XAU / PAXG-XAUT Pairs Stationarity Analysis

This notebook will provide arbitrage indications for spot to asset-backed cryptocurrency pairs. In particular, the XAUt/USDt and PAXG/USDt pairs will be analyzed against spot gold - taking advantage of low volume pair trading. In addition, this notebook will aim to run statistical tests to identify stationarity and mean reversion.

Data Consolidation

Create a helper function to fetch the data from the pyth price oracle.

# fetch data from the 'pyth' price oracle. import requests as req def getBarsPyth(symbol, resolution, start, end): return req.get("https://benchmarks.pyth.network/v1/shims/tradingview/history?symbol={0}&resolution={1}&from={2}&to={3}".format(symbol, resolution, start, end)).json()

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# fetch bars as JSON objecobject import time import datetime as dt interval = 60 # default = 60 start, end = dt.date(2025, 1, 6), dt.date(2025, 3, 13) # default = 1/4/2024 -> today XAU = getBarsPyth("Metal.XAU%2FUSD", interval, int(time.mktime(start.timetuple())), int(time.mktime(end.timetuple()))) XAUT = getBarsPyth("Crypto.XAUT%2FUSD", interval, int(time.mktime(start.timetuple())), int(time.mktime(end.timetuple()))) PAXG = getBarsPyth("Crypto.PAXG%2FUSD", interval, int(time.mktime(start.timetuple())), int(time.mktime(end.timetuple())))

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Data Cleaning

With the dataset from pyth, place the datasets into a dictionary to then be merged into one DataFrame, using time (in UNIX) as an index.

# create timeseries as dict import pandas as pd import matplotlib.pyplot as plt xau_pair = {'t': XAU['t'], 'c_xau': XAU['c']} xaut_pair = {'t': XAUT['t'], 'c_xaut': XAUT['c']} paxg_pair = {'t': PAXG['t'], 'c_paxg': PAXG['c']} # merge data into dataframe (data) df1, df2, df3 = pd.DataFrame(xau_pair), pd.DataFrame(xaut_pair), pd.DataFrame(paxg_pair) df1.set_index('t', inplace=True) df2.set_index('t', inplace=True) df3.set_index('t', inplace=True) merged = pd.concat([df1, df2, df3], axis=1, join='outer') merged.sort_values(by='t', inplace=True) merged = merged.reset_index() merged['time'] = pd.to_datetime(merged['t'],unit='s') merged = merged.set_index('t') data = merged

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Normalize XAU, XAUt, and PAXG by taking the percent difference of the closing price of each bar.

# adjust for price difference, normalize with pct change data['pct_c_xau'] = (data['c_xau'] / data['c_xau'].iloc[0] - 1) * 100 data['pct_c_xaut'] = (data['c_xaut'] / data['c_xaut'].iloc[0] - 1) * 100 data['pct_c_paxg'] = (data['c_paxg'] / data['c_paxg'].iloc[0] - 1) * 100

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Calculate the spread of each datapoint with respect to XAU (w/ a XAUt-PAXG spread).

# spread data['pct_c_xau_interpolated'] = data['pct_c_xau'].interpolate(method='linear') gold_i, paxg_i, xaut_i = data['pct_c_xau_interpolated'], data['pct_c_paxg'], data['pct_c_xaut'] aligned_goldGP, aligned_paxgGP = gold_i.align(paxg_i, join='outer') aligned_goldGX, aligned_xautGX = gold_i.align(xaut_i, join='outer') aligned_paxgPX, aligned_xautPX = paxg_i.align(xaut_i, join='outer') data['spread_xaupaxg'] = aligned_goldGP - aligned_paxgGP data['spread_xauxaut'] = aligned_goldGX - aligned_xautGX data['spread_xautpaxg'] = aligned_xautPX - aligned_paxgPX

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Data Analysis

Let us plot the graph of PAXG, XAUt, and XAU - both raw and normalized; additionally plot the correlation of PAXG and XAUt with respect to XAU.

