High-Frequency Financial Trading with Multivariate Direct Filtering Part Deux: Index Futures

optimisedstxe

Out-of-sample performance (cash, blue-to-pink line) of an MDFA low-pass filter built using the approach discussed in this article on the STOXX Europe 50 index futures with expiration March 18th (STXEH3) for 200 15-minute interval log-return observations. Only one small loss out-of-sample was recorded from the period of Jan 18th through February 1st, 2013.

Continuing along the trend of my previous installment on strategies and performances of high-frequency trading using multivariate direct filtering, I take on building trading signals for high-frequency index futures, where I will focus on the STOXX Europe 50 Index, S&P 500, and the Australian Stock Exchange Index. As you will discover in this article, these filters that I build using MDFA in iMetrica have yielded some of the best performing trading signals that I have seen using any trading methodology.  My strategy as I’ve been developing throughout my previous articles on MDFA has not changed much, except for one detail that I will discuss throughout and will be a major theme of this article, and that relates to an interesting structure found in index futures series for intraday returns. The structure is related to the close-to-open variation in the price, namely when the price at close of market hours significantly differs from the price at open. an effect I’ve mentioned in my previous two articles dealing with high(er)-frequency (or intraday) log-return data. I will show how MDFA can take advantage of this variation in price and profit from each one by ‘predicting’ with the extracted signal the jump or drop in the price at the open of the next trading day.

The frequency of observations on the index that are to be considered for building trading filters using MDFA is typically only a question of taste and priorities.  The beauty of MDFA lies in not only the versatility and strength in building trading signals for virtually any financial trading priorities, but also in the independence on the underlying observation frequency of the data. In my previous articles, I’ve considered and built high-performing strategies for daily, hourly, and 15 minute log returns, where the focus of the strategy in building the signal began with viewing the periodogram as the main barometer in searching for optimal frequencies on which one should set the low-pass cutoff for the extracting target filter \Gamma function.

Index futures, as in a futures contract on a financial index, as we will see in this article present no new challenges. With the success I had on the 15-minute return observation frequency that I utilized in my previous article in building a signal for the Japanese Yen, I will continue to use the 15 minute intervals for the index futures where I hope to shed some more light on the filter selection process. This includes deducing properties of the intrinsically optimal spectral peaks to trade on. To do this, I present a simple approach I take in these examples by first setting up a bandpass filter over the spectral peak in the periodogram and then study the in-sample and out-of-sample characteristics of this signal, both in performance and consistency. So without further ado, I present my experiments with financial trading on index futures using MDFA, in iMetrica.

STOXX Europe 50 Index (STXE H3, Expiration March 18 2013)

The STOXX Europe 50 Index, Europe’s leading Blue-chip index, provides a representation of sector leaders in Europe. The index covers 50 stocks from 18 European countries and has the highest trading volume of any European index. One of the first things to notice with the 15-minute log-returns of STXE are the frequent large spikes. These spikes will occur every 27 observations at 13:30 (UTC time zone) due to the fact that there are 26 15-minute periods during the trading hours. These spikes represent the close-to-open jumps that the STOXX Europe 50 index has been subjected to and then reflected in the price of the futures contract. With this ‘seasonal’ pattern so obviously present in the log-return data, the frequency effects of this pattern should be clearly visible in the periodogram of the data. The beauty of MDFA (and iMetrica) is that we have the ability to explicitly engineer a trading signal to take advantage of this ‘seasonal’ pattern by building an appropriate extractor \Gamma.

Figure 1: Log-returns of STXE for the 15-min observations from 1-4-2013 to 2-1-2013

Figure 1: Log-returns of STXE for the 15-min observations from 1-4-2013 to 2-1-2013

Regarding the periodogram of the data, Figure 2 depicts the periodograms for the 15 minute log-returns of STXE (red) and the explanatory series (pink) together on the same discretized frequency domain. Notice that in both log-return series, there is a principal spectral peak found between .23 and .32.   The trick is to localize the spectral peak that accounts for the cyclical pattern that is brought about by the close-to-open variation between 20:00 and 13:30 UTC.

Figure 2: The periodograms for the 15 minute log-returns of STXE (red) and the explanatory series (pink).

Figure 2: The periodograms for the 15 minute log-returns of STXE (red) and the explanatory series (pink).

In order to see the effects of the MDFA filter when localizing this spectral peak, I use my target \Gamma builder interface in iMetrica to set the necessary cutoffs for the bandpass filter directly covering both spectral peaks of the log-returns, which are found between .23 and .32. This is shown in Figure 3, where the two dashed red lines show indicate the cutoffs and spectral peak is clearly inside these two cutoffs, with the spectral peak for both series occurring in the vicinity of \pi/12.  Once the bandpass target \Gamma was fixed on this small frequency range, I set the regularization parameters for the filter coefficients to be \lambda_{smooth} = .86,   \lambda_{decay} = .11,  and \lambda_{decay2} = .11.

Figure 3: Choosing the cutoffs for the band pass filter to localize the spectral peak.

Figure 3: Choosing the cutoffs for the band pass filter to localize the spectral peak.

Pinpointing this frequency range that zooms in on the largest spectral peak generates a filter that acts on the intrinsic cycles found in the 15 minute log-returns of the STXE futures index. The resulting trading signal produced by this spectral peak extraction is shown in Figure 4, with the returns (blue to pink line) generated from the trading signal (green) , and the price of the STXE futures index in gray. The cyclical effects in the signal include the close-to-open variations in the data. Notice how the signal predicts the variation of the close-to-open price in the index quite well, seen from the large jumps or falls in price every 27 observations. The performance of this simple design in extracting the spectral peak of STXE yields a 4 percent ROI on 200 observations out-of-sample with only 3 losses out of 20 total trades (85 percent trade success rate), with two of them being accounted for towards the very end of the out-of-sample observations in an uncharacteristic volatile period occurring on January 31st 2013.

Figure : The performance in-sample and out-of-sample of the spectral peak localizing bandpass filter.

Figure 4: The performance in-sample and out-of-sample of the spectral peak localizing bandpass filter.

The two concurrent frequency response (transfer) functions for the two filters acting on the STXE log-return data (purple) and the explanatory series (blue), respectively, are plotted below in Figure 5. Notice the presence of the spectral peaks for both series being accounted for in the vicinity of the frequency \pi/12, with mild damping at the peak. Slow damping of the noise in the higher frequencies is aided by the addition of a smoothing expweight parameter that was set at \alpha = 4.

Figure : The performance in-sample and out-of-sample of the spectral peak localizing bandpass filter.

Figure 5: The concurrent frequency response functions of the localizing spectral peak band-pass filter.

With the ideal characteristics of a trading signal quite present in this simple bandpass filter, namely smooth decaying filter coefficients, in-sample and out-of-sample performance properties identical, and accurate, consistent trading patterns, it would be hard to imagine on improving the trading signal for this European futures index even more. But we can. We simply keep the spectral peak frequencies intact, but also account for the local bias in log-return data by extending the lower cutoff to frequency zero. This will provide improved systematic trading characteristics by not only predicting the close-to-open variation and jumps, but also handling upswings and downswings, and highly volatile periods much better.

In this new design, I created a low-pass filter by keeping the upper cutoff \omega_1 from the band-pass design and setting the lower cutoff to 0. I also increased the smoothing parameter to $\alpha = 32$. In this newly designed filter, we see a vast improvement in the trading structure. As before, the filter was able to deduce the direction of every single close-to-open jump during the 200 out-of-sample observations, but notice that it was also able to become much more flexible in the trading during any upswing/downswing and volatile period. This is seen in more detail in Figure 7, where I added the letter ‘D’ to each of the 5 major buy/sell signals occurring before close.

Figure : Performance of filter both in-sample (left of cyan line) and on 210 observations out-of-sample (right of cyan line).

Figure 6: Performance of filter both in-sample (left of cyan line) and on 210 observations out-of-sample (right of cyan line).

Notice that the signal predicted the jump correctly for each of these major jumps, resulting in large returns. For example, at the first “D” indicator,  the signal indicated sell/short (magenta dashed line) the STXE future index 5 observations before close, namely at 18:45 UTC, before market close at 20:00 UTC. Sure enough, the price of the STXE contract went down during overnight trading hours and opened far below the previous days close, with the filter signaling a buy (green dashed line) 45 minutes into trading. At the mark of the second “D”, we see that on the final observation before market close, the signal produced a buy/long indication, and indeed, the next day the price of the future jumped significantly.

Figure : Zooming in on the out-of-sample performance and showing all the signal responses that predicted the major close-to-open jumps.

Figure 7: Zooming in on the out-of-sample performance and showing all the signal responses that predicted the major close-to-open jumps.

Only two very small losses of less than .08 percent were accounted for.  One advantage of including the frequency zero along with the spectral peak frequency of STXE is that the local bias can help push-up or pull-down the signal resulting in a more ‘patient’ or ‘diligent’ filter that can better handle long upswings/downswings or volatile periods. This is seen in the improvement of the performance towards the end of the 200 observations out-of-sample, where the filter is more patient in signaling a sell/short after the previous buy. Compare this with the end of the performance from the band-pass filter, Figure 4.  With this trading signal out-of-sample, I computed a 5 percent ROI on the 200 observations out-of-sample with only 2 small losses. The trading statistics for the entire in-sample combined with out-of-sample are shown in Figure 8.

Figure 9: The total performance statistics of the STXEH3 trading signal in-sample plus out-of-sample.

Figure 8: The total performance statistics of the STXEH3 trading signal in-sample plus out-of-sample. The max drop indicates -0 due to the fact that there was a truncation to two decimal places. Thus the losses were less than .01.

S&P 500 Futures Index (ES H3, Expiration March 18 2013)

In this experiment trading S&P 500 future contracts (E-mini) on observations of 15 minute intervals from Jan 4th to Feb 1st 2013, I apply the same regimental approach as before.  In looking at the log-returns of ESH3 shown in Figure 10, the effect of close-to-open variation seem to be much less prominent here compared to that on the STXE future index. Because of this, the log-returns seem to be much closer to ‘white noise’ on this index. Let that not perturb our pursuit of a high performing trading signal however. The approach I take for extracting the trading signal, as always, begins with the periodogram.

The log-return data of ES H3 at 15 minute intervals from 1-4-2013 to 2-1-2013.

Figure 10: The log-return data of ES H3 at 15 minute intervals from 1-4-2013 to 2-1-2013.

As the large variations in the close-to-open price are not nearly as prominent, it would make sense that the same spectral peak found before at near \pi/12 is not nearly as prominent either. We can clearly see this in the periodogram plotted below in Figure 11. In fact, the spectral peak at \pi/12 is slightly larger in the explanatory series (pink), thus we should still be able to take advantage of any sort of close-to-open variation that exists in the E-min future index.

Figure 13: Periodograms of ES H3 log-returns (red) and the explanatory series (pink). The red dashed vertical lines are framing the spectral peak between .23 and .32.

Figure 11: Periodograms of ES H3 log-returns (red) and the explanatory series (pink). The red dashed vertical lines are framing the spectral peak between .23 and .32.

With this spectral peak extracted from the series, the resulting trading signal is shown in Figure 12 with the performance of the bandpass signal shown in Figure 13.

Figure 12: The signal built from the extracted spectral peak and the log-return ESH3 data.

Figure 12: The signal built from the extracted spectral peak and the log-return ESH3 data.

One can clearly see that the trading signal performs very well during the consistent cyclical behavior in the ESH3 price, However, when breakdown occurs in this stochastic structure and follows more prominently another frequency, the trading signal dies and no longer trades systematically taking advantage of the intrinsic cycle found near \pi/12. This can be seen in the middle 90 or so observations. The price can be seen to follow more closely a random walk and the trading becomes inconsistent. After this period of 90 or so observations however, just after the beginning of the out-of-sample period, the trajectory of the ESH3 follows back on its consistent course with a \pi/12 cyclical component it had before.

Figure 13: The performance in-sample and out-of-sample of the simple bandpass filter extracting the spectral peak.

Figure 13: The performance in-sample and out-of-sample of the simple bandpass filter extracting the spectral peak.

Now to improve on these results, we include the frequency zero by moving the lower cutoff of the previous band-pass filter to $\latex \omega_0 = 0$.  As I mentioned before, this lifts or pushes down the signal from the local bias and will trade much more systematically. I then lessened the amount of smoothing in the expweight function to \alpha = 24, down from \alpha = 36 as I had on the band-pass filter.  This allows for slightly higher frequencies than \pi/12 to be traded on. I then proceeded to adjust the regularization parameters to obtain a healthy dosage of smoothness and decay in the coefficients. The result of this new low-pass filter design is shown in Figure 14.

Figure 11: Performance out-of-sample (right of cyan line) of the ES H3 filter on 200 15 minute observations.

Figure 14: Performance out-of-sample (right of cyan line) of the ES H3 filter on 200 15 minute observations.

The improvement in the overall systematic trading and performance is clear. Most of the major improvements came from the middle 90 points where the trading became less cyclical. With 6 losses in the band-pass design during this period alone, I was able to turn those losses into two large gains and no losses.  Only one major loss was accounted for during the 200 observation out-of-sample testing of filter from January 18th to February 1st, with an ROI of nearly 4 percent during the 9 trading days. As with the STXE filter in the previous example, I was able to successfully build a filter that correctly predicts close-to-open variations, despite the added difficulty that such variations were much smaller. Both in-sample and out-of-sample, the filter performs consistently, which is exactly what one strives for thanks to regularization.

ASX Futures (YAPH3, Expiration March 18, 2013)

In the final experiment, I build a trading signal for the Australian Stock Exchange futures, during the same period of the previous two experiments. The log-returns show moderately large jumps/drops in price during the entire sample from Jan 4th to Feb 1st, but not quite as large as in the STXE index. We still should be able to take advantage of these close-to-open variations.

YAPH3

Figure 15: The log-returns of the YAPH3 in 15-minute interval observations.

In looking at the periodograms for both the YAPH3 15 minute log-returns (red) and the explanatory series (pink), it is clear that the spectral peaks don’t align like they did in the previous two exampls. In fact, there hardly exists a dominant spectral peak in the explanatory series, whereas the \pi/12 spectral peak in YAPH3 is very prominent. This ultimately might effect the performance of the filter, and consequently the trades. After building the low-pass filter and setting a high smoothing expweight parameter \alpha = 26.5, I then set the regularization parameters to be  \lambda_{smooth} = .85,   \lambda_{decay} = .11,  and \lambda_{decay2} = .11 (same as first example).

Figure : The periodograms for YAPH3 and explanatory series with spectral peak in YAPH3 framed by the red dashed lines.

Figure 16: The periodograms for YAPH3 and explanatory series with spectral peak in YAPH3 framed by the red dashed lines.

