WEAK ERROR RATES FOR OPTION PRICING UNDER LINEAR ROUGH VOLATILITY

International Journal of Theoretical and Applied Finance, vol. 25

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 [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.

Authors 3

1. Christian Bayer Aachen Chair of Mathematics for Uncertainty Quantification

   RWTH Aachen University · University of Dundee · King Abdullah University of Science and Technology · Weierstrass Institute for Applied Analysis and Stochastics

   Affiliation as printed

   Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany

   Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University Pontdriesch 14-16, 52062 Aachen, Germany

   Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST) Thuwal 23955-6900, Saudi Arabia

   Mathematics Division, School of Science and Engineering, University of Dundee Dundee, DD1 4HR, United Kingdom

2. Eric Joseph Hall Aachen Chair of Mathematics for Uncertainty Quantification

   RWTH Aachen University · University of Dundee · King Abdullah University of Science and Technology

   Affiliation as printed

   Mathematics Division, School of Science and Engineering, University of Dundee, Dundee, DD1 4HR, UK

   Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University Pontdriesch 14-16, 52062 Aachen, Germany

   Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST) Thuwal 23955-6900, Saudi Arabia

   Mathematics Division, School of Science and Engineering, University of Dundee Dundee, DD1 4HR, United Kingdom

3. Raul F. Tempone Aachen Chair of Mathematics for Uncertainty Quantification

   RWTH Aachen University · University of Dundee · King Abdullah University of Science and Technology

   Affiliation as printed

   Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University, Pontdriesch 14-16, 52062 Aachen, Germany

   Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia

   Mathematics Division, School of Science and Engineering, University of Dundee Dundee, DD1 4HR, United Kingdom