Parameter Sensitivity Analysis

In this post we demonstrate ideas to test for parameter sensitivity.

Here we have a strategy with 5x parameters. 3x being look back periods for a specific indiactor. The other 2x being an entry level threshold and an exit level threshold.

I decided to change the original parameters by up to 50% in either direction of the original. It might look something like this:

# Original 
param1 = 116
param2 = 10
param3 = 69
entry_level = 6
exit_level= 85

So if you multiply any of the params by 0.5… you obtain a value… and in this example lets multiply the first parameter, param1: 116 * 0.5 = 58. This provides the new upper and lower limit. The range now becomes:

58 to 174.

This has created a range of parameters up to a maximum of 50% change in either direction of the original. Do this for all parameters. Note if your indicator only goes to 100 for example then that would be your maximum versus the full 50% range. Same in the opposite direction. Parameter lower limit obviously 0.

Next run a monte carlo running randomly through the parameter ranges. I tend to do 5000 iterations then look at the distribution. An example output of this process is below:

1 (1)1 (2)1 (3)

Here we show some key metrics. Average trade profit. This is an important value because on average we want to know what we make per trade and it has to be high enough to withstand slippage and commissions.

Next we have the win % where depending on which money management system you are using, often this is an input variable in money management calculations. An example would be half Kelly formula.

And of course we have the net ending balance of the strategy itself less commissions (I didnt add slippage in this example).

The plots above show my original parameter setting results by marking a straight vertical red line on the plot.

This red line was some optimal combination again found through a random search. As we can see its not THE optimal as to the right side of the red line we have parameter sets which beat the original.

Besides this what can we learn – we see that by running 5000 iterations and randomly selecting a parameter from a range which includes up to a 50% change in either direction from the original. That we land in positive expectancy across our selected trading metrics. We may observe the mean values below:


Needless to say. The random parameter selection might not choose the most extreme 50% parameter change at any given iteration. Thus across the 5 parameter sets we would be combining a range of changes in either direction up to a 50% maximum. One iteration might might look like:

Param1 = +/- 10% change from original

Param2 = No change from original

Param 3 = +/- 40% change from original

Param 4 = +/- 12 % change from original

Param 5 = +/- 5% change from original

And run another:

Param1 = +/- 45% change from original

Param2 = +/- 8% change from original

Param 3 = +/- 2% change from original

Param 4 = +/- 22 % change from original

Param 5 = +/- 34% change from original

So on and so forth for 5000 iterations.

How do you obtain confidence that you can protect your initial strategy investment when going live?

This type of study essentially says we may place a strategy live and have a degree in confidence in changing parameters.

What would I do in real life?

I would place 3 parameters bands in live trading. Whichever parameter set triggered first is the trade that is taken. Thus I am covering a parameter band and I am covered if the market changes away from my original parameters…..

Food for thought!

If you have an interest in this type of research feel free to drop me a line.

Follow me: @flare9x / Github


SPY Mean Reversion With John Ehlers Adaptive RSI

It has been a busy few months. I have been exploring market indicators that John Ehlers has created which he publicly made available in his book: Cycle Analytics for Traders : Advanced Technical Trading Concepts.

The key theme of his book is applying digital signal processing filters to better process market data. He utilizes various high and low pass filters which only allow certain frequencies to pass at specified look back periods and thus make better attempts to reduce signal to noise and perhaps extract a dominant cycle from the data.

Key ideas from the book:

  1. Traditional indicators have lag and assume normal probability distribution
  2. His indicators reduce lag
  3. Extracting the dominant cycle over a range of periods can be used to calibrate traditional indicators such as RSI, stochastic etc….
  4. Has methods to convert indicator output to as close to normal probability distribution

I’m in the process of making his indicators from his Cycle analytics for traders book available on my github in a package called MarketCycles.jl. Please refer to the package for further information.

The package will be used in the next part of this post where we use an Adaptive RSI to time the mean reversion tendency of the daily S&P500.

First install the package:


The Adaptive RSI is adaptive because the indicator extracts the dominant cycle using the autocorrelation periodogram. At any given point in time [i] half the dominant cycle period is used to calibrate the lookback length of the RSI.

I want to point out a few stark differences with the below strategy back test code. First, the code is setup to BUY NEXT BAR AT OPEN. This means when a signal is triggered at the close of a bar. A trade is then entered at the next open price.

Secondly, rather than calculate the cumulative % close to close returns as so many do. We specify an initial starting capital and add the $ gain to a running total adjusting for trading commissions which provides us with a result that is more in line with how it might work in the real world.

Please see top of code snippet in order to adjust the following input variables for the strategy:

# Strategy input variables
starting_capital = 10000
comms = 10
entry_level = .3
exit_level = .8

The rest is best described by following the below julia language code. Note if you do not have a data source I have made this post easy to follow by providing a code to download data from the alphavantage api. You just need to edit the alphavantage link and add in your own API key:

# Ehlers mean reversion

using HTTP
using DataFrames
using CSV
using Gadfly
using MarketCycles

# Multiple symbols
adjusted = 1  # if want adjusted closes set to 1 else non adjusted set it to 0
t = 1
tickers = ["SPY"]
for t in 1:length(tickers)  # Note it might get stuck!! if it fails.. print the t value... then restart the for loop for t in 4:length(tickers)  instead of 1:length(tickers) for example
# Using HTTP package
sleep(10)  # sleep between API calls
if adjusted == 1
res = HTTP.get(joinpath(
    res = HTTP.get(joinpath(
mycsv = readcsv(res.body)
x = convert(DataFrame, mycsv)
x = x[2:nrow(x),:]  # subset remove header row
# Rename Columns
if adjusted == 1
    colnames = ["Date","Open","High","Low","Close","Adjusted_Close","Volume","Dividend_Amount","split_coefficient"]
colnames = ["Date","Open","High","Low","Close","Volume"]
names!(x.colindex, map(parse, colnames))
# Convert String Date to Date format
x[:Date] = Date.(x[:Date],Dates.DateFormat("yyy-mm-dd"))
# Sort Date Frame By Date
x = sort!(x, [order(:Date)], rev=(false))
# Convert OHLC to Float64 and Volume to Int64
if adjusted == 1
for i in 2:length(x)-2
    x[i] = convert(Array{Float64,1},x[i])
    x[7] = convert(Array{Int64,1},x[7])
    x[8] = convert(Array{Float64,1},x[8])
    for i in 2:length(x)-1
        x[i] = convert(Array{Float64,1},x[i])
        x[6] = convert(Array{Int64,1},x[6])

if adjusted == 1
CSV.write(joinpath("CSV_OUT_ADJUSTED_"tickers[t]".csv"), x)
CSV.write(joinpath("CSV_OUT_"tickers[t]".csv"), x)

# Load Data
SPY ="C:/Users/Andrew.Bannerman/Desktop/Julia/Indicators/CSV_OUT_ADJUSTED_SPY.csv",types=[String; fill(Float64, 5); Int;Float64],header=true)
SPY[:Date] = DateTime.(SPY[:Date],Dates.DateFormat("mm/dd/yyyy"))

# Pull arrays
Date_index = SPY[:Date]
Spy_close = SPY[:Close]
Spy_open = SPY[:Open]

# Eherls Adaptive RSI
Adaptive_RSI = AdaptiveRSI(Spy_close)

# Strategy input variables
starting_capital = 10000.00
comms = 10
entry_level = .3
exit_level = .8

# Cross under entry level and sell when cross over exit level
entry = zeros(Adaptive_RSI)
exit = zeros(Adaptive_RSI)
for i = 2:size(Adaptive_RSI,1)
    if (Adaptive_RSI[i] <= entry_level) && (Adaptive_RSI[i-1] > entry_level)
        entry[i] = 1
        entry[i] = 0
    if (Adaptive_RSI[i] >= exit_level) && (Adaptive_RSI[i-1] < exit_level)
        exit[i] = 2
        exit[i] = 0

# combine entry and exit
signal = exit .+ entry

# Fill the trade between entry and exit with 1's
all_signal = zeros(signal)
for i = 2:length(signal)
    if signal[i] == 1.0
        all_signal[i] = 1.0
    if signal[i] == 0.0 && all_signal[i-1] == 1.0
        all_signal[i] = 1
    if signal[i-1] == 2.0 && signal[i] == 0.0
        all_signal[i-1] = 1.0

# ID the start and end of the trade
trade_start = zeros(all_signal)
trade_end = zeros(all_signal)
for i = 2:size(trade_end,1)
if all_signal[i] == 1.0 && all_signal[i] != all_signal[i-1]
    trade_start[i] = 1
if all_signal[i-1] == 1.0 && all_signal[i-1] != all_signal[i]
    trade_end[i-1] = 2

# Calculate the % return buying at open and selling at the close
open_price = Float64[]
close_price = Float64[]
strategy_returns = zeros(size(all_signal,1))
for i = 1:size(all_signal,1)-1
    if trade_start[i] == 1
        open_price = Spy_open[i+1]
    elseif trade_end[i] == 2
        close_price = Spy_close[i]
        # place the close return on same index position as the end of the trade
    strategy_returns[i] = close_price /  open_price - 1

# Work out gain on starting investment amount
dollar_gain = zeros(size(strategy_returns,1))
dollar_gain[1] = starting_capital

for i=2:size(strategy_returns,1)
    # if the returns are 0 then use the previous $ amount
if strategy_returns[i] == 0.0
    dollar_gain[i] = dollar_gain[i-1]
elseif strategy_returns[i] != 0.0
    dollar_gain[i] = dollar_gain[i-1] + (dollar_gain[i-1] * strategy_returns[i]) - comms

# plot final result
white_panel = Theme(
p = plot(x=Date_index,y=dollar_gain,Geom.line,Guide.title("Ehlers Adaptive RSI"),white_panel,Scale.y_continuous(format=:plain))
         draw(PNG("C:/Users/Andrew.Bannerman/Desktop/Julia/ehlers_rsi.png", 1500px, 800px), p);

And we may plot the output which is the growth of $10,000 inclusive of $5 round trip commissions:


It should be noted that default settings from Ehlers book were used as a High and low pass filter as well as the range of lags used in the autocorrelation periodogram to calculate the dominant cycle.

The function I created has many inputs that may be calibrated:

AdaptiveRSI(x::Array{Float64}; min_lag::Int64=1, max_lag::Int64=48,LPLength::Int64=10, HPLength::Int64=48, AvgLength::Int64=3)::Array{Float64}

Thus the default settings might not be best tuned to SPY daily data. One would be advised to iterate through the input variables to learn how each variable affects the output as well as find a correct tuning of the indicator.

If anyone wishes to have a more professional back test script with varying entry / exits methods that accounts for slippage and commissions do not hesitate to contact me.

Follow me on twitter: @flare9x