The performance of the filter in-sample and out-of-sample is shown in Figure 18. This was one of the more challenging index futures series to work with as I struggled finding an appropriate explanatory series (likely because I was lazy since it was late at night and I was getting tired). Nonetheless, the filter still seems to predict the close-to-open variation on the Australian stock exchange index fairly well. All the major jumps in price are accounted for if you look closely at the trades (green dashed lines are buys/long and magenta lines are sells/shorts) and the corresponding action on the price of the futures contract.   Five losses out-of-sample for a trade success ratio of 72 percent and an ROI out-of-sample on 200 observations of 4.2 percent.  As with all other experiments in building trading signals with MDFA, we check the consistency of the in-sample and out-of-sample performance, and these seem to match up nicely.

Figure : The out-of-sample performance of the low-pass filter on YAPH3.

Figure 18 : The out-of-sample performance of the low-pass filter on YAPH3.

The filter coefficients for the YAPH3 log-difference series is shown in Figure 19. Notice the perfectly smooth undulating yet decaying structure of the coefficients as the lag increases. What a beauty.

Figure 16: Filter coefficients for the YAPH3 series.

Figure 19: Filter coefficients for the YAPH3 series.

Conclusion

Studying the trading performance of spectral peaks by first constructing band-pass filters to extract the signal corresponding to the peak in these index futures enabled me to understand how I can better construct the lowpass filter to yield even better performance. In these examples, I demonstrated that the close-to-open variation in the index futures price can be seen in the periodogram and thus be controlled for in the MDFA trading signal construction. This trading frequency corresponds to roughly \pi/12 in the 15 minute observation data that I had from Jan 4th to Feb 1st. As I witnessed in my empirical studies using iMetrica, this peak is more prominent when the close-to-open variations are larger and more often, promoting a very cyclical structure in the log-return data.  As I look deeper and deeper into studying the effects of extracting spectral peaks in the periodogram of financial data log-returns and the trading performance, I seem to improve on results even more and building the trading signals becomes even easier.

Stay tuned very soon for a tutorial using R (and MDFA) for one of these examples on high-frequency trading on index futures.  If you have any questions or would like to request a certain index future (out of one of the above examples or another) to be dissected in my second and upcoming R tutorial, feel free to shoot me an email.

Happy extracting!

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High-Frequency Financial Trading on FOREX with MDFA and R: An Example with the Japanese Yen

In-sample (observations 1-235) and out-of-sample (observations 236-455) performance of the trading signal built in this tutorial using MDFA. (Top) The log price of the Yen (FXY) in 15 minute intervals and the trades generated by the trading signal. Here black line is a buy (long), blue is sell (short position). (Bottom) The returns accumulated (cash)  generated by the trading, in percentage gained or lossed.

Figure 1: In-sample (observations 1-250) and out-of-sample performance of the trading signal built in this tutorial using MDFA. (Top) The log price of the Yen (FXY) in 15 minute intervals and the trades generated by the trading signal. Here black line is a buy (long), blue is sell (short position). (Bottom) The returns accumulated (cash) generated by the trading, in percentage gained or lost.

In my previous article on high-frequency trading in iMetrica on the FOREX/GLOBEX, I introduced some robust signal extraction strategies in iMetrica using the multidimensional direct filter approach (MDFA) to generate high-performance signals for trading on the foreign exchange and Futures market. In this article I take a brief leave-of-absence from my world of developing financial trading signals in iMetrica and migrate into an uber-popular language used in finance due to its exuberant array of packages, quick data management and graphics handling, and of course the fact that it’s free (as in speech and beer) on nearly any computing platform in the world.

This article gives an intro tutorial on using R for high-frequency trading on the FOREX market using the R package for MDFA (offered by Herr Doktor Marc Wildi von Bern) and some strategies that I’ve developed for generating financially robust trading signals. For this tutorial, I consider the second example given in my previous article where I engineered a trading signal for 15-minute log-returns of the Japanese Yen (from opening bell to market close EST).  This presented slightly new challenges than before as the close-to-open jump variations are much larger than those generated by hourly or daily returns. But as I demonstrated, these larger variations on close-to-open price posed no problems for the MDFA. In fact, it exploited these jumps and made large profits by predicting the direction of the jump. Figure 1 at the top of this article shows the in-sample (observations 1-250) and out-of-sample (observations 251 onward) performance of the filter I will be building in the first part of this tutorial. 

Throughout this tutorial, I attempt to replicate these results that I built in iMetrica and expand on them a bit using the R language and the implementation of the MDFA available in here.  The data that we consider are 15-minute log-returns of the Yen from January 4th to January 17th and I have them saved as an .RData file given by ld_fxy_insamp. I have an additional explanatory series embedded in the .RData file that I’m using to predict the price of the Yen. Additionally, I also will be using price_fxy_insamp which is the log price of Yen, used to compute the performance (buy/sells) of the trading signal. The ld_fxy_insamp will be used as the in-sample data to construct the filter and trading signal for FXY. To obtain this data so you can perform these examples at home, email me and I’ll send you all the necessary .RData files (the in-sample and out-of-sample data) in a .zip file. Taking a quick glance at the ld_fxy_insamp data, we see log-returns of the Yen at every 15 minutes starting at market open (time zone UTC). The target data (Yen) is in the first column along with the two explanatory series (Yen and another asset co-integrated with movement of Yen).

> head(ld_fxy_insamp)
[,1]           [,2]          [,3]
2013-01-04 13:30:00  0.000000e+00   0.000000e+00  0.0000000000
2013-01-04 13:45:00  4.763412e-03   4.763412e-03  0.0033465833
2013-01-04 14:00:00 -8.966599e-05  -8.966599e-05  0.0040635638
2013-01-04 14:15:00  2.597055e-03   2.597055e-03 -0.0008322064
2013-01-04 14:30:00 -7.157556e-04  -7.157556e-04  0.0020792190
2013-01-04 14:45:00 -4.476075e-04  -4.476075e-04 -0.0014685198

Moving on, to begin constructing the first trading signal for the Yen, we begin by uploading the data into our R environment, define some initial parameters for the MDFA function call, and then compute the DFTs and periodogram for the Yen.

load(paste(path.pgm,"ld_fxy_in15min.RData",sep=""))    #load in-sample log-returns of Yen
load(paste(path.pgm,"price_fxy_in15min.RData",sep="")) #load in-sample log-price of Yen

in_samp_lenprice_insample<-price_fxy_insamp

#setup some MDFA variables
x<-ld_fxy_insamp
len<-length(x[,1])
shift_constraint<-rep(0,length(x[1,])-1)
weight_constraint<-rep(0,length(x[1,])-1)
d<-0
plots<-T
lin_expweight<-F

# Compute DFTs and periodogram for initial analysis
spec_obj<-spec_comp(len,x,d)
weight_func<-spec_obj$weight_func
K<-length(weight_func[,1])-1
fxy_periodogram<-abs(spec_obj$weight_func[,1])^2

As I’ve mentioned in my previous articles, my step-by-step strategy for building trading signals always begin by a quick analysis of the periodogram of the asset being traded on. Holding the key to providing insight into the characteristics of how the asset trades, the periodogram is an essential tool for navigating how the extractor \Gamma is chosen. Here, I look for principal spectral peaks that correspond in the time domain to how and where my signal will trigger buy/sell trades. Figure 2 shows the periodogram of the 15-minute log-returns of the Japanese Yen during the in-sample period from January 4 to January 17 2013. The arrows point to the main spectral peaks that I look for and provides a guide to how I will define my \Gamma function. The black dotted lines indicate the two frequency cutoffs that I will consider in this example, the first being \pi/12 and the second at \pi/6. Notice that both cutoffs are set directly after a spectral peak, something that I highly recommend.  In high-frequency trading on the FOREX using MDFA, as we’ll see, the trick is to seek out the spectral peak which accounts for the close-to-open variation in the price of the foreign currency. We want to take advantage of this spectral peak as this is where the big gains in foreign currency trading using MDFA will occur.

Figure 2: Periodogram of FXY (Japanese Yen) along with spectral peaks and two different frequency cutoffs.

Figure 2: Periodogram of FXY (Japanese Yen) along with spectral peaks and two different frequency cutoffs.

In our first example we consider the larger frequency as the cutoff for \Gamma by setting it to \pi/6 (the right most line in the figure of the periodogram). I then initially set the timeliness and smoothness parameters, lambda and expweight to 0 along with setting all the regularization parameters to 0 as well. This will give me a barometer for where and how much to adjust the filter parameters. In selecting the filter length L, my empirical studies over numerous experiments in building trading signals using iMetrica have demonstrated that a ‘good’ choice is anywhere between 1/4 and 1/5 of the total in-sample length of the time series data.  Of course, the length depends on the frequency of the data observations (i.e. 15 minute, hourly, daily, etc.), but in general you will most likely never need more than L being greater than 1/4 the in-sample size. Otherwise, regularization can become too cumbersome to handle effectively. In this example, the total in-sample length is 335 and thus I set L= 82 which I’ll stick to for the remainder of this tutorial. In any case, the length of the filter is not the most crucial parameter to consider in building good trading signals. For a good robust selection of the filter parameters couple with appropriate explanatory series, the results of the trading signal with L= 80 compared with, say, L= 85 should hardly differ. If they do, then the parameterization is not robust enough.

After uploading both the in-sample log-return data along with the corresponding log price of the Yen for computing the trading performance, we the proceed in R to setting initial filter settings for the MDFA routine and then compute the filter using the IMDFA_comp function. This returns both the i_mdfa& object holding coefficients, frequency response functions, and statistics of filter, along with the signal produced for each explanatory series. We combine these signals to get the final trading signal in-sample. All this is all done in R as follows:


cutoff<-pi/6 #set frequency cutoff
Gamma<-((0:K)<(cutoff*K/pi)) #define Gamma

grand_mean<-F
Lag<-0
L<-82
lambda_smooth<-0
lambda_cross<-0
lambda_decay<-c(0.,0.) #regularization - decay

lambda<-0
expweight<-0
i1<-F
i2<-F
# compute the filter for the given parameter definitions
i_mdfa_obj<-IMDFA_comp(Lag,K,L,lambda,weight_func,Gamma,expweight,cutoff,i1,i2,weight_constraint,
lambda_cross,lambda_decay,lambda_smooth,x,plots,lin_expweight,shift_constraint,grand_mean)

# after computing filter, we save coefficients
bn<-i_mdfa_obj$i_mdfa$b

# now we build trading signal
trading_signal<-i_mdfa_obj$xff[,1] + i_mdfa_obj$xff[,2]

The resulting frequency response functions of the filter and the coefficients are plotted in the figure below.

Figure 3: The Frequency response functions of the filter and the filter coefficients

Figure 3: The Frequency response functions of the filter (top) and the filter coefficients (below)

Notice the abundance of noise still present passed the cutoff frequency. This is mollified by increasing the expweight smoothness parameter. The coefficients for each explanatory series show some correlation in their movement as the lags increase. However, the smoothness and decay of the coefficients leaves much to be desired. We will remedy this by introducing regularization parameters. Plots of the in-sample trading signal and the performance in-sample of the signal are shown in the two figures below. Notice that the trading signal behaves quite nicely in-sample. However, looks can be deceiving. This stellar performance is due in large part to a filtering phenomenon called overfitting. One can deduce that overfitting is the culprit here by simply looking at the nonsmoothness of the coefficients along with the number of freezed degrees of freedom, which in this example is roughly 174 (out of 174), way too high. We would like to get this number at around half the total amount of degrees of freedom (number of explanatory series x L).

Figure 4: The trading signal and the log-return data of the Yen.

Figure 4: The trading signal and the log-return data of the Yen.

The in-sample performance of this filter demonstrates the type of results we would like to see after regularization is applied.  But now comes for the sobering effects of overfitting. We apply these filter coeffcients to 200 15-minute observations of the Yen and the explanatory series from January 18 to February 1 2013 and compare with the characteristics in-sample. To do this in R, we first load the out-of-sample data into the R environment, and then apply the filter to the out-of-sample data that I defined as x_out.

load(paste(path.pgm,"ld_fxy_out15min.RData",sep=""))
load(paste(path.pgm,"price_fxy_out15min.RData",sep=""))
x_out<-rbind(ld_fxy_insamp,ld_fxy_outsamp) #bind the in-sample with out-of-sample data
xff<-matrix(nrow=out_samp_len,ncol=2)

#apply filter built in-sample
for(i in 1:out_samp_len)
{
  xff[i,]<-0
  for(j in 2:3)
  {
      xff[i,j-1]<-xff[i,j-1]+bn[,j-1]%*%x_out[335+i:(i-L+1),j]
  }
}
trading_signal_outsamp<-xff[,1] + xff[,2]     #assemble the trading signal out-of-sample
trade_outsamp<-trading_logdiff(trading_signal_outsamp,price_outsample,.0005)  #compute the performance

The plot in Figure 5 shows the out-of-sample trading signal. Notice that the signal is not nearly as smooth as it was in-sample. Overshooting of the data in some areas is also obviously present. Although the out-of-sample overfitting characteristics of the signal are not horribly suspicious, I would not trust this filter to produce stellar returns in the long run.

Figure : Filter applied to 200 15 minute observations of Yen out-of-sample to produce trading signal (shown in blue)

Figure 5 : Filter applied to 200 15 minute observations of Yen out-of-sample to produce trading signal (shown in blue)

Following the previous analysis of the mean-squared solution (no customization or regularization), we now proceed to clean up the problem of overfitting that was apparent in the coefficients along with mollifying the noise in the stopband (frequencies after \pi/6).  In order to choose the parameters for smoothing and regularization, one approach is to first apply the smoothness parameter first, as this will generally smooth the coefficients while acting as a ‘pre’-regularizer, and then advance to selecting appropriate regularization controls. In looking at the coefficients (Figure 3), we can see that a fair amount of smoothing is necessary, with only a slight touch of decay. To select these two parameters in R, one option is to use the Troikaner optimizer (found here) to find a suitable combination (I have a secret sauce algorithmic approach I developed for iMetrica for choosing optimal combinations of parameters given an extractor \Gamma and a performance indicator, although it’s lengthy (even in GNU C) and cumbersome to use, so I typically prefer the strategy discussed in this tutorial).   In this example, I began by setting the lambda_smooth to .5 and the decay to (.1,.1) along with an expweight smoothness parameter set to 8.5. After viewing the coefficients, it still wasn’t enough smoothness, so I proceeded to add more finally reaching .63, which did the trick. I then chose lambda to balance the effects of the smoothing expweight (lambda is always the last resort tweaking parameter).

lambda_smooth<-0.63
lambda_cross<-0.
lambda_decay<-c(0.119,0.099)
lambda<-9
expweight<-8.5

i_mdfa_obj<-IMDFA_comp(Lag,K,L,lambda,weight_func,Gamma,expweight,cutoff,i1,i2,weight_constraint,
lambda_cross,lambda_decay,lambda_smooth,x,plots,lin_expweight,shift_constraint,grand_mean)

bn<-i_mdfa_obj$i_mdfa$b    #save the filter coefficients

trading_signal<-i_mdfa_obj$xff[,1] + i_mdfa_obj$xff[,2]  #compute the trading signal
trade<-trading_logdiff(trading_signal[L:len],price_insample[L:len],0) #compute the in-sample performance

Figure 6 shows the resulting frequency response function for both explanatory series (Yen in red). Notice that the largest spectral peak found directly before the frequency cutoff at \pi/6 is being emphasized and slightly mollified (value near .8 instead of 1.0). The other spectral peaks below \pi/6 are also present. For the coefficients, just enough smoothing and decay was applied to keep the lag, cyclical, and correlated structure of the coefficients intact, but now they look much nicer in their smoothed form. The number of freezed degrees of freedom has been reduced to approximately 102.

