STOCHASTIC

392 GORDON PYE
be the vector of price differences between the actual price in period 0, 1, ••• , t and
the initial price. Let n,(p) be the n for the seller in period t. Let nb(p) be the n of
the buyer in period t. From the definition of n for the seller and buyer it follows that:
«»(p) - Pi — MinoSTS,Ipr] = MaxoSrS, [-pT] - (-p.) = n,(-p).
Since — p is the mirror image of the price differentials about zero, the stated proposition
has been proved.
It has now been established that the amount sold by a seller and the amount bought
by a buyer for any n and t are equal, and that nj(p) = n,( —p). Initially, a buyer will
want to buy the same amount of the asset as a seller. Subsequently, however, if the
price rises n, will continue to equal zero while ra& will rise. Thus, a buyer will want to
buy more while a seller will not want to sell more. The converse will hold if the price
falls. Behavior of buyers and sellers will be symmetrical. In the example in Figure 1 the
minimax behavior of a buyer will be given by the mirror image of each of the series.
In other words, the amounts by the nodes when the figure is rotated 180° about the
horizontal axis will give the quantities which a buyer will purchase. It is interesting to
note that sequential, minimax policies will tend to have a reinforcing effect on price
movements. As has been seen, price rises will make buyers want to buy more without
causing any increase in sales by sellers.
6. Policies under Reversibility
So far it has been assumed that the seller is prohibited from repurchasing any of the
asset. This assumption will now be relaxed to permit repurchases at any time before T.
Margin purchases and short sales, however, will continue to be prohibited. As discussed
in the introduction this has the effect of changing the flavor of the problem to one of
multiperiod portfolio selection. However, it is unconventional in that the objective
continues to be one of minimaxing regret. Under these conditions the problem does not
seem to replicate situations in which dollar averaging has been recommended. Nevertheless,
the problem is of interest because the answer is simple and forms an interesting
contrast to the results obtained previously.
The possibility of repurchase changes the maximum amount which can be obtained
for the asset, given the price series which occurs. Without repurchase the best one can
do is sell out everything at the highest price which occurs. In general, with repurchase
one can do better. The best that can be done is to be fully invested during every price
increase and completely disinvested during every price decrease. Thus, the amount
obtained for the asset will be equal to the initial price plus the sum of all the price increases
which occur. The expression for regret in (3) must be changed to read as follows:
(15) R = S+ Ap