More Robust Strategies

The previous article dealt with John Ehlers’ AM and FM demodulating technology for separating signal and noise in price curves. In the S&C June issue he described a practical example. Applying his FM demodulator makes a strategy noticeably more robust – at least with parameter optimization.

The simple example strategy is basically a short-term trend follower. The price curve is differentiated, then integrated again for removing the zero frequency part. The result is smoothed, and the system trades on zero crossings of its rate of change. Two parameters, the smoothing period and the rate-of-change period, are optimized. The code in C for Zorro:

void simple()
{
  int SigPeriod = optimize(8,6,14,1),
    ROCPeriod = optimize(1,1,6,1);
//Derivative of the price wave
  vars Deriv = series(priceClose(0)-priceClose(2));
//zeros at Nyquist and 2*Nyquist, i.e. Z3 = (1 + Z^-1)*(1 + Z^-2)
  vars Z3 = series(Sum(Deriv,4));
//Smooth Z3 for trading signal
  vars Signal = series(SMA(Z3,SigPeriod));
//Use Rate of Change to identify entry point
  vars Roc = series(Signal[0]-Signal[ROCPeriod]);
//If ROC Crosses Over 0 Then Buy Next Bar on Open;
//If Signal Crosses Under 0 Then Sell Next Bar on Open;
  if(crossOver(Roc,0))
    enterLong();
  else if(crossUnder(Signal,0))
    exitLong();
}

For replicating Ehlers’ results we use brute force in-sample optimization. This results in an impressive profit factor of almost 3 when applied to the S&P500 index (SPY) from 2009 to 2021. Still, the equity curve below shows that the main profit results from the last 2 years. So this is, as Ehlers admitted, not a very robust system.

Figure 1 – Simple system with no FM demodulator

Now we’re going to improve the system by inserting Ehler’s FM demodulator from the previous article. The code in C for Zorro:

void simpleFM()
{
  int i, SigPeriod = optimize(8,6,14,1),
  ROCPeriod = optimize(1,1,6,1);
//Derivative of the price wave
  vars Deriv = series(priceClose(0)-priceClose(2));
//FM filter: normalize Derive to half RMS and hard limit at +/- 1
  var RMS = 0;
  for(i=0; i<50; i++) RMS += Deriv[i]*Deriv[i];
  vars Clips = series(clamp(2*Deriv[0]/sqrt(RMS/50),-1,1));
//zeros at Nyquist and 2*Nyquist, i.e. Z3 = (1 + Z^-1)*(1 + Z^-2)
  vars Z3 = series(Sum(Clips,4));
//Smooth Z3 for trading signal
  vars Signal = series(SMA(Z3,SigPeriod));
//Use Rate of Change to identify entry point
  vars Roc = series(Signal[0]-Signal[ROCPeriod]);
//If ROC Crosses Over 0 Then Buy Next Bar on Open;
//If Signal Crosses Under 0 Then Sell Next Bar on Open;
  if(crossOver(Roc,0))
    enterLong();
  else if(crossUnder(Signal,0))
    exitLong();
}

The FM demodulator is basically the clamp() function, which removes the amplitude from the price curve derivative, in combination with the subsequent SMA, which acts as a lowpass filter. The result after brute force optimization:

Figure 2 – Simple system with FM demodulator

The profit factor is still about the same, slightly below 3, but the equity curve now looks a bit better and the total profit is higher. Ehlers attributed the increased robustness to the more linear parameter surface. For verifying that, we compare the 2D parameter charts from both versions:

Figure 3 – Parameter heatmap, no FM demodulator

Figure 4 – Parameter heatmap, with FM demodulator

The charts display the performance dependent on the parameter values on the y and x asis. We can see that the FM demodulator version has indeed the smoother heatmap with a top (green) in the center of the parameter ranges. The heatmap of the other system looks crumpled with a deep valley in the center and several tops apart from each other. Clearly the optimization of the FM demodulator system was more effective.

Further reading: John Ehlers, More Robust Strategies, Stocks & Commodities June 2021.

The script can be downloaded from the 2021 script repository. You will need Zorro S 2.37 or better for the brute force optimization and the 2D parameter charts.