Consider a firm which has a known demand for cash over time. To satisfy this cash demand, the firm must keep some cash on hand, assumed to be held in a checking account at a bank. If the firm keeps too much cash, it loses money in terms of opportunity cost, in that it can earn higher returns by buying securities such as bonds. On the other hand, if the cash balance is too small, the firm has to sell securities to meet the cash demand and thus incur a broker’s commission. The problem then is to find the tradeoff between the cash and security balances.
5.1.1 The Model
To formulate the optimal control problem we introduce the following notation:
T = the time horizon,
x(t) = the cash balance in dollars at timet, y(t) = the security balance in dollars at timet,
d(t) = the instantaneous rate of demand for cash;d(t) can be positive or negative,
u(t) = the rate of sale of securities in dollars; a negative sales rate means a rate of purchase,
r1(t) = the interest rate earned on the cash balance, r2(t) = the interest rate earned on the security balance,
α = the broker’s commission in dollars per dollar’s worth of securities bought or sold; 0< α <1.
The state equations are
˙
x=r1x−d+u−α|u|, x(0) =x0, (5.1)
˙
y=r2yưu, y(0) =y0, (5.2) and the control constraints are
−U2≤u(t)≤U1, (5.3)
where U1 and U2 are nonnegative constants. The objective function is:
max{J =x(T) +y(T)} (5.4) subject to (5.1)–(5.3). Note that the problem is in the linear Mayer form.
5.1. The Simple Cash Balance Problem 161 5.1.2 Solution by the Maximum Principle
Introduce the adjoint variables λ1 and λ2 and define the Hamiltonian function
H =λ1(r1xưd+uưα|u|) +λ2(r2yưu). (5.5) The adjoint variables satisfy the differential equations
λ˙1=−∂H
∂x =−λ1r1, λ1(T) = 1, (5.6) λ˙2=−∂H
∂y =−λ2r2, λ2(T) = 1. (5.7) It is easy to solve these, respectively, as
λ1(t) =etTr1(τ)dτ (5.8) and
λ2(t) =etTr2(τ)dτ. (5.9) The interpretations of these solutions are also clear. Namely,λ1(t) is the future value (at time T) of one dollar held in the cash account from timettoT and, likewise, λ2(t) is the future value of one dollar invested in securities from time t toT. Thus, the adjoint variables have natural interpretations as the actuarial evaluations of competitive investments at each point of time.
Let us now derive the optimal policy by choosing the control vari- able u to maximize the Hamiltonian in (5.5). In order to deal with the absolute value function we write the control variable u as the difference of two nonnegative variables, i.e.,
u=u1−u2, u1≥0, u2≥0. (5.10) Recall that this method was suggested in Remark 3.12 in Sect.3.7. In order to make u=u1 when u1 is strictly positive, andu=−u2 whenu2 is strictly positive, we also impose the quadratic constraint
u1u2 = 0, (5.11)
so that at most one of u1 andu2 can be nonzero. However, the optimal properties of the solution will automatically cause this constraint to be satisfied. The reason is that the broker’s commission must be paid on
162 5. Applications to Finance every transaction, which makes it not optimal to simultaneously buy and sell securities. Given (5.10) and (5.11) we can write
|u|=u1+u2. (5.12)
Also, sinceu∈[−U1, U2] from (5.3), we must haveu1≤U1andu2≤U2. In view of (5.10), the control constraints on the variablesu1 and u2 are 0≤u1≤U1 and 0≤u2≤U2. (5.13) We can now substitute (5.10) and (5.12) into the Hamiltonian (5.5) and reproduce the part that depends on control variablesu1 andu2,and denote it by W.Thus,
W =u1[(1−α)λ1−λ2]−u2[(1 +α)λ1−λ2]. (5.14) Maximizing the Hamiltonian (5.5) with respect to u ∈[−U1, U2] is the same as maximizing W with respect tou1 ∈[0, U1] andu2 ∈[0, U2].But W is linear inu1 and u2 so that the optimal strategy is bang-bang and is as follows:
u∗ =u∗1−u∗2, (5.15) where
u∗1= bang[0, U1; (1−α)λ1−λ2], (5.16) u∗2= bang[0, U2;−(1 +α)λ1+λ2]. (5.17) Since u1(t) represents the rate of sale of securities, (5.16) says that the optimal policy is: sell at the maximum allowable rate if the future value of a dollar less the broker’s commission (i.e., the future value of (1−α) dollars) is greater than the future value of a dollar’s worth of securities;
and do not sell if these future values are in reverse order. In case the future value of a dollar less the commission is exactly equal to the fu- ture value of a dollar’s worth of securities, then the optimal policy is undetermined. In fact, we are indifferent as to the action taken, and this is called singular control. Similarly,u2(t) represents the purchase of securities. Here we buy, do not buy, or are indifferent, if the future value of a dollar plus the commission is less than, greater than, or equal to the future value of a dollar’s worth of securities, respectively.
Note that if
(1−α)λ1(t)≥λ2(t), then
(1 +α)λ1(t)> λ2(t),
5.1. The Simple Cash Balance Problem 163
Figure 5.1: Optimal policy shown in (λ1, λ2) space so that if u1(t)>0,thenu2(t) = 0.Similarly, if
(1 +α)λ1(t)≤λ2(t), then
(1−α)λ1(t)< λ2(t),
so that if u2(t)>0,thenu1(t) = 0.Hence, with the optimal policy, the relation (5.11) is always satisfied.
Figure5.1illustrates the optimal policy at timet. The first quadrant is divided into three areas which represent different actions (including no action) to be taken. The dotted lines represent the singular control manifolds. A possible path of the vector (λ1(t), λ2(t)) of the adjoint variables is shown in Fig.5.1 also. Note that on this path, there is one period of selling, two periods of buying, and three periods of inactivity.
Note also that the final point on the path is (1,1), since the terminal values λ1(T) = λ2(T) = 1, and therefore, the last interval is always characterized by inactivity.
Another way to represent the optimal path is in the (t, λ2/λ1) space.
The path of (λ1(t), λ2(t)) shown in Fig.5.1 corresponds to the path of λ2(t)/λ1(t) over time shown in Fig.5.2.
164 5. Applications to Finance
Figure 5.2: Optimal policy shown in (t, λ2/λ1) space
Perhaps a more realistic version of the cash balance problem is to disallow overdraft on the bank account. This means imposing the pure state constraint x(t)≥0.In addition, if short selling of securities is not permitted, then we must also havey(t)≥0.These extensions give rise to pure state constraints treated in Chap.4. In Exercise 5.2 you are asked to formulate such an extension and write the indirect maximum principle (4.29) for it. Exercises5.3 and 5.4 present instances where it is easy to guess the optimal solutions. In Exercise5.5, you are asked to show if the guessed solution in Exercise 5.4satisfies the maximum principle (4.29).
It is in Chap.6 that we discuss in detail an application of the indirect maximum principle (4.29) for solving a problem called the wheat trading model.