Statistical Arbitrage. A Technical Guide.

A stochastic process $\{X_t\}$ is weak-sense stationary (or covariance-stationary) if its first two moments are time-invariant:

For statistical arbitrage purposes, weak-sense stationarity is sufficient: the process has a well-defined long-run mean to which it tends to return. Strict stationarity, invariance of the full joint distribution under time shifts, is rarely required and rarely tested.

Tests for stationarity in finite samples include the Augmented Dickey-Fuller test (Dickey and Fuller 1979; Said and Dickey 1984), the Phillips-Perron test (Phillips and Perron 1988) and the KPSS test (Kwiatkowski, Phillips, Schmidt and Shin 1992). The first two test the null of a unit root (non-stationarity); the third tests the null of stationarity. Pairs-trading implementations typically employ ADF as the primary qualifying test.

A process $\{X_t\}$ is integrated of order d, written $X_t \sim I(d)$, if its d-th difference $\Delta^d X_t$ is stationary and no lower-order difference is. Most financial price series are well-modelled as $I(1)$: the levels are non-stationary (random-walk like) but the first differences (returns) are approximately stationary.

The implication for trading: if $X_t$ is $I(1)$, then $X_t$ has no time-invariant mean to revert to. Strategies that bet on $X_t$ returning to a normal level without further qualification are statistically misspecified.

Cointegration was introduced by Granger (1981) and formalised by Engle and Granger (1987). Two $I(1)$ series $\{X_t\}$ and $\{Y_t\}$ are cointegrated if there exists $\beta \neq 0$ such that the linear combination

is stationary ($Z_t \sim I(0)$). The coefficient $\beta$ is the cointegrating coefficient (in trading terminology, the hedge ratio). The combination $Z_t$ is the cointegrating residual (in trading terminology, the spread).

The economic interpretation: although neither price has a stable mean, a specific linear combination does. The spread $Z_t$ fluctuates around its long-run mean $\mathbb{E}[Z]$ and exhibits the mean-reversion that pairs trading exploits.

Granger was awarded the 2003 Nobel Memorial Prize in Economic Sciences for methods of analysing economic time series with common trends (Royal Swedish Academy of Sciences 2003).

The Granger Representation Theorem (Engle and Granger 1987, Theorem 1) establishes that if $X_t$ and $Y_t$ are cointegrated, the system admits a Vector Error Correction Model (VECM) representation:

The term $(Y_{t-1} - \beta X_{t-1}) = Z_{t-1}$ is the error-correction term, the previous period s deviation from equilibrium. The coefficients $\alpha_X, \alpha_Y$ are the adjustment speeds governing how each series responds to disequilibrium. At least one of $\alpha_X, \alpha_Y$ is nonzero by the cointegration assumption.

For pairs trading, the practical significance is that the VECM provides predictability. Deviations from equilibrium contain information about future short-term returns of both legs. The adjustment-speed coefficients determine which leg is expected to do most of the correcting.

For an n-dimensional $I(1)$ process $\mathbf{Y}_t = (Y_{1,t}, \ldots, Y_{n,t})^\top$, Johansen (1988, 1991) and Johansen and Juselius (1990) developed the maximum-likelihood framework. The VECM is

where the rank of the long-run impact matrix $\Pi$ equals the number of cointegrating relationships. If $\operatorname{rank}(\Pi) = r$ with $0 < r < n$, then $\Pi = \alpha \beta^\top$ with $\alpha, \beta \in \mathbb{R}^{n \times r}$, where the columns of $\beta$ are the cointegrating vectors and the columns of $\alpha$ are the corresponding adjustment-speed vectors.

The Johansen trace statistic and maximum-eigenvalue statistic provide asymptotic tests for the rank r. Critical values were tabulated by Osterwald-Lenum (1992) and refined in subsequent work.

The cointegrating residual $Z_t$ is often modelled as an Ornstein-Uhlenbeck process (Uhlenbeck and Ornstein 1930):

where $\theta > 0$ is the mean-reversion rate, $\mu$ is the long-run mean, $\sigma > 0$ is the diffusion coefficient and $W_t$ is standard Brownian motion. The OU process has stationary distribution $Z_\infty \sim \mathcal{N}(\mu, \sigma^2/(2\theta))$.

The half-life of mean reversion is the expected time for a shock to decay to half its initial magnitude:

For a trading horizon to be commensurate with the mean-reversion timescale, holding periods should be on the order of one to several half-lives. Positions held substantially longer suffer from cumulative risk without commensurate edge. Conventional practice computes $\theta$ from a discrete-time AR(1) fit on the empirical spread:

with $\tau_{1/2} = -\ln 2 / \ln(1 + \phi)$ in the discretised form.

The Hurst exponent $H \in (0, 1)$ characterises the long-range dependence of a time series (Hurst 1951; Mandelbrot and Van Ness 1968). For financial-time-series applications:

• $H < 1/2$ indicates anti-persistence (mean-reverting behaviour)
• $H = 1/2$ indicates a random walk (no long-range dependence)
• $H > 1/2$ indicates persistence (trending behaviour)

Hurst-exponent screening is a useful auxiliary filter for candidate cointegrated pairs: a pair whose ADF test passes but whose spread has $H \geq 0.5$ is likely a false positive. Estimation methods include rescaled range analysis (R/S), the periodogram method and Detrended Fluctuation Analysis (Peng et al. 1994).

