One of a wide family of mathematical models that are used today in finance to determine the so-called fair value of options contracts.

An options contract - be it a call or a put, of the American or European variety - is a very simple example of what is more commonly known as a derivative; that is, a financial instrument that has no value on its own, instead it derives its value from another, underlying instrument.

The so-called Black-Scholes model, originally developed in 1973, is intended to allow traders and investors to calculate the fair value of an options contract. It was considered earth breaking (and in fact led to a Nobel Prize) since this problem (the valuation of options) had been attempted by various parties since the turn of the century.

It wasn't until the differential equations underlying the problem were recognized to be similar to the well known heat transfer problem from physics that sufficient progress was made.

In its basic form the Black-Scholes differential equation is able to value American and European options on non-dividend paying stocks.

During the intervening years since it's introduction, it has been extended to value other underlying instrument; for example, stock market indices (e.g., the Dow Jones Industrials, or the S&P 500) or various commodities.
Conceived in 1973 by Myron Scholes, Robert Merton, and Fischer Black, the Black-Scholes Formula is a way to find out how much a call option is worth at any given time. It led to a Nobel Prize for Scholes and Merton in 1997, and operated on the theory that that an investor can precisely replicate the payoff to a call option by buying the underlying stock and financing part of the stock purchase by borrowing. The formula was groundbreaking because it was the first of its kind that actually worked, due to the fact that it eliminated variables that were impossible to measure, such as ‘investor fear’, that other formulas carried.

The formula breaks down like this:

C: the current call-option value.

S: the current stock price.

N(d1): N(d) is the probability that a random draw taken from N will be less than d. d1 is derived from a different formula that utilizes the price of the stock, the exercise price, the risk-free interest rate, the time to maturity of the call option, and the volatility of the underlying stock price.

L: the exercise price.

e: 2.718, the base for the natural logarithm used for continuous compounding.

r: the risk free interest rate.

T: the time until the expiration of the call option.

ð: the volatility of the stock.

A few notes:

  • The only variable to Black-Sholes that cannot be measured from the market is the volatility of the underlying, as the volatilty in question is the volatility from the present to the expiration of the option.
  • The theory can be worked in reverse, using the market price of the option as an input, and getting out a volatility figure. This volatility is called implied volatility.
  • The theory is immensely popular. Some people believe it mostly works because people use it so much to determine what they want to buy. Another sign of its popularity is that in some markets, prices are usually quoted as implied volatility (using Black-Sholes) instead of directly.
  • The theory assumes that the distribution of future prices is a Gaussian curve around the present price. This is a simplification; securities prices have a "fat tailed" distribution, meaning that extreme price moves are more likely than in a gaussian distribution (extreme moves down are also more likely than extreme moves up.)
  • The theory assumes that dividends are paid continiously (which they are not.)

A lesser known fact about the Black-Scholes thing is that these two Nobel prize winner professors, Prof. Black and Prof. Scholes were both employed by the hedge fund LTCM (Long Term Capital Management) which went broke in 1998 through derivatives plays that went seriously wrong. LTCM had to be bailed out by a gathering of high powered banks galvanized by nothing short of the US Federal Reserve Bank.

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