The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 326 Wednesday, February 4, 2004 12:59 PM

326 The Mathematics of Financial Modeling and Investment Management

pt = pt – 1 ++μεt

A time series of this form is called an arithmetic random walk. It is a

generalization of the simple random walk that was introduced in Chap-

ter 6. The arithmetic random walk is the simplest example of an inte-

grated process.

Let’s go back to simple net returns. From the above definition, it is

clear that we can write

με

1 +Rt =e

+ t

If the white noise is normally distributed, then the returns Rt are lognor-

mally distributed. Recall that we found a simple correspondence

between a geometric Brownian motion with drift and an arithmetic

Brownian motion with drift. In fact, using Itô’s Lemma, we found that,

if the process St follows a geometric Brownian motion with drift

dS

------ =μdt +σdB

S

its logarithm st = log St then follows the arithmetic Brownian motion

with drift:

1 

ds =



μ–---σ

2

dt +σdB

 2 

In discrete time, there is no equivalent simple formula as we have to

integrate over a finite time step. If the logarithms of prices follow a discrete-

time arithmetic random walk with normal increments, the prices them-

selves follow a time series with lognormal multiplicative increments

written as

+ t

Pt =( 1 +Rt)Pt – 1 =e

με

Pt – 1

The arithmetic random walk model of log price processes is sug-

gested by theoretical considerations of market efficiency. As we have seen

in Chapter 3, it was Bachelier who first suggested Brownian motion as a

model of stock prices. Recall that the Brownian motion is the continu-

ous-time version of the random walk. Fama and Samuelson formally