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In addition to the data provided in the charts, a number of other tests were conducted using plasmodes. A plasmode is a set of data constructed to have the characteristics that are assumed to exist in the real data. The intention is to test the ability of some algorithm or analytic technique to properly extract, detect, or analyze those characteristics. A good cycle-based trading system should be able to do a good job trading a synthetic data series containing lots of noise and occasional embedded cycles. If it cannot, there could be no expectation of it trading well in any real market. The kind of filters used in the tests below perform very well in tests involving plasmodes. GENERATING BANKS CYCLE ENTRIES USING FILTER
One way to generate cycle entries is to set up a series of filters, each for a different frequency or periodicity (e.g., going down a certain percentage per filter throughout some range or spectrum that will be analyzed). If one of these filters shows strong resonance, while the others show little or no activity, there is presumably a strong cycle in the market. An entry is generated by looking at the pair of filter outputs and buying at the next bar, if the cycle phase is such that a cyclic bottom will occur on that bar, or selling at the next bar, if a cycle top is evident or would be expected on that bar, Since the strongest-responding filter should produce no undesirable lag or phase error, such cyclic entries should provide exceedingly timely signals if the market is evidencing cyclic behavior. Attempting to buy cycle bottoms and sell cycle tops is one of the traditional ways in which cycle information has been used to trade the markets. Cycle information derived from filter banks, or by other means, can also enhance other kinds of systems or adapt indicators to current market conditions. An example of how information regarding the signal-to-noise ratio and periodicity of the dominant cycle (if there is any) may be used within another system, or to adapt an indicator to current market conditions, can be found in Ruggiero (1997). CHARACTERISTICS OF CYCLE-BASED ENTRIES A cycle-based entry of the kind studied below (which attempts to buy bottoms and sell tops) has several strong characteristics: a high percentage of winning trades, low levels of slippage, and the ability to capture as much of each move as possible. This is the kind of trading that is a trader s dream. The assumption being made is that there are well-behaved cycles in the market that can be detected and, more importantly. extrapolated by this kind of technology. It has been said that the markets evidence cyclic activity up to 70% of the time. Even if clear cycles that lead to successful trades only occur some smaller percentage of the time, because of the nature of the model, the losses can be kept small on the failed trades by the use of tight stops. The main disadvantage of a cycle-based entry is that the market may
have become efftcient relative to such trading methodologies, thanks to the proliferation of fairly powerful cycle analysis techniques, e.g., maximum entropy. The trading away of well-behaved cycles may hamper all cycle detection approaches. Since cycle entries of the kind just discussed are countertrend in nature, if the cycles show no follow-through, but the trends do, the trader can get wiped out unless good money management practices (tight stops) are employed. Whether or not a sophisticated cycle analysis works, at least as implemented here, is the question to be answered by the tests that follow. TEST METHODOLOGY
In all tests of cycle-based entry models, the standard portfolio of 36 commodities is used. The number of contracts in any market at any time to buy or sell on entry was chosen to approximate the dollar volatility of two S&P 500 contracts at the end of 1998. Exits are the standard ones in which a money management stop closes out any trade that moves more than one volatility unit into the red, a profit target limit closes out trades that push more that four volatility units into the profit zone, and a market-at-close order ends any trade that has not yet been closed out by the stop-loss or profit target after 10 days has elapsed. Entry rules are specified in the discussion of the model code and the individual tests. All tests are performed using the standard C-Trader toolkit. Here is the code implementing the wavelet filter entry model along with the standard exit strategy:
The code above implements the model being tested. The first significant block of code specifically relevant to a cyclic trading model initializes the individual filters that make up the filter bank. This code is set up to run only on the fistpass, or when a parameter specifically affecting the computations involved in initializing the filter bank (e.g., the width parameter) has changed; if no relevant parameter has changed, there is no point in reinitializing the filters every time the Model function is called.
