Fractional Brownian motion and option pricing under rough volatility

Abstract

Forecasting volatility in financial markets has always been delicate and at the same time, essential and problematic given its non-deterministic nature. In the derivative world, particularly, different pricing models have been developed since the 20th century: the most famous and widely used is the Black-Sholes formula, proposed in their seminal work in the seventies. Since then, several other models have been developed to overcome its shortcomings that question its robustness such as the assumption of constant volatility (implied). Indeed, volatility surface is well-known to presents so-called smiles due to the option moneyness and ATM skews. If volatility were a constant as described by Black and Sholes, volatility surface would be flat instead of varying across moneyness and maturities. Hence, the need of models capable of capture consistently this feature, especially for exotic option with a market not as liquid as for plain vanilla European options, where the pricing and hedging can be done marked-to-market. Among stochastic volatility models, notable mentions are Heston model, SABR, Hull and White and Bergomi model. They differ from each other, essentially for the specific dynamics describing the volatility process. These models are considered time-homogenous, meaning that parameters are independent from price and time. This property assumes that the overall shape of the volatility surface does almost not change in the equity market, at least in first approximation, albeit level and orientation can. Despite these models were a notable improvement, they do not capture persistency and volatility clusters that instead are observable in time-series data. The limitation of the stochasticity term modeled as a Brownian motion process, was eliminated by Comte and Renault with the introduction of fractional stochastic volatility model, where a fractional Brownian motion with long dependency drives the model. A revisitation of that has been proposed by Gatheral, Jaisson and Rosenbaum, who defined volatility as rough and showed that volatility has short term dependence. Bayer, Friz and Gatheral also proposed an extension of the Bergomi model for option pricing purpose under rough volatility, called the Rough Bergomi model.