Natural Cubic Spline Approximation of Risk-Neutral Density

Abstract

The risk-neutral density is a fundamental concept in pricing financial derivatives, risk management, and assessing financial markets’ perceptions over significant political or economic events. In this paper, we propose a new nonparametric method for estimating the risk-neutral density using natural cubic splines (NCS). The estimated density is twice continuously differentiable with linear tails at both ends. Our method targets the logarithm of the underlying asset price, releasing the restriction to the positive domain. We theoretically prove the consistency of our NCS method. We conduct a comprehensive empirical study comparing the proposed NCS method with a piecewise constant method, a uniform quartic B-spline method, and a cubic spline method from the literature using 20 years of S&P 500 index option data. The empirical results show that our NCS method is more robust than the piecewise constant method, which can only produce a discontinuous density, especially for options with maturities longer than six months. Moreover, our NCS method outperforms other historical continuous methods in terms of optimization feasibility and option price estimation.

Keywords

• constrained optimization
• cubic splines
• option price
• risk-neutral density
• weighted least square loss