The Roll Spread Estimator
You observe only transaction prices $\{P_t\}$ at equally spaced times. Under the Roll model:
- The efficient ("true") price $M_t$ follows a random walk: $M_t = M_{t-1} + u_t$, where $u_t$ are i.i.d. mean-zero innovations.
- Each trade occurs at either the bid or the ask with equal probability, so the observed transaction price is $P_t = M_t + c \cdot Q_t$, where $Q_t \in \{-1, +1\}$ is an i.i.d. sign (each with probability
/2$) and $c > 0$ is the half-spread.
- The trade direction $Q_t$ is independent of the efficient price innovations $u_t$.
Let $\Delta P_t = P_t - P_{t-1}$ denote the price change.
1. Show that $\text{Cov}(\Delta P_t, \Delta P_{t-1}) = -c^2$.
2. Derive the Roll estimator $\hat{c} = \sqrt{-\widehat{\text{Cov}}(\Delta P_t, \Delta P_{t-1})}$, where the sample autocovariance replaces the population quantity.
3. Explain why this estimator can fail in practice and what modifications are used.
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