Larger than One Probabilities in Mathematical and Practical Finance

Traditionally probability is considered as a function that takes values in the interval [0, 1]. However, researchers found that negative, as well as larger than 1 probabilities could be a useful tool in making financial modeling more exact and flexible. Here we show how larger than 1 probabilities could be handy for financial modeling. First, we define and mathematically rigorously derive the properties of larger than 1 probabilities based on their frequency interpretation. We call these probabilities inflated probabilities because conventional probabilities are never larger than 1. It is transparently demonstrated that inflated probabilities emerge in various real life situations. We then explain how inflated 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, which assumes a lognormal distribution for the price of underlying assets. Since negative values are not defined in the lognormal framework, negative interest rates cannot be modeled. However interest rates have been negative several times in financial practice in the past. We show that applying inflated probabilities to the Black-Scholes-Merton model implies negative interest rates. Hence with this extension, Caps and Floors with negative interest rate can be conveniently modeled closed form.