The Black-Scholes Formula.

Beautiful Maths

Sorry but it’s a long one!

SINCE THE BEGINNING OF CAPITALISM THERE HAS BEEN ONE GOLDEN RULE; IF YOU WANT TO GET RICH YOU HAVE TO TAKE RISKS.

In the completely unsystematic and chaotic world of the financial markets, risk is perhaps the only certainty. Every individual asset carries its own risk, which makes predicting price movement extremely difficult.

I want to focus our attention on an era of mathematician lead economics. In the 1930s there was an influx of mathematician’s study, to see whether financial traders did actually know how to predict market movement, or if profit making was simply down to luck. In fact this study dates back even further. In the early 1900s a young and extremely intelligent French Graduate (Louis Bachelier) began to look at the financial markets from a mathematical perspective. He attempted to find an equation that could remove risk from a financial transaction. His vigorous studies lead him to first signs of an OPTIONS CONTRACT, Whereby traders could remove the downside of potential positions by placing trades in the future, (If the asset’s price doesn’t reach the level at which you have placed your trade, you can choose to opt out, and your only loss is simply the capital cost of the contract.)

Traders went mad for these options. However a new issue was raised; HOW DO YOU PRICE AN OPTION?

Mathematicians took it upon themselves to attempt to equate this problem; they tried to account for human emotions such as expectations. BUT their efforts were in vain. It’s simply impossible to make something like human emotion completely quantifiable.

SO essentially we’re back to square one.

ENTER: Myron Scholes and Fisher Black. In 1968 these two talented mathematicians built a model that satisfied every component of an options contract, they accounted for (price, volatility, maturity and interest rates.) Yet they were left with one variable; RISK.

HOW DO YOU ACCOUT FOR RISK?

They began by removing all equations that attempted to solve human emotions. They soon realised their predecessors were insane for even trying to solve this.

And soon they found the answer:

DYNAMIC HEDGING- By cancelling out market movements by placing trades in the opposite direction, uncertainty was eliminated. Furthermore the use of the notion “Continuous Time” (used to calculate trajectory of rocket ships) they could continuously recalculate prices, to keep up with market volatility and consequently RISK was completely removed.

With the help of Bob Merton, the Black-Scholes formula to price options was released onto the markets, and the effects were simply outrageous!

Traders around the world began to use the Black-Scholes formula, to great effect. Massive profits were made, and eventually Bob Merton and Myron Scholes were awarded the highest achievement; The Nobel Prize.

Greed soon took over…. Bob and Myron decided it was time to make some money. They set up Long Term Capital Management (LTCM) in 1995. Needless to say the oversubscribed hedge-fund found that raising capital was extremely easy. Everyone from the biggest Investment Banks to Pension Funds wanted in. In 3 short months LTCM had raised $3bn and it was ready to go!

LTCM outperformed any other investment firm. In its 1^st year it saw returns of 20%, in its 2^nd-40% and it’s 3^rd-30%. In context these returns were incredible.

HOWEVER…In 1997 the Asian recession hit global markets hard, market dynamics were changed and this proved to be a great threat to the Black-Scholes formula. BUT instead of easing up, LTCM began to expand. It took on debts of $100bn. To put this figure into context; 3% of losses would leave LTCM insolvent. Myron and Merton stuck to their guns and continued to trust their formula. UNTILL AUGUST 1998. Russia, the world’s largest country defaults on its debt, and leaves global markets in turmoil. The Black-Scholes formula didn’t account for systemic risk like this and consequently the model became useless, and LTCM began to lose, BIG TIME. Just 4 days after the Russian default LTCM recorded $500 million of losses…IN ONE DAY.

LTCM soon became insolvent, and consequently a massive gap in the global financial system began to open. All the investors who had trusted LTCM were recording massive losses. The US Federal Reserve had no choice but to bail out the firm for $3.5bn.

LTCM and Black-Scholes confirms that risk cannot be removed from any equation, you simply can’t prepare for systemic risk. Although the Black-Scholes formula was beautiful to look at…it proved to be disastrous when applied to the CRAZY WORLD OF FINANCE.

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