Linear Regression Curves vs Bollinger Bands

In my last post I showed what a linear regression curve was, this post will use it as part of a mean reverting trading strategy.

The strategy is simple:

  • Calculate a rolling ‘average’ and a rolling ‘deviation’
  • If the Close price is greater than the average+n*deviation go short (and close when you cross the mean)
  • If the Close price is less than the average-n*deviation go long (and close when you cross the mean)

Two cases will be analysed, one strategy will use a simple moving average(SMA), the other will use the linear regression curve(LRC) for the average. The deviation function will be Standard Devation, Average True Range, and LRCDeviation (same as standard deviation but replace the mean with the LRC).

Results (Lookback = 20 and Deviation Multiplier = 2:

mean reversion linear regression curves

Annualized Sharpe Ratio (Rf=0%)

  • GSPC = 0.05257118
  • Simple Moving Avg – Standard Deviation = 0.2535342
  • Simple Moving Avg – Average True Range = 0.1165512
  • Simple Moving Avg – LRC Deviation 0.296234
  • Linear Regression Curve – Standard Deviation = 0.2818447
  • Linear Regression Curve – Average True Range = 0.5824727
  • Linear Regression Curve – LRC Deviation = 0.04672071

Optimisation analysis:

Annoyingly the colour scale is different between the two charts, however the sharpe ratio is written in each cell. Lighter colours indicate better performance.

Over a 13year period and trading the GSPC the LRC achieved a sharpe of ~0.6 where as the SMA achieved a sharpe of ~0.3. The LRC appears superior to the SMA.

Mean Reversion LRC STDEV Mean Reversion SMA STDEVI will update this post at a later point in time when my optimisation has finished running for the other strategies.

?View Code RSPLUS
marketSymbol <- "^GSPC"
nLookback <- 20 #The lookback to calcute the moving average / linear regression curve / average true range / standard deviation
nDeviation <- 2
#Specify dates for downloading data, training models and running simulation
startDate = as.Date("2000-01-01") #Specify what date to get the prices from
symbolData <- new.env() #Make a new environment for quantmod to store data in
stockCleanNameFunc <- function(name){
getSymbols(marketSymbol, env = symbolData, src = "yahoo", from = startDate)
cleanName <- stockCleanNameFunc(marketSymbol)
mktData <- get(cleanName,symbolData)
linearRegressionCurve <- function(data,n){
    regression <- function(dataBlock){
           fit <-lm(dataBlock~seq(1,length(dataBlock),1))
    return (rollapply(data,width=n,regression,align="right",by.column=FALSE,na.pad=TRUE))
linearRegressionCurveStandardDeviation <- function(data,n){
    deviation <- function(dataBlock){
        fit <-lm(dataBlock~seq(1,length(dataBlock),1))
        quasiMean <- (last(fit$fitted.values))
        quasiMean <- rep(quasiMean,length(dataBlock))
        stDev <- sqrt((1/length(dataBlock))* sum((dataBlock - quasiMean)^2))
        return (stDev)
    return (rollapply(data,width=n,deviation,align="right",by.column=FALSE,na.pad=TRUE))
reduceLongTradeEntriesToTradOpenOrClosedSignal <- function(trades){
    #Takes something like
    #000011110000-1-1000011 (1 = go long, -1 = go short)
    #and turns it into
    #trades[] <- 0
    out <- trades #copy the datastructure over
    currentPos <-0
    for(i in 1:length(out[,1])){
      if((currentPos == 0) & (trades[i,1]==1)){
        currentPos <- 1
        out[i,1] <- currentPos
      if((currentPos == 1) & (trades[i,1]==-1)){
        currentPos <- 0
        out[i,1] <- currentPos
      out[i,1] <- currentPos
reduceShortTradeEntriesToTradOpenOrClosedSignal <- function(trades){
generateTradingReturns <- function(mktPrices, nLookback, nDeviation, avgFunction, deviationFunction,title,showGraph=TRUE){
    quasiMean <- avgFunction(mktPrices,n=nLookback)
    quasiDeviation <- deviationFunction(mktPrices,n=nLookback)
    colnames(quasiMean) <- "QuasiMean"
    colnames(quasiDeviation) <- "QuasiDeviation"
    price <- Cl(mktPrices)
    upperThreshold = quasiMean + nDeviation*quasiDeviation
    lowerThreshold = quasiMean - nDeviation*quasiDeviation
    aboveUpperBand <- price>upperThreshold
    belowLowerBand <- price<lowerThreshold
    aboveMAvg <- price>quasiMean
    belowMAvg <- price<quasiMean
    rawShort <- (-1)*aboveUpperBand+belowMAvg
    shortPositions <- reduceShortTradeEntriesToTradOpenOrClosedSignal(rawShort)
    rawLong <- (-1)*aboveMAvg+belowLowerBand
    longPositions <- reduceLongTradeEntriesToTradOpenOrClosedSignal(rawLong)
    positions = longPositions + shortPositions
    signal <- positions
      plot(Cl(mktPrices),type="l",main=paste(marketSymbol, "close prices"))
      legend('bottomright',c("Close",paste("Band - ",title),paste("Average - ",title)),lty=1, col=c('black', 'red', 'blue'), bty='n', cex=.75)
    mktReturns <- Cl(mktPrices)/Lag(Cl(mktPrices)) - 1
    tradingReturns <- Lag(signal)*mktReturns
    tradingReturns[] <- 0
    colnames(tradingReturns) <- title
    return (tradingReturns)
strategySMAandSTDEV <- function(mktData,nLookback,nDeviation){
       generateTradingReturns(mktData,nLookback,nDeviation,function(x,n) { SMA(Cl(x),n) },function(x,n) { rollapply(Cl(x),width=n, align="right",sd) },"Simple Moving Avg - Standard Deviation",FALSE)
strategySMAandATR <- function(mktData,nLookback,nDeviation){
       generateTradingReturns(mktData,nLookback,nDeviation,function(x,n) { SMA(Cl(x),n) },function(x,n) { atr <- ATR(x,n); return(atr$atr) },"Simple Moving Avg - Average True Range",FALSE)
strategySMAandLRCDev <- function(mktData,nLookback,nDeviation){
        generateTradingReturns(mktData,nLookback,nDeviation,function(x,n) { SMA(Cl(x),n) },function(x,n) { linearRegressionCurveStandardDeviation(Cl(x),n) },"Simple Moving Avg - LRC Deviation",FALSE)
strategyLRCandSTDEV <- function(mktData,nLookback,nDeviation){
       generateTradingReturns(mktData,nLookback,nDeviation,function(x,n) { linearRegressionCurve(Cl(x),n) },function(x,n) { rollapply(Cl(x),width=n, align="right",sd) },"Linear Regression Curve - Standard Deviation",FALSE)
strategyLRCandATR <- function(mktData,nLookback,nDeviation){
       generateTradingReturns(mktData,nLookback,nDeviation,function(x,n) { linearRegressionCurve(Cl(x),n) },function(x,n) { atr <- ATR(x,n); return(atr$atr) },"Linear Regression Curve - Average True Range",FALSE)
strategyLRCandLRCDev <- function(mktData,nLookback,nDeviation){
       generateTradingReturns(mktData,nLookback,nDeviation,function(x,n) { linearRegressionCurve(Cl(x),n) },function(x,n) { linearRegressionCurveStandardDeviation(Cl(x),n) },"Linear Regression Curve - LRC Deviation",FALSE)
bollingerBandsSMAandSTDEVTradingReturns <- strategySMAandSTDEV(mktData,nLookback,nDeviation)
bollingerBandsSMAandATRTradingReturns <- strategySMAandATR(mktData,nLookback,nDeviation)
bollingerBandsSMAandLRCDevTradingReturns <- strategySMAandLRCDev(mktData,nLookback,nDeviation)
bollingerBandsLRCandSTDEVTradingReturns <- strategyLRCandSTDEV(mktData,nLookback,nDeviation)
bollingerBandsLRCandATRTradingReturns <- strategyLRCandATR(mktData,nLookback,nDeviation)
bollingerBandsLRCandLRCDevTradingReturns <- strategyLRCandLRCDev(mktData,nLookback,nDeviation)
mktClClRet <- Cl(mktData)/Lag(Cl(mktData))-1
tradingReturns <- merge(as.zoo(mktClClRet),
charts.PerformanceSummary(tradingReturns,main=paste("Mean Reversion using nLookback",nLookback,"and nDeviation",nDeviation,"bands"),geometric=FALSE)
cat("Sharpe Ratio")
colorFunc <- function(x){
  x <- max(-4,min(4,x))
  if(x > 0){
  colorFunc <- rgb(0,(255*x/4)/255 , 0/255, 1)
  } else {
  colorFunc <- rgb((255*(-1*x)/4)/255,0 , 0/255, 1)
optimiseTradingStrat <- function(mktData,lookbackStart,lookbackEnd,lookbackStep,deviationStart,deviationEnd,deviationStep,strategy,title){
      lookbackRange <- seq(lookbackStart,lookbackEnd,lookbackStep)
      deviationRange <- seq(deviationStart,deviationEnd,deviationStep)
      combinations <- length(lookbackRange)*length(deviationRange)
      combLookback <- rep(lookbackRange,each=combinations/length(lookbackRange))
      combDeviation <- rep(deviationRange,combinations/length(deviationRange))
      optimisationMatrix <- t(rbind(t(combLookback),t(combDeviation),rep(NA,combinations),rep(NA,combinations),rep(NA,combinations)))
      colnames(optimisationMatrix) <- c("Lookback","Deviation","SharpeRatio","CumulativeReturns","MaxDrawDown")
        for(i in 1:length(optimisationMatrix[,1])){
            print(paste("On run",i,"out of",length(optimisationMatrix[,1]),"nLookback=",optimisationMatrix[i,"Lookback"],"nDeviation=",optimisationMatrix[i,"Deviation"]))
            runReturns <- strategy(mktData,optimisationMatrix[i,"Lookback"],optimisationMatrix[i,"Deviation"])
            optimisationMatrix[i,"SharpeRatio"] <- SharpeRatio.annualized(runReturns)
            optimisationMatrix[i,"CumulativeReturns"] <- sum(runReturns)
            optimisationMatrix[i,"MaxDrawDown"] <-  maxDrawdown(runReturns,geometric=FALSE)

