Black-Scholes Revisited


Came across this rather interesting paragraph from an essay in the TechnologyReview. You can read the full article here (requires registration)

Although I had learnt the Black-Scholes formula for option pricing somewhere in finance, I had no idea that the experssion for volatility had demonstrated properties of Brownian movement!!
I guess research must already exist that pares off Chaos Theory against the behaviour of financial markets.

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Though derivatives were simpler once, they were never very simple. The breakthrough in the valuation of derivatives in general, and options in particular, was the model and formula know as Black-Scholes, first proposed by Fischer Black and Myron Scholes in the 1970s and formalized by Robert Merton in 1973. (Merton, like so many of the best quants, came not out of Wall Street but out of aca­demia, earning a PhD in economics from MIT in 1970.)

In quantitative finance, the formal expression of Black-Scholes by Robert Merton is so important that everything that followed has been called a "footnote." The Black-­Scholes model assumes that a stock's price changes partly for predictable reasons and partly because of random events; the random element is called the stock's "volatility." The idea can be represented mathematically by a simple equation:

S_t is the value of the stock, and dS_t is the change in stock price. The symbol µS_tdt represents the stock's predictable change and its volatility. (View the results of Black-Scholes model using this interactive calculator.) That final, kabbalistic combination of letters, dW_t, is the mathematical expression for randomness, known as either Brownian motion or the ­Wiener process. (Chemically, Brownian motion is the random movement of particles in solution, identified by the botanist Robert Brown in 1828 and mathematically described by the great MIT mathematician Norbert Wiener. Black-Scholes shares some qualities with heat and diffusion equations, which describe everyday events like the flow of heat and the dispersion of populations. That some physical processes seem relevant to finance has inspired all kinds of far-out work, such as efforts to bend general relativity to a theory of finance.) Black-Scholes prices an option according to the amount of randomness in a stock's price; the greater the randomness, the higher the stock could climb, and thus the more expensive the option.