Figure : The frequency response functions and the coefficients after regularization and smoothing have been applied.

Figure 6: The frequency response functions and the coefficients after regularization and smoothing have been applied (top). The smoothed coefficients with slight decay at the end (bottom). Number of freezed degrees of freedom is approximately 102 (out of 172).

Along with an improved freezed degrees of freedom and no apparent havoc of overfitting, we apply this filter out-of-sample to the 200 out-of-sample observations in order to verify the improvement in the structure of the filter coefficients (shown below in Figure 7).  Notice the tremendous improvement in the properties of the trading signal (compared with Figure 5). The overshooting of the data has be eliminated and the overall smoothness of the signal has significantly improved. This is due to the fact that we’ve eradicated the presence of overfitting.

Figure : Out-of-sample trading signal with regularization.

Figure 7: Out-of-sample trading signal with regularization.

With all indications of a filter endowed with exactly the characteristics we need for robustness, we now apply the trading signal both in-sample and out of sample to activate the buy/sell trades and see the performance of the trading account in cash value. When the signal crosses below zero, we sell (enter short position) and when the signal rises above zero, we buy (enter long position).

The top plot of Figure 8 is the log price of the Yen for the 15 minute intervals and the dotted lines represent exactly where the trading signal generated trades (crossing zero). The black dotted lines represent a buy (long position) and the blue lines indicate a sell (and short position).  Notice that the signal predicted all the close-to-open jumps for the Yen (in part thanks to the explanatory series). This is exactly what we will be striving for when we add regularization and customization to the filter. The cash account of the trades over the in-sample period is shown below, where transaction costs were set at .05 percent. In-sample, the signal earned roughly 6 percent in 9 trading days and a 76 percent trading success ratio.

Figure : In-sample performance of the new filter  and the trades that are generated.

Figure 8: In-sample performance of the new filter and the trades that are generated.

Now for the ultimate test to see how well the filter performs in producing a winning trading signal, we applied the filter to the 200 15-minute out-of-sample observation of the Yen and the explanatory series from Jan 18th to February 1st and make trades based on the zero crossing. The results are shown below in Figure 9. The black lines represent the buys and blue lines the sells (shorts). Notice the filter is still able to predict the close-to-open jumps even out-of-sample thanks to the regularization. The filter succumbs to only three tiny losses at less than .08 percent each between observations 160 and 180 and one small loss at the beginning, with an out-of-sample trade success ratio hitting 82 percent and an ROI of just over 4 percent over the 9 day interval.

Figure : Out-of-sample performance of the regularized filter on 200 out-of-sample 15 minute returns of the Yen.

Figure 9: Out-of-sample performance of the regularized filter on 200 out-of-sample 15 minute returns of the Yen. The filter achieved 4 percent ROI over the 200 observations and an 82 percent trade success ratio.

Compare this with the results achieved in iMetrica using the same MDFA parameter settings. In Figure 10, both the in-sample and out-of-sample performance are shown. The performance is nearly identical.

Figure : In-sample and out-of-sample performance of the Yen filter in iMetrica. Nearly identical with performance obtained in R.

Figure 10: In-sample and out-of-sample performance of the Yen filter in iMetrica. Nearly identical with performance obtained in R.

Example 2

Now we take a stab at producing another trading filter for the Yen, only this time we wish to identify only the lowest frequencies to generate a trading signal that trades less often, only seeking the largest cycles. As with the performance of the previous filter, we still wish to target the frequencies that might be responsible to the large close-to-open variations in the price of Yen. To do this, we select our cutoff to be \pi/12 which will effectively keep the largest three spectral peaks intact in the low-pass band of \Gamma.

For this new filter, we keep things simple by continuing to use the same regularization parameters chosen in the previous filter as they seemed to produce good results out-of-sample. The \lambda and expweight customization parameters however need to be adjusted to account for the new noise suppression requirements in the stopband and the phase properties in the smaller passband. Thus I increase the smoothing parameter and decreased the timeliness parameter (which only affects the passband) to account for this change. The new frequency response functions and filter coefficients for this smaller lowpass design are shown below in Figure 11. Notice that the second spectral peak is accounted for and only slightly mollified under the new changes. The coefficients still have the noticeable smoothness and decay at the largest lags.

Figure : Frequency response functions of the two filters and their corresponding coefficients.

Figure 11: Frequency response functions of the two filters and their corresponding coefficients.

To test the effectiveness of this new lower trading frequency design, we apply the filter coefficients to the 200 out-of-sample observations of the 15-minute Yen log-returns. The performance is shown below in Figure 12. In this filter, we clearly see that the filter still succeeds in predicting correctly the large close-to-open jumps in the price of the Yen. Only three total losses are observed during the 9 day period. The overall performance is not as appealing as the previous filter design as less amount of trades are made, with a near 2 percent ROI and 76 percent trade success ratio. However, this design could fit the priorities for a trader much more sensitive to transaction costs.

Figure : Out-of-sample performance of filter with lower cutoff.

Figure 12: Out-of-sample performance of filter with lower cutoff.

Conclusion

Verification and cross-validation is important, just as the most interesting man in the world knows.

Verification and cross-validation is important, just as the most interesting man in the world will tell you.

The point of this tutorial was to show some of the main concepts and strategies that I undergo when approaching the problem of building a robust and highly efficient trading signal for any given asset at any frequency. I also wanted to see if I could achieve similar results with the R MDFA package as my iMetrica software package. The results ended up being nearly parallel except for some minor differences. The main points I was attempting to highlight were in first analyzing the periodogram to seek out the important spectral peaks (such as ones associate with close-to-open variations) and to demonstrate how the choice of the cutoff affects the systematic trading.  Here’s a quick recap on good strategies and hacks to keep in mind.

Summary of strategies for building trading signal using MDFA in R:

  • As I mentioned before, the periodogram is your best friend. Apply the cutoff directly after any range of spectral peaks that you want to consider. These peaks are what generate the trades.
  • Utilize a choice of filter length L no greater than 1/4. Anything larger is unnecessary.
  • Begin by computing the filter in the mean-square sense, namely without using any customization or regularization and see exactly what needs to be approved upon by viewing the frequency response functions and coefficients for each explanatory series.  Good performance of the trading signal in-sample (and even out-of-sample in most cases) is meaningless unless the coefficients have solid robust characteristics in both the frequency domain and the lag domain.
  • I recommend beginning with tweaking the smoothness customization parameter expweight and the lambda_smooth regularization parameters first. Then proceed with only slight adjustments to the lambda_decay parameters. Finally, as a last resort, the lambda customization. I really never bother to look at lambda_cross. It has seldom helped in any significant manner.  Since the data we are using to target and build trading signals are log-returns, no need to ever bother with i1 and i2. Those are for the truly advanced and patient signal extractors, and should only be left for those endowed with iMetrica 😉

If you have any questions, or would like the high-frequency Yen data I used in these examples, feel free to contact me and I’ll send them to you. Until next time, happy extracting!

High-Frequency Financial Trading with Multivariate Direct Filtering I: FOREX and Futures

Animation 1: Click to see animation of the Japanese Yen filter in action on 164 hourly out-of-sample observations.

Animation 1: Click to see animation of the Japanese Yen filter in action on 164 hourly out-of-sample observations.

I recently acquired over 300 GBs of financial data that includes tick data for over 7000 financial assets traded on multiple markets for the past 5 years up until February 1st 2013. This USB drive packed with nearly every detail of world financial markets coupled with iMetrica gave me an opportunity to explore at any fashion to my desire the ability of multivariate direct filtering to produce high performance financial trading signals on nearly any high-frequency. Let me begin this article with saying that I am more than ecstatic with the results, as I hope you will too after reading this article.  In this first article in a series of high-frequency trading with MDFA and iMetrica that I plan to write, I provide some initial experiments with building and extracting financial trading signals for high-frequency intraday observations on foreign exchange (FOREX) data, and by high-frequency in the context of this article, I mean higher frequencies than the daily log-returns I’ve been working with in my previous articles. In the first part of this high-frequency series, I begin by exploring hourly, 30 minute, and 15 minute log-returns, and test different strategies, mostly using low-pass and the recently introduced multi-bandpass (MBP) filter to deduce the best approach to tackle the problem of building successful trading signals in higher frequency data.

In my previous articles, I was working uniquely with daily log-return data from different time spans from a year to a year and a half. This enabled the in-sample period of computing the filter coefficients for the signal extraction to include all the most recent annual phases and seasons of markets, from holiday effects, to the transitioning period of August to September that is regularly highly influential on stock market prices and commodities as trading volume increases a significant amount. One immediate question that is raised in migrating to higher-frequency intraday data is what kind of in-sample/out-of-sample time spans should be used to compute the filter in-sample and then for how long do we apply the filter out-of-sample to produce the trades? Another question that is raised with intraday data is how do we account for the close-to-open variation in price? Certainly, after close, the after-hour bids and asks will force a jump into the next trading day. How do we deal with this jump in an optimal manner? As the observation frequency gets higher, say from one hour to 30 minutes, this close-to-open jump/fall should most likely be larger. I will start by saying that, as you will see in the results of this article, with a clever choice of the extractor \Gamma and explanatory series, MDFA can handle these jumps beautifully (both aesthetically and financially). In fact, I would go so far as to say that the MDFA does a superb job in predicting the overnight variation.

One advantage of building trading signals for higher intraday frequencies is that the signals produce trading strategies that are immediately actionable. Namely one can act upon a signal to enter a long or short position immediately when they happen. In building trading signals for the daily log-return, this is not the case since the observations are not actionable points, namely the log difference of today’s ending price with yesterday’s ending price are produced after hours and thus not actionable during open market hours and only actionable the next trading day. Thus trading on intraday observations can lead to better efficiency in trading.

In this first installment in my series on high-frequency financial trading using multivariate direct filtering in iMetrica, I consider building trading signals on hourly returns of foreign exchange currencies. I’ve received a few requests after my recent articles on the Frequency Effect in seeing iMetrica and MDFA in action on the FOREX sector. So to satisfy those curiosities, I give a series of (financially) satisfying and exciting results in combining MDFA and the FOREX. I won’t give all my secretes away into building these signals (as that would of course wipe out my competitive advantage), but I will give some of the parameters and strategies used so any courageously curious reader may try them at home (or the office). In the conclusion, I give a series of even more tricks and hacks.  The results below speak for themselves  So without further ado, let the games begin.

Japanese Yen

Frequency: One hour returns
30 day out-of-sample ROI: 12 percent
Trade success ratio: 92 percent

Yen Filter Parameters: \lambda = 9.2 \alpha = 13.2, \omega_0 = \pi/5
Regularization: smooth = .918, decay = .139, decay2 = .79, cross = 0

In the first experiment, I consider hourly log-returns of a ETF index that mimics the Japanese Yen called FXY. As for one of the explanatory series, I consider the hourly log-returns of the price of GOLD which is traded on NASDAQ. The out-of-sample results of the trading signal built using a low-pass filter and the parameters above are shown in Figure 1.  The in-sample trading signal (left of cyan line) was built using 400 hourly observations of the Yen during US market hours dating back to 1 October 2012. The filter was then applied to the out-of-sample data for 180 hours, roughly 30 trading days up until Friday, 1 February 2013.

Figure 3: Out-of-sample results for the Japanese Yen. The in-sample trading  signal was built using 400 hourly observations of the Yen during US market hours  dating back to November 1st, 2012. The out-of-sample portion passed the cyan line is on 180 hourly observations, about 30 trading days.

Figure 1: Out-of-sample results for the Japanese Yen. The in-sample trading signal was built using 400 hourly observations of the Yen during US market hours dating back to October 1st, 2012. The out-of-sample portion passed the cyan line is on 180 hourly observations, about 30 trading days.

This beauty of this filter is that it yields a trading signal exhibiting all the characteristics that one should strive for in building a robust and successful trading filter.

  1. Consistency: The in-sample portion of the filter performs exactly as it does out-of-sample (after cyan line) in both trade success ratio and systematic trading performance. 
  2. Dropdowns: One small dropdown out-of-sample for a loss of only .8 percent (nearly the cost of the transaction).
  3. Detects the cycles as it should: Although the filter is not able to pinpoint with perfect accuracy every local small upturn during the descent of the Yen against the dollar, it does detect them nonetheless and knows when to sell at their peaks (the magenta lines).
  4. Self-correction: What I love about a robust filter is that it will tend to self-correct itself very quickly to minimize a loss in an erroneous trade. Notice how it did this in the second series of buy-sell transactions during the only loss out-of-sample. The filter detects momentum but quickly sold right before the ensuing downfall. My intuition is that only frequency-based methods such as the MDFA are able to achieve this consistently. This is the sign of a skillfully smart filter.

The coefficients for this Yen filter are shown below. Notice the smoothness of the coefficients from applying the heavy smooth regularization and the strong decay at the very end.  This is exactly the type of smooth/decay combo that one should desire. There is some obvious correlation between the first and second explanatory series in the first 30 lags or so as well. The third explanatory series seems to not provide much support until the middle lags .

Coefficients of the Yen filter. Here we use three different explanatory series to extract  the trading signal shown in Figure 1.

Figure 2: Coefficients of the Yen filter. Here we use three different explanatory series to extract the trading signal.

One of the first things that I always recommend doing when first attempting to build a trading signal is to take a glance at the periodogram. Figure 2 shows the periodogram of the log-return data of the Japanese Yen over 580 hours.  Compare this with the periodogram of the same asset using log-returns of daily data over 580 days, shown in Figure 3.  Notice the much larger prominent spectral peaks at the lower frequencies in the daily log-return data. These prominent spectral peaks renders multibandpass filters much more advantageous and to use as we can take advantage of them by placing a band-pass filter directly over them to extract that particular frequency (see my article on multibandpass filters). However, in the hourly data, we don’t see any obvious spectral peaks to consider, thus I chose a low-pass filter and set the cutoff frequency at $\pi/5$, a standard choice, and good place to begin.

