Recovering Risk-Neutral Probability Density Functions Using Gaussian Mixture Distributions

This paper presents a new approach to recover the implied risk-neutral probability density function (PDF) using Gaussian mixture distribution from options prices. Suppose the risk-neutral PDF is subjected to Gaussian mixture distribution, by minimizing the distance between the risk-neutral PDF and the physical PDF, the risk-neutral PDF is obtained. This method can avoid non-negative of the implied distribution. We test our approach using options prices data and prove the effectiveness of our methodology. The results show that the risk neutral PDF has excess kurtosis, and a bimodal feature; a smaller peak of the left tail suggests that the approximation of lognormal distribution of underlying assets will underestimate the possibility of the loss.