Rough Volatility

Specialeforsvar ved Anna Sylvest Broberg

Titel:  Rough Volatility

 

Abstract: In this thesis we analyse if the volatility process can be well-modelled by a rough process driven by a fractional Brownian motion. A fractional Brownian motion is a self-similar, Gaussian process with stationary increments and non-zero correlation across time. Estimating realized volatility on the S\&P 500 Index from high frequency data, we find that the log-volatility process behaves essentially as a fractional Brownian motion with Hurst parameter less than 1/2. This implies, that the paths of the log-volatility process is more rough than what is obtained if it is modelled with a classical Brownian motion. This motivate us to present the Rough Fractional Stochastic Volatility (RFSV) model where volatility is driven by a fractional Brownian motion, and we demonstrate that this model is very consistent with financial time series data. We then show how the RFSV model can be used to price claims on the underlying, and under a deterministic change of measure we obtain the simple Rough Bergomi model. We find that this model is remarkably consistent with the shape of the volatility surface of the S\&P 500 Index and that it is able to capture the term structure of skew observed in equity markets - in particular, the exploding term structure of the at-the-money implied volatility skew when maturity goes to zero.

 

 

Vejleder:  Rolf Poulsen
Censor:    David Sloth Pedersen, Danske Bank