Table of contents |

2 The formula 3 Extensions of the formula 4 Formula derivation 5 Black-Scholes in practice 6 See also 7 External links and references |

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.

The above lead to the following formula for the price of a call on a stock currently trading at price *S*, where the option has an exercise price of K, i.e. the right to buy a share at price K, at T years in the future. The constant interest rate is *r* and the constant stock volatility is *v*:

- .
*N*is the cumulative Normal distribution function.

The above option pricing formula is used for pricing European put and call options on non-dividend paying stocks. The Black-Scholes model may be easily extended to options on instruments paying dividends. For options on indexes (such as the FTSE) where each of 100 constituent companies may pay a dividend twice a year and so there is a payment nearly every business day, it is reasonable to assume that the dividends are paid continuously. The dividend payment paid over the time period is then modelled as

Exactly the same formula is used to price options on foreign exchange rates, except now q plays the role of the foreign risk-free interest rate and S is the spot exchange rate. This is the **Garman-Kohlhagen model** (1983).

It is also possible to extend the Black-Scholes framework to options on instruments paying discrete dividends. This is useful when the option is struck on a single stock. A typical model is to assume that a proportion of the stock price is paid out at pre-determined times . The price of a stock is then modelled as

American options are more difficult to value, and a choice of models is available (for example Whaley, binomial options model).

In this section we derive the partial differential equation (PDE) at the heart of the Black-Scholes model via a no-arbitrage or delta-hedging argument. The presentation given here is informal and we do not worry about the validity of moving between dt meaning an small increment in time and dt as a derivative.

As in the model assumptions above we assume that the underlying (typically the stock) follows a geometric Brownian motion. That is,

**2) From the general Black-Scholes PDE to a specific valuation**

We now show how to get from the general Black-Scholes PDE to a specific valuation for this option. Consider as an example the Black-Scholes price of a call on a stock currently trading at price *S*. The option has an exercise price of K, i.e. the right to buy a share at price K, at T years in the future. The constant interest rate is *r* and the constant stock volatility is *v*(all as at top).
Now, for a call option the PDE above has boundary conditions:

- for all t
- as

Substituting v for u and the V for v, we finally obtain the value of a call option in terms of the Black-Scholes parameters:

- .
*N*is the cumulative Normal distribution function.

**3) Other derivations**

Above we used the method of arbitrage-free pricing ("delta-hedging") to derive a PDE governing option prices given the Black-Scholes model. It is also possible to use a risk neutrality argument. This latter method gives the price as the expectation of the option payoff under a particular probability measure, called the risk-neutral measure, which differs from the real world measure.

- The risk neutrality argument:
- Arbitrage-free pricing:
- Detailed solutions of the Black-Scholes equation or a further treatment.

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 unequivocably observable. This means there is one-to-one relationship between the option price and the volatility.) Traders prefer to think in terms of volatility.

However, the Black-Scholes model can not be modelling the real world completely accurately. 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 higher. The reason for this smile is still the subject of much speculation and research. A prominent proposed explanation is that the market in options away from the money is less liquid than at-the-money: traders demand a premium for these options because they know it may be more difficult to reverse an option position in illiquid markets. This view is consistent with the fact the smile was first observed shortly after the stock market crash of 1987. Before this crash, the first and most severe since the introduction of options, the Black-Scholes was more widely trusted.

- Binomial options model, which is able to handle a variety of conditions for which Black-Scholes cannot be applied.
- Black model a variant (and more general form) of the Black-Scholes option pricing model.
- Financial mathematics, which contains a list of related articles.

- Black, F. and M. Scholes, "The Pricing of Options and Corporate Liabilities" Journal of Political Economy 81, 1973, 637-654. Black and Scholes' original paper.
- Merton, Robert C., "Theory of rational option pricing", Bell Journal of Economics and Management Science 4 (1), 1973, 141-183.
- Trillion Dollar Bet - Companion Web site to a Nova episode originally broadcast on February 8, 2000.
*"The film tells the fascinating story of the invention of the Black-Scholes Formula, a mathematical Holy Grail that forever altered the world of finance and earned its creators the 1997 Nobel Prize in Economics."* - The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel for 1997
- Options pricing using the Black-Scholes Model, Investment Basics: XLII., The Investment Analysts Society of South Africa
- The Black Scholes Option Pricing Model, optiontutor
- Generalized Black-Scholes Calculator, Written by Espen Gaarder Haug (himself) 1998
- Further information on pricing options on continuous dividend-paying stocks
- Further information on pricing options on foreign exchange options.