Pathwise large deviations for the Rough Bergomi model
Pathwise large deviations for the Rough Bergomi model
This paper establishes a pathwise large deviations principle (LDP) for the rough Bergomi model, a stochastic volatility model where the log volatility is driven by a fractional Brownian motion with Hurst parameter H < 1/2. The model's volatility process involves an exponential function of a Gaussian process, which is superlinear and includes a drift term, making existing large deviations results inapplicable. The authors overcome these challenges by considering a small-noise rescaling of the model and leveraging the reproducing kernel Hilbert space (RKHS) associated with the Gaussian process. They prove that the rescaled log stock price satisfies an LDP with speed ε^{-β} and a rate function expressed as an infimum over paths in the RKHS. The main result is complemented by a corollary providing small-time asymptotics for the log stock price, which has potential applications in option pricing and variance reduction. The uncorrelated case is treated separately, yielding a simpler rate function. The paper contributes to the theoretical understanding of rough volatility models and provides a foundation for further asymptotic analysis in mathematical finance.
Highlights
- 1Proves a pathwise large deviations principle (LDP) for a small-noise version of the rough Bergomi model, overcoming challenges from exponential growth in the volatility drift and superlinear volatility function.
- 2Establishes the rate function explicitly via the reproducing kernel Hilbert space (RKHS) associated with the Gaussian process driving volatility.
- 3Provides a corollary for small-time asymptotics of the log stock price, with potential applications to implied volatility and variance reduction.
- 4Treats the uncorrelated case separately, where the RKHS degenerates, and derives an analogous LDP.
Methods
- MLarge deviations theory for Gaussian measures on Banach spaces, using the Schilder-type theorem for the driving Gaussian process.
- MReproducing kernel Hilbert space (RKHS) framework to characterize the rate function via the covariance structure of the Gaussian process.
- MContraction principle to map the LDP from the driving process to the log stock price via the rough Bergomi dynamics.
- MScaling analysis to translate small-noise into small-time asymptotics, with careful handling of the exponential volatility function and drift.
Results
- RThe rescaled log stock price process satisfies an LDP with speed ε^{-β} and rate function Λ^X defined via an infimum over paths in the RKHS.
- RThe rate function is zero at the origin, indicating that the most likely path is the deterministic one.
- RSmall-time asymptotics for the log stock price are obtained, with the rate function given by a variational problem.
- RThe uncorrelated case yields a simpler rate function due to the degeneracy of the RKHS.
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