          z <- matrix(optimisationMatrix[,"SharpeRatio"],nrow=length(lookbackRange),ncol=length(deviationRange),byrow=TRUE)
          colors <- colorFunc(optimisationMatrix[,"SharpeRatio"])
          rownames(z) <- lookbackRange
          colnames(z) <-deviationRange
          heatmap.2(z, key=TRUE,trace="none",cellnote=round(z,digits=2),Rowv=NA, Colv=NA, scale="column", margins=c(5,10),xlab="Deviation",ylab="Lookback",main=paste("Sharpe Ratio for Strategy",title))
  plot(Cl(mktData),type="l",main=paste(marketSymbol, "close prices"))
  legend('bottomright',c("Close",paste("Simple Moving Average Lookback=50"),paste("Linear Regression Curve Lookback=50")),lty=1, col=c('black', 'red', 'blue'), bty='n', cex=.75)
nLookbackStart <- 20
nLookbackEnd <- 200
nLookbackStep <- 20
nDeviationStart <- 1
nDeviationEnd <- 2.5
nDeviationStep <- 0.1
#optimiseTradingStrat(mktData,nLookbackStart,nLookbackEnd,nLookbackStep,nDeviationStart,nDeviationEnd,nDeviationStep,strategySMAandSTDEV,"AvgFunc=SMA and DeviationFunc=STDEV")
#optimiseTradingStrat(mktData,nLookbackStart,nLookbackEnd,nLookbackStep,nDeviationStart,nDeviationEnd,nDeviationStep,strategySMAandATR,"AvgFunc=SMA and DeviationFunc=ATR")
#optimiseTradingStrat(mktData,nLookbackStart,nLookbackEnd,nLookbackStep,nDeviationStart,nDeviationEnd,nDeviationStep,strategySMAandLRCDev,"AvgFunc=SMA and DeviationFunc=LRCDev")
#optimiseTradingStrat(mktData,nLookbackStart,nLookbackEnd,nLookbackStep,nDeviationStart,nDeviationEnd,nDeviationStep,strategyLRCandSTDEV,"AvgFunc=LRC and DeviationFunc=STDEV")
#optimiseTradingStrat(mktData,nLookbackStart,nLookbackEnd,nLookbackStep,nDeviationStart,nDeviationEnd,nDeviationStep,strategyLRCandATR,"AvgFunc=LRC and DeviationFunc=ATR")
#doptimiseTradingStrat(mktData,nLookbackStart,nLookbackEnd,nLookbackStep,nDeviationStart,nDeviationEnd,nDeviationStep,strategyLRCandLRCDev,"AvgFunc=LRC and DeviationFunc=LRCDev")
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High Probability Credit Spreads – Using Linear Regression Curves

