Option price data is often used to infer risk-neutral densities for future
prices of an underlying asset. Given the prices of a set of options on the same
underlying asset with different strikes and maturities, we propose a nonparametric
approach for estimating the evolution of the risk-neutral density in time. Our
method uses bicubic splines in order to achieve the desired smoothness for the
estimation and an optimization model to choose the spline functions that best fit the
price data. Semidefinite programming is employed to guarantee the nonnegativity of
the densities. We illustrate the process using synthetic option price data generated
using log-normal and absolute diffusion processes as well as actual price data for
options on the S&P500 index.
We also used the risk-neutral densities that we computed to price exotic options
and observed that this approach generates prices that closely approximate the
market prices of these options.FCT POCI/MAT/59442/2004, PTDC/MAT/64838/2006
