Estimation of risk-neutral measures using quartic B-spline cumulative distribution functions with power tails

Abstract

In this paper, we propose the B-spline (BSP) method, which overcomes problems with the smoothed implied volatility smile (SML) method for estimating option implied risk-neutral measures (RNMs). We model the risk-neutral cumulative distribution function (CDF) using quartic B-splines with power tails so that the resulting risk-neutral probability density function (PDF) has continuity and arbitrage-free properties. Since the number of knots is selected optimally in constructing the quartic B-spline risk-neutral CDF, our method avoids both overfitting and oversmoothing. To improve computational efficiency and accuracy, we introduce a three-step RNM estimation procedure that transforms a nonlinear optimization problem into a convex quadratic program. Monte-Carlo experiments and applications to S&P 500 index options suggest that the BSP method performs considerably better than the SML method. The BSP method always produces arbitrage-free RNM estimators and almost perfectly recovers the actual risk-neutral PDFs for various hypothetical distributions.