The trading algorithm “Sell in August and buy back in October” could have been realized with the Zorro software in just 5 lines of C:

asset("SPY");
if(month() == 8 && tdm() == 1) // sell 1st trading day of August
  exitLong();
else if(month() == 10 && tdm() == 1) // buy back 1st trading day of October
  enterLong();

Alas, this seasonality trading system was apparently way too simplistic for Markos Katsanos. His version is a veritable monster with many, many trade entry and exit conditions. The trading script below is a 1:1 translation from his AmiBroker code to C for the Zorro platform. I only moved his VFI indicator – a variant of the On Balance Volume, but with a lot more signals – in a separate indicator function because I didn’t like trading logic cluttered with indicator code.

var priceAvg(int Offset) {
  return (priceC(Offset)+priceH(Offset)+priceL(Offset))/3;
}

var VFI(var Period,var Coef, var VCoef)
{
  vars Inters = series(log(priceAvg(0))-log(priceAvg(1)));
  var Vinter = StdDev(Inters,30);
  var Cutoff = Coef * Vinter * priceC();
  vars Volumes = series(marketVol());
  var Vave = SMA(Volumes+1,Period);
  var Vmax = Vave * VCoef;
  var VC = min(Volumes[0],Vmax);
  var MF = priceAvg(0)-priceAvg(1);
  vars VCPs = series(ifelse(MF > Cutoff,VC,ifelse(MF < -Cutoff,-VC,0)));
  var VFI1 = Sum(VCPs,Period)/Vave;
  return EMA(VFI1,3);
}

function run()
{
  StartDate = 2006;
  EndDate = 2022;
  BarPeriod = 1440; // 1 day
  LookBack = 150;

  assetList("AssetsIB");
  MaxLong = 1;
  Capital = 100000;
  Margin = Equity; // invest all you have
  Leverage = 1;
  BarZone = EST;
  Fill = 3; // enter/exit at next day open
  set(PARAMETERS,TESTNOW,PLOTNOW);

  asset("VIX");
  var VIXdn = (priceC(0)/HH(25,0)-1)*100;
  var VIXup = (priceC(0)/LL(25,0)-1)*100;

  asset("SPY");
  int SellMonth = optimize(8,5,8,1);
  var VIXupMax = optimize(60,50,60,10);
  var Crit = -optimize(20,15,20,5); //VFI SELL
  var K = optimize(1.5,1.3,1.7,.2); // ATR/VIX RATIO
  vars ATRs = series(ATR(15));
  var ATRDn = (ATRs[0]/MaxVal(ATRs,25)-1)*100;
  var ATRUp = (ATRs[0]/MinVal(ATRs,25)-1)*100;
  vars VFIs = series(VFI(130,0.2,2.5));
  vars SMAVFIs = series(SMA(VFIs,10));
  bool VolCondition = (VIXup < VIXupMax || ATRUp < K*VIXupMax ) && VFIs[0] > Crit;
  bool Buy = (month() >= 10 || month() < SellMonth) && ref(VolCondition,1) != 0;
  bool SellSeasonal = month() == SellMonth ; //SEASONAL
  bool SellVolatility = VIXup > 2*VIXupMax ; //VOLATILITY EXIT
  bool SellMF = crossUnder(VFIs,Crit) && SMAVFIs[0] < SMAVFIs[1] ;
  bool Sell = SellSeasonal || ref(SellVolatility,1) != 0 || ref(SellMF,1) != 0;

  if(Sell)
    exitLong();
  else if(Buy)
    enterLong();
}

Phew. And who said that you need at least 30 historical trades per optimized parameter? We’re optimizing the heck out of sell month, threshold, ratio, and other parameters, and won’t care that this strategy trades only once per year. All is in-sample of course, since the low number of trades prevents walk-forward optimization. As can be expected, our reward is a splendid backtest:

We can see that the system managed to avoid the 2008 market drop, since it doubtlessly knew it in advance due to the in-sample optimization. So take the result with a grain – or better, a bag – of salt. For the audacious experimenter, the VFI indicator and the seasonality trading system can be downloaded from the 2022 script repository. You need Zorro S 2.47 or above for supporting volume and for the ref macro. And since the question recently came up: you can find the script repositories under “Links & Download” on the right side of this page.

