Negative Probabilities in Financial Modeling

Abstract

We first define and derive general properties of negative probabilities. We then show how negative probabilities can be applied to modeling financial options such as Caps and Floors. In trading practice, these options are typically valued in a Black-Scholes-Merton framework assuming a lognormal distribution for the underlying interest rate. However, in some cases, such as the 2008/2009 financial crisis, interest rates can get negative. Then the lognormal distribution is inapplicable. We show how negative probabilities associated with negative interest rates can be applied to value interest rate options. A model in VBA, which prices Caps and Floors with negative probabilities, is available upon request. A follow up paper will address bigger than unity probabilities in financial modeling.