Calculus Tutorial

Lecture 11

The Geometric Brownian Motion and the Black-Scholes formula

We are now ready to tackle the problem of solving the equation (2) in Lecture 10:

[pic 1]   (1)

If we divide by [pic 2] in (1) we see that the left hand-side is [pic 3] which leads us to think of the “derivative” of the function  [pic 4]. Therefore, we consider the process [pic 5] and compute its differential using Itô’s Lemma (Theorem 2 Lecture 10). To ease notation we will use [pic 6]. We have:

[pic 7]

Equation (2) gives an explicit solution to the equation (1). We can equivalently solve for [pic 8]  (3)

The model of stock price behavior described in (2) is known as the geometric Brownian motion. Notice that for each fixed t, the variable [pic 9] is normally distributed with mean [pic 10] and standard deviation [pic 11] since [pic 12] is normally distributed with mean zero and standard deviation [pic 13]. Therefore, from (2) we conclude that the natural logarithm of the stock price follows a normal distribution, that is, [pic 14] is normally distributed with mean [pic 15] and standard deviation [pic 16].

Definition 1. A random variable X for which [pic 17]is normally distributed is said to have a lognormal distribution.

In view of Definition 1, [pic 18] has a lognormal distribution.

Definition 2. A generalized Wiener process (or a Brownian motion with drift) is a process satisfying an equation of the form

[pic 19]  (4)

where a and b are real constants. Notice that if we integrate in (4) we get

[pic 20]

so [pic 21] is normally distributed with mean [pic 22] and standard deviation [pic 23]. So a affects the mean, reason for which it is called the drift rate, and b affects the standard deviation of the Brownian motion, reason for which [pic 24] is called the variance rate.

Example 1. Consider a stock for which [pic 25], the expected return is [pic 26] per year and a volatility [pic 27] per year. Let us find [pic 28] after one year.

Since we know the distribution of the logarithm of the stock price we write:

[pic 29].  (5)

Now, we know that [pic 30] follows a normal distribution with mean [pic 31] and standard deviation [pic 32]. Therefore we have to compute a normal probability in (5). If we denote by [pic 33] the cumulative distribution function for the standard normal distribution, we have: