Semi-Parametric Estimation of American Option Prices

Abstract

We introduce a novel semi-parametric estimator of the price of American options in a discrete time, Markovian framework. The estimator is based on a parametric specification of the stochasticdiscount factor and is non-parametric w.r.t. the historical dynamics of the state variables. The estimation method exploits the no-arbitrage conditions for the short-term risk-free bond, the underlying asset and a cross-section of observed prices of American options written on it. We use the dynamic programming representation of American option prices to make explicit the nonlinear restrictions on the model parameters coming from option data. We obtain an estimator of the transition density of the state variables process by minimizing a statistical measure based on the Kullback-Leibler divergence from a kernel-based transition density, subject to the no-arbitrage restrictions. We use the estimator to compute the price of American options not traded in the market by recursive valuation. Other functionals of the transition density interesting for financial applications can be estimated in a similar way.