# High-Frequency Financial Trading on Index Futures with MDFA and R: An Example with EURO STOXX50

Figure 1: In-sample and Out-of-sample performance (observations 240-457) of the trading signal for the Euro Stoxx50 index futures with expiration March 18th (STXE H3) during the period of 1-9-2013 and 2-1-2013, using 15 minute log-returns. The black dotted lines indicate a buy/long signal and the blue dotted lines indicate a sell/short (top).

In this second tutorial on building high-frequency financial trading signals using the multivariate direct filter approach in R, I focus on the first example of my previous article on signal engineering in high-frequency trading of financial index futures where I consider 15-minute log-returns of the Euro STOXX50 index futures with expiration on March 18th, 2013 (STXE H3).  As I mentioned in the introduction, I added a slightly new step in my approach to constructing the signals for intraday observations as I had been studying the problem of close-to-open variations in the frequency domain. With 15-minute log-return data, I look at the frequency structure 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 intraday log-return data. I will show (this time in R) 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. Seems to good to be true, right? I demonstrate in this article how it’s possible.

The first step after looking at the log-price and the log-return data of the asset being traded is to construct the periodogram of the in-sample data being traded on.  In this example, I work with the same time frame I did with my previous R tutorial by considering the in-sample portion of my data to be from 1-4-2013 to 1-23-2013, with my out-of-sample data span being from 1-23-2013 to 2-1-2013, which will be used to analyze the true performance of the trading signal. The STXE data and the accompanying explanatory series of the EURO STOXX50 are first loaded into R and then the periodogram is computed as follows.


#load the log-return and log-price SXTE data in-sample
#load the log-return and log-price SXTE data out-of-sample

len_price&lt;-557
out_samp_len&lt;-210
in_samp_len&lt;-347

price_insample&lt;-stxeprice_insamp
price_outsample&lt;-stxeprice_outsamp

#some mdfa definitions
x&lt;-stxe_insamp
len&lt;-length(x[,1])

#my range for the 15-min close-to-open cycle
cutoff&lt;-.32
ub&lt;-.32
lb&lt;-.23

#------------ Compute DFTs ---------------------------
spec_obj&lt;-spec_comp(len,x,0)
weight_func&lt;-spec_obj$weight_func stxe_periodogram&lt;-abs(spec_obj$weight_func[,1])^2
K&lt;-length(weight_func[,1])-1

#----------- compute Gamma ----------------------------
Gamma&lt;-((0:K)&lt;(K*ub/pi))&amp;((0:K)&gt;(K*lb/pi))

colo&lt;-rainbow(6)
xaxis&lt;-0:K*(pi/(K+1))
plot(xaxis, stxe_periodogram, main=&quot;Periodogram of STXE&quot;, xlab=&quot;Frequency&quot;, ylab=&quot;Periodogram&quot;,
xlim=c(0, 3.14), ylim=c(min(stxe_periodogram), max(stxe_periodogram)),col=colo[1],type=&quot;l&quot; )
abline(v=c(ub,lb),col=4,lty=3)


You’ll notice in the periodogram of the in-sample STXE log-returns that I’ve pinpointed a spectral peak between two blue dashed lines. This peak corresponds to an intrinsically important cycle in the 15-minute log-returns of index futures that gives access to predicting the close-to-open variation in the price. As you’ll see, the cycle flows fluidly through the 26 15-minute intervals during each trading day and will cross zero at (usually) one to two points during each trading day to signal whether to go long or go short on the index for the next day. I’ve deduced this optimal frequency range in a prior analysis of this data that I did using my target filter toolkit in iMetrica (see previous article). This frequency range will depend on the frequency of intraday observations, and can also depend on the index (but in my experiments, this range is typically consistent to be between .23 and .32 for most index futures using 15min observations). Thus in the R code above, I’ve defined a frequency cutoff at .32 and upper and lower bandpass cutoffs at .32 and .23, respectively.

Figure 2: Periodogram of the log-return STXE data. The spectral peak is extracted and highlighted between the two red dashed lines.