# graph of percent changes of gold, xaut, and paxg fig, axes = plt.subplots(nrows=2, ncols=2) data.loc[:,['c_xau', 'c_xaut', 'c_paxg', 'time']].plot(x = 'time', ax=axes[0,0]) data.loc[:,['pct_c_xau', 'pct_c_xaut', 'pct_c_paxg', 'time']].plot(x = 'time', ax=axes[0,1]) data.plot(kind='scatter', x='c_xau', y='c_xaut', s=16, alpha=0.4, ax=axes[1,0]) data.plot(kind='scatter', x='c_xau', y='c_paxg', s=16, alpha=0.4, ax=axes[1,1])

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data.loc[:,['pct_c_xau', 'pct_c_xaut', 'pct_c_paxg', 'time']].plot(x = 'time')

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Analytically, we can show that XAU, XAUt, and PAXG all exhibit strong correlation. Since these datasets are meant to be directly proportional to each other, any variation we exhibit in this dataset is a theoretical risk-free profit. However, before we put these claims into practice, let us perform statistical tests to prove direct proportionality, stationarity, and mean reversion.

print(data.loc[:,['c_xau', 'c_xaut', 'c_paxg']].corr())

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As this DataFrame is from January to Mid-May, on the assumption that this data isn't mean reverting [1], the expected loss for an unleveraged portfolio would be the spread between XAUt and PAXG. However, we see a strong positive correlation. To take advantage of spread trading, let us create three datasets - the spread between XAUt, PAXG w/ respect to XAU, and a XAUt-PAXG spread.

[1] A mean reverting series is a group of data that retains 'memory' - as a series deviate from a given mean, it tends to traverse back to the mean (hence 'reverting').

graph = data.copy() graph['root'] = aligned_goldGP - aligned_goldGP # this is such a bad way to initialize all zero values. graph.loc[:,['spread_xaupaxg', 'spread_xauxaut', 'spread_xautpaxg', 'root', 'time']].plot(x = 'time')

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Augmented Dickey-Fuller Test

import statsmodels.tsa.stattools as ts adf_xauxaut = ts.adfuller(data['spread_xauxaut'], 1) # perform adf on xau-xaut adf_xaupaxg = ts.adfuller(data['spread_xaupaxg'], 1) # perform adf on xau-paxg adf_xautpaxg = ts.adfuller(data['spread_xautpaxg'], 1) # perform adf on xaut-paxg print("adfuller for XAU-XAUT: {0}".format(adf_xauxaut)) print("adfuller for XAU-PAXG: {0}".format(adf_xaupaxg)) print("adfuller for XAUT-PAXG: {0}".format(adf_xautpaxg)) print(type(adf_xaupaxg))

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Hurst Exponent

# create hurst exponent function import numpy as np from numpy import cumsum, log, polyfit, sqrt, std, subtract # input is numpy array def hurst(ts): lags = range(2, 100) # create the range of lagging values tau = [sqrt(std(subtract(ts[lag:], ts[:-lag]))) for lag in lags] # variance m = polyfit(log(lags), log(tau), 1) # polynomial fit return m[0] * 2 # return exponent print("hurst(XAU-XAUT): {0}".format(hurst(data['spread_xauxaut'].values))) print("hurst(XAU-PAXG): {0}".format(hurst(data['spread_xaupaxg'].values))) print("hurst(XAUT-PAXG): {0}".format(hurst(data['spread_xautpaxg'].values)))

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Nice.

.css-15w88e5{color:var(--chakra-colors-fg-neutral-primary);font-weight:inherit;letter-spacing:-0.09px;}XAU / PAXG-XAUT Pairs Stationarity Analysis

Data Consolidation

Data Cleaning

Data Analysis

Augmented Dickey-Fuller Test

Hurst Exponent

XAU / PAXG-XAUT Pairs Stationarity Analysis