Bounds for Probabilities of Extreme Events Defined by Two Random Variables

This paper offers a methodology for calculating optimal bounds on tail risk probabilities by deriving upper and lower semiparametric bounds, given only the first two moments of the distribution.We apply this methodology to determine bounds for probabilities of two tail events. The first tail event occurs when two financial variables simultaneously have extremely low values. The second occurs when the sum of two financial variables takes a very low value. In both cases we are finding bounds for actual or physical probabilities of these events rather than probabilities for a pricing or risk neutral measure. We use sum of squares optimization programs to obtain the desired bounds. To illustrate our ideas, we present several numerical examples. This approach is suitable in the situations when it is difficult to make exact distributional assumptions due to, for instance, scarcity and/or high volatility of data. Even in the situations when distributional assumptions can be made, this approach can be used to check the consistency of those assumptions.