Rough stochastic volatility modeling

All empirical studies clearly show that the volatility of stock returns is not constant but varies with time. Numerous time-continuous models based on Brownian motions (such as the Heston model) have been proposed to model this volatility. However, recent studies show that a model based on fractional Brownian motions with a Hurst index close to 0.1 could reproduce parsimoniously the main empirical properties of this volatility. It can be observed that fractional Brownian motions with Hurst index H less than 0.5 have rougher sample paths than standard Brownian motions and smoother sample paths for H > 0.5. Hence, these recent models with H ≈ 0.1 are called rough volatility models. This master’s thesis studies general stochastic volatility models (Heston, Hull and White, Bates, etc) with a particular focus on those modeling the log-volatility using fractional Brownian motions (and more precisely fractional Ornstein-Uhlenbeck processes). This master’s thesis also includes practical applications with the pricing of financial options and life insurance contracts.