The Roll Spread Model

Suppose all you have is a list of prices a stock traded at — no bid, no ask, just the tape of executions. Can you figure out how wide the bid-ask spread was? Richard Roll showed in 1984 that you can, using one elegant observation: because trades randomly hit the bid or lift the offer, the traded price keeps bouncing between the two, and that bounce leaves a fingerprint in the numbers.

Picture a stock whose true value sits still at $50.00 with a spread of 4 cents, so the bid is $49.98 and the ask is $50.02. A buyer's trade prints at $50.02; the next trade, if a seller, prints at $49.98; then maybe a buyer again at $50.02. The value never moved, but the printed price zig-zags up and down. This is the bid-ask bounce, and it makes each price change tend to reverse the last one.

From bounce to spread

Roll modeled trades as equally likely to be buys or sells, independent of each other, around a fixed value. Under those assumptions the change in price from one trade to the next has a negative first-order autocovariance:

$$\operatorname{Cov}(\Delta p_t,\, \Delta p_{t-1}) = -\frac{s^2}{4},$$

where $\Delta p_t$ is the change in traded price from trade $t-1$ to trade $t$, and $s$ is the effective bid-ask spread (in the same units as price). The minus sign is the whole story: it says an up-tick is typically followed by a down-tick. Turning the equation around to solve for the spread gives Roll's estimator:

$$s = 2\sqrt{-\operatorname{Cov}(\Delta p_t,\, \Delta p_{t-1})} .$$

In words: take consecutive price changes, compute how much a move tends to reverse (the negative covariance), flip its sign, square-root it, and double it. Out pops the spread — from prices alone.

Worked example

You collect nine consecutive trade prices (in dollars): 50.02, 49.98, 50.02, 49.98, 49.98, 50.02, 49.98, 50.02, 49.98. The eight price changes $\Delta p$ are:

$$-0.04,\; +0.04,\; -0.04,\; 0,\; +0.04,\; -0.04,\; +0.04,\; -0.04 .$$

To get the first-order autocovariance, pair each change with the previous one and average the products. The seven pairs $(\Delta p_{t-1}, \Delta p_t)$ give products:

$$(-0.04)(+0.04),\, (+0.04)(-0.04),\, (-0.04)(0),\, (0)(+0.04),\, (+0.04)(-0.04),\, (-0.04)(+0.04),\, (+0.04)(-0.04)$$

which are $-0.0016,\, -0.0016,\, 0,\, 0,\, -0.0016,\, -0.0016,\, -0.0016$. Their average is

$$\operatorname{Cov} \approx \frac{-0.0080}{7} = -0.00114 .$$

Plug into Roll's formula:

$$s = 2\sqrt{0.00114} = 2 \times 0.0338 = 0.068 .$$

So the implied spread is about 6.8 cents — in the right neighborhood of the true 4-cent spread, and it would converge closer with a longer sample. The point is not the exact number from nine trades but that a pure price series handed us a spread estimate at all.

Where it sits

Roll launched the whole literature on inferring microstructure costs from transaction data, later refined by models that split the spread into its parts (order-processing, inventory, and adverse-selection components — see Bid-Ask Spread Decomposition). It remains a first-pass estimator when quote data is missing or unreliable, and its core insight — that the mechanical bounce between bid and ask hides inside the autocorrelation of prices — is one of the most quoted results in all of market microstructure.