Reconstructing the Joint Probability Distribution From Basket Prices: A Mildly Ill-Posed Problem

Abstract

In this thesis, we study basket options which are contracts on N assets. It is known that the risk neutral joint probability density of the assets is uniquely defined by the prices of basket options with positive weights. However, the proof of this fact requires a step involving the inversion of a Laplace transform. It is known that inversion of a Laplace transform is a severely ill-posed problem. The purpose of this thesis is to understand whether inversion of the basket transform is ill-posed or not.In our dissertation, we propose three methods to determine and prove the order of ill-posedness for the case when the basket contains only one asset and then map the problem to Radon transform to address two or higher dimensional case. Our proofs show that that problem is mildly ill-posed of order 1/2 (N -1) 2 given N dimensions.We also conduct numerical experiments under the Black-Scholes setting. First, using the reconstruction method from Cohen and Pagliacci, our results show that this method functions well in one-dimension but it becomes numerically intensive in our two-dimensional implementation, and we were unable to achieve convergence. Alternatively, we have looked at using an inverse Radon transform method to reconstruct the joint density numerically. As this requires knowledge of basket prices with negative weights, we have tried extending the basket price in this region as an odd function. For comparison, we compared this against the reconstruction from approximate basket prices calculated directly in the negative weight region. It is fascinating that these two approaches give similar results. This may indicate that the errors are due to the two-moment matching approximation rather than the odd extension. This intriguing possibility would certainly merit further investigation if more time were available.We believe that our mathematical particularly, but also our numerical results indicate there is cause for optimism that a stable numerical algorithm to reconstruct the joint density function from basket prices will be found in the future.