Developing a model-based strategy begins with the market inefficiency that you want to exploit. The inefficiency produces a price anomaly or price pattern that you can describe with a qualitative or quantitative model. Such a model predicts the current price y_t from the previous price y_t-1 plus some function f of a limited number of previous prices plus some noise term ε:

[pmath size=16]y_t ~=~ y_{t-1} + f(y_{t-1},~…, ~y_{t-n}) + epsilon[/pmath]

The time distance between the prices y_t is the time frame of the model; the number n of prices used in the function f is the lookback period of the model. The higher the predictive f term in relation to the nonpredictive ε term, the better is the strategy. Some traders claim that their favorite method does not predict, but ‘reacts on the market’ or achieves a positive return by some other means. On a certain trader forum you can even encounter a math professor who re-invented the grid trading system, and praised it as non-predictive and even able to trade a random walk curve. But systems that do not predict y_t in some way must rely on luck; they only can redistribute risk, for instance exchange a high risk of a small loss for a low risk of a high loss. The profit expectancy stays negative. As far as I know, the professor is still trying to sell his grid trader, still advertising it as non-predictive, and still regularly blowing his demo account with it. 

Trading by throwing a coin loses the transaction costs. But trading by applying the wrong model – for instance, trend following to a mean reverting price series – can cause much higher losses. The average trader indeed loses more than by random trading (about 13 pips per trade according to FXCM statistics). So it’s not sufficient to have a model; you must also prove that it is valid for the market you trade, at the time you trade, and with the used time frame and lookback period.

Not all price anomalies can be exploited. Limiting stock prices to 1/16 fractions of a dollar is clearly an inefficiency, but it’s probably difficult to use it for prediction or make money from it. The working model-based strategies that I know, either from theory or because we’ve been contracted to code some of them, can be classified in several categories. The most frequent are:

1. Trend 

Momentum in the price curve is probably the most significant and most exploited anomaly. No need to elaborate here, as trend following was the topic of a whole article series on this blog. There are many methods of trend following, the classic being a moving average crossover. This ‘hello world’ of strategies (here the scripts in R and in C) routinely fails, as it does not distinguish between real momentum and random peaks or valleys in the price curve.

The problem: momentum does not exist in all markets all the time. Any asset can have long non-trending periods. And contrary to popular belief this is not necessarily a ‘sidewards market’. A random walk curve can go up and down and still has zero momentum. Therefore, some good filter that detects the real market regime is essential for trend following systems. Here’s a minimal Zorro strategy that uses a lowpass filter for detecting trend reversal, and the MMI indicator for determining when we’re entering trend regime:

function run()
{
  vars Price = series(price());
  vars Trend = series(LowPass(Price,500));
	
  vars MMI_Raw = series(MMI(Price,300));
  vars MMI_Smooth = series(LowPass(MMI_Raw,500));
	
  if(falling(MMI_Smooth)) {
    if(valley(Trend))
      reverseLong(1);
    else if(peak(Trend))
      reverseShort(1);
  }
}

The profit curve of this strategy:

(For the sake of simplicity all strategy snippets on this page are barebone systems with no exit mechanism other than reversal, and no stops, trailing, parameter training, money management, or other gimmicks. Of course the backtests mean in no way that those are profitable systems. The P&L curves are all from EUR/USD, an asset good for demonstrations since it seems to contain a little bit of every possible inefficiency). 

2. Mean reversion

A mean reverting market believes in a ‘real value’ or ‘fair price’ of an asset. Traders buy when the actual price is cheaper than it ought to be in their opinion, and sell when it is more expensive. This causes the price curve to revert back to the mean more often than in a random walk. Random data are mean reverting 75% of the time (proof here), so anything above 75% is caused by a market inefficiency. A model:

[pmath size=16]y_t ~=~ y_{t-1} ~-~ 1/{1+lambda}(y_{t-1}- hat{y}) ~+~ epsilon[/pmath]

[pmath size=16]y_t[/pmath] = price at bar t
[pmath size=16]hat{y}[/pmath] = fair price
[pmath size=16]lambda[/pmath] = half-life factor
[pmath size=16]epsilon[/pmath] = some random noise term