I came across this video series over the weekend, an option trader discusses how he trades credit spreads (mainly looks for mean reversion). Most of you will be familiar with bollinger bands as a common mean reversion strategy, essentially you take the moving average and moving standard deviation of the stock. You then plot on to your chart the moving average and an upper and lower band(moving average +/- n*standard deviations).

It is assumed that the price will revert to the moving average hence any price move to the bands is a good entry point. A common problem with this strategy is that the moving average is a LAGGING indicator and is often very slow to track the price moves if a long lookback period is used.

Video 1 presents a technique called “linear regression curves” about 10mins in. Linear regression curves aim to solve the problem of the moving average being slow to track the price.

Linear Regression Curve vs Simple Moving Average

demo of linear regression curve good tracking


See how tightly the blue linear regression curve follows the close price, it’s significantly quicker to identify turns in the market where as the simple moving average has considerable tracking error. The MSE could be taken to quantify the tightness.

How to calculate the linear regression curve:

linear regression diagram

In this example you have 100 closing prices for your given stock. Bar 1 is the oldest price, bar 100 is the most recent price. We will use a 20day regression.

1. Take prices 1-20 and draw the line of best fit through them
2. At the end of your best fit line (so bar 20), draw a little circle
3. Take prices 2-21 and draw the line of best fit through them
4. At the end of your best fit line (so bar 21) draw a little circle
5. Repeat upto bar 100
6. Join all of your little circles, this is your ‘linear regression curve’
So in a nutshell you just join the ends of a rolling linear regression.

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Genetic Algorithm in R – Trend Following

This post is going to explain what genetic algorithms are, it will also present R code for performing genetic optimisation.

A genetic algo consists of three things:

  1. A gene
  2. A fitness function
  3. Methods to breed/mate genes

The Gene

The gene is typically a binary number, each bit in the binary number controls various parts of your trading strategy. The gene below contains 4 sub gene, a stock gene to select what stock to trade, a strategy gene to select what strategy to use, paramA sets a parameter used in your strategy and paramB sets another parameter to use in your strategy.

Gene = [StockGene,StrategyGene,ParamA,ParamB]

Stock Gene
00 Google
01 Facebook
10 IBM
11 LinkedIn

Strategy Gene
0 Simple Moving Average
1 Exponential Moving Average

ParamA Gene – Moving Average 1 Lookback
00 10
01 20
10 30
11 40

ParamB Gene – Moving Average 2 Lookback
00 15
01 25
10 35
11 45

So Gene = [01,1,00,11]

Would be stock=Facebook, strategy=Exponential Moving Average,paramA=10,paramB=45].

The strategy rules are simple, if the moving average(length=paramA) > moving average(length=paramB) then go long, and vice versa.

The fitness function

A gene is quantified as a good or bad gene using a fitness function. The success of a genetic trading strategy depends heavily upon your choice of fitness function and whether it makes sense with the strategies you intend to use. You will trade each of the strategies outlined by your active genes and then rank them by their fitness. A good starting point would be to use the sharp ratio as the fitness function.

You need to be careful that you apply the fitness function to statistically significant data. For example if you used a mean reverting strategy that might trade once a month (or what ever your retraining window is), then your fitness is determined by 1 or 2 datapoints!!! This will result in poor genetic optimisation (in my code i’ve commented out a mean reversion strategy test for yourself). Typically what happens is your sharpe ratio from 2 datapoints is very very high merely down to luck. You then mark this as a good gene and trade it the next month with terrible results.

Breeding Genes

With a genetic algo you need to breed genes, for the rest of this post i’ll assume you are breeding once a month. During breeding you take all of the genes in your gene pool and rank them according to the fitness function. You then select the top N genes and breed them (discard all the other genes they’re of no use).

Breeding consists of two parts:

Hybridisation – Take a gene and cut a chunk out of it, you can use whatever random number generator you want to determine the cut locations, swap this chunk with a corresponding chunk from another gene.

Old gene: 00110010 and 11100110 (red is the randomly select bits to cut)
New gene: 00100110 and 11110010

You do this for every possible pair of genes in your top N list.

Mutation – After hybridisation go through all your genes and randomly flip the bits with an fixed probability. The mutation prevents your strategy from getting locked into an every shrinking gene pool.

For a more detailed explanation with diagrams please see: scroll down to Genetic Algorithms and its Application in Trading

genetic algo sharpe 1.14

Annualized Sharpe Ratio (Rf=0%) 1.15

On to the code:

