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A spread from a covariance

Market makingMedium

Trades print at $p_{t} = m + (\frac{s}{2}) q_{t}$ , where the mid $m$ is fixed and each $q_{t}$ is $+ 1$ or $- 1$ with equal chance, independently across trades. The covariance between consecutive price changes $Δ p_{t}$ and $Δ p_{t - 1}$ is $- 2.25$ squared cents. What is the bid-ask spread $s$ in cents? Give a number.

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Hint

Write each price change in terms of $q$ and find the term that consecutive changes share.

Solution

Here $Δ p_{t} = (\frac{s}{2}) (q_{t} - q_{t - 1})$ .
Consecutive changes share only $q_{t - 1}$ , with opposite signs, and $Var (q_{t}) = 1$ , so the covariance is $- \frac{s ^{2}}{4}$ .
Then $s^{2} = 4 \times 2.25 = 9$ , so the spread is $s = 3$ .

Answer: $3$

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