The higher the half-life factor, the weaker is the mean reversion. The half-life of mean reversion in price series ist normally in the range of  50-200 bars. You can calculate λ by linear regression between y_t-1 and (y_t-1-y_t). The price series need not be stationary for experiencing mean reversion, since the fair price is allowed to drift. It just must drift less as in a random walk. Mean reversion is usually exploited by removing the trend from the price curve and normalizing the result. This produces an oscillating signal that can trigger trades when it approaches a top or bottom. Here’s the script of a simple mean reversion system:

function run()
{
  vars Price = series(price());
  vars Filtered = series(HighPass(Price,30));
  vars Signal = series(FisherN(Filtered,500));
  var Threshold = 1.0;

  if(Hurst(Price,500) < 0.5) { // do we have mean reversion?
    if(crossUnder(Signal,-Threshold))
      reverseLong(1); 
    else if(crossOver(Signal,Threshold))
      reverseShort(1);
  }
} 

The highpass filter dampens all cycles above 30 bars and thus removes the trend from the price curve. The result is normalized by the Fisher transformation which produces a Gaussian distribution of the data. This allows us to determine fixed thresholds at 1 and -1 for separating the tails from the resulting bell curve. If the price enters a tail in any direction, a trade is triggered in anticipation that it will soon return into the bell’s belly. For detecting mean reverting regime, the script uses the Hurst Exponent. The exponent is 0.5 for a random walk. Above 0.5 begins momentum regime and below 0.5 mean reversion regime.

3. Statistical Arbitrage

Strategies can exploit the similarity between two or more assets.  This allows to hedge the first asset by a reverse position in the second asset, and this way derive profit from mean reversion of their price difference:

[pmath size=16]y ~=~ h_1 y_1 – h_2 y_2[/pmath]

where y₁ and y₂ are the prices of the two assets and the multiplication factors h₁ and h₂ their hedge ratios. The hedge ratios are calculated in a way that the mean of the difference y is zero or a constant value. The simplest method for calculating the hedge ratios is linear regression between y₁ and y₂. A mean reversion strategy as above can then be applied to y.  

The assets need not be of the same type; a typical arbitrage system can be based on the price difference between an index ETF and its major stock. When y is not stationary – meaning that its mean tends to wander off slowly – the hedge ratios must be adapted in real time for compensating. Here is a proposal using a Kalman Filter by a fellow blogger.

The simple arbitrage system from the R tutorial:

require(quantmod)

symbols <- c("AAPL", "QQQ")
getSymbols(symbols)

#define training set
startT  <- "2007-01-01"
endT    <- "2009-01-01"
rangeT  <- paste(startT,"::",endT,sep ="")
tAAPL   <- AAPL[,6][rangeT]
tQQQ   <- QQQ[,6][rangeT]
 
#compute price differences on in-sample data
pdtAAPL <- diff(tAAPL)[-1]
pdtQQQ <- diff(tQQQ)[-1]
 
#build the model
model  <- lm(pdtAAPL ~ pdtQQQ - 1)
 
#extract the hedge ratio (h1 is assumed 1)
h2 <- as.numeric(model$coefficients[1])

#spread price (in-sample)
spreadT <- tAAPL - h2 * tQQQ
 
#compute statistics of the spread
meanT    <- as.numeric(mean(spreadT,na.rm=TRUE))
sdT      <- as.numeric(sd(spreadT,na.rm=TRUE))
upperThr <- meanT + 1 * sdT
lowerThr <- meanT - 1 * sdT
 
#run in-sample test
spreadL  <- length(spreadT)
pricesB  <- c(rep(NA,spreadL))
pricesS  <- c(rep(NA,spreadL))
sp       <- as.numeric(spreadT)
tradeQty <- 100
totalP   <- 0

for(i in 1:spreadL) {
     spTemp <- sp[i]
     if(spTemp < lowerThr) {
        if(totalP <= 0){
           totalP     <- totalP + tradeQty
           pricesB[i] <- spTemp
        }
     } else if(spTemp > upperThr) {
       if(totalP >= 0){
          totalP <- totalP - tradeQty
          pricesS[i] <- spTemp
       }
    }
}

4. Price constraints

A price constraint is an artificial force that causes a constant price drift or establishes a price range, floor, or ceiling. The most famous example was the EUR/CHF price cap mentioned in the first part of this series. But even after removal of the cap, the EUR/CHF price has still a constraint, this time not enforced by the national bank, but by the current strong asymmetry in EUR and CHF buying power. An extreme example of a ranging price is the EUR/DKK pair (see below). All such constraints can be used in strategies to the trader’s advantage. 

