The Black-Scholes model

The assumptions of the Black-Scholes model (Black and Sholes, 1973, see also Merton, 1973) are as follows:

• The price of the underlying asset (S) follows geometric Brownian motion:

  

  Here µ (drift) and σ (volatility) are constant parameters and W is a standard Wiener process.

• The market is arbitrage-free.

• The underlying is a stock paying no dividends.

• Buying and (short) selling the underlying asset is possible in any (even fractional) amount.

• There are no transaction costs.

• The short-term interest rate (r) is known and constant over time.

The main result of the model is that under these assumptions, the price of a European call option (c) has a closed form:

• 
• ,
• ,

Here X is the strike price, T-t is the time to maturity of the option, and N denotes the cumulative distribution function of the standard normal distribution. The equation giving the price of the option is usually referred to as the Black-Scholes formula. It is easy to see from put-call parity that the price of a European...