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Calculus Tutorial
Lecture 11The Geometric Brownian Motion and the Black-Scholes formulaWe 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:
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By: Aojun
Submitted: October 9, 2015
Essay Length: 633 Words / 3 Pages
Paper type: Term Paper Views: 339
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