Wednesday, November 16, 2011

The Black-Scholes Equation

New York Stock Exchange

Traditionally, higher returns have been associated, or rather demanded, with trades involving higher risk.  Traders since the dawn of time have endeavored to reduce or eliminate risk.  By the mid-twentieth century, many traders believed that they had done just that.  Building on the work of Paul Samuelson who, in turn, drew from Louis Bachelier's The Theory of Speculation.  The main way that risk has been reduced is through the Black-Scholes equation and a process known as dynamic hedging.

The Black-Scholes equation was first authored by Fischer Black and Myron Scholes.  They introduced a method of hedging by purchasing the option on an asset and the asset itself.  The equation creates a relationship between the asset and the option on the asset which aims for a risk free environment within the parameters of assumed volatility.  Around the same time, Robert Merton developed a more robust method of pricing options.

Chicago Board of Trade (photo: Jeremy Kemp)

Robert Merton and Myron Scholes won the Nobel Prize in economics for their idea in 1997.  The equation is used throughout financial markets as the basis for more complicated and tailored strategic financial models.  Because it is based on the random walk concept within markets, the model has an assumed parameter of volatility and performs very well within those boundaries.

The problem with the concept of eliminating risk is that it is impossible.  The model actually simply manages risk in a predictable way, when given a relatively predictable future (that the random walk will continue to be a random walk).  The model is incapable of detecting seismic shifts within the markets.  Traders that are overly reliant on this model can be disastrously ill prepared for stock market crashes and panics.  Nassim Nicholas Taleb and Benoît B. Mandelbrot have been critical of this model for understating the volatility of the stock market.  Here is one paper by Taleb arguing not to discount improbable events.

The Midas Formula is a BBC documentary on the Black-Scholes equation.  It is available in a series of YouTube videos below.

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