Derivatives Models on Models

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Why So Negative About Negative Probabilities?^*

What is the probability of the expected being neither expected nor unexpected?

1 The History of Negative Probability

In finance negative probabilities are considered nonsense, or at best an indication of model-breakdown. Searching the finance literature the comments I found on negative probabilities were all negative,¹ see for example Brennan and Schwartz (1978), Hull and White (1990), Derman, Kani and Chriss (1996), Chriss (1997), Rubinstein (1998), Jorgenson and Tarabay (2002) and Hull (2002). Why is the finance society so negative about negative probabilities? The most likely answer is simply that we “all” were taught that probabilities by definition must be between 0 and 1, as assumed in the Kolmogorov measure theoretical axioms. Our negativity about negative probabilities might be as short sighted as if we decided to limit ourselves to consider only positive money. I am not the first one to think that negative probabilities can be useful. Negative probabilities were to my knowledge first introduced by Paul Dirac. Dirac is probably best known for his mathematical prediction of antimatter, a feat for which he was awarded the Nobel prize in 1933. In 1942 Paul Dirac wrote a paper: “The Physical Interpretation of Quantum Mechanics” where he introduced the concept of negative energies and negative probabilities:

Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, ...