5.2 The Roll model: spread from bid–ask bounce

Motivation

Trades alternate between the bid and the ask. That alternation makes trade-price changes negatively autocorrelated even when the underlying value is a random walk. Roll (1984) turned this into an estimator of the spread that needs only trade prices. It is the first thing to compute when someone shows you a mean-reversion strategy on tick or minute data: much of the "reversion" may be bounce.

5.2.1The bid–ask bounce model#

Setup

Let the efficient price (the expected value of the asset given public information) follow a random walk, and trades occur at the bid or the ask:

m_t = m_{t-1} + \varepsilon_t, \qquad p_t = m_t + \tfrac{s}{2}\, q_t, \tag{5.2}

with assumptions:

(A1) $\varepsilon_t$ i.i.d., mean 0, variance $\sigma^2$ ;
(A2) $q_t \in \{-1, +1\}$ i.i.d. with $P(q_t = 1) = 1/2$ ;
(A3) $q$ and $\varepsilon$ are independent at all leads and lags.

The spread $s$ is constant; the dealer's quotes are $m_t \pm s/2$ .

Proposition 5.1Roll, 1984

Under (A1)–(A3), with $\Delta p_t = p_t - p_{t-1}$ ,

\begin{gathered} \operatorname{Var}(\Delta p_t) = \sigma^2 + \frac{s^2}{2}, \qquad \operatorname{Cov}(\Delta p_t, \Delta p_{t-1}) = -\frac{s^2}{4}, \\ \operatorname{Cov}(\Delta p_t, \Delta p_{t-k}) = 0 \;\; (k \ge 2). \end{gathered} \tag{5.3}

Hence $s = 2\sqrt{-\operatorname{Cov}(\Delta p_t, \Delta p_{t-1})}$ , and the Roll estimator is

\hat s = 2\sqrt{-\hat\gamma_1}, \qquad \hat\gamma_1 = \text{sample first-order autocovariance of } \Delta p. \tag{5.4}

Proof

From (5.2), $\Delta p_t = \varepsilon_t + \tfrac{s}{2}(q_t - q_{t-1})$ . Under (A2), $E q_t = 0$ and $\operatorname{Var}(q_t) = 1$ , and the $q$ 's are uncorrelated across time. By (A3) the $\varepsilon$ and $q$ terms are uncorrelated, so covariances add:

$\operatorname{Var}(\Delta p_t) = \sigma^2 + \frac{s^2}{4}\operatorname{Var}(q_t - q_{t-1}) = \sigma^2 + \frac{s^2}{4}\cdot 2 = \sigma^2 + \frac{s^2}{2}.$

$\operatorname{Cov}(\Delta p_t, \Delta p_{t-1}) = \operatorname{Cov}(\varepsilon_t, \varepsilon_{t-1}) + \frac{s^2}{4}\operatorname{Cov}(q_t - q_{t-1},\, q_{t-1} - q_{t-2})$ . The first term is 0 by (A1). Expanding the second, $\operatorname{Cov}(q_t, q_{t-1}) - \operatorname{Cov}(q_t, q_{t-2}) - \operatorname{Var}(q_{t-1}) + \operatorname{Cov}(q_{t-1}, q_{t-2}) = 0 - 0 - 1 + 0 = -1$ . So $\operatorname{Cov} = -s^2/4$ .

For $k \ge 2$ , $\Delta p_t$ involves $\varepsilon_t, q_t, q_{t-1}$ and $\Delta p_{t-k}$ involves $\varepsilon_{t-k}, q_{t-k}, q_{t-k-1}$ , which share no index when $k \ge 2$ , so the covariance is 0.∎

The first-order autocorrelation of trade-price changes is $\rho_1 = -\frac{s^2/4}{\sigma^2 + s^2/2}$ , which lies in $(-\tfrac12, 0)$ ; it is close to $-\tfrac12$ when the spread dominates the efficient-price noise, which is the typical situation at the tick level.

5.2.2Estimating the spread from trades#

Worked example

On one day, an NSE large-cap trading near ₹1,500 has 30,000 trades. The sample first-order autocovariance of trade-to-trade price changes is $\hat\gamma_1 = -0.0009$ (₹²). Then $\hat s = 2\sqrt{0.0009} = \text{₹}0.06$ , i.e. 0.4 bp of price. This is the effective spread implied by bounce, which can differ from the quoted spread (queue position and hidden orders mean many trades occur inside the quote).

Code Exercise m05_roll (in the app: Exercises → m05_roll) asks you to simulate (5.2), implement (5.4), and check that $\hat s$ recovers the true $s$ within a few percent on 300,000 simulated trades. A second test adds a proportional bounce to a GARCH, fat-tailed log-price path from toolkit.data.synthetic_prices and recovers a 100 bp spread from 20,000 observations within 20%. Problem 5.2 extends the model to persistent order flow, where the estimator is biased down.

5.2.3Limits and strategy hygiene#

Practitioner notes

When the estimator fails. If $\hat\gamma_1 > 0$ the estimator is undefined. This happens often in daily data, where the bounce is small relative to $\sigma$ and returns may have small positive autocorrelation from slow information diffusion. Use Roll on tick or one-minute data, or use Hasbrouck's Bayesian version (Hasbrouck 2007, Ch. 3).
Order-flow persistence. Real order flow is positively autocorrelated (large orders are split; M05 · 5.5), violating (A2). With $\operatorname{Corr}(q_t, q_{t-1}) = \rho$ and geometric decay, the autocovariance becomes $-\frac{s^2}{4}(1-\rho)^2$ (Problem 5.2), so Roll underestimates $s$ by the factor $(1 - \rho)$ .
Adverse selection. If trades carry information (the next sections), $m_t$ moves in the direction of $q_t$ , violating (A3); Roll then measures only part of the effective spread (Problem 5.3).
Strategy hygiene. A "mean-reversion" backtest on trade prices that buys after down-ticks and sells after up-ticks earns the bounce on paper but pays it in reality: you buy at the ask, not at the last trade. Always backtest intraday strategies on bid and ask, or on mids plus a spread.