Calibration, local and stochastic volatility

S. de Marco

Local volatility (LV) models: option replications, valuation PDEs. Some properties of prices and hedging in an LV model: monotonicity with respect to the volatility of convex payoff option prices and consequences for the hedging error.

Forward PDE for calls/puts in an LV model and applications to model calibration. Dupire’s formula and its proof. Extension to regular and arbitrage-free call/put price surfaces. Markov projection of a stochastic volatility model and link with Gyongy’s theorem.

Implied variance and volatility. Non-arbitrage conditions on the volatility surface, asymptotic properties. SVI and SSVI parameterizations.

Other volatility market instruments: variance swaps. Forward variance swaps and the notion of forward variance. The VIX index. First generation stochastic volatility models and their treatment. An example: the Heston model. Models of forward variance (according to Bergomi).

A precise bibliography will be given in class.