Ralph Vince - Portfolio Mathematics

ch01 JWBK035-Vince February 22, 2007 21 : 43 Char Count= 0

The Random Process and Gambling Theory 39

it.A trade can also be counted as a turningpointifitis for a lesser P&L

value than both the trade beforeit and the trade afterit.Notice that we are

usingtheindividual trades, not the equity curve (the cumulative values of

the trades).The number of turningpointsis totaled up for the entire stream

of trades.Note that we must start with the second trade and end with the

next to last trade, as we need a trade on either side of the trade we are

consideringas a turningpoint.

Consider now three values (1, 2, 3)in a random series, whereby each

of the six possible orderings are equally likely:

1 , 2 , 32 , 3 , 11 , 3 , 23 , 1 , 22 , 1 , 33 , 2 , 1

Of these six, four will resultin a turningpoint.Thus, for a random

stream of trades, the expected number of turningpointsisgiven as:

Expected number of turningpoints= 2 / 3 ∗(N−2) (1.06)

where: N=The total number of trades

We can derive the variancein the number of turningpoints of a random

series as:

Variance=(16∗N−29)/ 90 (1.07)

The standard deviationis the square root of the variance.Takingthe

difference between the actual number of turningpoints countedin the

stream of trades and the expected number and then dividingthe difference

by the standard deviation willgive us a Z score, whichis then expressed as

a confidence limit.The confidence limitisdiscerned from Equation (2.22)

for two-tailed Normal probabilities.Thus,if our stream of tradesis very far

away (very many standard deviations from the expected number),itis un-

likely that our stream of tradesis random;rather, dependencyis present.

If dependency appears to a high confidence limit (at least 95%) with the

turningpoints test, you can determine frominspection whether like begets

like (if there are fewer actual turningpoints than expected) or whether like

begets unlike (if there are more actual turningpoints than expected).

Another test for dependenceis thephase length test.Thisisastatistical

test similar to the turningpoints test.Rather than countingup the number

of turningpoints between (but notincluding) trade 1 and the last trade,

the phase length test looks at how many trades have elapsed between turn-

ingpoints.A“phase”is the number of trades that elapse between a turning

point high and a turningpoint low, or a turningpoint low and a turning

point high.It doesn’t matter which occurs first, the high turningpoint or

the low turningpoint.Thus,if trade number 4isaturningpoint (highor