Risk Neutral Valuations Based on Partial Probabilistic Information

Abstract

In a viable single-period model with one stock and k ≥ 2 scenarios the completeness of the market is equivalent to the uniqueness of the risk neutral probability; this equivalence allows to price every derivative security with a unique fair price. When the market is incomplete, the set of all possible risk neutral probabilities is not a singleton and for every non attainable derivative security we have a bid-ask interval of possible prices. In literature, different methods have been proposed in order to select a unique risk neutral probability starting with the real world probability p. Contrary to the complete case, in all these models $\textbf{p}$ is really used for the option pricing and its elicitation is a crucial point for every criterion used to select a risk neutral probability. We propose a method for the valuation problem in incomplete markets which can be used when p is a partial conditional probability assessment as well as when we have different expert opinions expressed through conditional probability assessments. In fact, it is not always possible to elicit a probability distribution p over all the possible states of the world: the information that we have could be partial, conditional or even not coherent. Therefore we will select a risk neutral probability by minimizing a discrepancy measure introduced in [2] and analized in [3] between p and the set of all possible risk neutral probability, where p can be a partial conditional probability assessments or it can be given by the fusion of different expert opinions.