152 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Recall that with optimalfwe are using the stream of past trade P&Ls
as a proxy for the distribution of expected trade P&Ls currently. Therefore,
we can preprocess the trade P&L data to reflect this by converting the past
trade P&L data to reflect a commensurate percentage gain or loss based
upon the current price.
For our first two trades, which occurred at a stock price of $20 per share,
the $2 gain corresponds to a 10% gain and the $3 loss corresponds to a 15%
loss. For the last two trades, taken at a stock price of $50 per share, the $10
gain corresponds to a 20% gain and the $5 loss corresponds to a 10% loss.
The formulas to convert raw trade P&Ls to percentage gains and losses
for longs and shorts are as follows:
P&L%=Exit Price/Entry Price− 1 (for longs) (4.10a)
P&L%=Entry Price/Exit Price− 1 (for shorts) (4.10b)or we can use the following formula to convert both longs and shorts:
P&L%= P&L in Points/Entry Price (4.11)Thus, for our four hypothetical trades, we now have the following
stream ofpercentagegains and losses (assuming all trades are long trades):
+.1,−.15,+.2,−. 1
We call this new stream of translated P&Ls theequalized data,because
it is equalized to the price of the underlying instrument when the trade
occurred.
To account for commissions and slippage, you must adjust the exit
price downward in Equation (4.10a) for an amount commensurate with the
amount of the commissions and slippage. Likewise, you should adjust the
exit price upward in (4.10b). If you are using (4.11), you must deduct the
amount of the commissions and slippage (in points again) from the numer-
ator P&L in Points.
Next, we determine our optimalfon these percentage gains and losses.
Thefthat is optimal is .09. We must now convert this optimalfof .09 into
a dollar amount based upon the current stock price. This is accomplished
by the following formula:
f$=Biggest % Loss*Current Price*$per Point/−f (4.12)Thus, since our biggest percentage loss was−.15, the current price is
$100 per share, and the number of dollars per full point is 1 (since we are
dealing with buying only one share), we can determine ourf$ as:
f$=−. (^15) (^100) 1/−.09
=−15/−.09
= 166. 67