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

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.

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

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

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

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.

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:

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.

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.

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 .

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:

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

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 .