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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 …