?View Code RSPLUS
topNToSelect <- 5   #Top n genes are selected during the mating, these will be mated with each other
mutationProb <- 0.05 #A mutation can occur during the mating, this is the probability of a mutation for individual chromes
symbolLst <- c("^GDAXI","^FTSE","^GSPC","^NDX","AAPL","ARMH","JPM","GS")
#Stock gene
stockGeneLength <- 3 #8stocks
#stockGeneLength<-6 #Allows 2^6 stocks (64)
#Strategy gene
#Paramter lookback gene
#Calculate the length of our chromozone, chromozone=[gene1,gene2,gene3...]
chromozoneLength <- stockGeneLength+strateyGeneLength+parameterLookbackGeneLength
signalMACross <- function(mktdata, paramA, paramB, avgFunc=SMA){
 signal = avgFunc(mktdata,n=paramA)/avgFunc(mktdata,n=paramB)
 signal[] <- 0
 signal <- (signal>1)*1 #converts bools into ints
 signal[signal==0] <- (-1)
 return (signal)
signalBollingerReversion <- function(mktdata, paramA, paramB){
  avg <- SMA(mktdata,paramB)
  std <- 1*rollapply(mktdata, paramB,sd,align="right")
  shortSignal <- (mktdata > avg+std)*-1
  longSignal <- (mktdata < avg-std)*1
  signal <- shortSignal+longSignal
  return (signal)
signalRSIOverBoughtOrSold <- function(mktdata, paramA, paramB){
  upperLim <- min(60*(1+paramB/100),90)
  lowerLim <- max(40*(1-paramB/100),10)
  rsisignal <- RSI(mktdata,paramB)
  signal <- ((rsisignal>upperLim)*-1)+((rsisignal<lowerLim)*1)
  return (signal)
#Gene = [StockGene,StrategyGene,ParamAGene,ParamBGene]
#The following functions extract specific parts of the gene
getStockGeneFromChromozone <- function(chrome){
getStrategyGeneFromChromozone <- function(chrome){
getParameterLookbackGeneFromChromozone <- function(chrome){
#Once parts of the gene have been extracted they are then converted into
#lookback values, what stocks to trade, or what strategy to use
getStockDataFromChromozone<- function(chrome){
      #Basically a binary number to decimal converter
      gene <- getStockGeneFromChromozone(chrome)
      index <-sum(chrome*(2^(seq(1,length(gene),1)-1)))+1 #The +1 is to stop 0 since not a valid index
      cleanName <- sub("^","",symbolLst[index],fixed=TRUE)
     return (get(cleanName,symbolData))
getStrategyFromChromozone <- function(chrome){
      gene <- matrix(getStrategyGeneFromChromozone(chrome))
       return (signalMACross)
       return (function(mktdata,paramA,paramB) {signalMACross(mktdata,paramA,paramB,avgFunc=EMA)})
       return (function(mktdata,paramA,paramB) {signalMACross(mktdata,paramA,paramB,avgFunc=ZLEMA)})
      # return (signalBollingerReversion)
       return (function(mktdata,paramA,paramB) {signalMACross(mktdata,paramA,paramB,avgFunc=WMA)})
      # return (signalRSIOverBoughtOrSold)
      print("nothing found")
getLookbackAFromChromozone <- function(chrome){
    gene <- getParameterLookbackGeneFromChromozone(chrome)
    gene <- gene[,1:3]
    gene <- matrix(gene)
        return (10)
        return (20)
        return (30)
        return (40)
        return (50)
        return (60)
        return (70)
        return (80)
getLookbackBFromChromozone <- function(chrome){
    gene <- getParameterLookbackGeneFromChromozone(chrome)
    gene <- gene[,4:6]
    gene <- matrix(gene)
        return (15)
        return (25)
        return (35)
        return (45)
        return (55)
        return (65)
        return (75)
        return (85)
#The more positive the fitness, the better the gene
calculateGeneFitnessFromTradingReturns <- function(tradingRet){
  tradingFitness <- SharpeRatio.annualized(tradingRet)
  #tradingFitness <- SharpeRatio.annualized(tradingRet) * (1/maxDrawdown(tradingRet))
  #tradingFitness <- max(cumsum(tradingRet))/maxDrawdown(tradingRet)
  #tradingFitness <- sum((tradingRet>0)*1)/length(tradingRet) #% of trades profitable
  #tradingFitness <- -1*maxDrawdown(tradingRet)
#This function performs the mating between two chromozones
genetricMating <- function(chromozoneFitness,useTopNPerformers,mutationProb){
        selectTopNPerformers <- function(chromozoneFitness,useTopNPerformers){
              #Ranks the chromozones by their fitness and select the topNPerformers
              orderedChromozones <- order(chromozoneFitness[,"Fitness"],decreasing=TRUE)
              orderedChromozones <- chromozoneFitness[orderedChromozones,]
              ##Often there are lots of overlapping strategies with the same fitness
              ##We should filter by unique fitness to stop the overweighting of lucky high fitness
              orderedChromozones <- subset(orderedChromozones, !duplicated(Fitness))
        hybridize <- function(topChromozones,mutationProb){
            crossoverFunc <- function(chromeA,chromeB){
            chromeA <- chromeA[,!colnames(chromeA) %in% c("Fitness")]
            chromeB <- chromeB[,!colnames(chromeB) %in% c("Fitness")]
                  #Takes a number of chromes from B and swaps them in to A
                  nCross <- runif(min=0,max=ncol(chromeA)-1,1) #the number of individual chromes to swap
                  swapStartLocation = round(runif(min=1,max=ncol(chromeA),1))
                  swapLocations <- seq(swapStartLocation,swapStartLocation+nCross) #Can run over the end of our vector, need to wrap around back to start
                  swapLocations <- swapLocations %% ncol(chromeA)+1 #Performs the wrapping
                  chromeA[1,swapLocations] <- chromeB[1,swapLocations] #Performs the swap
                  return (chromeA)
            mutateFunc <- function(chrome,mutationProb){
                return((round(runif(min=0,max=1,ncol(chrome))<mutationProb)+chrome) %% 2)
            #Take each chromozone and mate it with all the others (and it's self)
            a <- topChromozones[rep(seq(1,nrow(topChromozones)),each=nrow(topChromozones)),] #Repeat each row nrow times
            b <- topChromozones[rep(seq(1,nrow(topChromozones)),nrow(topChromozones)),] #Repeat whole matrix nrow times
            #Can this be vectorised (not huge amounts of data anyway so probs not an issue)?
            res <- matrix(nrow=0,ncol=ncol(a)-1) #The minus 1 is to drop the "Fitness" column
            for(i in 1:nrow(a)){
                res <- rbind(res,mutateFunc(crossoverFunc(a[i,],b[i,]),mutationProb))
            return (res)
        topChromozones <- selectTopNPerformers(chromozoneFitness,useTopNPerformers)
        #return ((hybridize(topChromozones,mutationProb))) #You may want duplicates to give more weight to 'good' genes
        return (unique(hybridize(topChromozones,mutationProb))) #Remove duplicate genes
#This function takes a chrome/gene and does the according trades
#It takes market data and a start and an end date
#It does not take responsibility for the mating and ranking of genes
doGeneticTrading <- function(mktdata,chrome, startDate, endDate){
    signalFunc <-getStrategyFromChromozone(chrome)
    paramA <- getLookbackAFromChromozone(chrome)
    paramB <- getLookbackBFromChromozone(chrome)
    signal <- signalFunc(Op(mktdata),paramA,paramB)
    opClRet <- (Cl(mktdata)/Op(mktdata)) - 1
    tradingReturns = opClRet * signal
    dataWin <- (paste(startDate,"::",endDate,sep=""))
    tradingReturns <- tradingReturns[dataWin]
    colnames(tradingReturns) <- c("TradingRet")
#This function mates genes every month
#It also passes those genes into the doGeneticTrading function
doTrading <- function(chromelist){
  #Function for taking a year and a month and spitting out a clean date
  cleanDate <- function(y,m){
      if(m == 13){
       m <- 1
       y <- y+1
       if(m < 10){
       } else {
  year <- 2002
  month <- 1
  totalRet <- 0
  fitnessEvoltion <- 0
  #Loop through many years and months
  for(y in 2002:2010){
    for(m in 1:12){
        chromeFitness <-,ncol=ncol(chromelist)))
        startD <- cleanDate(y-2,m) #Subtracting off 2 years to ensure we pass enough data in(should really be calculated from MA lookback)
        liveStart <- cleanDate(y,m)
        liveEnd <- cleanDate(y,m+1)
        dataWin <- (paste(startD,"::",liveEnd,sep=""))
        monthReturn <- data.frame()
        #Look through all the active chromes and use them for trading
        for(cn in 1:nrow(chromelist)){
        #USE a try catch just incase there are data issue etc...
            mktdata <- getStockDataFromChromozone(chromelist[cn,])
            tradingRet <- doGeneticTrading(mktdata[dataWin],chromelist[cn,],liveStart,liveEnd)
            tradingRet <- tradingRet*(1/nrow(chromelist)) #even money given to each strategy
            tradingFitness <- calculateGeneFitnessFromTradingReturns(tradingRet)
            if(!is.nan(tradingFitness) && !is.nan(max(tradingRet)) && !is.nan(min(tradingRet))){
              if(length(monthReturn) == 0 ){
               monthReturn <- tradingRet
              } else {
               monthReturn <- cbind(monthReturn,tradingRet)
            res <- cbind(chromelist[cn,],tradingFitness)
            colnames(res) <- c(colnames(chromelist[cn,]),"Fitness")
            chromeFitness <- rbind(chromeFitness,res)
        print("Month return")
        #Collapse all the trades from each chromozone into a single P&L for each day in the month
        monthReturn <- apply(monthReturn,1,sum,na.rm=TRUE)
        currentMonthFitness <- calculateGeneFitnessFromTradingReturns(monthReturn)
        #Update the running total of P&L
        totalRet <- c(totalRet,monthReturn)
        fitnessEvoltion <- c(fitnessEvoltion,currentMonthFitness)
        chromelist <- genetricMating(chromeFitness,topNToSelect,mutationProb)
        print(paste("There are",nrow(chromelist), "chromes active"))
        print(paste("Min Fitness:",min(chromeFitness[,"Fitness"])))
        print(paste("Max Fitness:",max(chromeFitness[,"Fitness"])))
        print(paste("Average Fitness:",mean(chromeFitness[,"Fitness"])))
        print(paste("Current Month Fitness:",currentMonthFitness))
   return (totalRet)
#Specify dates for downloading data, training models and running simulation
startDate = as.Date("2000-01-01") #Specify what date to get the prices from
symbolData <- new.env() #Make a new environment for quantmod to store data in
getSymbols(symbolLst, env = symbolData, src = "yahoo", from = startDate)
#Create some genes at random
#Make a diag matrix so that each chrome gets activated atleast once
startingChromozones <- diag(chromozoneLength)
rownames(startingChromozones) <- apply(t(seq(1,chromozoneLength)),2,function(x) { paste("Chrome",x,sep="") } )
fitness <- matrix(runif(min=-1,max=1,nrow(startingChromozones)),nrow=nrow(startingChromozones),ncol=1)
colnames(fitness) <- c("Fitness")
startingChromozones <-,fitness))
print("Before mating")
print("After mating")
startingChromozones <- genetricMating(startingChromozones,topNToSelect,mutationProb)
tradingReturns <- doTrading(startingChromozones)
tradingReturns <-[-1])))
charts.PerformanceSummary(tradingReturns,main=paste("Arithmetic Genetic Trading Returns"),geometric=FALSE)
cat("Sharpe Ratio")
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Is ‘risk’ rewarded in the equity markets?

