Robust Estimation of Shape-Constrained State Price Density Surfaces

Abstract

Given a theoretical pricing model, an implied volatility can be extracted from an option’s market price. Given a set of options with the same maturity and a range of strike prices, it is possible to extract (an approximation to) the entire risk-neutral probability density without having to assume a theoretical pricing model. There are a variety of related methods to do this, but all are subject to certain problems, including the fact that the data never exist to allow full estimation of the tails. Some methods produce improper densities with negative portions. In this article, Ludwig introduces a neural network approach to extract risk-neutral densities from option prices, imposing only a small number of constraints, such as probabilities must be nonnegative and an option’s price must be above intrinsic value. The resulting densities are smooth and sensible, even for days that other approaches find extremely difficult to handle.