In this first part of the tutorial, I extract this cycle responsible for marking the close-to-open variations and show how well it can perform. As I’ve mentioned in my previous articles on trading signal extraction, I like to begin with the mean-square solution (i.e. no customization or regularization) to the extraction problem to see exactly what kind of parameterization I might need. To produce the plain vanilla mean-square solution, I set all the parameters to 0.0 and then compute the filter by calling the main MDFA function (shown below). The function IMDFA returns an object with the filter coefficients and the in-sample signal. It also plots the concurrent transfer function for both of the filters along with the filter coefficients for increasing lag, shown in Figure 3.

L&lt;-86
lambda_smooth&lt;-0.0
lambda_cross&lt;-0.0
lambda_decay&lt;-c(0.00,0.0)
i1&lt;-F
i2&lt;-F
lambda&lt;-0
expweight&lt;-0
i_mdfa_obj&lt;-IMDFA(L,i1,i2,cutoff,lambda,expweight,lambda_cross,lambda_decay,lambda_smooth,weight_func,Gamma,x)


Figure 3: Concurrent transfer functions for the STXE (red) and explanatory series (cyan) (top). Coefficients for the STXE and explanatory series (bottom).

Notice the noise leakage past the stopband in the concurrent filter and the roughness of both sets of filter coefficients (due to overfitting). We would like to smooth both of these out, along with allowing the filter coefficients to decay as the lag increases. This ensures more consistent in-sample and out-of-sample properties of the filter. I first apply some smoothing to the stopband by applying an expweight parameter of 16, and to compensate slightly for this improved smoothness, I improve the timeliness by setting the lambda parameter to 1. After noticing the improvement in the smoothness of filter coefficients, I then proceed with the regularization and conclude with the following parameters.

lambda_smooth&lt;-0.90
lambda_decay&lt;-c(0.08,0.11)
lambda&lt;-1
expweight&lt;-16


Figure 4: Transfer functions and coefficients after smoothing and regularization.

A vast improvement over the mean-squared solution. Virtually no noise leakage in the stopband passed $\omega_1 =.32$ and the coefficients decay beautifully with perfect smoothness achieved. Notice the two transfer functions perfectly picking out the spectral peak that is intrinsic to the close-to-open cycle that I mentioned was between .23 and .32. To verify these filter coefficients achieve the extraction of the close-to-open cycle, I compute the trading signal from the imdfa object and then plot it against the log-returns of STXE. I then compute the trades in-sample using the signal and the log-price of STXE. The R code is below and the plots are shown in Figures 5 and 6.

bn&lt;-i_mdfa_obj$i_mdfa$b
trading_signal&lt;-i_mdfa_obj$xff[,1] + i_mdfa_obj$xff[,2]

plot(x[L:len,1],col=colo[1],type=&quot;l&quot;)


Figure 5: The in-sample signal and the log-returns of SXTE in 15 minute observations from 1-9-2013 to 1-23-2013

Figure 5 shows the log-return data and the trading signal extracted from the data. The spikes in the log-return data represent the close-to-open jumps in the STOXX Europe 50 index futures contract, occurring every 27 observations. But notice how regular the signal is, and how consistent this frequency range is found in the log-return data, almost like a perfect sinusoidal wave, with one complete cycle occurring nearly every 27 observations. This signal triggers trades that are shown in Figure 6, where the black dotted lines are buys/long and the blue dotted lines are sells/shorts. The signal is extremely consistent in finding the opportune times to buy and sell at the near optimal peaks, such as at observations 140, 197, and 240. It also ‘predicts’ the jump or fall of the EuroStoxx50 index future for the next trading day by triggering the necessary buy/sell signal, such as at observations 19, 40, 51, 99, 121, 156, and, 250.  The performance of this trading in-sample is shown in Figure 7.

Figure 6: The in-sample trades. Black dotted lines are buy/long and the blue dotted lines are sell/short.

Figure 7: The in-sample performance of the trading signal.

