Continuing with my suggested readings for Machine Learning for Trading, we reviewed this documentary on the beautiful formula called “Black-Scholes Model”, its Nobel Prize victory and how a company (Long Term Capital Management) made a bet of 1 trillion dollars using that formula and failed gloriously.
A 1999 BBC documentary which tells the story behind possibly the greatest options pricing formula ever created in finance, the Black-Scholes-Merton model which won the creators — three economists — a Nobel Prize. The formula had a far reaching effect on the financial markets.
The Holy Grail of the “Traders” trade
The Holy Grail of trading is to create a portfolio with no risk at all. The formula believed at its core that two risky positions taken together can eliminate risk itself. But , when the creators used the formula themselves, its dark side was revealed.
Capitalism is all about becoming wealthy by taking risks and it all began as a scientific project to find a formula to become wealthy with no risk at all. LTCM started to loose around 100M each day and finally it lost almost 500M in a single day.
Ben Schwartz is a Trader in Chicago Financial Exchange which trades in complex products such as Currency derivatives and Interest rate swaps. Their prices are constantly fluctuating based on market sentiment. His job as a trader is to guess a stock’s future price — 10 seconds or 1 year or 10 years from now, and do this thousand more times in a day.
Millions of dollars change hands everyday, even upto six figures. There exists a cut throat competition and potential to lose millions by a single mistake. So any formula invented should be capable to beat a Trader’s instincts and also compete with the knowledge of a legendary trader.
High above the trading floor are seated — the acknowledged experts. Leo Melamed works in Chicago Mercantile Exchange and has been a trader for 30 years. He has an assistant in the pit who hand signals to the broker all the orders, who executes and gives confirmation. He is privy to other information not usually available to other traders on the floor.
Leo notes that even body language of traders, such as anxiety, fear, trade volume (amount of noise on the floor) are also information. His usual routine every day morning involves, identifying how the market (dozen markets) will be reacting, analyzing newspapers to determine direction/attitude (like a psychiatrist), and determining if this information is going to make the markets bullish or not. Leo strongly believes that
Success is based on human judgement and intuition — qualities which cannot be reduced to a formula.
The Academia Strikes back
On the other hand, Academics who study Mathematics, believed this was sheer luck. Professor Zvi Bodie from Boston University explains that flipping a coin for long successful turns may result in a long list of heads, but that does not mean that the person flipping has the ability to do so. This is just the luck of the toss.
This multi decade battle all began in the1930s when the academics tried to verify if traders can actually predict the market. In one such experiment, they picked a set of random stocks and shares.
To make sure it is truly random, they threw darts at the stock index from Wall Street Journal blindfolded.
At the end of the year, the portfolio of these random choices outperformed the prediction from the traders. This meant that the prices themselves should be moving at random. Therefore by definition, it is not possible to predict the outcome.
The usual premise was that if some individual made a fortune in the stock market, we always assumed that they knew something and the individual reinforces that as well declaring that they were indeed a genius. However, what we don’t see are the failing individuals (hundreds and millions of people) who make similar predictions but which never worked successfully.
Professor Merton Miller from University of Chicago explains this eloquently. If 10,000 people are looking at stocks and try to pick winners, odds are one of them got it right just by chance. It is a chance operation, though people are thinking that they are doing something purposeful, which they are not.
This attitude of the academics disgusted traders and continues even today.
This discovery of randomness galvanized the academics, since Mathematics is used to study randomness in everything such as population growth, weather etc., So they began a quest (Economics equivalent of race to the moon) — To find a rational and scientific way to tame the market using Probability distribution, statistical models etc.,
Professor Paul Samuelson (MIT) believed that mathematics of probability and statistics can be a skeleton key to help understand the nature of chance, to predict or possibly even control the outcome. For the next 15 years, using the laws of probability, the academics would study how the price of a stock fluctuated in the past and tried to predict the future price using probabilities and attempt to control risk itself. These predictions were nothing close to being accurate than an average local weather forecast.
