Pricing Digital Outperformance Options with Uncertain Correlation

Multi-asset options exhibit sensitivity to the correlations between the underlying assets and these correlations are notoriously unstable. Moreover, some of these options such as the digital outperformance options, have a cross-gamma that changes sign depending on the relative evolution of the underlying assets. In this paper, I present a model to price digital outperformance options when there is uncertainty about correlation, but it is assumed to lie within a certain range. Under the assumption that assets prices follow a Geometric Brownian motion with constant instantaneous volatilities I present an analytic expression for the price of the digital outperformance option under the constant correlation assumption, as well as the partial differential equation corresponding to the uncertain correlation model. The comparison of the prices obtained using both models shows that there is no constant correlation which allows attaining the price obtained under the uncertain correlation model. This fact shows that it can be dangerous to assume a constant instantaneous correlation for products with a cross-gamma that changes sign