Now for the real litmus test in the performance of this extracted signal, we need to apply the filter out-of-sample to check for consistency in not only performance, but also in trading characteristics. To do this in R, we bind the in-sample and out-of-sample data together and then apply the filter to the out-of-sample set (needing the final L-1 observations from the in-sample portion). The resulting signal in shown in Figure 8.

x_out&lt;-rbind(stxe_insamp,stxe_outsamp)
xff&lt;-matrix(nrow=out_samp_len,ncol=2)

for(i in 1:out_samp_len)
{
xff[i,]&lt;-0
for(j in 2:3)
{
xff[i,j-1]&lt;-xff[i,j-1]+bn[,j-1]%*%x_out[in_samp_len+i:(i-L+1),j]
}
}

plot(stxe_outsamp[,1],col=colo[1],type=&quot;l&quot;)


The signal and log-return data Notice that the signal performs consistently out-of-sample until right around observation 170 when the log-returns become increasingly volatile. The intrinsic cycle between frequencies .23 and .32 has been slowed down due to this increased volatility and might affect the trading performance.

Figure 8: Signal produced out-of-sample on 210 observations and log-return data of STXE

The total in-sample plus out-of-sample trading performance is shown in Figure 9 and 10, with the final 210 points being out-of-sample.  The out-of-sample performance is very much akin to the in-sample performance we had, with a very clear systematic trading exposed by ‘predicting’ the next day close-to-open jump or fall in a consistent manner, by triggering the necessary buy/sell signal, such as at observations 310, 363, 383, and 413, with only one loss up until the final day trading.  The higher volatility during the final day of the in-sample period damages the cyclical signal and fails to trade systematically as it had been during the first 420 observations.

Figure 9: The total in-sample plus out-of-sample buys and sells.

Figure 10: Total performance over in-sample and out-of-sample periods.

With this kind of performance both in-sample and out-of-sample, and the beautifully consistent yet methodological trading patterns this signal provides, it would seem like attempting to improve upon it would be a pointless task. Why attempt to fix what’s not “broken”. But being the perfectionist that I am, I strive for an even “smarter” filter. If only there was a way to 1) keep the consistent cyclical trading effects as before 2)  ‘predict’ the next day close-to-open jump/fall in the Euro Stoxx50 index future as before, and 3) avoid volatile periods to eliminate erroneous trading, where the signal performed worse. After hours spent in iMetrica, I figured how to do it. This is where advanced trading signal engineering comes into play.

The first step was to include all the lower frequencies below .23, which were not included in my previous trading signal. Due to the low amount of activity in these lower frequencies, this should only provide the effect or a ‘lift’ or a ‘push’ or the signal locally, while still retaining the cyclical component. So after changing my $\Gamma$ to a low-pass filter with cutoff set at $\omega = .32$, I then computed the filter with the new lowpass design. The transfer functions for the filter coefficients are shown below in Figure 11, with the red colored plot the transfer function for the STXE. Notice that the transfer function for the explanatory series still privileges spectral peak between .23 and .32, with only a slight lift at frequency zero (compare this with the bandpass design in Figure 4, not much has changed).  The problem is that the peak exceeds 1.0 in the passband, and this will amplify the cyclical component extracted from the log-return. It might be okay, trading wise, but not what I’m looking to do.  For the STXE filter, we get slightly more of a lift at frequency zero, however this has been compensated with a decreased cycle extraction between frequencies .23 and .32.  Also, a slight amount of noise has entered in the stopband, another factor we must mollify.


#---- set Gamma to low-pass
cutoff&lt;-.32
Gamma&lt;-((0:K)&lt;(cutoff*K/pi))

#---- compute new filter ----------
i_mdfa_obj&lt;-IMDFA(L,i1,i2,cutoff,lambda,expweight,lambda_cross,lambda_decay,lambda_smooth,weight_func,Gamma,x)


Figure 11: The concurrent transfer functions after changing to lowpass filter.