This post looks to examine if the well known phrase “the higher the risk the higher the reward” applies to the FTSE 100 constituents. Numerous models have tried to capture risk reward metrics, the best known is the Capital Allocation Pricing Model (CAPM). CAPM tries to quantify the return on an investment an investor must receive in order to be adequately compensated for the risk they’ve taken.

The code below calculates the rolling standard deviation of returns, ‘the risk’, for the FTSE 100 constituents. It then groups stocks into quartiles by this risk metric, the groups are updated daily. Quartile 1 is the lowest volatility stocks, quartile 2 the highest. An equally weighted ($ amt) index is created for each quartile. According to the above theory Q4 (high vol) should produce the highest cumulative returns.

When using a 1 month lookback for the stdev calculation there is a clear winning index, the lowest vol index (black). Interestingly the 2nd best index is the highest vol index (blue). The graph above is calculated using arithmetic returns.

When using a longer lookback of 250 days, a trading year, the highest vol index is the best performer and the lowest vol index the worst performer.

For short lookback (30days) low vol index was the best performer

For long lookback (250days) high vol index was the best performer

One possible explanation (untested) is that for a short lookback the volatility risk metric is more sensitive to moves in the stock and hence on a news announcement / earnings the stock has a higher likelihood of moving from it’s current index into a higher vol index. Perhaps it isn’t unreasonable to assume that the high vol index contains only the stocks that have had a recent announcement / temporary volatility and are in a period of consolidation or mean reversion. Or to put it another way for short lookbacks the high vol index doesn’t contain the stocks that are permanently highly vol, whereas for long lookbacks any temporary vol deviations are smoothed out.

