The term Brownian Motion has been used to describe the motion of a particle that is subject to a large number of small molecular shocks. In financial calculus, Brownian Motion is used interchangeably with the term Wiener Process which is a particular type of Markov stochastic process. Markov stochastic processes are processes where only the present value of a variable is relevant for predicting the future. Models of stock price behavior are usually expressed as a Wiener Process or Brownian Motion (see Hull, 1997).
The price of a stock option is a function of the underlying stock's price and time. More generally, we can say that the price of any derivative is a function of the stochastic variables underlying the derivative and time. Ito's lemma states that if a variable x follows an Ito Process, that is, thatwhere dz is a Wiener Process and a and b are functions of x and t, then another function G of x and t follows the process where dz is the same Wiener Process. Thus, G also follows a Wiener Process.
To solve the Black and Scholes formula, one needs to calculate the cumulative normal distribution function, Θ. The function can be evaluated directly using numerical procedures. Alternatively, a polynomial approximation can be used that provides values for Θ(d) with a six-decimal-place accuracy. The following have been extracted from Hull (1997):where