Rough Volatility Models

Specialeforsvar: Maja Najbjerg Andersen og Jonatan Bjørndal Lychou

Titel: Rough Volatility Models
An investigation of the rough Bergomi model and the rough Heston model in relation to the implied volatility of the S&P500 index.

Abstract: In this thesis we will investigate the volatility of the S&P500-index with the aim of replicating volatility structures using the class of rough volatility models. We will show that the log-volatility increments exhibit roughness, unifractal scaling and Gaussianity, leading to the idea of modeling with fractional Brownian motions with a Hurst exponent H < 1/2. From this idea, we will develop the rough Bergomi model and the rough Heston model which are non-Markovian models leading to simulation difficulties. In the case of the rough Bergomi model, we will use an exact simulation scheme. To simulate the rough Heston model, we will introduce a non-rough multi-factor approximation model which has the ability to bring back both models to the Markovian world. We will show that this multi-factor model converges to the rough model in L^2.
The idea behind introducing these rough volatility models is to replicate the marketobserved power-law decay of the at-the-money implied volatility skew better than the classical models. For this, we will investigate how the model parameters affect the
implied volatility and thereby the at-the-money skew for both models. Finally, we will compare the rough Bergomi model and the rough Heston model based on the atthe-money skew to the extent of what is possible without calibrating the models and simulating the rough Heston model through the multi-factor approximation scheme.

Vejledere: Rolf Poulsen
Sigurd Emil Rømer
Censor: Thomas Kokholm, Aarhus Universitet