The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 324 Wednesday, February 4, 2004 12:59 PM

324 The Mathematics of Financial Modeling and Investment Management

σˆ XY,

ρˆ XY, = --------------

σˆ Xσˆ Y

Empirical standard deviations are defined as follows:

n

σˆ = ∑(Xi – X)

2

X

i = 1

n

σˆ = ∑(Yi – Y)

2

Y

i = 1

It can be demonstrated that the empirical covariance matrix is an

unbiased estimator of the variance-covariance matrix. If innovations are

jointly normally distributed, it is also an ML estimator.

LINEAR MODELS OF FINANCIAL TIME SERIES

Let’s now apply previous general theoretical considerations and those of

the previous chapter to modeling financial time series. This section

describes linear models of financial time series using the concepts intro-

duced in the previous sections. Linear financial models are regressive

and/or autoregressive models where a series is regressed over exogenous

variables and/or its own past under a number of constraints.

In the practice of asset and portfolio management, models of prices,

returns, and rates are used as inputs to asset selection methodologies

such as semiautomated investment processes, heuristic computational

procedures, or full-fledged optimization procedures. The following

chapters on methods for asset management will explain how the compu-

tational models described in this and the following chapter translate

into asset and portfolio management strategies. We will start with ran-

dom walk models and progressively introduce more complex factor-

based models.

RANDOM WALK MODELS

Consider a time series of prices Pt of a financial asset. Assume there are

no cash payouts. The simple net return of the asset between periods t –

1 and t is defined as