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| The Breakout
Bulletin
The following article was originally published in the April 2005 issue of
The Breakout Bulletin.
Optimizing
Over Multiple Markets
Most mechanical trading systems
have one or more parameters, such as the length of a moving average or the size
of the money management stop. Ideally, there should be a rational method for
choosing values for these parameters. Optimization is one such method. However,
the topic of system optimization tends to divide traders. On one side of the
debate are those who contend that any optimization is bad and will at best
produce parameter sets that are no better than ones chosen at random. On the
other side are traders, such as myself, who believe that we can do better than
picking numbers out of a hat.
In past issues of this
newsletter, I've tried to address some of the issues and possible solutions to
the problems of system optimization. For example, I've tried to make a
distinction between prudent optimization (a good thing) and over-fitting a
system to the market (a bad thing). In the
May 2003 issue,
for example, I looked at least-squares curve fitting and significance testing
for possible guidance. While significance tests, at least of the type I
discussed, need to be interpreted somewhat loosely, they do tell us one
important thing: the more trades in the optimization sample, the less likely the
system will be over-fit to the market.
As I discussed in the
November
2004 issue,
optimizing over a large sample of trades can sometimes be tricky because profits
over one specific time period can bias the results towards that time period if
you're optimizing for net profit or rate of return. To address this, I presented
a method (and EasyLanguage function) that makes it easier to optimize
effectively over large trade histories. But what if you don't have a large
sample of trades from your primary market?
One possibility is to optimize
the system simultaneously over multiple markets and choose the parameter values
that work best (in some sense) over all markets together. Not only will this
increase the number of trades in the optimization sample, but it will expose the
trading system to a wider variety of market conditions than if you had optimized
over just one market. The premise of this article is that a parameter set
that works well over several markets may be more likely to hold up well in the
future than a parameter set tailored to a single market.
In TradeStation, it's not
possible to optimize a trading system over more than one market. To make this
work, then, we need a sort-of hybrid approach. We'll do the optimizations in
TradeStation for each individual market, then bring them all together in Excel,
where we'll find the overall winner. In TradeStation, we'll optimize the system
for each market using the EqtyCorr function presented in the November 2004 issue
(referenced above). The EqtyCorr function records the system parameter values
and optimization objective function for each combination of parameters evaluated
during the optimization. Since TradeStation iterates through every possible
combination of parameter values, the output of the EqtyCorr function is a list
of every possible combination of parameter values with the corresponding
objective function values. An example is shown in Fig. 1.
Figure 1. Typical
output of the EqtyCorr function opened in Excel. The objective function
(combination of weighted profit and correlation coefficient) is shown in column
A, followed by the weighted profit in column B and the correlation coefficient
in column C. The remaining columns list the system parameter values. Each row
represents one trial or iterate of the
optimization.
The first
step, then, is to generate a file similar to that shown in Fig. 1 for each
market. We do this by optimizing the trading system over each market using the
EqtyCorr function. We need to make sure that each optimization uses the same
ranges and increments of the system parameters. This will be necessary for step
two.
Step two is
combining the spreadsheet files for each market into a single spreadsheet. We do
this by copying and pasting the columns for each market side-by-side. The
resulting spreadsheet should look something like Fig. 2. The first set of
columns are the same as shown in Fig. 1. To the right of those are three columns
copied from the EqtyCorr generated file for the NQ (E-mini Nasdaq) market. The
columns for the parameter values will be the same in each file (assuming we've
used the same ranges and increments for the parameters, as advised above), so
I've only copied the three columns related to the objective function. To the
right of the three columns for the NQ are the three columns for the ER2 (E-mini
Russell 2000). Other markets could be added to the right if
desired.
Figure 2. Combined
spreadsheet with EqtyCorr function results for all
markets.
To find the
parameter value set that works best over all markets, we need an objective
function for the combination of markets. The objective
function is a measure of how well the system performs. The objective function
produced by the EqtyCorr function is a weighted average of the net profit and
the correlation coefficient of the equity curve. It basically says we want a
high net profit but also a straight equity curve. This objective function is
listed in columns A, J, and N of Fig. 2. For the combined result, we'll take an
average of these individual objective functions and divide by the standard
deviation. The result is shown as column R in Fig. 3.
Figure 3. Combined
spreadsheet with overall objective function in column R. The overall objective
function is the average of the objective functions for each market divided by
their standard deviation.
The Excel
formula for column R is as follows:
R2=AVERAGE(A2,J2,N2)/STDEV(A2,J2,N2)
where columns
A, J, and N contain the individual objective functions for each
market. Cell R2 is "filled down" to the other cells in column R.
