Weak error rates for option pricing under linear rough volatility

TL;DR

This paper establishes new weak convergence rates for the Euler method in rough volatility models, improving understanding of numerical approximation accuracy in option pricing with fractional Brownian motion.

Contribution

It proves the weak convergence rate of the Euler method for the linear rough Stein-Stein model as H+1/2 and rate one for quadratic payoffs, advancing numerical analysis in rough volatility models.

Findings

Weak convergence rate of H+1/2 for the Euler method in the rough Stein-Stein model.

Rate one convergence for quadratic payoff functions.

Numerical experiments support the theoretical results.

Abstract

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 [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], 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, $H < 1/2$, over reasonable timescales. Both time series of asset prices and option-derived price data indicate that $H$ often takes values close to $0.1$ 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 $H$. We prove rate $H + 1/2$ 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. 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.