6.3.4 Quote Skewing

Basic Prerequisites: 03 · The Ho–Stoll Model and A–S: Inventory & Risk Aversion.

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1. Intuition & Practical Objective

This is the action page: how the reservation price becomes a quote. The practical objective is one mechanism - shift both quotes against the position - and a quantitative law for how far. The Ho–Stoll / A–S reservation price says the inventory-controlled centre is

$$r(I)=\bar{S}-\gamma {\sigma }^{2}I\tau ,$$

so the quote skew per unit of inventory is $\alpha =\gamma {\sigma }^2\tau$. A linear inventory-control rule makes this explicit:

$$\Delta s=-\alpha (I-I_{\text{target}}),\text{bid}=\text{mid}+\Delta s-\frac{s}{2},\text{ask}=\text{mid}+\Delta s+\frac{s}{2}.$$

Long inventory ⇒ both quotes shift down (attract sellers, discourage buyers); short ⇒ shift up. This is the same engine as a flat linear skew, and it is exactly A–S quoting around the reservation price.

The headline result of this page: skewing trades a little mean P&L for a large reduction in both P&L variance and inventory variance. The simulation below reproduces the A–S qualitative claim and adds the two production variants every desk actually runs - a linear skew and a linear skew plus a hard inventory cap.

The one-line takeaway. “Quote skewing converts an unbounded inventory random walk into a mean-reverting one: at the optimal strength $\alpha =\gamma {\sigma }^2\tau$, terminal-inventory std falls roughly $3\times$, and a hard cap $Q$ bounds it absolutely - at a small, known cost in mean P&L.”

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2. Mathematical Ground Truth & Derivations

The linear skew law. With target inventory $I_{\text{target}}$ (usually 0), the mid-quote skew is proportional to the inventory displacement:

$$\Delta s(q)=-\alpha (q-I_{\text{target}}),$$

giving inventory-controlled quotes

$$p^{\text{bid}}=\text{mid}+\Delta s-\frac{s}{2},p^{\text{ask}}=\text{mid}+\Delta s+\frac{s}{2}.$$

From the reservation-price view, $\Delta s(q)=r(q)-\bar{S}=-\gamma {\sigma }^2(q-I_{\text{target}})\tau$, so the optimal skew per share is

$$\alpha =\gamma {\sigma }^2\tau .$$

It is set by risk aversion, variance, and holding horizon - not by taste. Raising $\alpha$ makes the skew fight inventory harder (and, beyond a point, costs edge).

Nonlinear skew near a hard limit. When $|q|$ approaches a risk limit $Q$, production engines sharpen the skew quadratically (or pull the accumulating side’s quote out of the market):

$$\Delta s(q)=-\alpha \mathrm{sign}(q){\left(\frac{|q|}{Q}\right)}^2.$$

As $|q|\to Q$ the accumulating side is effectively withdrawn (infinite spread) while the unloading side is quoted aggressively to force a fill. This is the continuous version of the hard cap: the skew encourages flattening, the cap guarantees it.

Why variance is the right metric. Under CARA utility a smaller P&L variance is worth more than a larger mean whenever dispersion is high; a mean–variance proxy is $\mathbb{E}[W]-\frac{\gamma }{2}\mathrm{Var}[W]$. Skewing deliberately gives up a little mean for a large variance reduction - the correct trade for a risk-averse dealer.

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3. Computational Implementation - the headline experiment

We simulate a market maker filling at exponential rates $A{e}^{-k\delta }$ on each side, mid-price $dS=\sigma dW$, across 20 000 paths, comparing four strategies: symmetric (no skew), inventory skew (around the A–S reservation price), linear skew $\Delta s=-0.40q$, and skew + hard cap $Q=5$. Reported: mean P&L, P&L std, terminal-inventory std, max position.

Reading the numbers.

• Symmetric quoting earns the highest mean P&L ($60.30$) but with P&L std $17.72$ and terminal-inventory std $8.97$ - a max position of 48 lots. An unbounded inventory random walk.
• Inventory skew (the A–S reservation price) gives up $\sim$ $3 of mean ($56.95$) and in exchange cuts P&L std from $17.72$ to $5.82$ ($3.0\times$) and terminal-inventory std from $8.97$ to $3.19$ ($2.8\times$). This is A–S Table 1’s qualitative result.
• Linear skew $\alpha =0.40$ ($=\gamma {\sigma }^2\tau$) controls inventory hardest: final-q std $0.87$, max position $4$. It pays for that with lower mean ($51.46$) - over-strong for pure inventory control, but exactly the knife-edge $\alpha$ a desk trading off variance picks.
• Skew + hard cap $Q=5$ keeps mean P&L near the pure-skew level ($56.65$) and hard-bounds the position at $5$ - the production choice.

The qualitative law is exact and reproducible: skewing trades a little mean for a lot of variance, and a cap removes the tail.

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4. Failure Modes & First-Principles Breakdowns

1. Skewing is not a guarantee. It encourages flattening but never forbids a position; the symmetric strategy’s max position of 48 shows the tail skewing alone leaves. Only a hard cap removes it (06).
2. Over-skewing costs edge. At $\alpha =0.40$ the linear skew cuts mean P&L to $51.46$ (from $60.30$). The optimal $\alpha$ is $\gamma {\sigma }^2\tau$; larger values are a deliberate safety trade, not a free lunch.
3. The cap trades edge for safety. Bounding $|q|$ removes the tail but also removes the fills that earned the widest spreads; mean P&L falls. Choose $Q$ from a risk budget, not by taste.
4. Quote crossing / marketable quotes. For large $|q|$ the skewed quote can cross the mid or the opposite side; clamp $\Delta s$ and cap $|q|$ before publishing (05).
5. Adverse selection is untouched. Skewing manages the position, not informed flow. Combine with Glosten–Milgrom and Toxic Order Flow & VPIN.

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5. References

• Ho & Stoll (1981), Optimal dealer pricing under transactions and return uncertainty, JFE 9(1)
• Avellaneda & Stoikov (2008), High-frequency trading in a limit order book, Quantitative Finance 8(3)
• Guéant, Lehalle & Fernandez-Tapia (2013), Dealing with the inventory risk, Math. & Financial Econ. 7(4)
• Menkveld (2013), High frequency trading and the new market makers, Journal of Financial Markets 16(4)

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6. Connected Graph Bridges

• Back: 03 · The Ho–Stoll Model
• Forward: 05 · Failure Modes · Index Hub
• Sibling: A–S: Inventory & Risk Aversion · A–S: Advanced Extensions