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Inventory with a daily unwind

Market makingEasy

A market maker unwinds $10%$ of its inventory each day and then takes a new net fill: $I_{t} = 0.9 I_{t - 1} + N_{t}$ . Fills are independent with mean $0$ and variance $19$ . What is the long-run variance of the inventory? Give a number.

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Hint

Take variances on both sides and solve for the stationary value.

Solution

The stationary variance solves $V = 0.81 V + 19$ , because $0.9 \times 0.9 = 0.81$ .
So $1 - 0.81 = 0.19$ and $V = \frac{19}{0.19} = 100$ .

Answer: $100$

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