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...