The next block of code applies each of the filters in the bank to the input signal. In this block, two arrays are allocated to hold the filter bank outputs. The first array contains the in-phase outputs (inphase), and the second contains the in-quadrature outputs (inquad). The inputs to the filters are the raw closing prices. Because
the filters ;rre mathematiically optimal, and designed to eliminate offsets and trends,
there is no need to preprocess the closing prices before applying them, as might be necessary when using less sophisticated analysis techniques. Each row in the arrays represents the output of a single filter with a specified center frequency or periodicity. Each column represents a bar. The frequencies (or periodicities) at which the filters are centered are all spaced evenly on a logarithmic scale; i.e., the ratio between the center frequency of a given filter and the next has a fixed value. The selectivity or bandwidth (width) is the only adjustable parameter in the computation of the filter banks, the correct value of which may be sought by optimization. The usual bar-stepping loop is then entered and the actual trading signals generated. First, a good, pure cycle to trade is identified, which involves determining the power at the periodicity that has the strongest resonance with current market activity (peakpower). The cycle periodicity at which the peak power occurs is also assessed. If the periodicity is not at one of the end points of the range of periodicities being examined (in this case the range is 3 bars to 30 bars), one of the conditions for a potentially good cycle is met. A check is then made to see what the maximum power (peaknoise) is at periodicities at least 2 filters away from the periodicity at which peak power occurs If peakpower is more than 1.5 times thepeaknoise (a signal-to-noise ratio of 1.5 or greater), the second condition for a good cycle is met. The phase angle of that cycle is then determined (easy to do given the pair of filter outputs), making adjustments for the slice that occurs at 180 degrees in the plane of polar coordinates. The code then checks whether the phase is such that a cycle bottom or a cycle top is present. A small displacement term (disp) is incorporated in the phase assessments. It acts like the displacements in previous models, except that here it is in terms of phase angle, rather than bars. There is a direct translation between phase angle and number of bars; specifically, the period of the cycle is multiplied by the phase angle (in degrees), and the sum is then divided by 360, which is the number of bars represented by the phase angle. If the displaced phase is such that a bottom can be expected a certain number of degrees before or after the present bar, a buy is posted. If the phase angle is such that a top can be expected, a sell signal is issued. The limit and stop prices are then calculated, as usual. Finally, the necessary trading orders are posted. Many other blocks of code present in the above listing have not been discussed. These were used for debugging and testing. Comments embedded in the code should make their purpose fairly clear. TEST RESULTS Only one model was tested. Tests were performed for entry at the open (Test I), entry on a limit (Test 2), and entry on a stop (Test 3). The rules were simple: Buy
predicted cycle bottoms and sell predicted cycle tops. Exits took place when a cycle signal reversed an existing position or when the standard strategy closed out the trade, whichever came first. This simple trading model was first evaluated on a noisy sine wave that was swept from a period of about 4 bars to a period of about 20 bars to verify behavior of the model implementation. On this data, buy and sell signals appeared with clockwork precision at cycle tops and bottoms. The timing of the signals indicates that when real cycles are present, the model is able to detect and trade them with precision. Table 10-l contains the best in-sample parameters, as well as the performance of the portfolio on both the in-sample and verification sample data. In the table, SAMP = whether the test was on the optimization sample (IN or OUT); ROA% = the annualized return-on-account; ARRR = the annualized risk-toreward ratio; PROB = the associated probability or statistical significance; TRDS = the number of trades taken across all commodities in the portfolio: WIN% = the percentage of winning trades; $TRD = the average profit/loss per trade; BARS = the average number of days a trade was held; NETL = the total net profit on long trades, in thousands of dollars: and NETS = the total net profit on short trades, in thousands of dollars. Two parameters were optimized. The first (PI) represents the bandwidth for each filter in the filter bank. The second parameter (P2) represents the phase displacement in degrees. In all cases, the parameters were optimized on the in-sample data by stepping the bandwidth from 0.05 to 0.2 in increments of 0.05, and by stepping the phase angle displacement from -20 degrees to +20 degrees in increments of 10 degrees. Only the best solutions are shown. It is interesting that, overall, the cycle model performed rather poorly. This model was not as bad, on a dollars-per-trade basis, as many of the other systems tested, but it was nowhere near as good as the best. In-sample, the loss per trade was $1,329 with entry at open, $1,037 with entry on limit, and $1,245 with entry on stop. The limit order had the highest percentage of wins and the smallest average loss per
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