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The Black-Scholes model for a stock price assumes that the stock price follows geometric Brownian motion with constant drift and volatility. More precisely, if S(t) the stock price at time t, then

The Black-Scholes model for a stock price assumes that the stock price follows geometric Brownian motion with constant drift and volatility. More precisely, if S(t) the stock price at time t, then

Loosely speaking, this means that the return <math> \Delta S/S</math> of the stock over a very small time interval <math>\Delta t</math> can be viewed as a normal random variable with mean <math>\mu \Delta t</math> and variance <math>\sigma^2(\Delta t)^2 </math>. One can make this notion precise by invoking the concepts from the Ito calculus.

Loosely speaking, this means that the return <math> \Delta S/S</math> of the stock over a very small time interval <math>\Delta t</math> can be viewed as a normal random variable with mean <math>\mu \Delta t</math> and variance <math>\sigma^2(\Delta t)^2 </math>. One can make this notion precise by invoking the concepts from the Ito calculus.

The Black-Scholes pricing formula for a European call option can be deduced from the Black-Scholes model for a stock price. A European call option on a stock with strike price <math>K</math> and time to maturity <math>T</math> is a financial contract that gives the holder the option, but not the obligation, to purchase the stock for price <math>K</math> at time <math>T</math>. In other words, a European call on the stock S is a contract that provides a single pay-off of  

The Black-Scholes pricing formula for a European call option can be deduced from the Black-Scholes model for a stock price. A European call option on a stock with strike price <math>K</math> and time to maturity <math>T</math> is a financial contract that gives the holder the option, but not the obligation, to purchase the stock for price <math>K</math> at time <math>T</math>. In other words, a European call on the stock S is a contract that provides a single pay-off of