The "best"
parameter set will be the one with the highest value in column R. Dividing by
the standard deviation of the individual objective functions means that the
overall objective function will be higher when the individual objective
functions are similar in value. This helps prevent an unusually strong
result in one market from biasing the results towards that market at the
expense of the other markets. We want a parameter set that gives good results in
all markets, even if the results are not optimal in any one
market.
The final step is to sort the
columns by the values in column R (in descending order) using the Sort command
of the Data menu in Excel. This will put the row with the highest value of the
objective function at the top of the spreadsheet. The parameter values in
columns D through H in the top row will be our optimal parameter
set.
As an example, consider the
West Coast Index system. This system has one input parameter for entering trades
and four for exiting (two for longs and two for shorts), for a total of five
input parameters. We want to find the best set of parameter values for trading
the system on the four major E-mini stock index futures at the CME
exchange: ES (E-mini S&P 500), NQ (E-mini Nasdaq 100), ER2 (E-mini Russell
2000), and EMD (E-mini S&P MidCap 400).
The first parameter was varied
from 15 to 45 in increments of 5. The next two were varied from 3 to 9 in
increments of 1, and the last two from 1 to 5 in increments of 1. The same
ranges and increments were used for all markets. This resulted in a total of
8575 tests per market (and therefore 8575 rows of data in the spreadsheet).
Slippage and commission costs of $30 per round turn were deducted from each
trade. After running each individual optimization, the output files
from the EqtyCorr function were combined into a single spreadsheet and the
objective function was defined as shown above. After sorting the rows in
descending order of the objective function, the resulting spreadsheet is
as shown in Fig. 4.
Figure 4. Combined
spreadsheet for West Coast Index system with results sorted in descending order
of the overall objective function (column V). The optimal parameter values are
in columns D - H in row 2.
The overall objective function
is shown in column V, where the highest (optimal) value is at the top in row 2.
The optimal parameter values are shown in columns D - H in row 2: 25, 5, 5, 4,
5. These are the parameter values that maximize the objective function and
therefore give the best historical performance over all markets, as defined by
the objective function.
The individual equity curves
for each market obtained using the optimal parameter values are shown in Figs. 5
- 8. The number of trades generated by the system is not very large in any one
market, ranging from about 55 to 80 trades per market. However, when you combine
the trades into a single portfolio equity curve, as shown in Figs. 9 and 10,
there are 270 trades in all. That larger number of trades should give us
confidence that the system has not been over-optimized. Moreover, the fact that
the same set of parameter values works well in four different -- albeit
correlated -- markets further increases the chances of the system holding up
well in the future. Also note that the equity curve for the portfolio, as shown
in Fig. 9, is smoother and straighter than any of the individual equity curves
-- one of the virtues of diversification.
 Figure 5. One-contract
equity curve for the West Coast Index system on 30 min bars of the E-mini
S&P 500 using parameter values optimized over the ES, NQ, ER2 and EMD
markets simultaneously. $30 was deducted from each trade for slippage and
commissions.
Figure 6. One-contract
equity curve for the West Coast Index system on 30 min bars of the
E-mini Nasdaq using parameter values optimized over the ES, NQ, ER2
and EMD markets simultaneously. $30 was deducted from each trade for
slippage and commissions.
Figure 7. One-contract
equity curve for the West Coast Index system on 30 min bars of the
E-mini Russell 2000 using parameter values optimized over the ES, NQ,
ER2 and EMD markets simultaneously. $30 was deducted from each trade
for slippage and commissions.
Figure 8. One-contract
equity curve for the West Coast Index system on 30 min bars of the E-mini
S&P MidCap 400 using parameter values optimized over the ES, NQ, ER2
and EMD markets simultaneously. $30 was deducted from each trade for
slippage and commissions.
Figure 9. One-contract
portfolio equity curve for the West Coast Index system on the ES, NQ,
ER2 and EMD markets using parameter values optimized for all markets
together. $30 was deducted from each trade for slippage and commissions.
Figure 10. Portfolio
equity curve for the West Coast Index system with a variable number of
contracts on the ES, NQ, ER2 and EMD markets using parameter
values optimized for all markets together. 5% of account equity was risked on
each trade. $30 was deducted from each trade for slippage and commissions.
The method presented above can
be applied to any TradeStation system, provided you have access to the
EasyLanguage code (for adding the call to EqtyCorr). The spreadsheet
manipulations involve little more than copying and pasting columns of data.
There's only one spreadsheet formula (shown above) that needs to be entered. Of
course, the usual admonitions apply when optimizing a trading system: always
test out-of-sample and preferably in real-time before committing real
money.
Notes:
-
To learn more about the West Coast Index system or to order, please visit
[discontinued].
The portfolio equity curves were generated using Market System Analyzer.
For more information, please visit
http://www.adaptrade.com/MSA/index.htm.
That's all for
now. Good luck
with your trading.
Mike Bryant
Breakout Futures
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