Stationarity and Predictability
Stationarity forms the basis of mean reversion strategies. A stationo time series shows consistent statistical properties. Its mean, variance, and autocorrelation remain stable. Non-stationo series lack this consistency. Their statistical properties change. This makes future behavior unpredictable. Mean reversion relies on predictable returns to a long-term average. Without stationarity, this average shifts. This invalidates the strategy.
Consider a stock price, like Apple (AAPL). It generally trends upwards over decades. This trend makes it non-stationo. Its mean price constantly increases. A mean reversion strategy on raw AAPL prices would fail. The "mean" moves. However, AAPL's daily returns often show stationarity. The average daily return might stay around 0.05% with consistent variance. This stationarity in returns allows mean reversion on relative price movements, not absolute price levels.
Identifying Non-Stationarity: Tests and Transformations
Traders use statistical tests to find non-stationarity. The Augmented Dickey-Fuller (ADF) test is common. It tests the null hypothesis that a unit root exists. A unit root indicates non-stationarity. Rejecting the null hypothesis suggests stationarity.
Let's test a hypothetical stock price series, P.
P = [100, 102, 105, 103, 107, 109, 112, 110, 115, 118]
This series trends upwards. An ADF test would likely confirm non-stationarity. The p-value would be high. It would fail to reject the unit root hypothesis.
To achieve stationarity, traders apply transformations. Differencing is a primary method. First-differencing calculates the change between consecutive observations.
dP = P(t) - P(t-1)
For our hypothetical P:
dP = [2, 3, -2, 4, 2, 3, -2, 5, 3]
This dP series represents daily returns or price changes. It appears more stationo than the original price series. Its mean stays around 2.22. Its variance seems more stable. An ADF test on dP would likely yield a low p-value. It would reject the unit root hypothesis.
Other transformations include logarithmic differencing (calculating log returns) or detrending. Detrending removes a deterministic trend component. For mean reversion, traders often work with differenced series or cointegrated pairs.
Cointegration: Stationarity in Pairs
Individual asset prices often show non-stationarity. However, a linear combination of two or more non-stationo series can be stationo. This phenomenon is cointegration. Cointegration forms the basis for pairs trading strategies.
Consider two stocks, XYZ and ABC. Both may trend upwards independently.
XYZ = [50, 51, 52, 53, 54, 55]
ABC = [25, 25.5, 26, 26.5, 27, 27.5]
Individually, these series are non-stationo. They trend.
However, their spread might be stationo.
Define the spread S = XYZ - 2*ABC.
S = [50 - 2*25, 51 - 2*25.5, 52 - 2*26, 53 - 2*26.5, 54 - 2*27, 55 - 2*27.5]
S = [0, 0, 0, 0, 0, 0]
In this perfect example, the spread S is stationo. It maintains a constant mean (zero) and variance (zero). This implies a stable relationship. If XYZ deviates from 2*ABC, it suggests a temporary imbalance. A mean reversion strategy would bet on the spread returning to zero.
Real-world cointegration is seldom zero. It involves a stationo, but fluctuating, spread. Traders use the Engle-Granger test or Johansen test to determine cointegration. A low p-value from these tests indicates cointegration.
For example, consider the spread between two highly correlated ETFs, SPY (S&P 500) and IVV (iShares Core S&P 500). Both track the same index. From January 1, 2020, to December 31, 2020, their prices moved in tandem. Individual prices increased significantly. The ratio SPY/IVV remained relatively stable. It fluctuated around 1.0005. This ratio likely exhibited stationarity. A trader could short SPY and long IVV if the ratio deviated significantly above 1.0005. They would expect it to revert.
Implications for Mean Reversion Strategy Design
Stationarity determines the choice of mean reversion model.
If the asset itself is stationo (e.g., volatility index VIX), an Ornstein-Uhlenbeck process might model its behavior directly. This process describes a particle's movement towards a central mean, with random fluctuations.
dX(t) = \theta(\mu - X(t))dt + \sigma dW(t)
Here, X(t) is the asset price, \mu is the long-term mean, \theta is the speed of reversion, and \sigma is the volatility.
If the asset is non-stationo, traders must work with stationo transformations.
For example, a stock price P(t) is non-stationo. Its log returns r(t) = log(P(t)/P(t-1)) are often stationo. Mean reversion strategies would then target deviations in these returns. They would not target the absolute price.
A simple strategy might buy when r(t) is significantly negative (oversold). It would sell when r(t) is significantly positive (overbought). It expects a return to its mean return.
For cointegrated pairs, the stationo spread becomes the tradable instrument. The strategy involves monitoring the spread. When the spread deviates beyond a certain threshold (e.g., two standard deviations from its mean), a trader initiates a position. If the spread S = Asset1 - \beta * Asset2 moves to mean(S) + 2*std(S), the trader shorts Asset1 and longs \beta * Asset2. The expectation is for the spread to revert to its mean.*
Ignoring stationarity causes spurious regressions and unreliable trading signals. A strategy built on a non-stationo series might appear profitable during backtesting. Its performance would likely collapse out-of-sample. The "mean" it attempts to revert to would have shifted.
Practical implementation requires continuous monitoring of stationarity. Market regimes change. A series stationo last year might become non-stationo today. Re-evaluate stationarity tests regularly. Adjust transformation methods or cointegration relationships as needed.