ESTIMATING FORWARD LOOKING DISTRIBUTION WITH THE ROSS RECOVERY THEOREM

The payoff of option is determined by the future price of underlying asset and therefore the option prices contain the forward looking information. Implied distribution is a forward looking distribution of the underlying asset derived from option prices. There are a lot of studies estimating implied distribution in the risk neutral probability framework. However, a risk neutral probability generally differs from a real world probability, which represents actual investors view about asset return. Recently, Ross (2015) has showed remarkable theorem, named “Recovery Theorem”. It enables us to estimate the real world probability distribution from option prices under a particular assumption about representative investor's risk preferences. However, it is not easy to derive the appropriate estimators because it is necessary to solve an ill-posed problem in estimation process. This paper discusses about the method to estimate a real world distribution accurately with the Recovery Theorem. The previous studies propose the methods to estimate the real world distribution, whereas they do not investigate on the estimation accuracy. Hence, we test the effectiveness of the Tikhonov method used by Audrino et al. (2015) in the numerical analysis with hypothetical data. We propose a new method to derive the more accurate solution by configuring the regularization term considering prior information and compare it with the Tikhonov method. Moreover, we discuss regularization parameter selection to get the accurate real world distribution. We find the following three points through the numerical analysis. (1) To stabilize the solution by introducing regularization term is an effective method in terms of estimating a real world distribution with the Recovery Theorem. (2) Proposed method can estimate a real world distribution more accurately than the Tikhonov method. (3) We can offer the appropriate solutions even if the number of maturities is less than that of states.