Common Functional Implied Volatility Analysis

Trading, hedging and risk analysis of complex option portfolios depend on accurate pricing models. The modelling of implied volatilities (IV) plays an important role, since volatility is the crucial parameter in the Black-Scholes (BS) pricing formula. It is well known from empirical studies that the volatilities implied by observed market prices exhibit patterns known as volatility smiles or smirks that contradict the assumption of constant volatility in the BS pricing model. On the other hand, the IV is a function of two parameters: the strike price and the time to maturity and it is desirable in practice to reduce the dimension of this object and characterize the IV surface through a small number of factors. Clearly, a dimension reduced pricing-model that should reflect the dynamics of the IV surface needs to contain factors and factor loadings that characterize the IV surface itself and their movements across time.