Below are the same charts as above but for geometric returns.

On to the code:

?View Code RSPLUS
#Script parameters
#Specify dates for downloading data
startDate = as.Date("2000-01-01") #Specify what date to get the prices from
symbolData <- new.env() #Make a new environment for quantmod to store data in
clClRet <- new.env()
downloadedSymbols <- list()
for(i in 1:length(symbolLst)){
  #Download one stock at a time
    getSymbols(symbolLst[i], env = symbolData, src = "yahoo", from = startDate)
     cleanName <- sub("^","",symbolLst[i],fixed=TRUE)
     mktData <- get(cleanName,symbolData)
     print(paste("-Calculating close close returns for:",cleanName))
      ret <-(Cl(mktData)/Lag(Cl(mktData)))-1
      print("-There is a abs(return) > 50% the data is odd lets not use this stock")
      downloadedSymbols <- c(downloadedSymbols,symbolLst[i])
      assign(cleanName,ret,envir = clClRet)
    }, error = function(e) {
    print(paste("Couldn't download: ", symbolLst[i]))
#Combine all the returns into a zoo object (joins the returns by date)
#Not a big fan of this loop, think it's suboptimal
zooClClRet <- zoo()
for(i in 1:length(downloadedSymbols)){
  cleanName <- sub("^","",downloadedSymbols[i],fixed=TRUE)
  print(paste("Combining the close close returns to the zoo:",cleanName))
    zooClClRet <- as.zoo(get(cleanName,clClRet))
  } else {
    zooClClRet <- merge(zooClClRet,as.zoo(get(cleanName,clClRet)))
#This will take inzoo or data frame
#And convert each row into quantiles
#Quantile 1 = 0-0.25
#Quantile 2 = 0.25-0.5 etc...
quasiQuantileFunction <- function(dataIn){
    quantileFun <- function(rowIn){
        quant <- quantile(rowIn,na.rm=TRUE)
        a <- (rowIn<=quant[5])
        b <- (rowIn<=quant[4])
        c <- (rowIn<=quant[3])
        d <- (rowIn<=quant[2])
        rowIn[a] <- 4
        rowIn[b] <- 3
        rowIn[c] <- 2
        rowIn[d] <- 1
  return (apply(dataIn,2,quantileFun))
avgReturnPerQuantile <- function(returnsData,quantileData){
      q1index <- (clClQuantiles==1)
      q2index <- (clClQuantiles==2)
      q3index <- (clClQuantiles==3)
      q4index <- (clClQuantiles==4)
      q1dat <- returnsData
      q1dat[!q1index] <- NaN
      q2dat <- returnsData
      q2dat[!q2index] <- NaN
      q3dat <- returnsData
      q3dat[!q3index] <- NaN
      q4dat <- returnsData
      q4dat[!q4index] <- NaN
      avgFunc <- function(x) {
           #apply(x,1,median,na.rm=TRUE) #median is more resistant to outliers
      res <- returnsData[,1:4] #just to maintain the time series (there must be a better way)
      res[,1] <- avgFunc(q1dat)
      res[,2] <- avgFunc(q2dat)
      res[,3] <- avgFunc(q3dat)
      res[,4] <- avgFunc(q4dat)
      colnames(res) <- c("Q1","Q2","Q3","Q4")
nLookback <- 250 #~1year trading calendar
clClVol <- rollapply(zooClClRet,nLookback,sd,na.rm=TRUE)
clClQuantiles <- quasiQuantileFunction(clClVol)
returnPerVolQuantile <- avgReturnPerQuantile(zooClClRet,clClQuantiles)
colnames(returnPerVolQuantile) <- c("Q1 min vol","Q2","Q3","Q4 max vol")
returnPerVolQuantile[is.nan(returnPerVolQuantile)]<-0 #Assume if there is no return data that it's return is 0
#returnPerVolQuantile[returnPerVolQuantile>0.2] <- 0 #I was having data issues leading to days with 150% returns! This filters them out
cumulativeReturnsByQuantile <- apply(returnPerVolQuantile,2,cumsum)
charts.PerformanceSummary(returnPerVolQuantile,main=paste("Arithmetic Cumulative Returns per Vol Quantile - Lookback=",nLookback),geometric=FALSE)
cat("Sharpe Ratio")
for(i in seq(2012,2004,-1)){
  windowedData <- window(as.zoo(returnPerVolQuantile),start=as.Date(paste(i,"-01-01",sep="")),end=as.Date(paste(i+1,"-01-01",sep="")))
title(main=paste("Arithmetic Cumulative Returns per Vol Quantile - Lookback=",nLookback),outer=T)
charts.PerformanceSummary(returnPerVolQuantile,main=paste("Geometric Cumulative Returns per Vol Quantile - Lookback=",nLookback),geometric=TRUE)
cat("Sharpe Ratio")
for(i in seq(2012,2004,-1)){
  windowedData <- window(as.zoo(returnPerVolQuantile),start=as.Date(paste(i,"-01-01",sep="")),end=as.Date(paste(i+1,"-01-01",sep="")))
title(main=paste("Geometric Cumulative Returns per Vol Quantile - Lookback=",nLookback),outer=T)
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Analysis of returns after n consecutive up/down days – Predicting the Sign of Open to Close Returns

This weekend I was spammed for a “binary option trading system with 90% accuracy”. The advert caught my curiosity  in essence it detailed a method that was a variation of the well known roulette playing strategy that mathematically guarantees a profit (assuming infinite money, and no table limit).

Roulette Strategy

If you double your bet size after a loss and repeat the same bet you are guaranteed a profit, your next winning will cover all the preceding loses.

e.g Bet $1 on red, lose, Bet $2 on red, Win get $4 back ($2 is your stake, and $1 covers the loss from your first bet) giving $1 profit.

Exponential growth of lot size, no thanks :)

Binary options are analogous to betting on red, they offer virtually fixed odds for up or down directional bets. Naturally I want to know whats is the maximum number of consecutive up or down days in the market, how much pain would I have to suffer with this strategy.

