Inversion of option prices for implied risk-neutral probability density functions: general theory and its applications to the natural gas market

Abstract

This paper applies inverse theory to the estimation of the implied risk-neutral probability density function (PDF) from option prices. A general framework of inverting option prices for the implied risk-neutral PDF is formulated from the option pricing formula of Cox and Ross [J. Financial Econ., 1976, 3, 145–166]. To overcome the non-uniqueness and instability inherent in the option inverse problem, the smoothness requirement for the shape of the PDF and a prior model are introduced by a penalty function. Positivity constraints are included as a hard bond on the PDF values. The option inverse problem then becomes a non-negative least-squares problem that can be solved by classic methods such as the non-negative least-squares program of Lawson and Hanson [Solving Least Squares Problems, 1974 (Prentice-Hall: Englewood Cliffs, NJ)]. The best solution is not the one that gives the best fit to the observed option prices or provides the smoothest PDF, but the one that gives the optimal trade-off between the goodness-of-fit and smoothness of the estimated risk-natural PDF. The proposed inversion technique is compared with the models of Black–Scholes (BS), a mixture of two lognormals (MLN), Jarrow and Rudd's Edgeworth expansion (JR), and jump diffusion (JD) for the estimation of the PDF from the option prices associated with the September 2007 NYMEX natural gas futures. It is found that the inversion technique not only gives the best goodness-of-fit, but also a significantly better model resolution. An empirical study for the last three months of the September 2007 futures contract shows that the shapes of the estimated PDFs become more symmetric as the futures contract becomes closer to the expiration date. The dispersion of the estimated PDFs decreases with decreasing time to expiry, indicating the resolution of uncertainty with passing time.

Keywords:

• Risk neutral distributions
• Option pricing
• Parameter estimation techniques
• Goodness of fit tests
• Inverse theory

JEL Classification::

• C1
• C5
• C13
• C14
• C53
• G1
• G13

Acknowledgements

We would like to thank three anonymous referees for their critical and constructive reviews of the manuscript. Yijun Du and Christine Wang thank Dan Kuehn and Jim Gleitman for their support throughout the course of the project.