Stochastic volatility models are known to capture features observed in the markets and have been widely used in the financial industry. In particular, rough volatility models, where the instantaneous volatility is driven by a fractional Brownian motion, provide a more accurate fit of the power-law decay observed on volatility smiles close to maturity.
This thesis is concerned with deriving the asymptotic behaviour of these two classes of stochastic volatility models using large deviations techniques, in order to understand the behaviour of the implied volatility. Indeed, when pricing options, a closed-form solution to the pricing problem is often not available. Practitioners are using implied volatility as a measure of option prices that is consistent for different options with different maturities and strike prices. As this implied volatility cannot be expressed in closed form either, finding asymptotic formulas is crucial. We therefore focus on ob- taining the asymptotic behaviour of two rough volatility models, namely a generalised version of the Stein-Stein model where the volatility starts from a random distribution and the (multi-factor) rough Bergomi model for realised variance options. We also study a particular complete model with stochas- tic volatility, which can be fitted to market data perfectly via the leverage function. In each case, we obtain the asymptotic behaviour of the stock price process and the implied volatility. We produce numerical schemes to compute the implied volatility smiles.