Estimation of risk‐neutral density surfaces

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 risk‐neutral densities associated with several maturities. 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&P 500 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.