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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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(2015, 10). Calculus Tutorial. EssaysForStudent.com. Retrieved 10, 2015, from
“Calculus Tutorial” EssaysForStudent.com. 10 2015. 2015. 10 2015 < "Calculus Tutorial." EssaysForStudent.com. EssaysForStudent.com, 10 2015. Web. 10 2015. < "Calculus Tutorial." EssaysForStudent.com. 10, 2015. Accessed 10, 2015. Essay Preview By: Aojun Submitted: October 9, 2015 Essay Length: 633 Words / 3 Pages Paper type: Term Paper Views: 339 Report this essay Tweet Related Essays Tutorial Six Not long ago a friend of mine, a successful entrepreneur, was crying on my shoulder. "Fred, he said to me, "when I started my company 3,823 Words  |  16 Pages History of Calculus History of Calculus The history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods. The ancient period introduced 1,147 Words  |  5 Pages Regional Institute for Tutorial Education Alan has been a volunteer for five years for the Regional Institute for Tutorial Education (RITE) at the University of Missouri, St. Louis. RITE serves 630 Words  |  3 Pages Pokemon Gba Scripting Tutorial Getting Started #ORG $StartScript Lock Faceplayer #ORG : beginning of a script Lock : Locks down the character you activate in the game so that 504 Words  |  3 Pages Get Access to 89,000+ Essays and Term Papers Join 209,000+ Other Students High Quality Essays and Documents Sign up © 2008–2020 EssaysForStudent.comFree Essays, Book Reports, Term Papers and Research Papers Essays Sign up Sign in Contact us Site Map Privacy Policy Terms of Service Facebook Twitter

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