Risk neutral and real world densities for the S&P 500 index during the crisis period from 2008 to 2009

Abstract

Risk neutral and real world densities derived from option prices provide rich source of information for future asset price forecast. Three approaches (mixtures of two lognormals, jump diffusion models and implied volatility function models) are used to estimate risk neutral densities. Both power utility function and beta function are used to transform mixtures of two lognormal risk neutral densities into real world densities. Transformations are estimated by maximizing the likelihood of observed index levels. Results for the S&P 500 index indicate that two parametric methods, especially the jump diffusion models are preferable than implied volatility function methods. The log-likelihood tests cannot reject the hypothesis that there is no risk premium for both year 2008 and year 2009.