The Black-Scholes Model

"Once a model has been developed, we are able to improve the realism of its assumptions step by step. But unlike physics, which is a science with constant (if poorly understood) laws, the "laws" of economics and finance change constantly, even as we discover them. Sometimes they change because we have discovered them." - Charles Sanford: The Risk Management Revolution

Origins of the Black-Scholes Formula

The roots of the Black-Scholes formula go back to the nineteenth century. In the 1820s, a Scottish scientist, Robert Brown, observed the motion of pollen suspended in water and noticed that the movements followed no distinct pattern, moving randomly, independent of any current in the water. This phenomenon came to be known as Brownian motion. Similar versions of Brownian motion were subsequently discovered by other scientists studying other natural phenomena. In 1990, a French doctoral student Louis Bachelier, wrote his dissertation on the pricing of options in the Paris market and developed a model strikingly similar to the Black-Scholes model. Unfortunately, his dissertation advisor was disappointed because Bachelier's work was orientated toward such a practical issue as pricing a financial instrument. Although Bachelier received his degree, the less than enthusiastic support of his advisor damaged his career, and nothing further was heard from him.

In the early twentieth century, Albert Einstein, working on the foundations of his theories relativity, used the principles of Brownian motion to explain movements in molecules. This work led to several research papers that earned Einstein the Nobel Prize and world renown. By that time, a fairly well developed branch of mathematics, often attributed to American mathematician Norbert Wiener, proved useful for explaining the movements of random particles. Other contributions to the mathematics were made by Japanese mathematician Kiyoshi Ito. In 1951, Ito developed an extremely important result called Ito's Lemma that 20 years later made it possible to find an option price. Keep in mind, however, that these people were working complex problems in physics and mathematics, not finance.

The mathematics that was used to model random movements had now evolved into its own subdiscipline, which came to be known as stochastic calculus. While ordinary calculus defined the rates of change of known functions, stochastic calculus defined the rates of change of functions in which one or more terms were random but behaved according to well-defined rules of profitability.

In the late 1960s, Fischer Black finished his doctorate in mathematics at Harvard. Passing up a career as a mathematician, he went to work for Arthur Little, a management consulting firm in Boston. Black met a young MIT finance professor named Myron Scholes, and the twon began an interchange of ideas on how financial markets worked. Soon Black and Scholes then began studying options, which at that time were traded only on the OTC market. They first reviewed the attempts of previous researchers to find the elusive option pricing formula.

Black and Scholes took two approaches to finding the price. One approach assumed that all assets were priced according to Capital Asset Pricing Theory, a well-accepted model in finance. The other approach used stochastic calculus. They obtained an equation using the first approach, but the second method left them with a differential equation they were unable to solve. The more mathematical approach was considered more important, so they continued to work on the problem, looking for a solution. Black eventually found that the differential equation could be transformed into the same one that described the movement of heat as it travels across an object. There was already a known solution, and Black and Scholes simply looked it up, applied it to their problem, and obtained using the first method. Their paper reporting their findings was rejected by two academic journals before eventually being published in the Journal of Political Economy, which reconsidered an earlier decision to reject the paper.

At the same time, another young financial economist at MIT, Robert Merton, was also working on option pricing. Merton discovered many of the arbitrage rules. In addition, Merton more or less simultaniously derived the formula. Merton's modesty, however, compelled him to ask a journal editor that his paper not be published before that of Black and Scholes. As it turned out, both papers were published, with Merton's paper appearing in Bell Journal of Economics and Management Science at about the same time. Merton, however, did not initially receive as much credit as Black and Scholes, whose names became permanently associated with the model.

Fischer Black left academia in 1983 and went back to work for the Wall Street firm Goldman Sachs. Unfortunately, he died in 1995 at the age of 57. Scholes and Merton have remained in academia but have been extensively involved in real world derivatives applications.

In 1997, the Nobel Committee awarded the Nobel Prize for Economic Science to Myron Scholes and Robert Merton, while recognising Black's contributions.

The model has been one of the most significant developments in the history of pricing of financial instruments. It has generated considerable research attempting to test the model and improve on it. A new industry of derivative products based on the Black-Scholes model has developed. Even if one does not agree with everything the model says, knowing something about it is important for surviving in the finance markets.


Black-Scholes Model - The Theory Behind it

The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. The Black-Scholes formula is a mathematical formula for the theoretical value of European put and call stock options that may be derived from the assumptions of the model.

The key assumptions of the Black-Scholes model are:

• The price of the underlying instrument is a geometric Brownian motion, in particular with constant drift and volatility.
  
• It is possible to short sell the underlying stock.
  
• There are no riskless arbitrage opportunities.
  
• Trading in the stock is continuous.
  
• There are no transaction costs.
  
• All securities are perfect divisible (e.g. it is possible to buy 1/100th of a share).
  
• The risk free interest rate is constant, and the same for all maturity dates.

Black-Scholes in practice

The use of the Black-Scholes formula is pervasive in the markets. In fact the model has become such an integral part of market conventions that it is common practice for the implied volatility rather than the price of an instrument to be quoted. (All the parameters in the model other than the volatility - that is the time to expiry, the strike, the risk-free rate and current underlying price—are unequivocally observable. This means there is one-to-one relationship between the option price and the volatility.). Traders prefer to think in terms of volatility as it allows them to evaluate and compare options of different maturities , strikes, etc...

However, the Black-Scholes model can not be modelling the real world exactly. If the Black-Scholes model held, then the implied volatility of an option on a particular stock would be constant, even as the strike and maturity varied. In practice, the volatility surface (the two-dimensional graph of implied volatility against strike and maturity ) is not flat. In fact, in a typical market, the graph of strike against implied volatility for a fixed maturity is typically smile-shaped (see volatility smile). That is, at-the-money (the option for which the underlying price and strike co-incide) the implied volatility is lowest; out-of-the-money or in-the-money the implied volatility tends to be different, usually higher on the put side (low strikes), and call side (high strikes).

Practically, the volatility surface of a given underlying instrument depends among other things on its historical distribution, and is constanty re-shaping as investors, market-makers, and arbitragists re-evaluate the probability of the underlying reaching a given strike and the risk-reward associated to it.

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Information taken from Wikipedia - the free encyclopedia