Lognormal Stock-Price Models

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Making sense of . . .

LogNormal stock-price models in Exams MFE/3 and C/4
James W. Daniel Austin Actuarial Seminars http://www.actuarialseminars.com June 26, 2008

c Copyright 2007 by James W. Daniel; reproduction in whole or in part without the express permission of the author is forbidden.

Foreword
This document briefly describes the ideas behind the use of LogNormal models for stock prices in some of the material for Exams MFE and C of the Society of Actuaries and Exams 3 and 4 of the Casualty Actuarial Society. Not a traditional exam-prep study manual, it concentrates on explaining key ideas so that you can then understand the details presented in the textbooks or study manuals. It can be especially useful to anyone taking Exam C/4 without having studied the material for Exam MFE/3.

2

Chapter 1

LogNormal stock-price models
1.1 Why LogNormal models?

Why learn about and use LogNormal models for stock prices? I could answer “Because it’s on the exam syllabi” or “Why not?”, but that wouldn’t be helpful. Instead, I’ll take a little space to motivate this. Suppose that the price of a stock or other asset at time 0 is known to be S(0) and we want to model its future price S(10) at time 10—note that some texts use the notation S0 and S10 instead. Let’s break the time interval from 0 to 10 into 10,000 pieces of length 0.001, and let’s let Sk stand for S(0.001k), the price at time 0.001k. I know the price S0 = S(0) and want to model the price S10000 = S(10). I can write (1.1) S(10) = S10000 = S2 S1 S10000 S9999 ··· S0 . S9999 S9998 S1 S0

Now suppose that the ratios Rk = SSk that appear in Equation 1.1 that represent the growth factors k−1 in price over each interval of length 0.001 are random variables, and—to get a simple model—are all independent of one another. Then Equation 1.1 writes S(10) as a product of a large number of independent random…...

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