The first published academic study of pairs trading on a large scale was Gatev, Goetzmann and Rouwenhorst (2006). Their methodology: over a 12-month formation window, normalise each asset s cumulative total return to start at 1.0. For each pair $(i, j)$ in the universe, compute the sum of squared differences of the normalised price paths:

Sort pairs in ascending order of SSD; select the top k pairs (the original study used the top 20). Over a 6-month trading window, open a divergence position whenever the normalised paths differ by more than 2 historical standard deviations of their formation-period distance. Close when the paths re-converge.

Empirical results on 1962 to 2002 US equities: about 11% annualised excess return, declining over time. Do and Faff (2010, 2012) document substantial post-publication decline in distance-method profitability, consistent with the strategy being arbitraged away as it became widely known.

The cointegration approach replaces the distance criterion with the formal statistical test for a stable long-run relationship. The two main estimation procedures are due to Engle and Granger (1987) and Johansen (1988, 1991).

Engle-Granger two-step procedure. Step 1: estimate the cointegrating regression by OLS,

Step 2: apply the ADF test to the residuals $\{\hat{\varepsilon}_t\}$ using the Engle-Granger critical values (Engle and Yoo 1987; MacKinnon 1991, 2010), which differ from the standard ADF values because the residuals are estimated rather than observed. If the null of unit root is rejected, $(X_t, Y_t)$ are cointegrated with hedge ratio $\hat{\beta}$.

Johansen procedure. For $\mathbf{Y}_t = (Y_{1,t}, \ldots, Y_{n,t})^\top$, estimate the VECM by maximum likelihood under Gaussian innovations. Test $H_0: \operatorname{rank}(\Pi) \leq r$ using the trace statistic:

or the maximum-eigenvalue statistic. The Johansen procedure is symmetric, handles $n > 2$ assets and provides a formal test for the number of cointegrating relationships, but is more sensitive to model specification (lag length, deterministic terms).

The time-series approach models the spread $Z_t$ directly as a stochastic process, most commonly the Ornstein-Uhlenbeck process, and derives trading rules from its properties. Common variants include exponential OU for spreads with multiplicative mean reversion, Cox-Ingersoll-Ross (Cox, Ingersoll, Ross 1985) for non-negative spreads and Lévy-driven OU (Endres and Stübinger 2019) for spreads with non-Gaussian jump behaviour.

Optimal entry/exit thresholds. Bertram (2010) derives analytically the entry and exit z-score thresholds that maximise the long-run expected profit per unit time under the OU model. For symmetric thresholds a (entry) and b (exit) with $a > b$, the expected return per unit time admits a closed-form expression involving the imaginary error function and the Mills ratio. Bertram (2010, eq. 17 to 18) provides the first-order conditions for the optimum. Zeng and Lee (2014) extend the framework to incorporate transaction costs explicitly.

Stochastic optimal control. The time-series approach generalises to the stochastic optimal control problem: given the OU spread, find the optimal portfolio weight $\pi_t \in [-1, 1]$ that maximises expected utility. Mudchanatongsuk, Primbs and Wong (2008) solve this under constant relative risk aversion (CRRA); Jurek (2007) provides a similar treatment with consumption decisions. The optimal weight is typically a smooth function of the spread s distance from equilibrium rather than the discrete in/out trading rules of threshold approaches.

The cointegration approach assumes the dependence between two assets is linear and stationary. The copula approach (Liew and Wu 2013; Stander, Marais and Botha 2013; Krauss and Stübinger 2017) relaxes the linearity assumption by modelling the joint distribution of returns directly using copula functions.

By Sklar s theorem (Sklar 1959), any continuous bivariate distribution F admits the unique decomposition

where $F_X, F_Y$ are the marginal distributions and $C: [0, 1]^2 \to [0, 1]$ is the copula, a function fully characterising the dependence structure independently of the marginals.

The trading signal is constructed from the conditional copula probability (the mispricing index of Liew and Wu 2013):

where $u = F_X(x_t)$, $v = F_Y(y_t)$. Extreme values of MPI indicate that the joint observation is unusual under the historical dependence structure, interpreted as a signal that asset X is mispriced relative to asset Y given the latter s value.

Common copula families include elliptical (Gaussian, Student-t), Archimedean (Clayton, Gumbel, Frank, Joe) and vine constructions (Bedford and Cooke 2002). Selection is typically by AIC, BIC or hypothesis tests (Genest and Rivest 1993; Chen and Fan 2006). The principal limitation is opacity of fitted copulas and the higher dimensionality of the parameter space.

The role of machine learning in pairs trading is principally in pair selection rather than signal generation. Sarmento and Horta (2020) provide the canonical treatment: principal component analysis on returns to reduce dimensionality, followed by density-based clustering (OPTICS, Ankerst et al. 1999, or DBSCAN, Ester et al. 1996) to group assets with similar return profiles. Candidate pairs are formed within clusters and subjected to cointegration testing as a final filter.

Avellaneda and Lee (2010) use PCA differently: they extract synthetic industry factors from the return covariance matrix, then model the idiosyncratic residual of each stock as an OU process. The strategy goes long stocks whose residual is below its equilibrium and short stocks whose residual is above. This is a multi-asset extension of pairs trading implemented at the residual level.

Neural-network forecasting of spread returns (Krauss, Do and Huck 2017; Fischer and Krauss 2018) is highly sensitive to the training window and exhibits poor out-of-sample stability relative to the structural cointegration approach.