Rough Volatility - Financial modelling with fractional Brownian motions

Specialeforsvar ved Troels Lund Jensen

Titel: Rough Volatility - Financial modelling with fractional Brownian motions  

Abstract: We analyse the possibilities of option price modelling using fractional Brownian motions. A fractional Brownian motion is a self-similar, Gaussian process with stationary increments and non-zero correlation across time. These properties are found to match the log-volatility process on the S&P 500 index, using estimates of realized volatility at high frequencies. The primary finding is, that the log-volatility process behaves as a fractional Brownian motion with a Hurst-parameter less than 1/2. This implies, that the paths of the log-volatility process is more rough than what is obtainable if the log-volatility is modelled with a standard Brownian motion. We propose a model where the volatility is driven by a fractional Brownian motion, and we make a short-maturity expansion of the price and implied volatility to analyse the model dynamics. Especially, the term structure of the at-the-money implied volatility skew is proven to decay as a power-law of maturity. We argue, that such a skew is not possible to obtain in a conventional local stochastic volatility model. Finally, observed implied volatilities on the SPX is analysed. We find that both the term structure of the at-the-money implied volatility skew and the skew-stickiness-ratio matches the quantities we would obtain theoretically in a model where volatility is rough. 

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