Closed-Form Transformations from Risk-Neutral to Real-World Distributions

Abstract

Risk-neutral (RN) and real-world (RW) densities are derived from option prices and risk assumptions, and are compared with historical densities obtained from time series. Two parametric methods that adjust from RN to RW densities are developed, firstly a CRRA risk aversion transformation and secondly a statistical calibration. Both risk transformations are estimated using likelihood techniques, for two flexible but tractable parametric density families. Results for the FTSE-100 index show that parametric densities derived from option prices have more explanatory power than historical densities. The parametric densities also have higher likelihoods than nonparametric densities estimated by spline methods. Furthermore, the pricing kernel between RN & historical densities is only incompatible with a risk averse representative agent when spline methods provide the RN densities.