Pricing Bounds for Bivariate Options with Super- and Submodular Payoff FunctionPublic Deposited
We first introduce the concept of copulas and advocate its use for multivariate option pricing. We focus on four types of bivariate options: basket, rainbow-max, rainbow-min, and spread options. We derive expressions for these options as a function of the copula. We then construct pricing bounds for these bivariate options without imposing any structure at all on the bivariate risk-neutral distribution. Specifically, we introduce the concept of extreme pricing bounds, which are the bounds a bivariate option has to satisfy in order to avoid arbitrage with the univariate options and futures that are trading in the market place. We show that these four types of bivariate options have a super- or submodular payoff function, and that as a result the extreme pricing bounds correspond the Frechet-Hoeffding bounds on the copula. We then proceed to investigate how we can extract partial information about the bivariate risk-neutral distribution from products that are trading in the market place. It is important that when we price a new bivariate option its price is consistent with the products that are currently being traded. We introduce both non-parametric and parametric methods to extract this partial information. Finally, we show how this partial information about the dependence between two assets translates into tighter pricing bounds.