The Empirical Performance of Option-Based Densities of Foreign Exchange

Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to stimulate discussion and critical comment on research in progress. They may not have been subject to the formal editorial review accorded official Federal Reserve Bank of Cleveland publications. The views stated herein are those of the authors and are not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System, the Bundesbank, or Oestereichische Nationalbank. In this paper, we calculate risk-neutral densities (RND) by estimating the daily diffusion process of the underlying futures contract for foreign exchange, based on the price of the American puts and calls reported on the Chicago Mercantile Exchange for the end of the day. Our quick and accurate method of calculating the prices of the American options uses higher-order lattices and smoothing of the option's value function at the boundaries to mitigate the nondifferentiability of the payoff boundary at expiration and the early exercise boundary. We estimate the diffusion process by minimizing the squared distance between the calculated prices and the observed prices in the data. We also test whether the densities provided from American options provide a good forecasting tool. We use a nonparametric test of the densities that depends on inverse probabilities. We modify the test to compensate for an inherent problem that arises from the time-series nature of the transformed variables when the forecasting windows overlap. We find that the densities based on the American option prices for foreign exchange do considerably well for the longer time horizons.