Sieve estimation of option-implied state price density

This paper proposes a nonparametric estimator for the state price density implied by a single cross-section of European options with different strikes and the same maturity. The proposed estimator has two distinctive features. First, it extracts information from both call and put options, as opposed to only call options. Second, it does not require estimating any second-order derivative; instead, it solves a constrained and penalized linear regression. The asymptotic analysis faces two challenges because the state price density is defined by a Fredholm integral equation of the first kind with an unbounded support, and the kernel function is unbounded and non-differentiable. We address these challenges by exploiting the structure of the option pricing problem. After establishing the estimator’s consistency and convergence rate, we apply it to estimate the state price densities implied by the S&P500 index options and those by the VIX options. The sample period includes the recent financial crisis and the Great Recession, during which the turbulent market conditions imposed substantial challenges on the estimation. We show that the estimator can work with both daily and high-frequency observations. We also study whether the various features of this density can predict future asset returns and obtain positive findings. Finally, we apply the method to examine the causal effects of monetary policy announcements on the financial market, using high-frequency data.