Security MarketsSecurity MarketsVIIVII

Miloslav S. Vosvrda

Teorie financnich trhu

SiXdXXCvEiCv

i i ,...,1,111

Matrix expression

We can also simplify

to show that the price matrix is given by a simpleequation , where the matrix has a useful interpretation.

d S S

Let A denote the diagonal SxS matrix whose i-th

diagonal element is . Then

is equivalent to

Let , yielding for any time T:

SiXdXXCvEiCv

i i ,...,1,111

iCv

dPAA 1

B A PA 1

T

t

Tt BdB

BBdBBd

BBd

1

We see that converges to the zero matrix as T

goes to infinity, leaving

By a series calculation,

Equivalently

BT

B d dt

t

1

IAPIA 11

SiXdXCvEiCv

i ttt

ti ,...,1,

1

The current value of a security is the expected

discounted infinite horizon sum of its dividends,

discounted by the marginal utility of consumption

at the time the dividends occur, all divided by the

current marginal utility for consumption. This

extends the single period pricing model suggested

by relation

.''

1

10sn

is

S

sisi

in dxv

xvq

This multiperiod pricing model extends easily to the

case of state dependent utility for consumption:

to an infinite state-space; and even to continuous-

time. In fact, in continuous-time, one extend

Consumption-Based Capital Asset Pricing Model

to non-quadratic utility functions.

;,,0

LcXcvEcut tt

Under regularity conditions, that is, the increment of

a differentiable function can be approximated by the

first two terms of its Taylor series expansion, a

quadratic function, and this approximation becomes

exact in expectation as the time increment shrinks to

zero under the uncertainty generated by Brownian

Motion. This idea is formalized as Ito‘s Lemma,

and leads to many additional results that depend on

gradual transitions in time and state.

A Standard Brownian Motion

An illustrative model of continuous „perfectly

random„ fluctuation is a Standard Brownian

Motion, a stochastic process, that is, a family of

random variables,

on some probability space, with the defining

properties:

,,0: tBB t

a) for any and is normally distributed with zero mean and variance t - s,

b) for any times the increments for are independent,

c) almost surely.B( )0 0

s 0 sBtBst ,

0 0 1 t t tl... , 10 , kk tBtBtB 1 k l,

We will illustrate the role of Brownian Motion in

governing the motion of a Markov state process X.

For any times let ,

and

for . A stochastic difference equation for the

motion of X might be:

,

where and are given functions.

0 0 1 t t ..., t t tk k k 1

1 kkk tXtXX 1 kkk tBtBB

k 1

kkkkk BtXttXX 11 k 1

A stochastic difference equation

A stochastic differential equation

For the moment, we assume that and are

bounded and Lipschitz continuous (an existence of a

bounded derivative is sufficient.) Given ,

the properties defining the Brownian Motion B

imply that has conditional mean

and conditional variance .

A continuous-time analogue to a stochastic

differential equation is the stochastic differential

equation .

1ktX

X k kk ttX 1

kk ttX 2

1

dX X dt X dB tk t t t , 0

A Diffusion ProcessX is an example of a diffusion process. By analogy

with the difference equation, we may heuristically

treat and . The stochastic

differential equation has the following form

for some starting point . By the properties of the

(as yet undefined) Ito integral

we have:

dtX t dtX t2

0,000 tdBXdsXXX s

t

s

t

st

X 0

,tt dBX

ITO‘S LEMMAIf f is a twice continuously differentiable function,

then for any time

where

T 0

,000 tt

T

t

T

tT dBXXfdtXfDXfXf

.2

1 2xxfxxfxfD

If is bounded, the fact that B has increments of zero expectation implies that

It then follows that

In other words, Ito‘s Lemma tells us that the

expected rate of change of f at any point x is

.

.00

ttt

TdBXXfE

00

0lim XfD

T

XfXfE T

T

xfD

f '