Figure 1: Periodogram of hourly log-returns of the Japanese Yen over 580 hours.

Figure 3: Periodogram of hourly log-returns of the Japanese Yen over 580 hours.

Figure 3: Periodogram of Japanese Yen using 580 daily log-return observations.

Figure 4: Periodogram of Japanese Yen using 580 daily log-return observations. Many more spectral peaks are present in the lower frequencies.

Japanese Yen

Frequency: 15 minute returns
7 day out-of-sample ROI: 5 percent
Trade success ratio: 82 percent

Yen Filter Parameters: \lambda = 3.7 \alpha = 13, \omega_0 = \pi/9
Regularizationsmooth = .90, decay = .11, decay2 = .09, cross = 0

In the next trading experiment, I consider the Japanese Yen again, only this time I look at trading on even high-frequency log-return data than before, namely on 15 minute log-returns of the Yen from the opening bell to market close.  This presents slightly new challenges than before as the close-to-open jumps are much larger than before, but these larger jumps do not necessarily pose problems for the MDFA. In fact, I look to exploit these and take advantage to gain profit by predicting the direction of the jump.  For this higher frequency experiment, I considered 350 15-minute in-sample observations to build and optimize the trading signal, and then applied it over the span of 200 15-minute out-of-sample observations. This produced the results shown in the Figure 5 below. Out of 17 total trades out-of-sample, there were only 3 small losses each less than .5 percent drops and thus 14 gains during the 200 15-minute out-of-sample time period.  The beauty of this filter is its impeccable ability to predict the close-to-open jump in the price of the Yen. Over the nearly 7 day trading span, it was able to correctly deduce whether to buy or short-sell before market close on every single trading day change. In the figure below, the four largest close-to-open variation in Yen price is marked with a “D” and you can clearly see how well the signal was able to correctly deduce a short-sell before market close. This is also consistent with the in-sample performance as well, where you can notice the buys and/or short-sells at the largest close-to-open jumps (notice the large gain in the in-sample period right before the out-of-sample period begins, when the Yen jumped over 1 percent over night.  This performance is most likely aided by the explanatory time series I used for helping predict the close-to-open variation in the price of the Yen. In this example, I only used two explanatory series (the price of Yen, and another closely related to the Yen).

Figure : Out-of-sample performance of the Japanese Yen filter on 15 minute log-return data.

Figure 5: Out-of-sample performance of the Japanese Yen filter on 15 minute log-return data.

We look at the filter transfer functions to see what frequencies they are being privileged in the construction of the filter. Notice that some noise leaks out passed the frequency cutoff at \pi/9, but this is typically normal and a non-issue. I had to balance for both timeliness and smoothness in this filter using both the customization parameters \lambda and \alpha. Not much at frequency 0 is emphasized, with more emphasis stemming from the large spectral peak found right at \pi/9.

Figure : The filter transfer functions.

Figure 6: The filter transfer functions.

British Pound

Frequency: 30 minute returns
14 day out-of-sample ROI: 4 percent
Trade success ratio: 76 percent

British Pound Filter Parameters: \lambda = 5 \alpha = 15, \omega_0 = \pi/9
Regularizationsmooth = .109, decay = .165, decay2 = .19, cross = 0

In this example we consider the frequency of the data to 30 minute returns and attempt to build a robust trading signal for a derivative of the British Pound (BP) on this higher frequency. Instead of using the cash value of the BP, I use 30 minute returns of the BP Futures contract expiring in March (BPH3). Although I don’t have access to tick data from the FOREX, I do have tick data from GLOBEX for the past 5 years.  Thus the futures series won’t be an exact replication of the cash price series of the BP, but it should be quite close due to very low interest rates.

The results of the out-of-sample performance of the BP futures filter are shown in Figure 7. I constructed the filter using an initial in-sample size of 390 30 minute returns dating back to 1 December 2012. After pinpointing a frequency cutoff in the frequency domain for the \Gamma that yielded decent trading results in-sample, I then proceeded to optimize the filter in-sample on smoothness and regularization to achieve similar out-of-sample performance. Applying the resulting filter out-of-sample on 168 30-minute log-return observations of the BP futures series along with 3 explanatory series, I get the results shown below. There were 13 trades made and 10 of them were successful. Notice that the filter does an exquisite job at triggering trades near local optimums associated with the frequencies inside the cutoff of the filter.

Figure 5: The out-of-sample results of the British Pound using 30-minute return data.

Figure 7: The out-of-sample results of the British Pound using 30-minute return data.

In looking at the coefficients of the filter for each series in the extraction, we can clearly see the effects of the regularization: the smoothness of the coefficients the fast decay at the very end. Notice that I never really apply any cross regularization to stress the latitudinal likeliness between the 3 explanatory series as I feel this would detract from the predicting advantages brought by the explanatory series that I used.

Figure 6: The coefficients for the 3 explanatory series of the BP futures,

Figure 8: The coefficients for the 3 explanatory series of the BP futures,

Euro

Frequency: 30 min returns
30 day out-of-sample ROI: 4 percent
Trade success ratio: 71 percent

Euro Filter Parameters: \lambda = 0, \alpha = 6.4, \omega_0 = \pi/9
Regularizationsmooth = .85, decay = .27, decay2 = .12, cross = .001

Continuing with the 30 minute frequency of log-returns, in this example I build a trading signal for the Euro futures contract with expiration on 18 March 2013 (UROH3 on the GLOBEX). My in-sample period, being the same as my previous experiment, is from 1 December 2012 to 4 January 2013 on 30 minute returns using three explanatory time series.  In this example, after inspecting the periodogram, I decided upon a low-pass filter with a frequency cutoff of \pi/9. After optimizing the customization and applying the filter to one month of 30 minute frequency return data out-of-sample (month of January 2013, after cyan line) we see the performance is akin to the performance in-sample, exactly what one strives for. This is due primarily to the heavy regularization of the filter coefficients involved. Only four very small losses of less than .02 percent are suffered during the out-of-sample span that includes 10 successful trades, with the losses only due to the transaction costs. Without transaction costs, there is only one loss suffered at the very beginning of the out-of-sample period.

Figure : Out-of-sample performance on the 30-min log-returns of Euro futures contract UROH3.

Figure 9 : Out-of-sample performance on the 30-min log-returns of Euro futures contract UROH3.

As in the first example using hourly returns, this filter again exhibits the desired characteristics of a robust and high-performing financial trading filter. Notice the out-of-sample performance behaves akin to the in-sample performance, where large upswings and downswings are pinpointed to high-accuracy. In fact, this is where the filter performs best during these periods. No need for taking advantage of a multibandpass filter here, all the profitable trading frequencies are found at less than \pi/9.  Just as with the previous two experiments with the Yen and the British Pound, notice that the filter cleanly predicts the close-to-open variation (jump or drop) in the futures value and buys or sells as needed.  This can be seen from many of the large jumps in the out-of-sample period (after cyan line).

One reason why these trading signals perform so well is due to their approximation power of the symmetric filter. In comparing the trading signal (green) with a high-order approximation of the symmetric filter (gray line) transfer function \Gamma shown in Figure 10, we see that trading signal does an outstanding job at approximating the symmetric filter uniformly. Even at the latest observation (the right most point), the asymmetric filter hones in on the symmetric signal (gray line) with near perfection. Most importantly, the signal crosses zero almost exactly where required.  This is exactly what you want when building a high-performing trading signal.

Figure : Plot of approximation of the real-time trading signal for UROH3 with a high order approximation of the symmetric filter transfer function.

Figure 10: Plot of approximation of the real-time trading signal for UROH3 with a high order approximation of the symmetric filter transfer function.

In looking at the periodogram of the log-return data and the output trading signal differences (colored in blue), we see that the majority of the frequencies were accounted for as expected in comparing the signal with the symmetric signal. Only an inconsequential amount of noise leakage passed the frequency cutoff of \pi/9 is found.  Notice the larger trading frequencies, the more prominent spectral peaks, are located just after \pi/6. These could be taken into account with a smart multibandpass filter in order to manifest even more trades, but I wanted to keep things simple for my first trials with high-frequency foreign exchange data.  I’m quite content with the results that I’ve achieved so far.

Figure : Comparing the periodogram of the signal with the log-return data.

Figure 11: Comparing the periodogram of the signal with the log-return data.

Conclusion

I must admit, at first I was a bit skeptical of the effectiveness that the MDFA would have in building any sort of successful trading signal for FOREX/GLOBEX high frequency data. I always considered the FOREX market rather ‘efficient’ due to the fact that it receives one of the highest trading volumes in the world.  Most strategies that supposedly work well on high-frequency FOREX all seem to use some form of technical analysis or charting (techniques I’m particularly not very fond of), most of which are purely time-domain based. The direct filter approach is a completely different beast, utilizing a transformation into the frequency domain and a ‘bending and warping’ of the metric space for the filter coefficients to extract a signal within the noise that is the log-return data of financial assets.  For the MDFA to be very effective at building timely trading signals, the log-returns of the asset need to diverge from white noise a bit, giving room for pinpointing intrinsically important cycles in the data. However, after weeks of experimenting, I have discovered that building financial trading signals using MDFA and iMetrica on FOREX data is as rewarding as any other.

As my confidence has now been bolstered and amplified even more after my experience with building financial trading signals with MDFA and iMetrica for high-frequency data on foreign exchange log-returns at nearly any frequency, I’d be willing to engage in a friendly competition with anyone out there who is certain that they can build better trading strategies using time domain based methods such as technical analysis or any other statistical arbitrage technique.  I strongly believe these frequency based methods are the way to go, and the new wave in financial trading.  But it takes experience and a good eye for the frequency domain and periodograms to get used to. I haven’t seen many trading benchmarks that utilize other types of strategies, but i’m willing to bet that they are not as consistent as these results using this large of an out-of-sample to in-sample ratio (the ratios in these experiments were between .50 and .80).  If anyone would like to take me up on my offer for a friendly competition (or know anyone that would), feel free to contact me.

After working with a multitude of different financial time series and building many different types of filters, I have come to the point where I can almost eyeball many of the filter parameter choices including the most important ones being the extractor \Gamma along with the regularization parameters, without resorting to time consuming, and many times inconsistent, optimization routines.  Thanks to iMetrica, transitioning from visualizing the periodogram to the transfer functions and to the filter coefficients and back to the time domain to compare with the approximate symmetric filter in order to gauge parameter choices is an easy task, and an imperative one if one wants to build successful trading signals using MDFA.

Here are some overall tips and tricks to build your own high performance trading signals on high-frequency data at home:

  • Pay close attention to the periodogram. This is your best friend in choosing the extractor \Gamma. The best performing signals are not the ones that trade often, but trade on the most important frequencies found in the data. Not all frequencies are created equal. This is true when building either low-pass or multibandpass frequencies. 
  • When tweaking customization, always begin with \alpha, the parameter for smoothness. \lambda for timeliness should be the last resort. In fact, this parameter will most likely be next to useless due to the fact that the log-return of financial data is stationary. You probably won’t ever need it.
  • You don’t need many explanatory series. Like most things in life, quality is superior to quantity. Using the log-return data of the asset you’re trading along with one and maybe two explanatory series that somewhat correlate with the financial asset you’re trading on is sufficient. Anymore than that is ridiculous overkill, probably leading to over-fitting (even the power of regularization at your fingertips won’t help you).

In my next article, I will continue with even more high-frequency trading strategies with the MDFA and iMetrica where I will engage in the sector of Funds and ETFs. If any curious reader would like even more advice/hints/comments on how to build these trading signals on high-frequency data for the FOREX (or the coefficients built in these examples), feel free to get in contact with me via email. I’ll be happy to help.

Happy extracting!

Realizing the Future with iMetrica and HEAVY Models

In this article we steer away from multivariate direct filtering and signal extraction in financial trading and briefly indulge ourselves a bit in the world of analyzing high-frequency financial data, an always hot topic with the ever increasing availability of tick data in computationally convenient formats. Not only has high-frequency intraday data been the basis of higher frequency risk monitoring and forecasting, but it also provides access to building ‘smarter’ volatility prediction models using so-called realized measures of intraday volatility. These realized measures have been shown in numerous studies over the past 5 years or so to provide a solidly more robust indicator of daily volatility.   While daily returns only capture close-to-close volatility, leaving much to be said about the actual volatility of the asset that was witnessed during the day, realized measures of volatility using higher frequency data such as second or minute data provide a much clearer picture of open-to-close variation in trading.

In this article, I briefly describe a new type of volatility model that takes into account these realized measures for volatility movement called  High frEquency bAsed VolatilitY (HEAVY) models developed and pioneered by Shephard and Sheppard 2009. These models take as input both close-to-close daily returns r_t as well as daily realized measures to yield better forecasting dynamics. The models have been shown to be endowed with the ability to not only track momentum in volatility, but also adjust for mean reversion effects as well as adjust quickly to structural breaks in the level of the volatility process.  As the authors (Sheppard and Shephard, 2009) state in their original paper, the focus of these models is on predictive properties, rather than on non-parametric measurement of volatility. Furthermore, HEAVY models are much easier and more robust estimation wise than single source equations (GARCH, Stochastic Volatility) as they bring two sources of volatility information to identify a longer term component of volatility.

The goal of this article is three-fold. Firstly, I briefly review these HEAVY models and give some numerical examples of the model in action using a gnu-c library and Java package called heavy_model that I develped last year for the iMetrica software. The heavy_model package is available for download (either by this link or e-mail me) and features many options that are not available in the MATLAB code provided by Sheppard (bootstrapping methods, Bayesian estimation, track reparameterization, among others). I will then demonstrate the seamless ability to model volatility with these High frEquency bAsed VolatilitY models using iMetrica, where I also provide code for computing realized measures of volatility in Java with the help of an R package called highfrequency (Boudt, Cornelissen, and Payseur 2012).  

HEAVY Model Definition

Let’s denote the daily returns as r_1, r_2, \ldots, r_T, where T is the total amount of days in the sample we are working with. In the HEAVY model, we supplement information to the daily returns by a so-called realized measure of intraday volatility based on higher frequency data, such as second, minute or hourly data. The measures are called daily realized measures and we will denote them as RM_1, RM_2, \ldots, RM_T for the total number of days in the sample.  We can think of these daily realized measures as an average of variance autocorrelations during a single day. They are supposed to provide a better snapshot of the ‘true’ volatility for a specific day t. Although there are numerous ways of computing a realized measure, the easiest is the realized variance computed as RM_t = \sum_j (X_{t+t_{j,t}} - X_{t+t_{j-1,t}})^2 where t_{j,t} are the normalized times of trades on day t. Other methods for providing realized measures includes using Kernel based methods which we will discuss later in this article (see for example http://papers.ssrn.com/sol3/papers.cfm?abstract_id=927483).

