Skewing your quotes for inventory risk

Question

You are a market maker holding an unwanted inventory of $q = +100$ shares (long) in a $\$ 50 $stock. Holding inventory is risky, and you're mildly risk-averse. The stock's variance is$ \sigma^2 = 0.25 $(in$ $^2 $per day), your risk-aversion is$ \gamma = 0.001 $, and the horizon is$ T = 1$ day.

Where should you center your quotes, at the $\$ 50$ mid, or somewhere else, and by how much?

Accepted Answer

When you carry inventory, your personal fair value isn't the market mid, it's shifted to reflect the risk of holding that position. This shifted value is the reservation price (the Avellaneda–Stoikov idea): r = S - q\,\gamma\,\sigma^2\,(T-t). The correction q\gamma\sigma^2(T-t) is the risk penalty for holding q shares: it grows with the size of your inventory, your risk-aversion, the stock's variance, and the time you'll be exposed. Plug in: r = 50 - (100)(0.001)(0.25)(1) = 50 - 0.025 = \49.975. So you should center your quotes around \49.975, about 2.5 cents below the market mid, rather than…

Skewing your quotes for inventory risk

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