For the most of the 2024, I was battling an illness. As a result, this blog and all other projects didn’t get as much love as they deserved. Hopefully, things will improve in 2025.

Let’s get started.

Roll’s model of spreads

This estimator of spreads is based solely on the covariance of returns. Suppose a midquote follows a random walk:

\[m_t = m_{t-1} + \epsilon_t\]

where $\epsilon_t $ has 0 expectation and the $E\lbrace\epsilon_t\epsilon_s\rbrace=0, s\neq t.$ Now suppose, bid-ask spread $S$ is constant. The ask and bid prices become $a_t = m_t + \frac{S}{2},$ and $b_t = m_t - \frac{S}{2}.$ The transaction happening at time $t$ will happen at price

\[p_t = m_t + d_t\frac{S}{2}\]

where $d_t\in\lbrace-1,1\rbrace$ indicates the direction of the initiator of the trade and is assumed to have 0 expectation. Using the above, the monetary return of the back-to-back transactions is

\[\begin{align*} p_t - p_{t-1} =& m_t + \frac{S}{2}d_t - \left(m_{t-1} + \frac{S}{2}d_{t-1}\right)\\ =& \frac{S}{2}d_t - \frac{S}{2}d_{t-1}+\epsilon_t \end{align*}\]

Under the assumptions of the model, $E(m_t - m_{t-1})=0$, $E(\epsilon_t \epsilon_s) = 0$ for $s\neq t$, $E(d_s d_t)=0$ for $s\neq t$ as well as $E(d_t \epsilon_t)=E(d_t\epsilon_{t+1})=0$, we have

\[\begin{align} cov(p_{t+1}-p_t, p_t - p_{t-1}) =&\\ =& E\left\lbrace-\frac{1}{4} S^2 d_t^2+\frac{1}{4} S^2 d_{t-1} d_t-\frac{1}{4} S^2 d_{t-1} d_{t+1}+ \frac{1}{4} S^2 d_t d_{t+1}-\frac{1}{2} S d_t \epsilon_t+\\ + \frac{1}{2} S d_{t+1} \epsilon_t-\frac{1}{2} S d_{t-1} \epsilon _{t+1}+\frac{1}{2} S d_t \epsilon_{t+1}+\epsilon_t \epsilon_{t+1}\right\rbrace\\ =& \frac{1}{4} \left(S^2 E\left(d_{t-1} d_t\right)-S^2 E\left(d_t^2\right)-S^2 E\left(d_{t-1} d_{t+1}\right)+S^2 E\left(d_t d_{t+1}\right)\right)\\ =& -\frac{S^2}{4} \end{align}\]

Finally, we obtain the Roll’s spread estimator

\[S_R = 2\sqrt{-cov(p_{t+1}-p_t, p_t - p_{t-1})}\]

There are extensions possible to this model that account for

  • unbalanced order flow
  • autocorrelation of orders
  • midquote effect of orders
  • varying expected return

All of these bias or tilt the Roll’s estimator and require corrections. Those are tedious but largely mechanical to derive.