Once the realized measures have been computed for T days, the HEAVY model is given by:

Var(r_t | \mathcal{F}_{t-1}^{HF}) = h_t = \omega_1 + \alpha RM_{t-1} + \beta h_{t-1} + \lambda r^2_t

E(RM_t | \mathcal{F}_{t-1}^{HF}) = \mu_t = \omega_2+ \alpha_R RM_{t-1} + \beta_R \mu_{t-1},

where the stability constraints are  \alpha, \omega_1 \geq 0, \beta \in [0,1] and \omega_2, \alpha_R \geq 0 with \lambda + \beta \in [0,1] and \beta_R + \alpha_R \in [0,1]. Here, the \mathcal{F}_{t-1}^{HF} denotes the high-frequency information from the previous day t-1. The first equation models the close-to-close conditional variance and is akin to a GARCH type model, whereas the second equation models the conditional expectation of the open-to-close variation.  

With the formulation above, one can easily see that slight variations to the model are perfectly plausible. For example, one could consider additional lags in either the realized measure RM_t (akin to adding additional moving average parameters) or the conditional mean/variance variable (akin to adding autoregression parameters). One could also leave out the dependence on the squared returns r^2_t by setting \lambda to zero, which is what the original others recommended. A third variation is adding yet another equation to the pack that models a realized measure that takes into account negative and positive momentum to yield possibly better forecasts as it tracks both losses and gains in the model. In this case, one would add the third component by introducing a new equation for a realized semivariance to parametrically model statistical leverage effects, where falls in asset prices are associated with increases in future volatility.  With realized semivariance computed for the T days as RMS_1, \ldots RMS_T, the third equation becomes

E(RMS_t | \mathcal{F}_{t-1}^{HF}) = \phi_t = \omega_3 + \alpha_{RS} RMS_{t-1} + \beta_{RS} \phi_{t-1}

where \alpha_{RS} + \beta_{RS} < 1 and both positive.

HEAVY modeling in C and Java

To incorporate these HEAVY models into iMetrica, I began by writing a gnu-c library for providing a fast and efficient framework for both quasi-likelihood evaluation and a posteriori analysis of the models. The structure of estimating the models follows very closely to the original MATLAB code provided by Sheppard. However, in the c library I’ve added a few more useful tools for forecasting and distribution analysis. The Java code is essentially a wrapper for the c heavy_model library to provide a much cleaner approach to modeling and analyzing the HEAVY data such as the parameters and forecasts.  While there are many ways to declare, implement, and analyze HEAVY models using the c/java toolkit I provide, the most basic steps involved are as follows.

heavyModel heavy = new heavyModel();
heavy.setForecastDimensions(n_forecasts, n_steps);
heavy.setParameterValues(w1, w2, alpha, alpha_R, lambda, beta, beta_R);
heavy.setTrackReparameter(0);
heavy.setData(n_obs, n_series, series);
heavy.estimateHeavyModel();

The first line declares a HEAVY model in java, while the second line sets the number of forecasts samples to compute and how many forecast steps to take. Forecasted values are provided for both the return variable r_t (using a bootstrapping methodology) and the h_t, \mu_t variables. In the next line, the parameter values for the HEAVY model are initialized. These are the initial points that are utilized in the quasi-maximum likelihood optimization routine and can be set to any values that satisfy the model constraints.   Here, w1 = \omega_1, w2 = \omega_2.

The fourth line is completely optional and is used for toggling (0=off, 1=on) a reparameterization of the HEAVY model so the intercepts of both equations in the HEAVY model are explicitly related to the unconditional mean of squared returns r^2 and realized measures RM_t. The reparameterization of the model has the advantage that it eliminates the estimation of \omega_1, \omega_2 and instead uses the unconditional means, leaving two less degrees of freedom in the optimization. See page 12 of the Shephard and Sheppard 2009 paper for a detailed explanation of the reparameterization. After setting the initial values, the data is set for the model by inputting the total number of observation T, the number of series (normally set to 2 and the data in column-wise format (namely a double array of length n_obs x n_series, where the first column is the return data r_t and the second column is the daily realized measure data.  Finally, with the data set and the parameters initialized  we estimate the model in the 6th line. Once the model is finished estimating (should take a few seconds, depending on the number of observations), the heavyModel java object stores the parameter values, forecasts, model residuals, likelihood values, and more. For example, one can print out the estimated model parameters and plot the forecasts of h_t using the following:


heavy.printModelParameters();
heavy.plotForecasts();
Output:
w_1 = 0.063 w_2 = 0.053
beta = 0.855 beta_R = 0.566
alpha = 0.024 alpha_R = 0.375
lambda = 0.087

Figure 1 shows the plot of the filtered h_t, \mu_t values for 300 trading days from June 2011 to June 2012 of AAPL with the final 20 points being the forecasted values. Notice that the multistep ahead forecast shows momentum which is one of the attractive characteristics of the HEAVY models as mentioned in the original paper by Shephard and Sheppard.

Figure 1: Plots of the filtered returns and realized measures with 20 step forecasts for Verizon for 300 trading days.

Figure 1: Plots of the filtered returns and realized measures with 20 step forecasts for Verizon for 300 trading days.

We can also easily plot the estimated joint distribution function F_{\zeta, \eta} by simply using the filtered h_t, \mu_t and computing the devolatilized values \zeta_t = r_t/ \sqrt{h_t}, \eta_t = (RM_t/\mu_t)^{1/2}, leading to the innovations for the model for t = 2,\ldots,T.

Figure 2 below shows the empirical distribution of F_{\zeta, \eta} for 600 days (nearly two years of daily observations from AAPL).  The $\zeta_t$ sequence should be roughly a martingale difference sequences with unit variance and the $\eta_t$ sequence  should have unit conditional means and of course be uncorrelated.  The empirical results validate the theoretical values.

Figure 2: Scatter plot of the empirical distribution of devolatilized values for h and mu.

Figure 2: Scatter plot of the empirical distribution of devolatilized values for h and mu.

In order to compile and run the heavy_model library and the accompanying java wrapper, one must first be sure to meet the requirements for installation. The programs were extensively tested on a 64bit Linux machine running Ubuntu 12.04. The heavy_model library written in c uses the GNU Scientific Library (GSL) for the matrix-vector routines along with a statistical package in gnu-c called apophenia (Klemens, 2012) for the optimization routine. I’ve also included a wrapper for the GSL library called multimin.c which enables using the optimization routines from the GSL library, but were not heavily tested.  The first version (version 00) of the heavy_model library and java wrapper can be downloaded at sourceforge.net/projects/highfrequency.  As a precautionary warning, I must confess that none of the files are heavily commented in any way as this is still a project in progress. Improvements in code, efficiency, and documentation will be continuously coming.

After downloading the .tar.gz package, first ensure that GSL and Apophenia are properly installed and the libraries are correctly installed to the appropriate path for your gnu c compiler. Second, to compile the .c code, copy the makefile.test file to Makefile and then type make. To compile the heavyModel library and utilize the java heavyModel wrapper (recommended), copy makefile.lib to Makefile, then type make. After it constructs the libheavy.so, compile the heavyModel.java file by typing javac heavyModel.java. Note that the java files were complied successfully using the Oracle Java 7 SDK.  If you have any questions about this or any of the c or java files, feel free to contact me. All the files were written by me (except for the optional multimin.c/h files for the optimization) and some of the subroutines (such as the HEAVY model simulation) are based on the MATLAB code by Sheppard. Even though I fully tested and reproduced the results found in other experiments exploring HEAVY models, there still could be bugs in the code. I have not fully tested every aspect (especially the Bayesian estimation components, an ongoing effort) and if anyone would like to add, edit, test, or comment on any of the routines involved in either the c or java code, I’d be more than happy to welcome it.

HEAVY Modeling in iMetrica

The Java wrapper to the gnu-c heavy_model library was installed in the iMetrica software package and can be used for GUI style modeling of high-frequency volatility. The HEAVY modeling environment is a feature of the BayesCronos module in iMetrica that also features other stochastic models for capturing and forecasting volatility such as (E)GARCH, stochastic volatility, mutlivariate stohastic factor modeling, and ARIMA modeling, all using either standard (Q)MLE model estimation or a Bayesian estimation interface (with histograms showing the MCMC results of the parameter chains).

Modeling volatility with HEAVY models is done by first uploading the data into the BayesCronos module (shown in Figure 3) through the use of either the BayesCronos Menu (featured on the top panel) or by using the Data Control Panel (see my previous article on Data Control).

Figure 3: BayesCronos interface in iMetrica for HEAVY modeling.

Figure 3: BayesCronos interface in iMetrica for HEAVY modeling.

In the BayesCronos control panel shown above, we estimate a HEAVY model for the uploaded data (600 observations of r_t, RM_t) that were simulated from a model with omega_1 = 0.05, omega_2 = 0.10, beta = 0.8 beta_R = 0.3, alpha = 0.02, alpha_R = 0.3 (the simulation was done in the Data Control Module).

The model type is selected in the panel under the Model combobox. The number of forecasting steps and forecasting samples (for the r_t variable) are selected in the Forecasting panel. Once those values are set, the model estimates are computed by pressing the “MLE” button in the bottom lower left corner. After the computing is done, all the available plots to analyze the HEAVY model are available by simply clicking the appropriate plotting checkboxes directly below the plotting canvas.   This includes up to 5 forecasts, the original data, the filtered h_t, \mu_t values,  the residuals/empirical distributions of the returns and realized measures, and the pointwise likelihood evaluations for each observation. To see the estimated parameter values, simply click the “Parameter Values” button in the “Model and Parameters” panel and pop-up control panel will appear showing the estimated values for all the parameters.

Realized Measures in iMetrica

Figure : Computing Realized measures in iMetrica using a convenient realized measure control panel.

Figure 4: Computing Realized measures in iMetrica using a convenient realized measure control panel.

Importing and computing realized volatility measures in iMetrica is accomplished by using the control panel shown in Figure 4. With access to high frequency data, one simply types in the ticker symbol in the “Choose Instrument” box, sets the starting and ending date in the standard CCYY-MM-DD format, and then selects the kernel used for assembling the intraday measurements. The Time Scale sets the frequency of the data (seconds, minutes  hours) and the period scrollbar sets the alignment of the data. The Lags combo box determines the bandwidth of the kernel measuring the volatility. Once all the options have been set, clicking on the “Compute Realized Volatility” button will then produce three data sets for the time period given between start date and end data: 1) The daily log-returns of the asset r_1, \ldots, r_T 2) The log-price of the asset, and 3) The realized volatility measure RM_1, \ldots, RM_T. Once the Java-R highfrequency routine has finished computing the realized measures, the data sets are automatically available in the Data Control Module of iMetrica. From here, one can annualize the realized measures using the weight adjustments in the Data Control Module (see Figure 5). Once content with the weighting, the data can then be exported to the MDFA module or the BayesCronos module for estimating and forecasting the volatility of GOOG using HEAVY models.

Figure 4: The log-return data (blue) and the (annualized) realized measure data using 5 minute returns (pink).

Figure 5: The log-return data (blue) and the (annualized) realized measure data using 5 minute returns (pink) for Google from 1-1-2011 to 6-19-2012.

The Realized Measure uploading in iMetrica utilizes a fantastic R package for studying and working with high frequency financial data called highfrequency (Boudt, Cornelissen, and Payseur 2012). To handle the analysis of high frequency financial data in java, I began by writing a Java wrapper to the R functions of the highfrequency R package to enable GUI interaction shown above in order to download the data into java and then iMetrica. The java environment uses library called RCaller that opens a live R kernel in the Java runtime environment from which I can call and R routines and directly load the data into Java. The initializing sequence looks like this.


caller.getRCode().addRCode("require (Runiversal)");
caller.getRCode().addRCode("require (FinancialInstrument)");
caller.getRCode().addRCode("require (highfrequency)");
caller.getRCode().addRCode("loadInstruments('/HighFreqDataDirectoryHere/Market/instruments.rda')");
caller.getRCode().addRCode("setSymbolLookup.FI('/HighFreqDataDirectoryHere/Market/sec',use_identifier='X.RIC',extension='RData')");

Here, I’m declaring the R packages that I will be using (first three lines) and then I declare where my HighFrequency financial data symbol lookup directory is on my computer (next two lines). This
then enables me to extract high frequency tick data directly into Java. After loading in the desired intrument ticker symbol names, I then proceed to extract the daily log-returns for the given time frame, and then compute the realized measures of each asset using the rKernelCov function in highfrequency R package. This looks something like

for(i=0;i<n_assets;i++)
{
String mark = instrum[i] + "<-" + instrum[i] + "['T09:30/T16:00',]";

caller.getRCode().addRCode(mark);

String rv = "rv"+i+"<-rKernelCov("+instrum[i]+"$Trade.Price,kernel.type ="+kernels[kern]+", kernel.param="+lags+",kernel.dofadj = FALSE, align.by ="+frequency[freq]+", align.period="+period+", cts=TRUE, makeReturns=TRUE)"

caller.getRCode().addRCode(rv);

caller.getRCode().addRCode("names(rv"+i+")<-'rv"+i+"'");
rvs[i] = "rv_list"+i;

caller.getRCode().addRCode("rv_list"+i+"<-lapply(as.list(rv"+i+"), coredata)");
}

In the first line, I’m looping through all the asset symbols (I create Java strings to load into the RCaller as commands). The second line effectively retrieves the data during market hours only (America/New_York time), then creates a string to call the rKernelCov function in R. I give it all the user defined parameters defined by strings as well. Finally, in the last two lines, I extract the results and put them into an R list from which the java runtime environment will read.

Conclusion

In this article I discussed a recently introduced high frequency based volatility model by Shephard and Sheppard and gave an introduction to three different high-performance tools beyond MATLAB and R that I’ve developed for analyzing these new stochastic models. The heavyModel c/java package that I made available for download gives a workable start for experimenting in a fast and efficient framework the benefit of using high frequency financial data and most notably realized measures of volatility to produce better forecasts. The package will continuously be updated for improvements in both documentation, bug fixes, and overall presentation. Finally, the use of the R package highfrequency embedded in java and then utilized in iMetrica gives a fully GUI experience to stochastic modeling of high frequency financial data that is both conveniently easy to use and fast.

Happy Extracting and Volatilitizing!

Building a Multi-Bandpass Financial Portfolio

Animation 1: The changing periodogram for different in-sample sizes and selecting an appropriate band-pass component to the multi-bandpass filter.

