Inventory Risk and Quote Skew
When a market maker holds inventory, the standard move is to skew quotes: nudge the fair value you quote around to attract flow that flattens you. The simplest version treats this as a linear function of inventory. Twice the position, twice the adjustment. Clean, intuitive, and wrong the moment real constraints exist.
Why skew is superlinear
A linear skew assumes constant marginal cost: every extra unit of inventory is exactly as costly to hold as the last. That only holds in a world with no limits. The instant you have a risk limit you actually care about, the marginal cost of carrying more stops being constant and rises as you approach the edge. The danger of forced liquidation — or liquidity thinning out exactly when you need to exit — grows faster than the position does. Rising marginal cost is convexity, and convex cost means a superlinear, convex-up skew. So your fair value adjustment curves upward: lean gently near flat, and increasingly hard as inventory builds toward the limit. The curve does not need to be steep. It only needs to bend.
Soft limits vs hard limits
Not every limit is the same shape. A hard limit is a wall. Cross it and you simply cannot operate: no margin, no borrow, no capacity left. The cost goes effectively vertical at the boundary, so the skew has to ramp almost asymptotically as you approach.
A soft limit is a line you are allowed to touch but not live beyond. Nothing physically stops you. But as you push against it you can only quote one side safely, so your two-sided presence and uptime decay — and that is the obligation you are actually trying to uphold. The cost is real and continuous, gentler than a wall but still convex, so the skew still bends up as you approach.
The real insight
The shape of your skew is a picture of how you believe risk grows with size. If you believe a large position is only proportionally riskier, you lean linearly. If you believe it is disproportionately dangerous — which most desks do once thin liquidity and gap risk enter the picture — you lean with a convex curve.
What actually holds the line
A subtlety worth stating: skew alone should not enforce a limit. Past a point, leaning harder on your quotes costs you more than simply trading out of the position directly, so the curve should cap there and the boundary should be enforced separately. Skew prices the cost of carrying inventory. A separate mechanism — pulling quotes or actively reducing — enforces the boundary. Blur the two and you either over-skew and bleed edge, or lean too softly and quickly breach your limit.
Staying flat is not a formula you apply. It is a set of beliefs about your own risk, made visible in where you choose to quote.
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