STOCHASTIC

M1NIMAX POLICIES FOB SELLING AN ASSET AND DOLLAR AVERAGING 381
situation occurs on financial markets where a regret criterion may be applicable.
Frequently, a customer gives a broker an asset to sell for him by a certain date at the
broker's "discretion". This means the customer sets the time by which sales must be
completed but leaves to the broker when within this time sales are actually to be made.
The broker may well have no particular feeling as to whether the price of the asset is
going to move up or down over the relatively short time available for selling it. He has
a strong feeling, however, that the customer will interpret a failure to get a good price
as a lack of skill on his part. In fact, if the price obtained by the broker is far from
the best price which could have been obtained, the customer may well take his business
elsewhere the next time. Suppose the broker were to sell all of the asset initially. In
this case it may be rational to assume that the customer will not be consoled much ex
post by the fact that he was guaranteed a certain outcome ex ante, if the price rises
significantly. Hence, it will be rational for the broker to hedge against large regrets.
The applicability of a regret criterion in situations in which the decision maker is subject
to blame has been mentioned by other writers, for instance, Borch [2, p. 82].
In general one might posit the existence of a strictly convex disutility of regret
function. The regret criterion would then require the minimization of the expected
value of this function. However, a minimax criterion will be adopted here. For a symmetric
type of stochastic price behavior such a criterion may give a good approximation
to the policies arising from many risk averse disutility functions. The minimax
criterion, though, has the distinct advantage of being more tractable. The price at
which the asset may be converted will be assumed to follow an arithmetic random walk.
Use of such an assumption in an asset sale problem goes back as far as the early work
of Bachelier [1]. Such an assumption has been criticized by Samuelson [10] and others
in favor of a geometric random walk. In the geometric random walk changes in the
logarithm of price follow an arithmetic walk. In the problem here, the horizon is short,
and for short periods of time an arithmetic random walk is a good approximation to
the geometric walk. This same justification for using an arithmetic walk has been given
by Taylor [11, p. 12].
The motivation of the problem is now complete. The plan of the remainder of the
paper is as follows. In §2 it is shown that dollar averaging is a nonsequential mimmax
strategy if the largest possible price increase in each period is equal to the largest
possible decrease. It is shown that dollar averaging cannot be a multiperiod, nonsequential,
expected utility maximizing strategy for any strictly concave utility function
and any arithmetic random walk. In §3 the minimax sequential strategy is shown
to be of the following form. At any point of time there exists a critical value which
depends only on n, the difference between the current price and the maximum price
since sales began. Any asset held in excess of the critical value will be sold. If less than
the critical value is held none will be sold. The critical values are shown to be decreasing
functions of n. When n becomes so large that it is impossible to exceed the previous
maximum before time is up, the critical value becomes equal to zero. Under a sequential
policy more will be sold when the price decreases than when it increases. This seems
in accord with the second thoughts of those embarking on nonsequential plans of the
dollar averaging type. These second thoughts might, of course, also be explained by
extrapolative expectations. The variable minimax regret at any point in the sale is
shown to be a convex and decreasing function of the two state variables, n and z.
The variable z is the amount of the asset left to be sold. In §4 the minimax sequential
policies are calculated explicitly for a simple example. In §5 it is shown that the minimax
policies for buying a given amount of an asset are symmetric with the selling
3. MODELS OF OPTION STRATEGY 579