Animation 1: Click the image to view the animation. The changing periodogram for different in-sample sizes and selecting an appropriate band-pass component to the multi-bandpass filter.

In my previous article, the third installment of the Frequency Effect trilogy, I introduced the multi-bandpass (MBP) filter design as a practical device for the extraction of signals in financial data that can be used for trading in multiple types of market environments.  As depicted through various examples using daily log-returns of Google (GOOG) as my trading platform, the MBP demonstrated a promising ability to tackle the issue of combining both lowpass filters to include a local bias and slow moving trend while at the same time providing access to higher trading frequencies for systematic trading during sideways and volatile market trajectories. I identified four different types of market environments and showed through three different examples how one can attempt to pinpoint and trade optimally in these different environments.

After reading a well-written and informative critique of my latest article, I became motivated to continue along on the MBP bandwagon by extending the exploration of engineering robust trading signals using the new design. In Marc’s words (the reviewer) regarding the initial results of this latest design in MDFA signal extraction for financial trading : “I tend to believe that some of the results are not necessarily systematic and that some of the results – Chris’ preference – does not match my own priority. I understand that comparisons across various designs of the triptic may require a fixed empirical framework (Google/Apple on a fixed time span).  But this restricted setting does not allow for more general inference (on other assets and time spans). And some of the critical trades are (at least in my perspective) close to luck.”

As my empirical framework was fixed in that I applied the designed filters to only one asset throughout the study and for a fixed time span of a year worth of in-sample data applied to 90 days out-of-sample, results showing the MBP framework applied to other assets and time frames might have made my presentation of this new design more convincing. Taking this relevant issue of limited empirical framework into account, I am extending my previous article many steps further by presenting in this article the creation of a collection of financial trading signals based entirely on the MBP filter.  The purpose of this article is to further solidify the potential for MBP filters and extend applications of the new design to constructing signals for various types of financial assets and in-sample/out-of-sample time frames. To do this I will create a portfolio of assets comprised of a group of well known companies coupled with two commodity ETFs (exchange traded funds) and apply the MBP filter strategy to each of the assets using various out-of-sample time horizons. Consequently, this will generate a portfolio of trading signals that I can track over the next several months.

Portfolio selection

In choosing the assets for my portfolio, I arranged a group of companies/commodities whose products or services I use on a consistent basis (as arbitrary as any other portfolio selection method, right?). To this end, I chose  Verizon (VZ) (service provider for my iPhone5), Microsoft (MSFT) (even though I mostly use Linux for my computing needs), Toyota (TM) ( I drive a Camry), Coffee (JO) (my morning espresso keeps the wheels turning), and Gold (GLD) (who doesn’t like Gold, a great hedge to any currency).  For each of these assets, I built a trading signal using various in-sample time periods beginning summer of 2011 and ending toward the end of summer 2012, to ensure all seasonal market effects were included. The out-of-sample time period in which I test the performance of the filter for each asset ranges anywhere from 90 days to 125 days out-of-sample. I tried to keep the selection of in-sample and out-of-sample points as arbitrary as possible.

Portfolio Performance

And so here we go. The performance of the portfolio.

Coffee (NYSEARCA:JO)

  • Regularization: smooth = .22, decay = .22, decay2 = .02, cross = 0
  • MBP = [0, .2], [.44,.55]
  • Out-of-sample performance: 32 percent ROI in 110 days

In order to work with commodities in this portfolio, the easiest way is through the use of ETFs that are traded in open markets just as any other asset. I chose the Dow Jones-UBS Coffee Subindex JO which is intended to reflect the returns that are potentially available through an unleveraged investment in one futures contract on the commodity of coffee as well as the rate of interest that could be earned on cash collateral invested in specified Treasury Bills.  To create the MBP filter for the JO index, I used JO and USO (a US Oil ETF) as the explanatory series from the dates of 5-5-2011 until 1-13-2013 (just a random date I picked from mid 2011, cinqo de mayo) and set the initial low-pass portion for the trend component of the MBP filter to [0, .17]. After a significant amount of regularization was applied, I added a bandpass portion to the filter by initializing an interval at [.4, .5]. This corresponded to the principal spectral peak in the periodogram which was located just below \pi/6 for the coffee fund. After setting the number of out-of-sample observations to 110,  I then proceeded to optimize the regularization parameters in-sample while ensuring that the transfer functions of the filter were no greater than 1 at any point in the frequency domain. The result of the filter is plotted below in Figure 1, with the transfer functions of the filters plotted below it. The resulting trading signal from the MBP filter is in green and the out-of-sample portion after the cyan line, with the cumulative return on investment (ROI) percentage in blue-pink and the daily price of JO the coffee fund in gray.

Figure : The MBP filter for JO applied 110 Out-of-sample points (after cyan line).

Figure 1: The MBP filter for JO applied 110 Out-of-sample points (after cyan line).

Figure : Transfer function for the JO and USO MBP filters.

Figure 2: Transfer function for the JO and USO MBP filters.

Notice the out-of-sample portion of 110 observations behaving akin to the in-sample portion before it, with a .97 rank coefficient of the cumulative ROI resulting from the trades. The ROI in the out-of-sample portion was 32 percent total and suffered only 4 small losses out of 18 trades. The concurrent transfer functions of the MBP filter clearly indicate where the principal spectral peak for JO (blue-ish line) is directly under the bandpass portion of the filter. Notice the signal produced no trades during the steepest descent and rise in the price of coffee, while pinpointing precisely at the right moment the major turning point (right after the in-sample period). This is exactly what you would like the MBP signal to achieve.

Gold (SPDR Gold Trust, NYSEARCA:GLD)

As one of the more difficult assets to form a well-performing signal both in-sample and out-of-sample using the MBP filter, the GLD (NYSEARCA:GLD) ETF proved to be quite cumbersome in not only locating an optimal bandpass portion to the MBP, but also finding a relevant explaining series for GLD. In the following formulation, I settled upon using a US dollar index given by the PowerShares ETF UUP (NYSEARCA:UUP), as it ended up giving me a very linear performance that is consistent both in-sample and out-of-sample. The parameterization for this filter is given as follows:

  • Regularization: smooth = .22, decay = .22, decay2 = .02, cross = 0
  • MBP = [0, .2], [.44,.55]
  • Out-of-sample performance: 11 percent ROI in 102 days
Figure : Out-of-sample results of the MBP applied to the GLD ETF for 102 observations

Figure 3 : Out-of-sample results of the MBP applied to the GLD ETF for 102 observations

Figure : The Transfer Functions for the GLD and DIG filter.

Figure 4 : The Transfer Functions for the GLD and DIG filter.

Figure : Coefficients for the GLD and DIG filters. Each are of length 76.

Figure 5: Coefficients for the GLD and DIG filters. Each are of length 76.

The smoothness and decay in the coefficients is quite noticeable along with a slight lag correlation along the middle of the coefficients between lags 10 and 38.  This trio of characteristics in the above three plots is exactly what one strives for in building financial trading signals. 1) The smoothness and decay of the coefficients, 2) the transfer functions of the filter not exceeding 1 in the low and band pass, and 3) linear performance both in-sample and out-of-sample of the trading signal.

Verizon (NYSE:VZ)

  • Regularization: smooth = .22, decay = 0, decay2 = 0, cross = .24
  • MBP = [0, .17], [.58,.68]
  • Out-of-sample performance: 44 percent ROI in 124 days trading

The experience of engineering a trading signal for Verizon was one of the longest and more difficult experiences out of the 5 assets in this portfolio. Strangely a very difficult asset to work with. Nevertheless, I was determined to find something that worked. To begin, I ended up using AAPL as my explanatory series (which isn’t a far fetched idea I would imagine. After all, I utilize Verizon as my carrier service for my iPhone 5).  After playing around with the regularization parameters in-sample, I chose a 124 day out-of-day horizon for my Verizon to apply the filter to and test the performance. Surprisingly, the cross regularization seemed to produce very good results both out-of-sample. This was the only asset in the portfolio that required a significant amount of cross regularization, with the parameter touching the vicinity of .24. Another surprise was how high the timeliness parameter \lambda was (40) in order to produce good in-sample and out-of-sample trading results. By far the highest amount of the 5 assets in this study. The amount of smoothing from the weighting function $W(\omega; \alpha)$ was also relatively high, reaching a value of 20.

The out-of-sample performance is shown in Figure 6. Notice how dampened the values of the trading signal are in this example, where the local bias during the long upswings is present, but not visible due to the size of the plot. The out-of-sample performance (after the cyan line) seems to be superior to that of the in-sample portion. This is most likely due to the fact that the majority of the frequencies that we were interested in, near \pi/6, failed to become prominent in the data until the out-of-sample portion (there were around 120 trading days not shown in the plot as I only keep a maximum of 250 plotted on the canvas).  With 124 out-of-sample observations, the signal produced a performance of 44 percent ROI. The filter seems to cleanly and consistently pick out local turning points, although not always at their optimal point, but the performance is quite linear, which is exactly what you strive for.

Figure : The out-of-sample performance on 124 observations from 7-2012 to 1-13-2013.

Figure 6: The out-of-sample performance on 124 observations from 7-2012 to 1-13-2013.

Figure : Coefficients of lag up to 76 of the Verizon-Apple filter,

Figure 7: Coefficients of lag up to 76 of the Verizon-Apple filter,

In the coefficients for the VZ and AAPL data shown in Figure 7, one can clearly see the distinguishing effects of the cross regularization along with the smooth regularization. Note that no decay regularization was needed in this example, with the resulting number of effective degrees of freedom in the construction of this filter being 48.2 an important number to consider when applying regularization to filter coefficients (filter length was 76),

Microsoft (NASDAQ:MSFT) 

  • Regularization: smooth = .42, decay = .24, decay2 = .15, cross = 0
  • MBP = [0, .2], [.59,.72]
  • Out-of-sample performance: 31 percent ROI in 90 days trading

In the Microsoft data I used a time span of a year and three months for my in-sample period and a 90 day out-of-sample period from August through 1-13-2012. My explanatory series was GOOG (the search engine Bing and Google seem to have quite the competition going on, so why not) which seemed to correlate rather cleanly with the share price of MSFT. The first step in obtaining a bandpass after setting my lowpass filter to [0, .2] was to locate the principal spectral peak (shown in the periodogram figure below). I then adjusted the width until I had near monotone performance in-sample. Once the customization and regularization parameters were found, I applied the MSFT/AAPL filter to the 90 day out-of-sample period and the result is shown below. Notice that the effect of the local bias and slow moving trends from the lowpass filter are seen in the output trading signal (green) and help in identifying the long down swings found in the share price. During the long down swings, there are no trades due to the local bias from frequency zero.

Figure : Microsoft trading signal for 90 out-of-sample observations. The ROI out-of-sample is 31 percent.

Figure 8: Microsoft trading signal for 90 out-of-sample observations. The ROI out-of-sample is 31 percent.

Figure : Aggregate periodogram of MSFT and Google showing the principal spectral peak directly inside the bandpass.

Figure 9: Aggregate periodogram of MSFT and Google showing the principal spectral peak directly inside the bandpass.

Figure : The coefficients for the MSFT and GOOG series up to lag 76.

Figure 10: The coefficients for the MSFT and GOOG series up to lag 76.

With a healthy amount of regularization applied to the coefficient space, we can clearly see the smoothness and decay towards the end of the coefficient lags. The cross regularization parameter provided no improvement to either in-sample or out-of-sample performance and was left set to 0.

Despite the superb performance of the signal out-of-sample with a 31 percent ROI in 90 days in a period which saw the share price descend by 10 percent, and relatively smooth decaying coefficients with consistent performance both in and out-of-sample, I still feel like I could improve on these results with a better explanatory series than AAPL. That is one area of this methodology in which I struggle, namely finding “good” explanatory series to better fortify the in-sample metric space and produce more even more anticipation in the signals. At this point it’s a game of trial and error. I suppose I should find a good market economist to direct these questions to.

Toyota (NYSE:TM)

  • Regularization: smooth = .90, decay = .14, decay2 = .72, cross = 0
  • MBP = [0, .21], [.49,.67]
  • Out-of-sample performance: 21 percent ROI in 85 days trading

For the Toyota series, I figured my first explanatory series to test things with would be an asset pertaining to the price of oil. So I decided to dig up some research and found that DIG ( NYSEARCA:DIG), a ProShares ETF, provides direct exposure to the global price of oil and gas (in fact it is leveraged so it corresponds to twice the daily performance of the Dow Jones U.S. Oil & Gas Index).  The out-of-sample performance, with heavy regularization in both smooth and decay, seems to perform quite consistently with in-sample, The signal shows signs of patience during volatile upswings, which is a sign that the local bias and slow moving trend extraction is quietly at work. Otherwise, the gains are consistent with just a few very small losses. At the end of the out-of-sample portion, namely the past several weeks since Black Friday (November 23rd), notice the quick climb in stock price of Toyota. The signal is easily able to deduce this fast climb and is now showing signs of slowdown from the recent rise (the signal is approaching the zero crossing, that’s how I know).  I love what you do for me, Toyota! (If you were living in the US in the1990s, you’ll understand what I’m referring to).

Figure : Out-of-sample performance of the Toyota trading signal on 85 trading days.

Figure 11: Out-of-sample performance of the Toyota trading signal on 85 trading days.

Figure : Coefficients for the  TM and DIG log-return series.

Figure 12: Coefficients for the TM and DIG log-return series.

Figure : The transfer functions for the TM and DIG filter coefficients.

Figure 13: The transfer functions for the TM and DIG filter coefficients.

The coefficients for the TM and DIG series depicted in Figure 12 show the heavy amount of smooth and decay (and decay2) regularization, a trio of parameters that was not easy to pinpoint at first without significant leakage above one in the filter transfer functions (shown in Figure 13). One can see that two major spectral peaks are present under the lowpass portion and another large one in the bandpass portion that accounts for the more frequent trades.

Conclusion

With these trading signals constructed for these five assets, I imagine I have a small but somewhat diverse portfolio, ranging from tech and auto to two popular commodities. I’ll be tracking the performance of these trading signals together combined as a portfolio over the next few months and continuously give updates. As the in-sample periods for the construction of these filters ended around the end of last summer and were already applied to out-of-sample periods ranging from 90 days to 124 (roughly one half to one third of the original in-sample period), with the significant amount of regularization applied, I am quite optimistic that the out-of-sample performance will continue to be the same over the next few months, but of course one can never be too sure of anything when it comes to market behavior. In the worse case scenario, I can always look into digging though my dynamic adaptive filtering and signal extraction toolkit.

