Cointegration and statistical arbitrage

Definition Dates are integers. A process is integrated of order zero, written I(0), if it is covariance stationary with an autocovariance function $\gamma$ that satisfies $\sum_{h} \lvert \gamma(h) \rvert < \infty$ and has positive long-run variance

Under the summability condition, $n \operatorname{Var}(\bar x_n) \to \omega^2$ for the mean $\bar x_n$ of $n$ consecutive values, which is why $\omega^2$ is called the long-run variance. A process $X$ is integrated of order one, I(1), if its first difference $\Delta X_t = X_t - X_{t-1}$ is I(0), and more generally I($d$) if $d$ differences give an I(0) process. The positive long-run variance makes the order unique. The difference of an I(0) process has autocovariances $2\gamma(h) - \gamma(h - 1) - \gamma(h + 1)$, whose sum is $0$, so it is not I(0), and in particular the level of an I(1) process is not I(0). These conditions fix the short-memory meaning used in this lesson, and other texts use other conventions.

Definition A common trend is an I(1) component shared by several series. In the first constructed model of this lesson, $Z$ is a random walk started at $Z_0 = 0$ with independent, identically distributed increments $\eta_t$ of mean $0$ and variance $\sigma_\eta^2 > 0$, and $(e_t)$ and $(v_t)$ are independent, identically distributed sequences with mean $0$ and variances $\sigma_e^2 > 0$ and $\sigma_v^2 > 0$, the three sequences being mutually independent. The log series are

Both carry the trend $Z$, with loadings $1$ and $2$, and $\operatorname{Var}(X_t) = t\,\sigma_\eta^2 + \sigma_e^2$ and $\operatorname{Var}(Y_t) = 4t\,\sigma_\eta^2 + \sigma_v^2$ grow linearly in $t$, so neither level is covariance stationary. The difference $\Delta X_t = \eta_t + e_t - e_{t-1}$ has $\gamma(0) = \sigma_\eta^2 + 2\sigma_e^2$, $\gamma(\pm 1) = -\sigma_e^2$ and $\gamma(h) = 0$ for $\lvert h \rvert \ge 2$, so its long-run variance is $\sigma_\eta^2 > 0$, and in the same way $\Delta Y_t$ has long-run variance $4\sigma_\eta^2$. Both $X$ and $Y$ are therefore I(1). Systems of more series can have several common trends and several independent cointegrating relations, and in the standard representation of a linear I(1) system of $k$ series with $r$ independent cointegrating vectors there are $k - r$ common trends. That representation is further study.

Definition Two I(1) series $X$ and $Y$ are cointegrated if some nonzero vector $(a, b)$ makes $aX_t + bY_t$ an I(0) process. Such a vector is a cointegrating vector. It is defined only up to a nonzero factor, and a normalization such as $b = 1$ fixes it. In the common-trend model,

an independent, identically distributed sequence, hence I(0), with variance $\sigma_v^2 + 4\sigma_e^2$ by independence. So $(-2, 1)$ is a cointegrating vector, and up to a factor it is the only one. Indeed $aX_t + bY_t = (a + 2b)Z_t + a e_t + b v_t$, and when $a + 2b \ne 0$ its difference has long-run variance $(a + 2b)^2\sigma_\eta^2 > 0$, so the combination is I(1). Cointegration is a statement about the levels and their stochastic trend. A high sample correlation of two levels does not establish it, and two independent random walks, which are not cointegrated, have a sample correlation of levels that does not settle at zero as the sample grows.

Definition In a second constructed model, $X$ is a random walk of log prices with independent, identically distributed increments of positive, finite variance, $Y_t = 2X_t + z_t$, and the spread follows

The innovations $u_t$ are independent and identically distributed with mean $0$ and variance $\sigma_u^2 > 0$, and the whole sequence $(u_t)$ is independent of $X$. Let $\mathcal{F}_t$ be the information generated by $X_s$ and $Y_s$ for $s \le t$, so that $z_t = Y_t - 2X_t$ is known at $t$ and $u_t$ is independent of $\mathcal{F}_{t-1}$. The stationary solution is

which converges in mean square because $\sigma_u^2 \sum_j \rho^{2j} < \infty$, and its variance is

The spread is I(0), and $\Delta Y_t = 2\,\Delta X_t + \Delta z_t$ has long-run variance $4\operatorname{Var}(\Delta X_t) > 0$, so $X$ and $Y$ are I(1) and cointegrated with the vector $(-2, 1)$. For a horizon $h \ge 1$, the forecast and its conditional variance are

When $\rho \ne 0$ the conditional variance is below $\bar\sigma_z^2$ for every $h$ and approaches it as $h$ grows. At $\rho = 1$ the spread is a random walk, the denominator $1 - \rho^2$ vanishes and no stationary solution exists.

Definition In the stationary-spread model, subtract the level identity $Y_{t-1} = 2X_{t-1} + z_{t-1}$ from $Y_t = 2X_t + z_t$, and subtract $z_{t-1}$ from both sides of the spread recursion, which gives $\Delta z_t = (\rho - 1)z_{t-1} + u_t$. Replacing $z_{t-1}$ by $Y_{t-1} - 2X_{t-1}$ gives the error-correction relation

The coefficient $\rho - 1$ of the lagged deviation is the adjustment coefficient, and $\lvert \rho \rvert < 1$ means it lies strictly between $-2$ and $0$. The relation follows from the two identities alone, with no separate assumption that prices move toward each other. Because $u_t$ has mean $0$ and is independent of $\mathcal{F}_{t-1}$ and $\Delta X_t$ together,

When $z_{t-1} > 0$ the expected change of the spread is negative, and when $z_{t-1} < 0$ it is positive. For $-1 < \rho < 0$ the expected next spread $\rho z_{t-1}$ also has the opposite sign, so the correction overshoots. A realized innovation can still push the spread further from zero. In larger systems the link between cointegration and error correction is the Granger representation theorem, which is further study.

Definition Statistical arbitrage is a name for trading strategies whose expected profit rests on a statistical model, for example positions that bet on a stationary spread moving back toward its mean. It is not an arbitrage in the pricing sense, which needs zero or negative initial cost, a nonnegative payoff in every state and a positive payoff with positive probability, because a spread position can lose money. Assessing such a strategy takes its wealth process, built from holdings in shares of instruments with currency prices $P^X_t$ and $P^Y_t$, a cash account with its financing, rebalancing that satisfies the self-financing condition, and costs and margin. A coefficient in a relation between log prices is not a number of shares. With $X_t = \ln P^X_t$ and $Y_t = \ln P^Y_t$, a position short one currency unit of the second instrument and long two currency units of the first, rebalanced to these values every period, earns over one period, before dividends, financing and costs,

where the approximation is the first-order Taylor expansion of the exponential. The exact result depends on both log changes, not only on the spread.