The Equations That Broke Wall Street in 2008

Before 2008, Wall Street’s most prized assets were maths. Researchers developed computational models that can assess risks and manage portfolios. The consensus was that finance had become too smart to fail.

The 1970s saw Fischer Black and Myron Scholes develop the Black-Scholes equation. It can determine the fair price of an option, which is a contract that gives investors the right to buy or sell an asset at a specific price:

This equation can determine the perfect price for the option where an investor can perfectly hedge the option by buying and selling in just the right time to completely eliminate risk. The model during peaceful market conditions allowed investors to trade derivatives, financial contracts that receive their values from another security (e.g, stock, bonds) rather than a physical asset. By 2007, the world was trading on trillions of dollars worth of derivatives that are created on one formula.

At the same time, the entire profitability framework of the financial system was built on a simple concept of exponential growth:

As Nick Bingham reflects on the model, “Capitalism is built on exponential growth. Exponential growth is highly unstable. So, therefore, is capitalism.” For at extremely large values of A₀ and t which are billions of dollars in the span of years, a minor change in rates can significantly alter the outcome of an investment. Banks utilized this to bundle more financial products, especially home mortgages as derivatives (mortgage-backed securities). During the housing bubble, banks sold these bundles to Wall Street banks, which advertised them to consumers as low-risk investments. Like compound growth, the system between investors and banks grew exponentially, more mortgages were being bundled as securities, more money was generated and so was the risk.

But how are banks not aware of the increasing risks of mortgage securities and still advertised them to investors as “low-risk” investments?

Perhaps the best model that can explain this behavior was the Gaussian copula by David Li. The formula calculates the probability of mass defaulting from borrowers at the same time:

Each u in the model accounts for a borrower while combining many borrowers together using correlation ρ, which is how often things move together. The approach allowed banks to group mortgages into securities effectively.

The fateful twist was that all of the mathematical models had converged together under one blissfully ignorant assumption—home prices never drop. The Black-Scholes equation had encouraged trillions of dollars of trading in derivatives, assuming that the market would continue growing and stay liquid. The exponential growth model pushed borrowing and financial bundling to extreme highs. On the latter, securities were rated as the safest (AAA) using the Gaussian copula without any defensives against price shocks. It gave bankers and investors a false sense of security, encouraging even more investments into unstable securities that were heading into uncertainty.

When interest rates rose to 5.25% in 2006, home prices started to fall, spiking correlation up to 0.8 instantly turning many safe securities into toxic investments. Homes were now worth less than what owners had originally paid for, forcing them to incur debt to lenders when they try to sell their homes. This relationship shows that while the cost for a home was rising, its value was falling. When many U.S. homeowners defaulted on their debt, banks and companies that issued the mortgage securities were left in a financial strain as they had no money to pay back investors. Suddenly, all the mathematical models shattered, losses multiplied as banks defaulted on their payments to companies, forcing them to default their debt to investors. Significant losses from the investments caused major panic among banks as many of them acquired “toxic-debt” which cannot be traced on traditional balance sheets. The money that was flowing between banks froze completely due to distrust, placing even more monetary strain on banks and inevitably caused many to file for bankruptcy even with aid. The death of many big financial institutions created ripple effects throughout the economy causing 3.8 million Americans to lose their homes and unemployment to reach highs of 10%.

The equations weren’t evil. It just had its own limitations and we happened to turn a blind eye to them. Investors disregarded the model’s weakness of capturing human emotions like fear and greed, that they have assumed that past patterns guaranteed future stability, and that people will exit rationally when panic settles in. Finance runs on the trust that everyone can fulfill their roles and when those erode, the market no longer follows the rules of mathematics.

After the 2008 crisis, modern financial models became more sophisticated, with some accounting for scenarios like price shocks. Yet, the lesson still remains: Numbers can never replace human judgement. When that distinction falls, it happens all over again.