Splitting a quote into width and skew

Question

The Avellaneda–Stoikov framework says an optimal market maker quotes around a reservation price (fair value shifted for inventory), with a symmetric width set by risk and competition:

$r = s - q\,\gamma\,\sigma^2 (T-t), \qquad \text{spread} = \gamma\,\sigma^2 (T-t) + \frac{2}{\gamma}\ln\!\left(1 + \frac{\gamma}{k}\right).$

Take mid $s = 100$ , inventory $q = +3$ units, risk aversion $\gamma = 0.5$ , volatility $\sigma = 2$ (so $\sigma^2 = 4$ ), time remaining $T - t = 0.25$ , and order-flow parameter $k = 1.5$ (constants stylized for a clean plug-in).

Compute the reservation price, the optimal spread, and the resulting bid and ask.

Accepted Answer

Reservation price (the skew). Being long (q = +3) drags fair value down: r = 100 - (3)(0.5)(4)(0.25) = 100 - 1.5 = 98.5. So even though the mid is 100, you behave as if the asset is worth 98.5 to you, because you're carrying too much of it. Optimal spread (the width). Two additive pieces, an inventory/risk term and a competition term: \underbrace{\gamma\sigma^2(T-t)}{	ext{risk}} = (0.5)(4)(0.25) = 0.5, \underbrace{\frac{2}{\gamma}\ln\!\left(1+\frac{\gamma}{k}ight)}{	ext{competition}} = \frac{2}{0.5}\ln\!\left(1+\frac{0.5}{1.5}ight) = 4\ln(1.3333) = 4(0.2877) = 1.1507. 	ext{spread} = 0.…

Splitting a quote into width and skew

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