In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [C. Bayer, P. K. Friz & J. Gatheral (2016) Pricing under rough volatility, Quantitative Finance 16 (6), 887–904, doi:10.1080/14697688.2015.1099717], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, [Formula: see text], over reasonable timescales. Both time series of asset prices and option-derived price data indicate that [Formula: see text] often takes values close to [Formula: see text] or less, i.e. rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only [Formula: see text]. We prove rate [Formula: see text] for the weak convergence of the Euler method for the rough Stein–Stein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Indeed, the problem of weak convergence for rough volatility models is very subtle; we provide examples demonstrating the rate of convergence for payoff functions that are well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof uses Talay–Tubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.
|Original language||English (US)|
|Journal||International Journal of Theoretical and Applied Finance|
|State||Published - Jan 19 2023|
Bibliographical noteKAUST Repository Item: Exported on 2023-01-23
Acknowledged KAUST grant number(s): OSR2019-CRG8-4033
Acknowledgements: This work was supported by the KAUST Office of Sponsored Research (OSR) under Award No. OSR2019-CRG8-4033 and the Alexander von Humboldt Foundation. R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. A portion of this work was carried out while E. Hall was a Postdoctoral Research Scientist in the Chair of Mathematics for Uncertainty Quantification at RWTH Aachen University. C. Bayer gratefully acknowledges support by the German Research Council (DFG) via the Research Unit FOR 2402. We are grateful to Andreas Neuenkirch for pointing out the simpler proof in the case of quadratic payoffs presented in Section 4.2.
ASJC Scopus subject areas
- Economics, Econometrics and Finance(all)