Determination of the Probability Distribution Measures from Market Option Prices Using the Method of Maximum Entropy in the Mean

Abstract

We consider the problem of recovering the risk-neutral probability distribution of the price of an asset, when the information available consists of the market price of derivatives of European type having the asset as underlying. The information available may or may not include the spot value of the asset as data. When we only know the true empirical law of the underlying, our method will provide a measure that is absolutely continuous with respect to the empirical law, thus making our procedure model independent. If we assume that the prices of the derivatives include risk premia and/or transaction prices, using this method it is possible to estimate those values, as well as the no-arbitrage prices. This is of interest not only when the market is not complete, but also if for some reason we do not have information about the model for the price of the underlying.

Key Words:

• Risk-neutral measure reconstruction
• maximum entropy in the mean
• inverse problems

Acknowledgements

This research was funded by the grant MCyE CO 2009-107960 from the Department of Business Administration, Universidad Carlos III de Madrid, Madrid, Spain. We want to thank the referees for their comments which contributed to improve our presentation considerably.