A Nonparametric Estimate of the Risk-Neutral Density and Its Applications

The risk-neutral density for a future payoff of an asset can be estimated from market option prices that expire on the same date. We reformulate the estimation problem into a double-constrained optimization problem to determine its parameters, which can be efficiently solved using numerical implementations in R. Our proposed nonparametric approach for estimating the risk-neutral density using a step function shows promising results. Firstly, it can recover the risk-neutral density very well with market option prices. Secondly, it provides accurate estimates for option prices with any strike, which further presents a practical way to identify profitable investment opportunities in financial markets. We evaluate our method using options written on S&P 500 over twenty years. The cross-validation study shows that our method performs much better than the cubic spline method proposed in the literature. As an application, our approach can reproduce the market prices of long-term variance swaps reasonably well.