Black–Scholes model

The Black-Scholes model, also known as the Black-Scholes-Merton model, is a mathematical model of financial derivative markets. It was developed in the early-1970s by economists Fischer Black and Myron Scholes, with contributions from Robert Merton.

The model serves as the basis for the Black-Scholes formula, which is widely used to calculate the theoretical price of European-style options – ie, those which are exercisable only at a specific single future date.

In its standard form, the Black-Scholes formula has five parameters, each representing a particular asset dynamic or aspect of market conditions. These are: the price of the underlying stock, the option's strike price, the time until expiration, the risk-free interest rate, and the expected stock price volatility. If using the Merton modification, there is a sixth input, namely the level of the maintainable dividend on a continuously compounding basis.

Due to its solid foundation as a mathematical proof, the Black-Scholes model has served as the basis for several other models and formulae, created by other academics for alternative uses. Examples include Margrabe's formula for currency futures, and various put option models used for measuring a lack of liquidity. International Valuation Standards advocate for the use of option models when valuing certain categories of complex share classes.

In practice, the Black-Scholes option pricing model is often used within a spreadsheet. There are various ‘ready-made’ Black-Scholes spreadsheets available on the internet, as well as numerous resources which explain how to create an option pricing spreadsheet incorporating the Black-Scholes formula (including some of the eBook chapters listed below). If using such a spreadsheet, you should take care to ensure that it faithfully replicates the formula, and that it is appropriate for your purposes. Here, it might be helpful to check its output against another internet source.

The Library enquiry team can provide ICAEW members, ACA students and registered ICAEW affiliates with an example Black-Scholes spreadsheet, though it should be noted that this spreadsheet is not approved by ICAEW. For more information, please contact us on +44 (0)20 7920 8620, or email library@icaew.com.

• View a list of books and articles in our collection on the Black–Scholes model

To find out how you can borrow books from the Library please see our guide to borrowing books.

You can obtain copies of articles or extracts of books and reports through our document supply service.