Two unconditionally implied parameters and volatility smiles and skews

Abstract

The paper studies estimation of parameters of diffusion market models from historical data. The standard definition of implied volatility for these models presents its value as an implicit function of several parameters, including the risk-free interest rate. In reality, the risk free interest rate is unknown and need to be forecasted, because the option price depends on its future curve. Therefore, the standard implied volatility is {\it conditional}: it depends on the future values of the risk free rate. We study two implied parameters: the implied volatility and the implied average cumulative risk free interest rate. They can be found unconditionally from a system of two equations. We found that very simple models with random volatilities (for instance, with two point distributions) generate various volatility smiles and skews with this approach.