Estimation of Risk-Neutral Densities Using Positive Convolution Approximation

This paper proposes a new nonparametric method for estimating the conditional risk-neutral density (RND) from a cross-section of option prices. The idea of the method is to fit option prices by finding the optimal density in a special admissible set. The admissible set consists of functions, each of which may be represented as a convolution of a positive kernel with another density. The method is termed the Positive Convolution Approximation (PCA). The important properties of PCA are that it 1) is completely agnostic about the data generating process, 2) controls against overfitting while allowing for small samples, 3) always produces arbitrage-free estimators, and 4) is computationally simple. In a Monte-Carlo experiment, PCA is compared to several popular methods: mixtures of lognormals (with one, two, and three lognormals), Hermite polynomials, two regularization methods (for the RND and for implied volatilities), and sigma shape polynomials. PCA is found to be a promising alternative to the competitors.

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