Quantitative Trading Systems

On a forum I regularly follow, Yahoo AmiBroker Forum, a message complained that the AmiBroker trading system development platform was unable to handle an optimization that had 10 parameters, each with 20 values. Whether that poster was serious about the numbers 10 and 20 or not, the question is worth considering.

A large number of parameters, say 10, each with a large number of evaluation points, say 20, leads directly to the “curse of dimensionality.” Assuming that it takes one nano second to evaluate each alternative, and that everything works as it should and the run does finish (no power failure, memory failure, cpu failure, hard disk failure, flood, earthquake, premature death, etc), exhaustive evaluation of the n = 10^20 points will have taken about 3100 years. At 50 lines per 12 inch page, the listing of the results will be 65 light years long. (The nearby universe is described as being those bodies within about 15 light years.)

Each result gives the value of an objective function. Sorting the results into ascending or descending order according to the value of the objective function puts the set of values that are “best” at the top of the list. Assuming a sorting algorithm that takes Order (n*ln(n)) is used, it will take an additional 46 times 3100 years to sort them.

Now make an individual run using each of the sets of values near the top to gain some confidence that the alternative ranked first is the one that is preferred. Maybe an hour or two more?

Just in time to give a signal for tomorrow’s trade?

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What is a reasonable number of alternatives to evaluate?

I recommend beginning by giving a considerable amount of thought and experimentation to design and testing of the objective function. Have confidence that the alternative with the highest objective function score is the one that is preferred. Or at least that it is acceptable — not letting the search for perfect get in the way of finding a satisfactory solution.

Run some tests using your system, your objective function, your data, on your computer. How long does it take to generate and sort 1000 or 10,000 alternatives? How many of these test runs do you expect to make before deciding on the logic and parameters that will be used? How many hours are you willing to allocate to the process? Working the arithmetic backwards will give an estimate of the number of alternatives that can realistically be tested. Continuing to work backward, you can compute the size of the parameter space that can be exhaustively processed in that amount of time. Make some trial runs using both exhaustive and non-exhaustive methods. Compare the results that rank best from each set of runs. Decide whether you can accept the results of the non-exhaustive method. If so, use it; if not, modify the afl code so that the number of parameters and number of evaluation points for each create an examination space that can be processed in the time you have allocated.

Thanks to Tomasz for creating a superb toolset that enables us to have these choices.

Thanks for listening,

Equity curve feedback uses the equity curve of a trading system to determine when to take trades from that system and when to just watch them.

More information, including AmiBroker code, can be found on the Equity curve feedback static page.

Many applications, including Excel and AmiBroker, have built-in functions that compute the standard deviation of a series of numeric values. The definition of standard deviation is square root of variance, where variance is the average of the squared deviation of each element in the series from the mean. (The Mean is the first moment of the distribution of the series, and Variance is the second moment of the distribution of the series.) The most common formula for variance is
Var = summation for all elements [(x[i] – xbar)^2]
where x[i] is the ith element of the series and xbar is the mean of the series.
This is straight forward, but requires that the mean be known before computing the summation of squared deviations.

When the data points are being received or processed sequentially, it may be inconvenient or even impossible to make the two passes necessary to compute the mean before computing the summation of squared deviations. By using a little algebra, it is possible to compute the mean and variance in a single pass. The AmiBroker code for doing this is provided on the Standard deviation in a single pass static page.