Occurrences of n Consecutive Up or Down Days

Analysing the last 12 years of returns data for the S&P 500, the maximum consecutive number of up days is 9 (occurred in 2004-2005), the maximum consecutive number of down days is -8 which, you guessed it, occurred in 2008-2009.

So 9 days of pain should we always short the direction of the market.

Instead of enduring the 9 days of draw down, it is interesting to see what the consecutive number of up/down days says about the probability the next day is an up day. The maximum likelihood probability of an up day is count(up days)/count(up and down days). Naturally we will condition this data on the consecutive number of up/down days.

Consecutive Up or Down Days vs Maximum Likelihood Probability the next day is up

This data is fairly nice looking, for example in 2012-2013 there is a clear relationship. The more down days in a row the higher the likelihood of an up day. 6 down days implied the probability of an up day is 80%! I must raise a note of caution here, 6 down days in a row was seen less than 5 times in the year. Hence the probability estimate is based on 5 points and not statistically significant, perhaps looking at 5 years of returns might be better.

I appreciate that most people don’t trade binary options can we trade the index/stock out right, it is interesting to see what the consecutive number of up/down days says about future Open to Close Returns. The image below regresses Open to Close Returns (time t) with Consecutive Up/Down days (time t-1).

Consecutive Up or Down Days vs Next Day Open to Close Returns

Very disappointing chart, doesn’t really show much relationship between returns and consecutive up/down days. For some of the data points the up move is more probable than a down move but the magnitude of the up moves are significantly smaller than that of the down moves. These charts vary greatly by asset class and by security, single stocks have much more favorable plots.

Prediction Accuracy

The plot below shows the accuracy of using this maximum likelihood estimate approach. The model takes the last 250 days of returns and calculates the probability of an up move given that the current day has seen n consecutive days of trading. If the prob of an up move or is over a certain threshold go long, if its below a certain threshold go short.

Heatmap of Accuracy vs Model Parameters


The histogram on the heatmap shows that approximately half of the parameter combinations can predict the direction with accuracy greater than 50%.

Final Comments

The beauty of this approach is that it’s simple, it can be applied to any asset class but most importantly it can be applied across different time frames.

Onto the code:

