1. Theory of Capital Markets
Security Markets VIII
Miloslav S. Vosvrda
Theory of Capital Markets

2. Matrix expression We can also simplify
to show that the price matrix is given by a simple
equation , where the matrix
has a useful interpretation.

3. Let A denote the diagonal SxS matrix whose i-th
diagonal element is Then
is equivalent to
Let , yielding for any time T:

4. We see that converges to the zero matrix as T
goes to infinity, leaving
By a series calculation,
Equivalently

5. Or 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

6. 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.

7. 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.

8. 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:

9. 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.

10. A stochastic difference equation
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.

11. 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

12. A Diffusion Process X 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:

13. ITO‘S LEMMA If f is a twice continuously differentiable function,
then for any time
where

14. 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 .