The need of considering price dynamics alternative to the classical Black-Scholes model for derivatives pricing is widely known. The stochastic variability of market parameters and in particular the empirical evidence of non constant surfaces of implied volatility in real markets require more realistic models for the assets dynamics. Many approaches are available to obtain a better fitting of m...

Implied volatility expansions allow calibration of sophisticated volatility models. They provide an accurate fit and parametrization of implied volatility surfaces that is consistent with empirical observations. Fine-grained higher order expansions offer a better fit but pose the challenge of finding a robust, stable and computationally tractable calibration procedure due to a large number of m...

This thesis consists of two parts, one concerning implied volatility and one concerning local volatility. The SABR model and SVI model are investigated to model implied volatility. The performance of the two models were tested on the Eurcap market in March 2008. Two ways of extracting local volatility are reviewed by a test performed on data from European options based on the S&P 500 index. The...

We derive the real Profit&Loss(P&L) that accrues in what option traders do every day: continuously Delta (∆)-hedge a call, by substituting the option’s running implied volatility (generally stochastic) into the Black-Scholes(BS) ∆-formula. The result provides formal justification for a heuristic rule-of-thumb that a trader’s P&L on continuously ∆-hedged positions is his vega times times the dai...