To improve the concurrent filter properties for both, I increase the smoothing expweight to 26, which will in turn affect the lambda_smooth, so I decrease it to .70. This gives me a much better transfer function pair, shown in Figure 12.  Notice the peak in the explanatory series transfer function is now much closer to 1.0, exactly what we want.

Figure 12: The concurrent transfer functions after changing to lowpass filter, increasing expweight to 26, and decreasing lambda_smooth to .70.

I’m still not satisfied with the lift at frequency zero for the STXE series. At roughly .5 at frequency zero, the filter might not provide enough push or pull that I need. The only way to ensure a guaranteed lift in the STXE log-return series is to employ constraints on the filter coefficients so that the transfer function is one at frequency zero. This can be achieved by setting i1 to true in the IMDFA function call, which effectively ensures that the sum of the filter coefficients at $\omega = 0$ is one. After doing this, I get the following transfer functions and the respective filter coefficients.

#---- Update the regularization parameters
lambda_smooth&lt;-0.68
lambda_cross&lt;-0.0
lambda_decay&lt;-c(0.083,0.11)

#---- update customization parameters
lambda&lt;-0
expweight&lt;-28

#---- set filter constraint -------
i1&lt;-T
weight_constraint[1]&lt;-1


Figure 13: Transfer function and filter coefficients after setting the coefficient constraint i1 to true.

Now this is exactly what I was looking for. Not only does the transfer function for the explanatory series keep the important close-to-open cycle intact, but I have also enforced the lift I need for the STXE series. The coefficients still remain smooth with a nice decaying property at the end.  With the new filter coefficients, I then applied them to the data both in-sample and out-of-sample, yielding the trading signal shown in Figure 14.  It posses exactly the properties that I was seeking. The close-to-open cyclical component is still being extracted (thanks in part to the explanatory series), and is still relatively consistent, although not as much as the pure bandpass design. The feature that I like is the following: When the log-return data diverges away from the cyclical component, with increasing volatility, the STXE filter reacts by pushing the signal down to avoid any erroneous trading. This can be seen in observations 100 through 120 and then at observations 390 through the end of trading. Figure 15 (same as Figure 1 at the top of the article) show the resulting trades and performance produced in-sample and out-of-sample by this signal. This is the art of meticulous signal engineering folks.

Figure 14: In-sample and out-of-sample signal produced from the low-pass with i1 coefficient constraints.

With only two losses suffered out-of-sample during the roughly 9 days trading, the filter performs much more methodologically than before. Notice during the final two days trading, when volatility picked up, the signal ceases to trade as it is being pushed down. It even continues to ‘predict’ the close-to-open jump/fall correctly, such as at observations 288, 321, and 391. The last trade made was a sell/short sell position, with the signal trending down at the end. The filter is in position to make a huge gain from this timely signaling of a short position at 391, correctly determining a large fall the next trading day, and then waiting out the volatile trading. The gain should be large no matter what happens.

Figure 15: In-sample and out-of-sample performance of the i1 constrained filter design.

One thing I mention before concluding is that I made a slight adjustment to my filter design after employing the i1 constraint to get the results shown in Figure 13-15. I’ll leave this as an exercise for the reader to deduce what I have done. Hint: Look at the freezed degrees of freedom before and after applying the i1 constraint. If you still have trouble finding what I’ve done, email me and I’ll give you further hints.

### Conclusion

The overall performance of the first filter built, in regards to total return on investment out-of-sample, was superior to the second. However, this superior performance comes only with the assumption that the cycle component defined between frequencies .23 and .32 will continue to be present in future observations of STXE up until the expiration. If volatility increases and this intrinsic cycle ceases to exist in the log-return data, the performance will deteriorate.

For a better more comfortable approach that deals with changing volatile index conditions, I would opt for ensuring that the local-bias is present in the signal, This will effectively push or pull the signal down or up when the intrinsic cycle is weak in the increasing volatility, resulting in a pullback in trading activity.

As before, you can acquire the high-freq data used in this tutorial by requesting it via email.

Happy extracting!

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

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.

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

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).

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.

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 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 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 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 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 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.

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 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.

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.

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 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.

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 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 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 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!