# The mathematical equation that caused the banks to crash

They were priced and considered to be assets in their own right.

He had realised that stock prices moved at random and that it was impossible to make exact predictions about them, but Bachelier said he had also found a solution - through the pricing of a financial contract called an option.

It could do the unthinkable - it took the risk out of playing the money-markets.

It applies to the simplest and oldest derivatives: There are two main kinds. A put option gives its buyer the right to sell a commodity at a specified time for an agreed price. A call option is similar, but it confers the right to buy instead of sell.

The equation provides a systematic way to calculate the value of an option before it matures. Then the option can be sold at any time. The equation was so effective that it won Merton and Scholes the Nobel prize in economics.

Black had died by then, so he was ineligible. If everyone knows the correct value of a derivative and they all agree, how can anyone make money?

The formula requires the user to estimate several numerical quantities. But the main way to make money on derivatives is to win your bet — to buy a derivative that can later be sold at a higher price, or matures with a higher value than predicted. The winners get their profit from the losers. The world's banks lost hundreds of billions when the sub-prime mortgage bubble burst. In the ensuing panic, taxpayers were forced to pick up the bill, but that was politics, not mathematical economics.

The Black-Scholes equation relates the recommended price of the option to four other quantities. Three can be measured directly: This is the theoretical interest that could be earned by an investment with zero risk, such as government bonds. The fourth quantity is the volatility of the asset.

This is a measure of how erratically its market value changes. The equation assumes that the asset's volatility remains the same for the lifetime of the option, which need not be correct. Volatility can be estimated by statistical analysis of price movements but it can't be measured in a precise, foolproof way, and estimates may not match reality. The idea behind many financial models goes back to Louis Bachelier in , who suggested that fluctuations of the stock market can be modelled by a random process known as Brownian motion.

At each instant, the price of a stock either increases or decreases, and the model assumes fixed probabilities for these events. They may be equally likely, or one may be more probable than the other. It's like someone standing on a street and repeatedly tossing a coin to decide whether to move a small step forwards or backwards, so they zigzag back and forth erratically. Their position corresponds to the price of the stock, moving up or down at random.

The most important statistical features of Brownian motion are its mean and its standard deviation. The mean is the short-term average price, which typically drifts in a specific direction, up or down depending on where the market thinks the stock is going.

The standard deviation can be thought of as the average amount by which the price differs from the mean, calculated using a standard statistical formula.

For stock prices this is called volatility, and it measures how erratically the price fluctuates. On a graph of price against time, volatility corresponds to how jagged the zigzag movements look. Black-Scholes implements Bachelier's vision. It does not give the value of the option the price at which it should be sold or bought directly. It is what mathematicians call a partial differential equation, expressing the rate of change of the price in terms of the rates at which various other quantities are changing.

Fortunately, the equation can be solved to provide a specific formula for the value of a put option, with a similar formula for call options. The early success of Black-Scholes encouraged the financial sector to develop a host of related equations aimed at different financial instruments. Conventional banks could use these equations to justify loans and trades and assess the likely profits, always keeping an eye open for potential trouble.

But less conventional businesses weren't so cautious. Soon, the banks followed them into increasingly speculative ventures. Any mathematical model of reality relies on simplifications and assumptions. The Black-Scholes equation was based on arbitrage pricing theory, in which both drift and volatility are constant. This assumption is common in financial theory, but it is often false for real markets.

The equation also assumes that there are no transaction costs, no limits on short-selling and that money can always be lent and borrowed at a known, fixed, risk-free interest rate.

Again, reality is often very different. When these assumptions are valid, risk is usually low, because large stock market fluctuations should be extremely rare. An event this extreme is virtually impossible under the model's assumptions.

In his bestseller The Black Swan , Nassim Nicholas Taleb , an expert in mathematical finance, calls extreme events of this kind black swans. In ancient times, all known swans were white and "black swan" was widely used in the same way we now refer to a flying pig. But in , the Dutch explorer Willem de Vlamingh found masses of black swans on what became known as the Swan River in Australia.

So the phrase now refers to an assumption that appears to be grounded in fact, but might at any moment turn out to be wildly mistaken. Large fluctuations in the stock market are far more common than Brownian motion predicts.

The reason is unrealistic assumptions — ignoring potential black swans. But usually the model performed very well, so as time passed and confidence grew, many bankers and traders forgot the model had limitations. In theory options are a perfect way to get rid of risk, but there was a problem.

How much would someone pay for such absolute peace of mind? Bachelier believed that if someone could discover a formula that would allow option contracts to be widely used, they would be able to tame the markets completely, but he died before he could find it. By the end of the 60s, academics were no nearer to pricing options than they'd ever been.

But all this was about to change when Myron Scholes and his colleague Fischer Black set out to tackle the problem of options… At its simplest level, the Black Scholes formula could be used to hedge against losing any bet, by working out how to place another bet in the opposite direction. That way, you couldn't lose.

The formula had the almost magical ability to allow you to make a fortune with the minimum of risk. But there was one problem. In the time it took to make the calculation, the fast moving markets had moved on and the calculation would effectively be out-of-date.

However, unbeknown to them, the problem had already been solved by a financial genius called Bob Merton.

Using an idea taken from rocket science, the value of an option could now be constantly recalculated and the risk eliminated continually. Relying on mathematics, the company traded and borrowed on a scale never seen before. But the mathematical model was based on normal market behaviour and unforeseen events were about to send the markets wild.

## Related Documentaries

Finance, money and stock market documentary - The Midas Formula. The Black--Scholes or Black--Scholes--Merton is a mathematical model of a financial market c.

It is the insights of the model, as exemplified in the Black–Scholes formula, that are frequently used by market participants, as distinguished from the actual prices. These insights include no-arbitrage bounds and risk-neutral . The Black Scholes formula is a model for pricing only, just like DCF. As with any model/theory, the assumptions are everything. Often, the assumptions (or the market's inputs as far as assumptions) are wrong, so you can find many different ways to profit if you understand the model. I love options, so useful.