Model Uncertainty, Fréchet Bounds and Applications in Option Pricing
We consider the problem of finding arbitrage bounds for option prices of multi-asset options (i.e. options on multiple underlyings) in the case when partial information of the assets' probability distribution is available. We focus on the case in which the one-dimensional marginal distribution of each individual asset is known while also partial information on the dependence structure between the assets is available. This is in the literature often referred to as dependence uncertainty. The problem has been extensively studied in the two-asset case for which solutions were given by Tankov (2011) and Bernard et al. (2012). We generalize these results for options that depend on more than two assets. The solution is based on an improvement of the classical Fréchet-Hoeffding bounds that allows for a representation of partial information of the dependence structure. By an extension of the results of Müller and Stoyan (2003) on multivariate stochastic dominance we are able to show that the improved bounds can be interpreted as minimal or maximal distributions with respect to the lower orthant order. The link between the lower orthant order on the set of distribution functions and the prices of multi-asset options is established via a multivariate partial integration formula.