spectrum, we could observe the spread at regular time intervals and put on
a position whenever we spot a deviation, regardless of the current holdings
in inventory relying on the statistics of the spread series to control our in-
ventory. On the other extreme, we could limit our exposure to one spread
unit;^1 that is, if we currently have one unit of spread in inventory, then even
if we observe a deviation in the same direction, we do not add on to our po-
sition. However, if the spread deviates in the opposite direction, we close out
our current position and enter into a new spread position in the opposite di-
rection of our original holdings. Practical trading is probably somewhere be-
tween the two extremes.
Given the different trading styles, it would be natural to require that
trading rule design be tailored to the specifics of each trading style. Fortu-
nately for us, though, the specifics of the trading style do not matter in the
determination of ∆to maximize profits. To see that more clearly, let us now
walk through the process of determining the value of ∆when the spread is a
Gaussian white noise series.
The Gaussian white noise series is a series of drawings from a Gaussian
distribution. We buy one unit of the spread whenever we observe that the
spread has a value less than or equal to –∆. Similarly, we sell one unit of the
spread when we observe a value greater than or equal to ∆. The probability
that a white noise process at any time instant deviates by an amount greater
than or equal to ∆(∆being positive) is determined by the integral of the
Gaussian process, which is 1–N(∆). Therefore, in Ttime steps we can expect
to have T instances greater than ∆. Similarly, the probability of
the value being less than or equal to –∆is given by N(–∆). Now, owing to the
symmetry of the Gaussian process N(–∆) = 1 – N(∆) and therefore the num-
ber of instances, we expect the value of the spread to be less than or equal
to –∆is also T(1–N(∆). Thus, in a time span of Tunits we can expect to have
bought and sold the spread on an average of T times. The profit
on each buy and sell is 2∆. A measure of profitability for trading in the
time period Tis therefore (profit per trade ×number of trades); that is,
.^2 Also, note that even if we were to liquidate our positions at
equilibrium value, the measure of profitability would remain the same.
Now the problem of band design boils down to determining the value of
∆that maximizes. Figures 8.1a and b are plots of such a func-
tion. On the x-axis is the value of ∆as measured in terms of the standard de-
viation of the normal density about the mean. The y-axis is a plot of the
∆∆() 1 −N()
21 TN∆∆−()()
()^1 −N()∆
()^1 −N()∆
120 STATISTICAL ARBITRAGE PAIRS
(^1) A unit of spread is determined by the average volume per trade in the two stocks and
the ratio between them.
(^2) Note that this is not the actual percentage profit as measured in conventional terms.
If we do assume that we have relatively deep pockets and wish to maximize the dol-
lar yield, then the measure as described here is appropriate.