5. Cycles

Non-seasonal cycles are caused by feedback from the price curve. When traders believe in a ‘fair price’ of an asset, they often sell or buy a position when the price reaches a certain distance from that value, in hope of a reversal. Or they close winning positions when the favorite price movement begins to decelerate. Such effects can synchronize entries and exits among a large number of traders, and cause the price curve to oscillate with a period that is stable over several cycles. Often many such cycles are superposed on the curve, like this:

[pmath size=16]y_t ~=~ hat{y}  ~+~ sum{i}{}{a_i sin(2 pi t/C_i+D_i)} ~+~ epsilon[/pmath]

When you know the period C_i and the phase D_i of the dominant cycle, you can calculate the optimal trade entry and exit points as long as the cycle persists. Cycles in the price curve can be detected with spectral analysis functions – for instance, fast Fourier transformation (FFT) or simply a bank of narrow bandpass filters. Here is the frequency spectrum of the EUR/USD in October 2015:

Exploiting cycles is a little more tricky than trend following or mean reversion. You need not only the cycle length of the dominant cycle of the spectrum, but also its phase (for triggering trades at the right moment) and its amplitude (for determining if there is a cycle worth trading at all). This is a barebone script:

function run()
{
  vars Price = series(price());
  var Phase = DominantPhase(Price,10);
  vars Signal = series(sin(Phase+PI/4));
  vars Dominant = series(BandPass(Price,rDominantPeriod,1));
  ExitTime = 10*rDominantPeriod;
  var Threshold = 1*PIP;
	
  if(Amplitude(Dominant,100) > Threshold) {
    if(valley(Signal))
      reverseLong(1); 
    else if(peak(Signal))
      reverseShort(1);
  }
}

The DominantPhase function determines both the phase and the cycle length of the dominant peak in the spectrum; the latter is stored in the rDominantPeriod variable. The phase is converted to a sine curve that is shifted ahead by π/4. With this trick we’ll get a sine curve that runs ahead of the price curve. Thus we do real price prediction here, only question is if the price will follow our prediction. This is determined by applying a bandpass filter centered at the dominant cycle to the price curve, and measuring its amplitude (the a_i in the formula). If the amplitude is above a threshold,  we conclude that we have a strong enough cycle. The script then enters long on a valley of the run-ahead sine curve, and short on a peak. Since cycles are shortlived, the duration of a trade is limited by ExitTime to a maximum of 10 cycles. 

We can see from the P&L curve that there were long periods in 2012 and 2014 with no strong cycles in the EUR/USD price curve. 

6. Clusters

The same effect that causes prices to oscillate can also let them cluster at certain levels. Extreme clustering can even produce “supply” and “demand” lines (also known as “support and resistance“), the favorite subjects in trading seminars. Expert seminar lecturers can draw support and resistance lines on any chart, no matter if it’s pork belly prices or last year’s baseball scores. However the mere existence of those lines remains debatable: There are few strategies that really identify and exploit them, and even less that really produce profits. Still, clusters in price curves are real and can be easily identified in a histogram similar to the cycles spectogram.

7. Curve patterns

They arise from repetitive behavior of traders. Traders not only produce, but also believe in many curve patterns; most – such as the famous ‘head and shoulders’ pattern that is said to predict trend reversal – are myths (at least I have not found any statistical evidence of it, and heard of no other any research that ever confirmed the existence of predictive heads and shoulders in price curves). But some patterns, for instance “cups” or “half-cups”, really exist and can indeed precede an upwards or downwards movement. Curve patterns – not to be confused with candle patterns – can be exploited by pattern-detecting methods such as the Fréchet algorithm. 

A special variant of a curve pattern is the Breakout – a sudden momentum after a long sidewards movement. Is can be caused, for instance, by trader’s tendency to place their stop losses as a short distance below or above the current plateau. Triggering the first stops then accelerates the price movement until more and more stops are triggered. Such an effect can be exploited by a system that detects a sidewards period and then lies in wait for the first move in any direction.

