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Options are a financial instrument giving the holder the right to buy or sell an underlying stock or commodity at a future point in time, at an agreed upon price. The Black-Scholes model, for which Fischer Black, Myron Scholes and Robert Merton were awarded the Nobel Prize in Economics, is a tool for pricing equity options. Prior to its development there was no standard way to price options; in a very real sense, the Black-Scholes model marks the beginning of the modern era of financial derivatives.
There are several assumptions underlying the Black-Scholes model. The most significant is that volatility, a measure of how much a stock can be expected to move in the near-term, is a constant over time. The Black-Scholes model also assumes stocks move in a manner referred to as a random walk; at any given moment, they are as likely to move up as they are to move down. By combining these assumptions with the idea that the cost of an option should provide no immediate gain to either seller or buyer, a set of equations can be formulated to calculate the price of any option.
The Black-Scholes model takes as input current prices, length of time until the option expires worthless, an estimate of future volatility known as implied volatility, and the so-called risk free rate of return, generally defined as the interest rate of short term US treasury notes. The model also works in reverse: instead of calculating a price, an implied volatility for a given price can be calculated.
Options traders often refer to "the greeks", especially Delta, Vega, and Theta. These are mathematical characteristics of the Black-Scholes model named after the greek letters used to represent them in equations. Delta measures how much an option price will move relative to the underlying, Vega is the sensitivity of the option price to changes in implied volatility, and Theta is the expected change in option price due to the passage of time.
There are known problems with the Black-Scholes model; markets often move in ways not consistent with the random walk hypothesis, and volatility is not, in fact, constant. A Black-Scholes variant known as ARCH, Autoregressive Conditional Heteroskedasticity, was developed to deal with these limitations. The key adjustment is the replacement of constant volatility with stochastic, or random, volatility. After ARCH came an explosion of different models; GARCH, E-GARCH, N-GARCH, H-GARCH, etc, all incorporating ever more complex models of volatility. In everyday practice, however, the classic Black-Scholes model remains dominant with options traders.