The Empirical Performance of Option Based Densities of Foreign Exchange
Risk neutral densities (RND) can be used to forecast the price of the underlying basis for the option, or it may be used to price other derivates based on the same sequence. The method adopted in this paper to calculate the RND is to firts estimate daily the diffusion process of the underlying futures contract for foreign exchange, based on the price of the American puts and calls reported on the Chicago Mercentile Exchange for the end of the day. This process implies a risk neutral density for each point of time in the future on each day. I order to estimate the diffusion process we need methods of calculating the prices of American options that are fast and accurate. The numercial problems posed by American options are tough. We solve the pricing of American options by using higher order lattices combined with smoothing the value function of the American Option at the boundaries in order to mitigate the non-differentiability of both the payoff boundary at expiration and the early exercise boundary. By calculating the price of an American option quickly, we can estimate the diffusion process by minimizing the squared distance between the calculated prices and the observed prices in the data. This paper also tests wheter the densities provided from American options provide a good forecasting tool. We use a non-parametric test of the densities that depends on inverse probabilities. A problem with the use of these tests in the past has been the time series nature of the transformed variables when the forecasting windows overlap. The inverse probability of the realized thirty day ahead spot at time t is correlated with the corresponding inverse probability at time t-1, because the development of the spot rate untill t shares twenty-nine days of history. We modify the tests based on the inverse probability function to account for this correlation between our random variables that are uniform distributed under the null hypothesis. We find that the densities based on the American option prices for foreign exchange do considerably well for the thirty to sixty day time horizon, but less well for the shorter horizons. The most sophisticated single state model of the diggusion process did best at the one-hundred-eighty day horizon.
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