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03 April 2013

The Misapplication of Gaussian Math in the Financial Sector

Business is about mathematics. Macro-economics is not, it is about people and reality. This is a fundamental flaw in economic thinking. And one that causes economic disasters one after another.

This is the formula for the Gaussian Distribution, better known as the “bell curve”. Most financial model up until very recently used this model. Many still do. It is used in the ubiquitous Black-Scholes option pricing model. 

However, because it is a Gaussian distribution, it does not properly account for risk (events) outside of 5σ, therefore pricing for far-out-of-the-money options are mis-priced.

In 2000, David Li published his “Li’s Function”, a type of Gaussian copula function.

Pr – Probability.. as in default. This is the variable that is solved for. In other words, The rest of the equation gives the answer for Pr
T – Time between now and when default is expected.
= - Equality used to eliminate uncertainty

F – Probability of survival 
Φ – Used to sum the probabilities of A and B 
γ – Used to reduce correlation.

Li’s Function was quickly adapted for use in pricing Collateralized Debt Obligations (CDOs) because it allowed investors to quantify risk. Unfortunately, it suffered from the same type of flaw as the Gaussian Distribution – it failed to account for unlikely events1.

Whatever other factors involved in the financial meltdown of 2008, the use of this formula was the singular major cause of investor losses. Because it failed to assign proper risk values, the actual risks were far greater than expected.

Even after the fiasco of 2008, modified versions of Li’s Function and other Gaussian Copula functions are still being used in financial circles to price multi-instrument products such as currency swaps.

Economists and financial advisers are still attempting to modify Gaussian mathematics and Game Theory to fit economic reality. But all attempts suffer the same weakness. It is impossible to account for all possibilities since the bounds of all possible things that could happen are infinite. This isn’t the case with Game Theory. The roll of a roulette Wheel or the deal of a card hand may be random, but they are bounded. There are a known number of slots on the roulette wheel, and a known number of possible card hands. The bounds of possibility in the real world, and by implication the financial world, are infinite and unknown. We do not know what possibly can happen to affect financial markets in the future. Therefore, if no one knows, how can risk be calculated? It can’t. And so the next big financial catastrophe is just around the corner.

Mainstream financial “gurus” (like Dave Ramsey and most financial advisers) mislead their clients by not taking into account the huge losses that happen regularly, but are not predicted or accounted for in their models. The fact that a lot of retirement accounts and other wealth were wiped out in 2008-12 is NOT unusual or dramatic. It is guaranteed to happen. Putting money into non-cash, non-good based instruments is reckless; no difference in essence than gambling.

Business is about mathematics. Macro-economics is not, it is about people and reality.

1. In fairness to Mr. Li, his function was theoretical, and may have never meant to be applied to real-world trades. We do not know, since Mr. Li returned to his native China and refuses to discuss the matter.