependent renormalization by its mean.
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