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 equation was derived by Fisher Black and Myron Scholes; the paper that contains the result was published in 1973. They built on earlier research by Paul Samuelson and Robert Merton. The fundamental insight of Black and Scholes was that the call option is implicitly priced if the stock is traded. The use of the Black-Scholes model and formula is pervasive in financial markets.

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.