Company S Cash Position Measured In Millions Of Dollars Follows A Generalized Wiener Process

where the e,- (i = 1,2, ..., N) are random drawings from a standardized normal distribution. From property 2 the e, 's are independent of each other. It follows from Equation (9.2) that z(T) — z(0) is normally distributed with2

mean of [;z(T) - z(0)] = 0 variance of [z(T) - z(0)] = N At = T standard deviation of [z(T) — z(0)] = Vt

Thus in any time interval of length T, the increase in the value of a variable that follows a Wiener process is normally distributed with a mean of zero and a standard deviation of <JT. It should now be clear why Az is defined as the product of e and -s/A7 rather than as the product of e and At. Variancesare additive for independentnormaldistributions; standard deviations are. not. It makes sense to define the stochastic process so that the variance rather than the standard deviation of changes is proportional to the length of the time interval considered.

Example 9.1

Suppose that the value, z, of a variable which follows a Wiener process is initially 25 and that time is measured in years. At the end of 1 year the value of the variable is normally distributed with a mean of 25 and a standard deviation of 1.0. At the end of 2 years it is normally distributed with a mean of 25 and a standard deviation of V2, or 1.414. Note that our uncertainty about the value of the variable at a certain time in the future, as measured by its standard deviation, increases as the square root of how far we are looking ahead.

In ordinary calculus, it is usual to proceed from small changes to the limit as the small changes become closer to zero. Thus Ay/Ax becomes dy/dx in the limit, and so on. We can proceed similarly when dealing with continuous-time stochastic processes. A Wiener process is the limit as At —► 0 of the process described above for z. Figure 9.1 illustrates what happens to the path followed by z as the limit At —► 0 is taken. Analogously to ordinary calculus, we write the limiting case of Equation (9.1) as dz = e -Jdt

Generalized Wiener Process

The basic Wiener process that has been developed so far has a drift rate of zero and a variance rate of 1.0. The drift rate of zero means that the expected

2This result is based on the following well-known property of normal distributions. If a variable Y is equal to the sum of N independent normally distributed variables X, (1 < ; < N), Y is itself normally distributed. The mean of Y is equal to the sum of the means of the X, 's. The variance of Y is equal to the sum of the variances of the X('s.

Figure 9.1 Illustration of How a Wiener Process is Obtained when At —> 0 in Equation (9.1)

/value of z at any future time is equal to its current value. The variance rate of 1.0 means that the variance of the change in z in a time interval of length T is 1.0 x T. A generalized Wiener process for a variable x can be defined in terms of dz as follows:

where a and b are constants.

To understand Equation (9.3) it is useful to consider the two components on the right-hand side separately. The a dt term implies that x has an expected drift rate of a per unit time. Without the b dz term, the equation is dx = a dt which implies that or x = xo + at where xq is the value of x at time zero. In a time interval of length T, x increases by an amount aT. The bdz term on the right-hand side of Equation (9.3) can be regarded as adding noise or variability to the path followed by x. The amount of this noise or variability is b times a Wiener process. In a small time interval Af, the change in the value of x, Ax, is from equations (9.1) and (9.3) given by:

Ax = a At + be y/~At where, as before, e is a random drawing from a standardized normal distribution. Thus Ax has a normal distribution with mean of Ax = a At standard deviation of Ax = b -J~At variance of Ax — b2 At

Similar arguments to those just given show that the change in the value of x in any time interval T is normally distributed with mean of change in x = aT standard deviation of change in x = bVT variance of change in x = b2T

Thus the generalized Wiener process given in Equation (9.3) has an expected drift rate (i.e., average drift per unit time) of a and a variance rate (i.e., variance per unit of time) of b2. It is illustrated in Figure 9.2.

Example 9.2

Consider the situation where the cash position of a company, measured in thousands of dollars, follows a generalized Wiener process with a drift of 20 per year and a variance rate of 900 per year. Initially, the cash position is 50. At the end of 1 year the cash position will have a normal distribution with a mean of 70 and a standard deviation of V900 or 30. At the end of 6 months it will have a normal distribution with a mean of 60 and a standard dx dt

Generalized Wiener Process
Figure 9.2 Generalized Wiener Process; a=0.3, ¿>=1.5

deviation of 30 V0?5 = 21.21. Note that our uncertainty about the cash position at some time in the future, as measured by its standard deviation, increases as the square root of how far ahead we are looking. Also, note that the cash position can become negative (which we can interpret as a situation where the company is borrowing funds).

Ito Process

A further type of stochastic process can be defined. This is known as an Ito process. It is a generalized Wiener process where the parameters a and b are functions of the value of the underlying variable, x, and time, t. Algebraically, an Ito process can be written dx = a(x, t)dt + b(x, t)dz (9.4)

■^Both the expected drift rate and variance rate of an Ito process are liable to change over time.

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