8. Seasonality

“Season” does not necessarily mean a season of a year. Supply and demand can also follow monthly, weekly, or daily patterns that can be detected and exploited by strategies. For instance, the S&P500 index is said to often move upwards in the first days of a month, or to show an upwards trend in the early morning hours before the main trading session of the day. Since seasonal effects are easy to exploit, they are often shortlived, weak, and therefore hard to detect by just eyeballing price curves. But they can be found by plotting a day, week, or month profile of average price curve differences.

9. Gaps

When market participants contemplate whether to enter or close a position, they seem to come to rather similar conclusions when they have time to think it over at night or during the weekend. This can cause the price to start at a different level when the market opens again. Overnight or weekend price gaps are often more predictable than price changes during trading hours. And of course they can be exploited in a strategy. On the Zorro forum was recently a discussion about the “One Night Stand System“, a simple currency weekend-gap trader with mysterious profits.

10. Autoregression and heteroskedasticity

The latter is a fancy word for: “Prices jitter a lot and the jittering varies over time”. The ARIMA and GARCH models are the first models that you encounter in financial math. They assume that future returns or future volatility can be determined with a linear combination of past returns or past volatility. Those models are often considered purely theoretical and of no practical use. Not true: You can use them for predicting tomorrow’s price just as any other model. You can examine a correlogram – a statistics of the correlation of the current return with the returns of the previous bars – for finding out if an ARIMA model fits to a certain price series. Here are two excellent articles by fellow bloggers for using those models in trading strategies:  ARIMA+GARCH Trading Strategy on the S&P500 and Are ARIMA/GARCH Predictions Profitable?

11. Price shocks

Price shocks often happen on Monday or Friday morning when companies or organizations publish good or bad news that affect the market. Even without knowing the news, a strategy can detect the first price reactions and quickly jump onto the bandwagon. This is especially easy when a large shock is shaking the markets. Here’s a simple Forex portfolio strategy that evaluates the relative strengths of currencies for detecting price shocks:

function run()
{
  BarPeriod = 60;
  ccyReset();
  string Name;
  while(Name = (loop(Assets)))
  {
    if(assetType(Name) != FOREX) 
      continue; // Currency pairs only
    asset(Name);
    vars Prices = series(priceClose());
    ccySet(ROC(Prices,1)); // store price change as strength
  }
  
// get currency pairs with highest and lowest strength difference
  string Best = ccyMax(), Worst = ccyMin();
  var Threshold = 1.0; // The shock level

  static char OldBest[8], OldWorst[8];	// static for keeping contents between runs
  if(*OldBest && !strstr(Best,OldBest)) { // new strongest asset?
    asset(OldBest);
    exitLong();
    if(ccyStrength(Best) > Threshold) {
      asset(Best);
      enterLong();
    }
  } 
  if(*OldWorst && !strstr(Worst,OldWorst)) { // new weakest asset?
    asset(OldWorst);
    exitShort();
    if(ccyStrength(Worst) < -Threshold) {
      asset(Worst);
      enterShort();
    }
  }

// store previous strongest and weakest asset names  
  strcpy(OldBest,Best);
  strcpy(OldWorst,Worst);
}

The equity curve of the currency strength system (you’ll need Zorro 1.48 or above):

The blue equity curve above reflects profits from small and large jumps of currency prices. You can clearly identify the Brexit and the CHF price cap shock. Of course such strategies would work even better if the news could be early detected and interpreted in some way. Some data services provide news events with a binary valuation, like “good” or “bad”. Especially of interest are earnings reports, as provided by data services such as Zacks or Xignite. Depending on which surprises the earnings report contains, stock prices and implied volatilities can rise or drop sharply at the report day, and generate quick profits.

For learning what can happen when news are used in more creative ways, I recommend the excellent Fear Index by Robert Harris – a mandatory book in any financial hacker’s library.

This was the second part of the Build Better Strategies series. The third part will deal with the process to develop a model-based strategy, from inital research up to building the user interface. In case someone wants to experiment with the code snippets posted here, I’ve added them to the 2015 scripts repository. They are no real strategies, though. The missing elements – parameter optimization, exit algorithms, money management etc. – will be the topic of the next part of the series.

⇒ Build Better Strategies – Part 3