Some general comments as I conclude this article. What I truly enjoy about these trading signals constructed for this portfolio experiment (and robust trading signals in general per my other articles on financial trading) is that when any losses out-of-sample or even in-sample occur, they tend to be extremely small relative to the average size of the gains. That is the sign of a truly robust signal I suppose; that not only does it perform consistently both in-sample and out-of-sample, but also that when losses do arrive, they are quite small. One characteristic that I noticed in all robust and high performing trading signals that I tend to stick with is that no matter what type of extraction definition you are targeting (lowpass, bandpass, or MBP), when an erroneous trade is executed (leading to a loss), the signal will quickly correct itself to minimize the loss. This is why the losses in robust signals tend to be small (look at any of the 5 trading signals produced for the portfolio in this article).  Of course, all these good trading signal characteristics are in addition to the filter characteristics (smooth, slightly decaying coefficients with minimal effective degrees of freedom, transfer functions less than or equal to one everywhere, etc.)

Overall, although I’m quite inspired and optimistic with these results. there is still slight room for improvement in building these MBP filters, especially for low volatility sideways markets (for example, the one occurring in the Toyota stock price in the middle of the plot in Figure 11). In general, this is a difficult type of stock price movement in which any type of signal will have success. With low volatility and no trending movements, the log-returns are basically white noise – there is no pertinent information to extract. The markets are currently efficient and there is nothing you can do about it. Only good luck will win (in that case you’re as well off building a signal based on a coin flip). Typically the best you can do in these types of areas is prevent trading altogether with some sort of threshold on the signal, which is an idea I’ve had in my mind recently but haven’t implemented, or make sure any losses are small, which is exactly what my signal achieved in Figure 11 (and which is what any robust signal should do in the first place.)

Lastly, if you have a particular financial asset for which you would like to build a trading signal (similar to the examples shown above), I will be happy to take a stab at it using iMetrica (and/or give you pointers in the right direction if you would prefer to pursue the endeavor yourself). Just send me what asset you would like to trade on, and I’ll build the filter and send you the coefficients along with the parameters used. Offer holds for a limited time only!

Happy extracting.

The Frequency Effect Part III: Revelations of Multi-Bandpass Filters and Signal Extraction for Financial Trading

Animation of the out-of-sample performance of one of the multibandpass filters built in this article for the daily returns of the price of Google. The resulting trading signal was extracted and yielded a trading performance near 39 percent ROI during an 80 day out-of-sample period on trading shares of Google.

Animation of the out-of-sample performance of one of the multibandpass filters built in this article for the daily returns of the price of Google. The resulting trading signal was extracted and yielded a trading performance near 39 percent ROI during an 80 day out-of-sample period on trading shares of Google.

To conclude the trilogy on this recent voyage through various variations on frequency domain configurations and optimizations in financial trading using MDFA and iMetrica, I venture into the world of what I call multi-bandpass filters that I recently implemented in iMetrica.  The motivation of this latest endeavor in highlighting the fundamental importance of the spectral frequency domain in financial trading applications was wanting to gain better control of extracting signals and engineering different trading strategies through many different types of market movement in financial assets. There are typically four different basic types of movement a price pattern will take during its fractalesque voyage throughout the duration that an asset is traded on a financial market. These patterns/trajectories include

  1. steady up-trends in share price
  2. low volatility sideways patterns (close to white noise)
  3. highly volatile sideways patterns (usually cyclical)
  4. long downswings/trends in share price.

Using MDFA for signal extraction in financial time series, one typically indicates an a priori trading strategy through the design of the extractor, namely the target function \Gamma(\omega) (see my previous two articles on The Frequency Effect). Designating a lowpass or bandpass filter in the frequency domain will give an indication of what kind of patterns the extracted trading signal will trade on. Traditionally one can set a lowpass with the goal of extracting trends (with the proper amount of timeliness prioritized in the parameterization), or one can opt for a bandpass to extract smaller cyclical events for more systematic trading during volatile periods. But now suppose we could have the best of both worlds at the same time. Namely, be profitable in both steady climbs and long tumbles, while at the same time systematically hacking our way through rough sideways volatile territory, making trades at specific frequencies embedded in the share price actions not found in long trends. The answer is through the construction of multi-band pass filters. Their construction is relatively simple, but as I will demonstrate in this article with many examples, they are a bit more difficult to pinpoint optimally (but it can be done, and the results are beautiful… both aesthetically and financially).

With the multi-bandpass defined as two separate bands given by A := 1_{[\omega_0, \omega_1]}B := 1_{[\omega_2, \omega_3]} with 0 \leq \omega_0 and \omega_1 < \omega_2, zero everywhere else, it is easy to see that the motivation here is to seek a detection of both lower frequencies and low-mid frequencies in the data concurrently. With now up to four cutoff frequencies to choose from, this adds yet another few wrinkles in the degrees of freedom in parameterizing the MDFA setup. If choosing and optimizing one cutoff frequency for a simple low-pass filter in addition to customization and regularization parameters wasn’t enough, now imagine extracting signals with the addition of up to three more cutoff frequencies. Despite these additional degrees of freedom in frequency interval selection, I will later give a couple of useful hacks that I’ve found helpful to get one started down the right path toward successful extraction.

With this multi-bandpass definition for \Gamma comes the responsibility to ensure that the customization of smoothness and timeliness is adjusted for the additional passband. The smoothing function W(\omega; \alpha) for \alpha \geq 0 that acts on the periodogram (or discrete Fourier transforms in multivariate mode) is now defined piecewise according to the different intervals [0,\omega_0], [\omega_1, \omega_2], and [\omega_3, \pi].  For example, \alpha = 20 gives a piecewise quadratic weighting function (an example shown in Figure 1) and for \alpha = 10, the weighting function is piecewise linear. In practice, the piecewise power function smooths and rids of unwanted frequencies in the stop band much better than using a piecewise constant function. With these preliminaries defined, we now move on to the first steps in building and applying multiband pass filters.

Figure 1: Plot of the Piecewise Smoothing Function for alpha = 15 on a mutli-band pass filter.

Figure 1: Plot of the Piecewise Smoothing Function for alpha = 15 on a mutli-band pass filter.

To motivate this newly customized approach to building financial trading signals, I begin with a simple example where I build a trading signal for the daily share price of Google. We begin with a simple lowpass filter defined by \Gamma(\omega) = 1 if \omega \in [0,.17], and 0 otherwise. This formulation, as it includes the zero frequency, should provide a local bias as well as extract very slow moving trends. The trick with these filters for building consistent trading performance is ensure a proper grip on the timeliness characteristics of the filter in a very low and narrow filter passage. Regularization and smoothness using the weighting function shouldn’t be too much of a problem or priority as typically just only a small fraction of the available degrees of freedom on the frequency domain are being utilized, so not much concern for overfitting as long as you’re not using too long of a filter.  In my example, I maxed out the timeliness \lambda parameter and set the \lambda_{smooth} regularization parameter to .3. Fortunately, no optimization of any parameter was needed in this example, as the performance was spiffy enough nearly right after gauging the timeliness parameter \lambda. Figure 2 shows the resulting extracted trend trading signal in both the in-sample portion (left of the cyan colored line) and applied to 80 out-of-sample points (right of the cyan line, the most recent 80 daily returns of Google, namely 9-29-12 through today, 1-10-13). The blue-pink line shows the progression of the trading account, in return-on-investment percentage. The out-of-sample gains on the trades made were 22 percent ROI during the 80 day period.

Figure 1: The in-sample and out-of-sample gains made by constructing a low-pass filter employing a very high timeliness parameter and small amount of regularization in smoothness. The out-of-sample gains are nearly 30 percent and no losses on any trades.

Figure 2: The in-sample and out-of-sample gains made by constructing a low-pass filter employing a very high timeliness parameter and small amount of regularization in smoothness. The out-of-sample gains are nearly 30 percent and no losses on any trades.

Although not perfect, the trading signal produces a monotonic performance both in-sample and out-of-sample, which is exactly what you strive for when building these trend signals for trading. The performance out-of-sample is also highly consistent (in regards to trading frequency and no losses on any trades) with the in-sample performance. With only 4 trades being made, they were done at very interesting points in the trajectory of the Google share price. Firstly, notice that the local bias in the largest upswing is accounted for due to the inclusion of frequency zero in the low pass filter. This (positive) local bias continues out-of-sample until, interestingly enough, two days before one of the largest losses in the share price of Google over the past couple years. A slightly earlier exit out of this long position (optimally at the peak before the down turn a few days before) would have been more strategic; perhaps further tweaking of various parameters would have achieved this, but I happy with it for now. The long position resumes a few days after the dust settles from the major loss, and the local bias in the signal helps once again (after trade 2). The next few weeks sees shorter downtrending cyclical effects, and the signal fortunately turns positively increasingly right before another major turning point for an upswing in the share price. Finally, the third transaction ends the long position at another peak (3), perfect timing. The fourth transaction (no loss or gain) was quickly activated after the signal saw another upturn, and thus is now in the long position (hint: Google trending upward).  Figure 3 shows the transfer functions \hat{\Gamma} for both the sets of explanatory log-return data and Figure 4 depicts the coefficients for the filter. Notice that in the coefficients plot, much more weight is being assigned to past values of the log-return data with extreme (min and max values) at around lags 15 and 30 for the GOOG coefficients (blue-ish line). The coefficients are also quite smooth due to the slight amount of smooth regularization imposed.

Figure 3: Transfer functions for the concurrent trend filter applied to GOOG.

Figure 3: Transfer functions for the concurrent trend filter applied to GOOG.

Figure 4: The filter coefficients for the log-return data.

Figure 4: The filter coefficients for the log-return data.

Now suppose we wish to extract a trading signal that performs like a trend signal during long sweeping upswings or downswings, and at the same time shares the property that it extracts smaller cyclical swings during a sideways or highly volatile period. This type of signal would be endowed with the advantage that we could engage in a long position during upswings, trade systematically during sideways and volatile times, and on the same token avoid aggressive long-winded downturns in the price. Financial trading can’t get more optimistic then that, right? Here is where the magic of the multi-bandpass comes in. I give my general “how-to” guidelines in the following paragraphs as a step-by-step approach. As a forewarning, these signals are not easy to build, but with some clever optimization and patience they can be done.

In this new formulation, I envision not only being able to extract a local bias embedded in the log-return data but also gain information on other important frequencies to trade on while in sideways markets. To do this, I set up the lowpass filter as I did earlier on [0,\omega_0]. The choice of \omega_0 is highly dependent on the data and should be located through a priori investigations (as I did above, without the additional bandpass).

Animation 2: Example of constructing a multiband pass using the Target Filter control panel in iMetrica. Initially, a low-pass filter is set, then the additional bandpass is added by clicking "Multi-Pass" checkbox. The location is then moved to the desired location using the scrollbars. The new filters are computed automaticall if "Auto" is checked on (lower left corner).

Click on the Animation 2: Example of constructing a multiband pass using the Target Filter control panel in iMetrica. Initially, a low-pass filter is set, then the additional bandpass is added by clicking “Multi-Pass” checkbox. The location is then moved to the desired location using the scrollbars. The new filters are computed automaticall if “Auto” is checked on (lower left corner).

Before setting any parameterization regarding customization, regularization, or filter constraints, I perform a quick scan of the periodogram (averaged periodogram if in multivariate mode) to locate what I call principal trading frequencies in the data. In the averaged periodogram, these frequencies are located at the largest spectral peaks, with the most useful ones for our purposes of financial trading typically before \pi/4. The largest of these peaks will be defined from here on out as the principal spectral peak (PSP). Figure 6 shows an example of an averaged periodogram of the log-return for GOOG and AAPL with the PSP indicated. You might note that there exists a much larger spectral peak located at 7\pi/12, but no need to worry about that one (unless you really enjoy transaction costs). I locate this PSP as a starting point for where I want my signal to trade.

Figure 5: Principal spectral peak in the log-return data of GOOG and AAPL.

Figure 5: Principal spectral peak in the log-return data of GOOG and AAPL.

In the next step, I place a bandpass of width around .15 so that the PSP is dead-centered in the bandpass. Fortunately with iMetrica, this is a seamlessly simple task with just the use of a scrollbar to slide the positioning of this bandpass (and also adjust  the lowpass) to where I desire. Animation 2 above (click on it to see the animation) shows this process of setting a multi-passband in the MDFA Target Filter control panel. Notice as I move the controls for the location of the bandpass, the filter is automatically recomputed and I can see the changes in the frequency response functions \hat{\Gamma} instantaneously.

With the bandpass set along with the lowpass, we can now view how the in-sample performance is behaving at the initial configuration. Slightly tweaking the location of the bandpass might be necessary (width not so much, in my experience between .15 and .20 is sufficient).  The next step in this approach is now to not only adjust for the location of the bandpass while keeping the PSP located somewhat centered, but also adding the effects of regularization to the filter as well. With this additional bandpass, the filter has a tendency to succumb to overfitting if one is not careful enough.

In my first filter construction attempt, I placed my bandpass at [.49,.65] with the PSP directly under it. I then optimized the regularization controls in-sample (a feature I haven’t discussed yet) and slightly tweaked the timeliness parameter (ended up setting it to 3) and my result (drumroll…)  is shown in Figure 6.

Figure 6: The trading performance and signal for the initial attempt at a building a multiband pass fitler.

Figure 6: The trading performance and signal for the initial attempt at a building a multiband pass fitler.

Not bad for a first attempt. I was actually surprised at how few trades there were out-of-sample. Although there are no losses during the 80 days out-of-sample (after cyan line), and the signal is sort of what I had in mind a priori, the trades are minimal and not yielding any trading action during the period right after the large loss in Google when the market was going sideways and highly volatile. Notice that the trend signal gained from the lowpass filter indeed did its job by providing the local bias during the large upswing and then selling directly at the peak (first magenta dotted line after the cyan line).  There are small transactions (gains) directly after this point, but still not enough during the sideways market after the drop.  I needed to find a way to tweak the parameters and/or cutoff to include higher frequencies in the transactions.

In my second attempt, I kept the regularization parameters as they were but this time increased the bandpass to the interval [.51, .68], with the PSP still underneath the bandpass, but now catching on to a few more higher frequencies then before.  I also slightly increased the length of the filter to see if that had any affect. After optimizing on the timeliness parameter \lambda in-sample, I get a much improved signal. Figure 7 shows this second attempt.

Figure 7: The trading performance and signal for the second attempt at construction a multiband pass filter. This one included a few more higher frequencies.

Figure 7: The trading performance and signal for the second attempt at construction a multiband pass filter. This one included a few more higher frequencies.

