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I'd like to try a strategy using the z-score of a ratio of two assets, but it seems we don't have a built-in for this, and the built-in function for standard deviation will only accept a singular security. Anybody here happen to have an example of how to go about this one?
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I was able to get this working, but in a rather hacky way. I'm hoping someone will be kind enough to reply with a better solution than this.
Python is not my forte, and I'm not trained in data science, so probably this is not the best solution, and I wonder about the memory impact. All the example code I could find for rolling z-score, involved pandas, so I just imported it and used a pandas series object.
import pandas as pd
import math
asset_class = 'stocks'
pairs_list = {
'stocks': [
('CVX', 'XOM'),
('EGBN', 'FMBI')
],
'forex': [
('EURUSD', 'GBPUSD')
]
}
(symbol1, symbol2) = pairs_list[asset_class][0]
class MeanReversionResearch(QCAlgorithm):
def Initialize(self):
self.SetStartDate(2020, 12, 3)
self.SetEndDate(2020, 12, 5)
self.SetCash(100_000)
self.ZScoreSeries = Series("Z Score", SeriesType.Line, 0)
chart = Chart("Z Score")
chart.AddSeries(self.ZScoreSeries)
self.AddChart(chart)
self.Ratios = pd.Series()
if asset_class == 'stocks':
self.AddEquity(symbol1, Resolution.Minute)
self.AddEquity(symbol2, Resolution.Minute)
elif asset_class == 'forex':
self.AddForex(symbol1, Resolution.Minute)
self.AddForex(symbol2, Resolution.Minute)
def OnData(self, data):
if data.ContainsKey(symbol1) and data.ContainsKey(symbol2):
price1 = data[symbol1].Close
price2 = data[symbol2].Close
self.ZScore(price1 / price2)
def ZScore(self, ratio):
self.Debug(f'ratio: {ratio}')
self.Ratios = self.Ratios.append(pd.Series([ratio]))
x = self.Ratios.rolling(window = 1).mean()
w = self.Ratios.rolling(window = 20)
score = (x - w.mean()) / w.std()
if score.size > 0:
s = score.iat[-1]
if math.isnan(s):
return
else:
self.ZScoreSeries.AddPoint(self.Time, s)
Â
0
The method that Robert shared looks good, but has some computational overhead (via pandas.Series.rolling() which requires repeatedly computing cumulants on the past N observations). As N grows, the cost becomes more expensive - though the example limits the "window size" to 20 which alleviates this at the cost of precision of the sample estimates.
Another way would be to do this more incrementally like:
import math
class MyAlgorithm(QCAlgorithm):
def Initialize(self):
self.AAPL_SPY_SpreadRatio = SpreadRatio('AAPL_SPY')
def OnData(self, data):
# Latest observation of AAPL/SPY
AAPL_Close = data['AAPL'].Close
SPY_Close = data['SPY'].Close
AAPL_SPY_Ratio = AAPL_Close / SPY_Close
# Update sample estimates of mean/variance
self.AAPL_SPY_SpreadRatio.UpdateEstimates( AAPL_SPY_Ratio )
# Get the latest Z-score of the spread
AAPL_SPY_ZScore = self.AAPL_SPY_SpreadRatio.GetZScore( AAPL_SPY_Ratio )
# Plot stuff
self.Plot('Z Score of Ratios', 'AAPL/SPY', AAPL_SPY_ZScore)
class SpreadRatio:
def __init__(self, tickers):
self.tickers = tickers
self.N = 1 # Number of observations
self.mean = 0 # Sample mean of observed x's
self.var = 0 # Sample variance of observed x's
self.squared_differences = 0 # Sum of squares of deviations of x from mean
def UpdateEstimates(x):
# Incrementally update sample estimates via Welford (1962)
new_mean = self.mean + ( ( x - self.mean ) / self.N )
self.squared_differences += (x - self.mean) * (x - new_mean)
new_var = self.squared_differences / self.N
self.N += 1
self.mean = new_mean
self.var = new_var
def GetZScore(x):
z_score = ( x - self.mean ) / math.sqrt(self.var)
return z_score
For K symbols, you can add a for-loop then collect all the computed z-scores into a dict.
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This looks great, thank you! But I'm getting a division by zero error. It looks like new_mean will equal the first ratio passed in:
new_mean = 0 + ((x - 0) / 1)
# new_mean = x
squared_differences = 0 + (x - 0) * (x - x)
# squared_differences = 0
var = 0 / 1
# var = 0
z_score = (x - 0) / math.sqrt(0)
# x / 0: division by zero error
What is the correct way to fix this?
Thanks again.
0
I tried making the division by square root conditional, and this resolved the error for me. I don't know if the math is right, but the chart looks right.
sqrt = math.sqrt(self.var)
if sqrt == 0.0:
return x - self.mean
else:
return (x - self.mean) / sqrtÂ
0
Good catch - typed that up pretty quickly. It would throw an 0-division error during the first run since the variance is 0 (or if the spread ratio is exactly the same consecutively for whatever reason).
Many different ways to fix it, the conditional division looks good or adding some boolean flag like:
class MyAlgorithm(QCAlgorithm):
def OnData(self, data):
# if data.ContainsKey(etc)
# Check if sufficient variance is observed
if self.AAPL_SPY_SpreadRatio.isReady:
AAPL_SPY_ZScore = self.AAPL_SPY_SpreadRatio.GetZScore( AAPL_SPY_Ratio )
class SpreadRatio:
def __init__(self, tickers):
self.tickers = tickers
self.N = 1 # Number of observations
self.mean = 0 # Sample mean of observed x's
self.var = 0 # Sample variance of observed x's
self.squared_differences = 0 # Sum of squares of deviations of x from mean
self.isReady = False # Check if variance is non-zero
def UpdateEstimates(self, x):
# Incrementally update sample estimates via Welford (1962)
new_mean = self.mean + ( ( x - self.mean ) / self.N )
self.squared_differences += (x - self.mean) * (x - new_mean)
new_var = self.squared_differences / self.N
self.N += 1
self.mean = new_mean
self.var = new_var
# Catch 0-variance case
if math.sqrt(self.var) != 0.0:
self.isReady = True
def GetZScore(self, x):
z_score = ( x - self.mean ) / math.sqrt(self.var)
return z_scoreÂ
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Hi Robert and Adam,
We can simplify this further by utilizing the SimpleMovingAverage and StandardDeviation indicators. If we update the indicators with the price ratio at each time step, then we can compute the z-score with
z_score = (ratio - self.sma.Current.Value) / self.std.Current.ValueSee the attached backtest for reference.
Best,
Derek Melchin
1
The material on this website is provided for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation or endorsement for any security or strategy, nor does it constitute an offer to provide investment advisory services by QuantConnect. In addition, the material offers no opinion with respect to the suitability of any security or specific investment. QuantConnect makes no guarantees as to the accuracy or completeness of the views expressed in the website. The views are subject to change, and may have become unreliable for various reasons, including changes in market conditions or economic circumstances. All investments involve risk, including loss of principal. You should consult with an investment professional before making any investment decisions.
0
The material on this website is provided for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation or endorsement for any security or strategy, nor does it constitute an offer to provide investment advisory services by QuantConnect. In addition, the material offers no opinion with respect to the suitability of any security or specific investment. QuantConnect makes no guarantees as to the accuracy or completeness of the views expressed in the website. The views are subject to change, and may have become unreliable for various reasons, including changes in market conditions or economic circumstances. All investments involve risk, including loss of principal. You should consult with an investment professional before making any investment decisions.
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