Sieve estimation of option implied state price density

Abstract

The state price density, as a central concept in asset pricing, embodies rich information about market expectations and risk attitudes. The paper develops a nonparametric estimator for this density using a single cross section of European option prices. The estimator has two features that di erentiate it from other methods in the literature. First, it uses information from both call and put option prices. Second, it does not require estimating any second order derivative. The estimator is characterized by the solution to a constrained and penalized linear regression. The technical analysis faces two challenges because the density is de ned by the Fredholm integral equation of the rst kind with an unbounded support, and the kernel functions are unbounded and non-di erentiable. We address these challenges by exploiting the structure of the option pricing problem. After establishing the consistency and the convergence rate of the estimator, we apply it to estimate the state price densities implied by S&P500 index options and by the VIX options. The sample periods include the recent nancial crisis and the Great Recession, during which the market turbulence imposes substantial challenges for adaptive estimation. We show that the procedure can work with both daily and high frequency observations. We also study whether quantiles of this density have predictive power for future return distributions.