This course provides an introduction to models of randomly fluctuating asset price volatility. In particular extensions of the Black-Scholes (BS) model to continuous-time local and stochastic volatility models. We discuss various volatility models, including: Heston, Hull-White. We explore extensions to the BS model where the underlying includes jumps and their effect on implied volatility.
Different notions of volatility: spot, realised and implied volatility; stylised facts of asset returns; deterministic volatility model; local volatility (LV) models and the Dupire equation; stochastic volatility (SV) models; delta and vega hedging; robustness of the BS hedging paradigm. Jumps in prices and the effect on implied volatility.