A Model-Free Fourier Cosine Method for Estimating the Risk-Neutral Density

Abstract

In this article, we present a new nonparametric method to extract the risk-neutral density from market-observed options prices. The method is based on novelly combining the Fourier cosine series method and the Carr-Madan spanning formula. In contrast to the seminal Breeden-Litzenberger formula, which is based on twice differentiating the options prices with respect to the strikes, our method is based on integrating the options prices with respect to available strikes at a given maturity. We employ the Black-Scholes model, constant elasticity of variance model, and the Heston model as data-generating processes in the numerical experiments, and real market data in the empirical experiments. They demonstrate that the proposed method is accurate, is highly efficient to evaluate, and compares favorably with existing methods in the literature.

TOPICS: Derivatives, options, quantitative methods, statistical methods, performance measurement

Key Findings

• ▪ We present a new model-free method to extract the risk-neutral density from options prices.

• ▪ The formula is explicit and does not involve interpolation or intermediate numerical procedures.

• ▪ Numerical and empirical experiments confirm the accuracy of the formula. It also compares favorably with existing literature.