OPTION SURFACE STATISTICS WITH APPLICATIONS

At each maturity a discrete return distribution is inferred from option prices. Option pricing models imply a comparable theoretical distribution. As both the transformed data and the option pricing model deliver points on a simplex, the data is statistically modeled by a Dirichlet distribution with expected values given by the option pricing model. The resulting setup allows for maximum likelihood estimation of option pricing model parameters with standard errors that enable the testing of hypotheses. Hypothesis testing is then illustrated by testing for the consistency of risk neutral return distributions being those of a Brownian motion with drift time changed by a subordinator. Models mixing processes of independent increments with processes related to solutions of Ornstein–Uhlenbeck (OU) equations are also tested for the presence of the OU component. Solutions to OU equations may be viewed as processes of perpetual motion responding continuously to their past movements. The tests support the rejection of Brownian subordination and the presence of a perpetual motion component.