The Missing Link
What was missing all through this time, was a more exact way to ensure someone against risk. In 1951, Prof. Paul Samuelson (MIT) found an earlier thesis by a French graduate “Louis Bachelier” from University of Paris student in 1900. He created a series of equations representing the first complete mathematical model of the market. His observations too noted that the markets moved at random and its impossible to predict their exact future.
But he had the most innovative spark for an obscure, almost magical financial contract called OPTION (to completely eliminate risk). If someone could create a formula that would allow this rare contract to be widely used, they would be able to tame the market completely. Unfortunately, he died before it could be completed.
- So the Academics decided that the next step needed was “to get the perfect formula to evaluate and price OPTIONS” and started investigating the instrument called OPTIONS in the market
What are these Options
In theory, Options are a form of financial insurance. With the ever present risk in the market, if you buy a stock today and the price drops tomorrow, you lose money. However, an OPTION contract gives the buyer the option to wait and buy a stock in the future if it reaches an agreed price. There is no obligation. If the price never reaches the predicted price, you can opt out and you lose only the cost of the OPTION.
In theory, OPTIONs are a great way to eliminate risk. However, how much would someone pay for such absolute peace of mind and this varies based on personal confidence of each individual investor. At that time, no one could agree on a standardized way to price these OPTION contracts. Traders hated this problem, while Academic relished such a bewildering problem.
Throughout 1960s, Academics attempted to build the mathematical model to represent the emotional confidence of the investors. They kept adding more and more symbols — Level of satisfaction, reasonableness and aggressiveness. Even symbols for the guesses of other traders, defensiveness, safety — creating a giant mathematical edifice with no positive outcome
But these are unobservable inputs in the real world — like expectations of investors. They even attempted to add people utility structure, Risk aversion etc., which made the final model look like psychotherapy than real science.
Black and Scholes
In 1968, Fischer Black and Myron Scholes tried to tackle the problem of OPTIONS. Though every stock price moves up and down and similarly the value of an OPTIONS contract also changes, there is no predictable relationship between the fluctuating stock price and the options price.
They wanted to find a formula to find the price of an OPTIONS contract for any given stock price at any given time. The existing literature was dissatisfactory as there were assumptions which did not always make sense.
Starting with the behemoth of the earlier built mathematical model, they started dropping unmeasurable symbols one by one, which the academics have been adding over the past years. Surprisingly, they did not affect the calculations at all. The bare bones of the formula which affects the pricing of an OPTIONS contract was finally visible.
The key ingredients of the Black-Scholes formula were Stock price, volatility, duration of the contract, interest rate and level of risk
Every element of the formula were all quantifiable except “Level of risk”
Thinking out of the box
So they started thinking laterally and in a moment of genius, decided that instead of finding the measure of risk exactly, they attempted to make it less significant by using the age old idea of hedging (hedging their bets in the opposite direction). They created a theoretical portfolio consisting of stocks and option contract. Whenever, either fluctuated up or down, they tried to cancel the movement by making another risky move in the opposite direction. Their aim was to keep the overall value of the portfolio “in perfect balance”
Since everything moves at random it was very difficult, in their initial experiments, only a small portion can be cancelled. But eventually using complex algebra and massive calculations, they found that they can balance the movement precisely. They called this “Dynamic Hedging”. At the end they were able to eliminate the uncertainty in the movement of the stock completely and were able to create a portfolio in perfect equilibrium where risk cancels themselves out.
Using “Dynamic Hedging” they were able to eliminate the unmeasurable risk component in determining the value of an option contract.
If I have the proper recipe for dynamically hedging, the net position is risk free and if we know the price of the stock, we can fix the price of the option.
There remained one last obstacle. The miracle breakthrough of the Black-Scholes formula required extremely fast calculations to perform Dynamic Hedging. Since they cannot keep up-to-date with the fast moving markets, their calculated position would be stale and irrelevant in comparison to the market’s pace.
There was a need to instantaneously recalculate the dynamic hedging to keep eliminating the risk based on stock price changes continuously. Unknown to Black-Scholes, someone has found the way.
Part 2 available here