?View Code RSPLUS
#Control Parameters
dataStartDate = as.Date("2000-01-01")
symbol<- "^GSPC"
longThreshold <- 0.70  #If the probability of an upday is greater than this limit go long
shortThreshold <- 1-longThreshold  #If the probability of an upday is lower than this limit go short
nLookback<-250 #Days to look back when generating the rolling probability distribution (approx 250 trading days in a year)
#Function to turn a boolean vector into a vector containing the consecutive num of trues or falses seen
#Will be used to calculate the consecutive number of up and down days
consecutiveTruesExtractor <- function(data){
        genNumOfConsecutiveTrues <- function(x, y) { (x+y)*y  } #Y is either 0 or 1
        upDaysCount <- Reduce(genNumOfConsecutiveTrues,data,accumulate=TRUE)
        upDaysCount <- as.vector(Lag(upDaysCount))
        upDaysCount[] <- 0
        downDaysCount <- Reduce(genNumOfConsecutiveTrues,!data,accumulate=TRUE)
        downDaysCount <- as.vector(Lag(downDaysCount))
        downDaysCount[] <- 0
        consecutiveTruesExtractor <- upDaysCount-downDaysCount
#Function to plot data and add regression line
doPlot <- function(x,y,title,xlabel,ylabel,ylimit){
  boxplot(y~x,main=title, xlab=xlabel,ylab=ylabel,ylim=ylimit)
#Function to calculate the percentage of updays from returns set
calcPercentageOfUpdaysFromReturnsSet <- function(data){
  calcPercentageOfUpdaysFromReturnsSet <- sum(data>0)/length(data)
#Function takes a set of returns and consecutive up down day data and aggregates it into a probability distribution
#Generated a matrix of consecutive Direction vs prob of up move
generateProbOfUpDayDistribution <- function(dataBlock){
 y <- as.matrix(by(dataBlock[,"OpClRet"],list(ConsecutiveDir = dataBlock[,"ConsecutiveDir"]),calcPercentageOfUpdaysFromReturnsSet)) #Prob of upmove
 x <- as.matrix(as.numeric(as.matrix(rownames(y))))
 res <- cbind(x,y)
 colnames(res) <- c("ConsecutiveDir","Prob")
 generateProbOfUpDayDistribution <- res
#Given current consecutive up down day data, what is the probability the current day is an up day
#For use with the rollapply function (since needs to use the past n days worth of data for generating probability distribution)
probOfUpDayForUseWithRollApply <- function(dataBlock){
  dist <- generateProbOfUpDayDistribution(head(dataBlock,-1)) #Use head to drop the last row, prevents a lookforward issue
  currentConsecutiveRun <- last(dataBlock[,"ConsecutiveDir"])
  probOfUpDay <- dist[dist[,"ConsecutiveDir"] == rep(coredata(currentConsecutiveRun), length(dist[,"ConsecutiveDir"])),"Prob"]
  if(!is.numeric(probOfUpDay)) {probOfUpDay <- 0.5 } #Never this many consecutive days before, dont know what will happen make up and down events equally likely
  #print(paste("Current Run:",coredata(currentConsecutiveRun),"Prob of up day:",probOfUpDay ))
  probOfUpDayForUseWithRollApply <- probOfUpDay
#Just a quick test to check that the consecutiveTruesExtractor is working as expected
#Define the input data, and define the expected output
#Check that the output of the function equals the expected output
data <-           c(0, 0, 0, 1,0, 1,1,0,0)  #0 is down day, 1 is up day
expectedOutput <- c(0,-1,-2,-3,1,-1,1,2,-1)
res <- consecutiveTruesExtractor(data)
if( identical(res,expectedOutput)){
  print("Match consecutiveTruesExtractor is correct")
} else {
  print("Error consecutiveTruesExtractor contains bugs")
#Download the data
symbolData <- new.env() #Make a new environment for quantmod to store data in
getSymbols(symbol, env = symbolData, src = "yahoo", from = dataStartDate)
mktdata <- eval(parse(text=paste("symbolData$",sub("^","",symbol,fixed=TRUE))))
opClRet <- (Cl(mktdata)/Op(mktdata))-1
consecutiveDir <- consecutiveTruesExtractor(as.matrix(opClRet>0))
completeData<- cbind(opClRet,consecutiveDir)
colnames(completeData) <- c("OpClRet","ConsecutiveDir")
#Plot of consecutive up down days vs next day Open Close Returns
for(i in seq(2012,2004,-1)){
  windowedData <- window(completeData,start=as.Date(paste(i,"-01-01",sep="")),end=as.Date(paste(i+1,"-01-01",sep="")))
  doPlot(windowedData$ConsecutiveDir,windowedData$OpClRet,paste("Consecutive Up / Down days vs Open close Return (",i,",",i+1,")"),"Consecutive Up or Down days","Open Close Returns",c(-0.07,0.07))
#Plot of consecutive up down days vs the maximum likelihood probability that the next day is an up day
for(i in seq(2012,2004,-1)){
  windowedData <- window(completeData,start=as.Date(paste(i,"-01-01",sep="")),end=as.Date(paste(i+1,"-01-01",sep="")))
  y <-as.matrix(by(as.vector(windowedData$OpClRet),list(ConsecutiveDir = windowedData$ConsecutiveDir),calcPercentageOfUpdaysFromReturnsSet)) #Prob of upmove
  x <- as.matrix(as.numeric(as.matrix(rownames(y)))) #Consecutive up or down days
  plot(cbind(x,y),main=paste("Consecutive Up / Down days vs Next day Dir (",i,",",i+1,")"), xlab="Consecutive Up or Down days",ylab="Conditional Probability of an Up day")
#Plot of consecutive up down days vs the number of occurences seen
for(i in seq(2012,2004,-1)){
  windowedData <- window(completeData,start=as.Date(paste(i,"-01-01",sep="")),end=as.Date(paste(i+1,"-01-01",sep="")))
  y <-abs(as.matrix(by(windowedData$ConsecutiveDir,list(ConsecutiveDir = windowedData$ConsecutiveDir),sum))) #Count the number of occurences of each consecutive run
  x <- as.matrix(as.numeric(as.matrix(rownames(y))))
  plot(y,xaxt="n",main=paste("Consecutive Up / Down days vs Next day Dir (",i,",",i+1,")"), xlab="Consecutive Up or Down days",ylab="Occurences of consecutive run",type="l")
  axis(1, at=1:length(x),labels=x)
predictionPerformance <- function(completeData,longThreshold,shortThreshold,nLookback,displayPlot){
    #Calcuate the probabiliy of an up day using a nLookback day window
    rollingProbOfAnUpday <- rollapply(completeData,FUN=probOfUpDayForUseWithRollApply,align="right",fill=NA,width=nLookback,by.column=FALSE)
    rollingProbOfAnUpday <- as.matrix(rollingProbOfAnUpday)
    colnames(rollingProbOfAnUpday) <- "ProbTodayIsAnUpDay"
    completeData <- cbind(completeData,rollingProbOfAnUpday)
    suggestedTradeDir <- rollingProbOfAnUpday #Just to copy the structure
    colnames(suggestedTradeDir) <- "SuggestedTradeDir"
    suggestedTradeDir[rollingProbOfAnUpday>longThreshold] <- 1  #Long Trade
    suggestedTradeDir[rollingProbOfAnUpday<shortThreshold] <- -1 #Short Trade
    suggestedTradeDir[rollingProbOfAnUpday<longThreshold & rollingProbOfAnUpday>shortThreshold] <- 0 #Do nothing
    completeData <- cbind(completeData,suggestedTradeDir)
    isPredictionCorrect <- suggestedTradeDir #Just to copy structure
    isPredictionCorrect <- sign(completeData$SuggestedTradeDir * completeData$OpClRet) #sign(0) is 0 so will capture no trades as well
    isPredictionCorrect[] <- 0
    isPredictionCorrect[is.nan(isPredictionCorrect)] <- 0
      plot(cumsum(isPredictionCorrect), main=paste("Market Direction Prediction Performance for",symbol,"(Probability Threshold, Long=",longThreshold,"Short=",shortThreshold,"Lookback=",nLookback,")"),xlab="Date",ylab="Cumulative Sum of Correct (+1) and Wrong(-1) Predictions")
      msgIncorrectPred <- (paste("Incorrect Predictions (out of the days when a prediction was made)",100*abs(sum(isPredictionCorrect[isPredictionCorrect==-1]))/length(isPredictionCorrect[isPredictionCorrect!=0]),"%"))
      msgCorrectPred <- (paste("Correct Predictions (out of the days when a prediction was made)",100*sum(isPredictionCorrect[isPredictionCorrect==1])/length(isPredictionCorrect[isPredictionCorrect!=0]),"%"))
      msgPercOfDaysWithPred <- (paste("Percent of days when a prediction was made",100*sum(abs(isPredictionCorrect[isPredictionCorrect!=0]))/length(isPredictionCorrect),"%"))
    predictionPerformance <- sum(isPredictionCorrect[isPredictionCorrect==1])/length(isPredictionCorrect[isPredictionCorrect!=0])
#Parameter search
resultsMat <- matrix(numeric(0), 0,3)
colnames(resultsMat) <- c("Lookback","LongProbThreshold","Accuracy")
for(nLookback in seq(30,240,30)){
  for(longThreshold in seq(0.55,1,0.05)){
    shortThreshold <- 1-longThreshold
    accuracy <- predictionPerformance(completeData,longThreshold,shortThreshold,nLookback*2,FALSE)
    resultsMat <- rbind(resultsMat,as.matrix(cbind(nLookback,longThreshold,accuracy)))
tt <-(as.matrix(,Lookback~LongProbThreshold))))
heatmap.2(tt[,-1],key=TRUE,Rowv=FALSE,Colv=FALSE,xlab="Prob of Up Day Threshold",ylab="Lookback",trace="none",main=paste("Prediction Accuracy (correct predictions as % of all predictions) for ",symbol))


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