Black-Scholes Model

From Planipedia

Black-Scholes Model

The Black–Scholes model is a mathematical description of financial markets and derivative investment instruments. The model develops partial differential equations whose solution, the Black–Scholes formula, is widely used in the pricing of European-style options. The model was first articulated by Fischer Black and Myron Scholes in their 1973 paper, "The Pricing of Options and Corporate Liabilities." The foundation for their research relied on work developed by scholars such as Jack L. Treynor, Paul Samuelson, A. James Boness, Sheen T. Kassouf, and Edward O. Thorp. The fundamental insight of Black–Scholes is that the option is implicitly priced if the stock is traded. Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term Black–Scholes options pricing model. Merton and Scholes received the 1997 Nobel Prize in Economics (The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) for their work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy.

Model Assumptions

The Black–Scholes model of the market for a particular equity makes the following explicit assumptions:

• It is possible to borrow and lend cash at a known constant risk-free interest rate. This restriction has been removed in later extensions of the model.

• The price follows a Geometric Brownian motion with constant drift and volatility. It follows from this that the return is a Log-normal distribution. This often implies the validity of the efficient-market hypothesis.

• There are no transaction costs or taxes.

• The stock does not pay a dividend (see below for extensions to handle dividend payments).

• Securities are perfectly divisible (i.e. it is possible to buy any fraction of a share).

• There are no restrictions on short selling.

• There is no arbitrage opportunity.

• Options use the European exercise terms, which dictate that options may only be exercised on the day of expiration.