Upon inspection, this signal behaves more consistently with what I had in mind. Notice that directly out-of-sample during the long upswing, the signal (barely) shows signs of the local bias, but enough not to make any trades fortunately. However, in this signal, we see that filter is much too late in detecting the huge loss posted by Google, and instead sells immediately after (still a profit however). Then during the volatile sideways market, we see more of what we were wishing for; timely trades to the earn the signal a quick 9 percent in the span of a couple weeks. Then the local bias kicks in again and we see not another trade posted during this short upswing, taking advantage of the local trend. This signal earned a near 22 percent ROI during the 80 day out-of-sample trading period, however not as good as the previous signal at  32 percent ROI.

Now my priority was to find another tweak that I could perform to change the trading structure even more. I’d like it to be even more sensitive to quick downturns, but at the same time keep intact the sideways trading from the signal in Figure 7. My immediate intuition was to turn on the i2 filter constraint and optimize the time-shift, similar to what I did in my previous article, part deux of the Frequency Effect. I also lessened the amount of smoothing from my weighting function W(\omega; \alpha), turned off any amount of decay regularization that I had and voila, my final result in Figure 8.

Figure 8: Third attempt at building a multiband pass filter. Here, I turn on i2 filter constraint and optimize the time shift.

Figure 8: Third attempt at building a multiband pass filter. Here, I turn on i2 filter constraint and optimize the time shift.

While the consistency with the in-sample performance to out-of-sample performance is somewhat less than my previous attempts, out-of-sample performs nearly exactly how I envisioned. There are only two small losses of less than 1 percent each, and the timeliness of choosing when to sell at the tip of the peak in the share price of Google couldn’t have been better. There is systematic trading governed by the added multiband pass filter during the sideways and slight upswing toward the end. Some of the trades are made later than what would be optimal (the green lines enter a long position, magenta sells and enters short position), but for the most part, they are quite consistent.  It’s also very quick in pinpointing its own erronous trades (namely no huge losses in-sample or out of sample). There you have it, a near monotonic performance out-of-sample with 39 percent ROI.

In examining the coefficients of this filter in Figure 9, we see characteristics of a trend filter as coefficients are largely weighting the middle lags much more than than initial or end lags (note that no decay regularization was added to this filter, only smoothness) . While at the same time however, the coefficients also weight the most recent log-return observations unlike the trend filter from Figure 4, in order to extract signals for the more volatile areas. The undulating patterns also assist in obtaining good performance in the cyclical regions.

Figure 9: The coefficients of the final filter depicting characteristics of both a trend and bandpass filter, as expected.

Figure 9: The coefficients of the final filter depicting characteristics of both a trend and bandpass filter, as expected.

Finally, the frequency response functions of the concurrent filters show the effect of including the PSP in the bandpass (figure 10). Notice, the largest peak in the bandpass function is found directly at the frequency of the PSP, ahh the PSP. I need to study this frequency with more examples to get a more clear picture to what it means. In the meantime, this is the strategy that I would propose. If you have any questions about any of this, feel free to email me. Until next time, happy extracting!

Figure 10: The frequency response functions of the multiband filter.

Figure 10: The frequency response functions of the multi-bandpass filter.

The Frequency Effect Part Deux: Shifting Time at Frequency Zero For Better Trading Performance

Animation 1: The out-of-sample performance over 60 trading days of signal built using a custom i2 criterion. With 5 trades and 4 successful, the ROI is nearly 40 percent.

Animation 1: The out-of-sample performance over 60 trading days of a signal built using an optimized time-shift criterion. With 5 trades and 4 successful, the ROI is nearly 40 percent over 3 month.

What is an optimized time-shift? Is it important to use when building successful financial trading signals? While the theoretical aspects of the frequency zero and vanishing time-shift can be discussed in a very formal and mathematical manner,  I hope to answer these questions in a more simple (and applicable) way in this article. To do this, I will give an informative and illustrated real world example in this unforeseen continuation of my previous article on the frequency effect a few days ago. I discovered something quite interesting after I got an e-mail from Herr Doktor Marc (Wildi) that nudged me even further into my circus of investigations in carving out optimal frequency intervals for financial trading (see his blog for the exact email and response).  So I thought about it  and soon after I sent my response to Marc, I began to question a few things even further at 3am in the morning while sipping on some Asian raspberry white tea (my sleeping patterns lately have been as erratic as fiscal cliff negotiations), and came up with an idea. Firstly, there has to be a way to include information about the zero-frequency (this wasn’t included in my previous article on optimal frequency selection). Secondly, if I’m seeing promising results using a narrow band-pass approach after optimizing the location and distance, is there anyway to still incorporate the zero-frequency and maybe improve results even more with this additional frequency information?

Frequency zero is an important frequency in the world of nonstationary time series and model-based time series methodologies as it deals with the topic of unit roots, integrated processes,  and (for multivariate data) cointegration. Fortunately for you (and me), I don’t need to dwell further into this mess of a topic that is cointegration since typically, the type of data we want to deal with in financial trading (log-returns) is closer to being stationary (namely close to being white noise, ehem, again, close, but not quite). Nonetheless, a typical sequence of log-return data over time is never zero-mean, and full of interesting turning points at certain frequency bands. In essence, we’d somehow like to take advantage of that and perhaps better locate local turning points intrinsic to the optimal trading frequency range we are dealing with.

The perfect way to do this is through the use of the time-shift value of the filter. The time-shift is defined by the derivative of the frequency response (or transfer) function at zero. Suppose we have an optimal bandpass set at (\omega_0, \omega_1) \subset [0,\pi] where \omega_0 > 0. We can introduce a constraint on the filter coefficients so as to impose a vanishing time-shift at frequency zero. As Wildi says on page 24 of the Elements paper: “A vanishing time-shift is highly desirable because turning-points in the filtered series are concomitant with turning-points in the original data.” In fact, we can take this a step further and even impose an arbitrary time-shift with the value s at frequency zero, where s is any real number. In this case, the derivative of the frequency response function (transfer function) \hat{\Gamma}(\omega) at zero is s. As explained on page 25 of Elements,  this is implemented as \frac{d}{d \omega} |_{\omega=0} \sum_{l=0}^{L-1} b_j \exp(-i j \omega) = s, which implies b_1 + 2b_2 + \cdots + (L-1) b_{L-1} = s.

This constraint can be integrated into the MDFA formulation, but then of course adds another parameter to an already full-flight of parameters.  Furthermore, the search for the optimal s with respect to a given financial trading criterion is tricky and takes some hefty computational assistance by a robust (highly nonlinear) optimization routine, but it can be done. In iMetrica I’ve implemented a time-shift turning point optimizer, something that works well so far for my taste buds, but takes a large burden of computational time to find.

To illustrate this methodology in a real financial trading application, I return to the same example I used in my previous article, namely using daily log-returns of GOOG and AAPL from 6-3-2011 to 12-31-2012 to build a trading signal. This time to freshen things up a but, I’m going to target and trade shares of Apple Inc. instead of Google.  Quickly, before I begin, I will swiftly go through the basic steps of building trading signals. If you’re already familiar, feel free to skip down two paragraphs.

As I’ve mentioned in the past, fundamentally the most important step to building a successful and robust trading signal is in choosing an appropriate preliminary in-sample metric space in which the filter coefficients for the signal are computed. This preliminary in-sample metric space represents by far the most critically important aspect of building a successful trading signal and is built using the following ingredients:

  • The target and explanatory series (i.e. minute, hourly, daily log-returns of financial assets)
  • The time span of in-sample observations (i.e. 6 hours, 20 days, 168 days, 3 years, etc.)

Choosing the appropriate preliminary in-sample metric space is beyond the scope of this article, but will certainly be discussed in a future article.  Once this in-sample metric space has been chosen, one can then proceed by choosing the optimal extractor (the frequency bandpass interval) for the metric space. While concurrently selecting the optimal extractor, one must  begin warping and bending the preliminary metric space through the use of the various customization and regularization tools (see my previous Frequency Effect article, as well as Marc’s Elements paper for an in-depth look at the mathematics of regularization and customization). These are the principle steps.

Now let’s look at an example. In the .gif animation at the top of this article, I featured a signal that I built using this time-shift optimizer and a frequency bandpass extractor heavily centered around the frequency \pi/12, which is not a very frequent trading frequency, but has its benefits, as we’ll see. The preliminary metric space was constructed by an in-sample period using the daily log-returns of GOOG and AAPL and AAPL as my target is from 6-4-2011 to 9-25-2012, nearly 16 months of data. Thus we mention that the in-sample includes many important news events from Apple Inc. such as the announcement of the iPad mini, the iPhone 4S and 5, and the unfortunate sad passing of Steve Jobs. I then proceeded to bend the preliminary metric space with a heavy dosage of regularization, but only a tablespoon of customization¹. Finally, I set the time-shift constraint and applied my optimization routine in iMetrica to find the value s that yields the best possible turning-point detector for the in-sample metric space. The result is shown in Figure 1 below in the slide-show. The in-sample signal from the last 12 months or so (no out-of-sample yet applied) is plotted in green, and since I have future data available (more than 60 trading days worth from 9-25 to present), I can also approximate the target symmetric filter (the theoretically optimal target signal) in order to compare things (a quite useful option available with the click of a button in iMetrica I might add). I do this so I can have a good barometer of over-fitting and concurrent filter robustness at the most recent in-sample observation. Figure 1 in the slide-show below, the trading signal is in green, the AAPL log-return data in red, and the approximated target signal in gray (recall that if you can approximate this target signal (in gray) arbitrarily well, you win, big).

Notice that at the very endpoint (the most challenging point to achieve greatness) of the signal in Figure 1, the filter does a very fine job at getting extremely close. In fact, since the theoretical target signal is only a Fourier approximation of order 60, my concurrent signal that I built might even be closer to the ‘true value’, who knows. Achieving exact replication of the target signal (gray) elsewhere is a little less critical in my experience. All that really matters is that it is close in moving above and below zero to the targeted intention (the symmetric filter) and close at the most recent in-sample observation. Figure 2 above shows the signal without the time-shift constraint and optimization. You might be inclined to say that there is no real big difference. In fact, the signal with no time-shift constraint looks even better. It’s hard to make such a conclusion in-sample, but now here is where things get interesting.

We apply the filter to the out-of-sample data, namely the 60 tradings days. Figure 3 shows the out-of-sample performance over these past 60 trading days, roughly October, November, and December, (12-31-2012 was the latest trading day), of the signal without the time-shift constraint. Compare that to Figure 4 which depicts the performance with the constraint and optimization. Hard to tell a difference, but let’s look closer at the vertical lines. These lines can be easily plotted in iMetrica using the plot button below the canvas named Buy Indicators. The green line represents where the long position begins (we buy shares) and the exit of a short position. The magenta line represents where selling the shares occurs and the entering of a short position. These lines, in other words, are the turning point detection lines. They determine where one buys/sells (enter into a long/short position). Compare the two figures in the out-of-sample-portion after the light cyan line (indicated in Figure 4 but not Figure 3, sorry).

Figure 3: Out-of-sample performance of the signal built without time-shift constraint The out-of-sample period beings where the light cyan line is from Figure 4.

Figure 3: Out-of-sample performance of the signal built without time-shift constraint The out-of-sample period beings where the light cyan line is from Figure 4 below.

Figure 4: Out-of-sample performance of the signal built with time-shift constraint and optimized for turning point-detection,  The out-of-sample period beings where the light cyan is.

Figure 4: Out-of-sample performance of the signal built with time-shift constraint and optimized for turning point-detection, The out-of-sample period beings where the light cyan is.

Notice how the optimized time-shift constraint in the trading signal in Figure 4 pinpoints to a close perfection where the turning points are (specifically at points 3, 4,and 5).  The local minimum turning point was detected exactly at 3, and nearly exact at 4 and 5. The only loss out of the 5 trades occurred at 2, but this was more the fault of the long unexpected fall in the share price of Apple in October. Fortunately we were able to make up for those losses (and then some) at the next trade exactly at the moment a big turning point came (3).  Compare this to the non optimized time-shift constrained signal (Figure 3), and how the second and third turning points are a bit too late and too early, respectively. And remember, this performance is all out-of-sample, no adjustments to the filter have been made, nothing adaptive. To see even more clearly how the two signals compare, here are gains and losses of the 5 actual trades performed out-of-sample (all numbers are in percentage according to gains and losses in the trading account governed only by the signal. Positive number is a gain, negative a loss)

                       Without Time-Shift Optimization              With Time-Shift Optimization

Trade 1:                              29.1 -> 38.7 =  9.6                          14.1 -> 22.3  =  8.2
Trade 2:                              38.7 -> 32.0  = -6.7                         22.3 -> 17.1  = -5.2
Trade 3:                              32.0 -> 40.7  =  8.7                         17.1  -> 30.5  = 13.4
Trade 4:                              40.7 -> 48.2  =  7.5                         30.5 -> 41.2   = 10.7
Trade 5:                              48.2 -> 60.2  = 12.0                        41.2 -> 53.2   = 12.0

The optimized time-shift signal is clearly better, with an ROI of nearly 40 percent in 3 months of trading. Compare this to roughly 30 percent ROI in the non-constrained signal. I’ll take the optimized time-shift constrained signal any day. I can sleep at night with this type of trading signal. Notice that this trading was applied over a period in which Apple Inc. lost nearly 20 percent of its share price.

Another nice aspect of this trading frequency interval that I used is that trading costs aren’t much of an issue since only 10 transactions (2 transaction each trade) were made in the span of 3 months, even though I did set them to be .01 percent for each transaction nonetheless.

To dig a bit deeper into plausible reasons as to why the optimization of the time-shift constraint matters (if only even just a little bit), let’s take a look at the plots of the coefficients of each respective filter. Figure 5 depicts the filter coefficients with the optimized time-shift constraint, and Figure 6 shows the coefficients without it.  Notice how in the filter for the AAPL log-return data (blue-ish tinted line) the filter privileges the latest observation much more, while slowly modifying the others less. In the non optimized time-shift filter, the most recent observation has much less importance, and in fact, privileges a larger lag more. For timely turning point detection, this is (probably) not a good thing.  Another interesting observation is that the optimized time-shift filter completely disregards the latest observation in the log-return data of GOOG (purplish-line) in order to determine the turning points. Maybe a “better” financial asset could be used for trading AAPL? Hmmm…. well in any case I’m quite ecstatic with these results so far.  I just need to hack my way into writing a better time-shift optimization routine, it’s a bit slow at this point.  Until next time, happy extracting. And feel free to contact me with any questions.

Figure 5: The filter coefficients with time-shift optimization.

Figure 5: The filter coefficients with time-shift optimization.

Figure 6: The filter coefficients without the time-shift optimization.

Figure 6: The filter coefficients without the time-shift optimization.

¹ I won’t disclose quite yet how I found these optimal parameters and frequency interval or reveal what they are as I need to keep some sort of competitive advantage as I presently look for consulting opportunities 😉 .