Recovering Risk-Neutral Probability Density Functions from Options Prices using Cubic Splines

We present a new approach to estimate the risk-neutral probability density function (pdf) of the future prices of an underlying asset from the prices of options written on the asset. The estimation is carried out in the space of cubic spline functions, yielding appropriate smoothness. The resulting optimization problem, used to invert the data and determine the corresponding density function, is a convex quadratic or semidefinite programming problem, depending on the formulation. Both of these problems can be efficiently solved by numerical optimization software. In the quadratic programming formulation the positivity of the risk-neutral pdf is heuristically handled by posing linear inequality constraints at the spline nodes. In the other approach, this property of the risk-neutral pdf is rigorously ensured by using a semidefinite programming characterization of nonnegativity for polynomial functions. We tested our approach using data simulated from Black-Scholes option prices and using market data for options on the S&P 500 Index. The numerical results we present show the effectiveness of our methodology for estimating the risk-neutral probability density function. Faculdade de Economia, Universidade de Coimbra, Av. Dias da Silva, 165, 3004-512 Coimbra, Portugal ([email protected]). Department of Mathematical Sciences, 6113 Wean Hall, Carnegie Mellon University, Pittsburgh, PA 15213, USA ([email protected]). Support for this author was provided by the National Science Foundation under grants CCR-9875559 and DMS-0139911. Departamento de Matemática, Universidade de Coimbra, 3001-454 Coimbra, Portugal ([email protected]). Support for this author was provided by Centro de Matemática da Universidade de Coimbra, by FCT under grant POCTI/35059/MAT/2000, by the European Union under grant IST-2000-26063, and by Fundação Calouste Gulbenkian. The author would also like to thank the IBM T.J. Watson Research Center and the Institute for Mathematics and Its Applications for their local support.