6.2.4 Inventory, Risk Aversion and the Role of Gamma

Basic Prerequisites: 03 · The AS Model.

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1. Intuition & Practical Objective

The AS model has exactly one behavioural parameter: the risk aversion $\gamma$. This page shows what $\gamma$ does - how it sets the skew, the spread, and the trade-off between profit and inventory control - and reproduces the paper’s central numerical result: the inventory strategy earns slightly less than naive symmetric quoting but with dramatically lower variance of P&L and of terminal inventory.

The paper’s claim, in one line: the AS strategy produces a P&L profile and final inventory with significantly less variance than the symmetric benchmark. We reproduce it from scratch below.

Two limits are worth holding in mind:

• $\gamma \to 0$ (risk-neutral): the skew $q\gamma {\sigma }^2\tau \to 0$, the inventory strategy becomes the symmetric strategy, and the two are identical (AS §3.3).
• $\gamma$ large (very risk-averse): the dealer will go to great lengths to avoid inventory; profits fall but dispersion collapses.

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2. Mathematical Ground Truth & Derivations

Where $\gamma$ enters. Three places, all monotone:

Object	Formula	Effect of raising $\gamma$
Reservation skew	$r-s=-q\gamma {\sigma }^2(T-t)$	larger skew per unit inventory
Inventory-risk spread term	$\gamma {\sigma }^2(T-t)$	wider spread
Stationary book spread	$\frac{2}{\gamma }\ln (1+\frac{\gamma }{k})$	narrower (falls monotonically: $\to 2/k$ as $\gamma \to 0$, $\to 0$ as $\gamma \to \infty$)

The stationary component is decreasing in $\gamma$ - a dealer who cares more about inventory quotes tighter in this component to trade out faster (a volume/edge trade-off), which partly offsets the elsewhere-increasing inventory-risk term $\gamma {\sigma }^2(T-t)$. The paper’s tables show exactly this: spread $1.33$ at $\gamma =0.01$, $1.29$ at $\gamma =0.1$, $1.15$ at $\gamma =0.5$.

The skew per unit inventory is $2{\theta }_2$ in the $\theta$ language - concretely, moving inventory by one lot shifts the reservation price by $\gamma {\sigma }^2(T-t)$. With the paper’s simulation parameters ($\gamma =0.1,\sigma =2,\tau =1$) that is $0.40 per lot, so a +10$lotpositionshadesthequotes$$4 down.

Why variance is the right metric. Under CARA utility $\mathbb{E}[-\exp (-\gamma W)]$, a smaller P&L variance is worth more than a larger mean whenever dispersion is high - a mean–variance proxy is $\mathbb{E}[W]-\frac{\gamma }{2}\mathrm{Var}[W]$. The AS strategy deliberately trades a small amount of mean for a large reduction in variance, which is the correct trade for a risk-averse dealer.

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3. Computational Implementation - inventory vs symmetric (the headline experiment)

We simulate the paper’s protocol: mid-price $dS=\sigma dW$ (arithmetic BM), fills arriving at Poisson rates $A{e}^{-k\delta }$ on each side, run to time $T$, and compare the AS “inventory” strategy against a “symmetric” strategy that quotes the same spread around the mid. NumPy, 20 000 paths.

Reading the numbers.

• $\gamma =0.01$ (near risk-neutral): the two strategies are nearly identical in mean P&L ($59.97$ vs $60.01$) - the paper’s “in the limit $\gamma \to 0$ the two strategies coincide”. The AS final-inventory std ($5.67$) is about $60\%$ of symmetric ($9.39$).
• $\gamma =0.1$: AS gives up $\sim$ $3.3 of mean P&L ($56.95$ vs $60.30$) and in exchange cuts P&L std from $17.72$ to $5.82$ - a $3.0\times$ reduction - and terminal-inventory std from $8.97$ to $3.19$ ($2.8\times$). This is AS Table 1’s result ($62.94/5.89$ vs $67.21/13.43$ for their 1000-path run; our seed/step-size differ, ratios match).
• $\gamma =0.5$ (very risk-averse): AS std falls to $4.83$ but mean P&L to $43.13$ - the dealer pays a real price for safety. Terminal inventory std $2.14$ vs symmetric $7.46$.

The qualitative law is exact and reproducible: AS trades a little mean for a lot of variance.

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4. Failure Modes & First-Principles Breakdowns

1. Calibrating $\gamma$ is an art, not a science. $\gamma$ is a preference, not a market parameter - but it directly sets position size and P&L variance. Too small and inventory runs (the random walk of 01); too large and the desk quotes itself out of the market. Production desks bound $|q|$ directly rather than relying on $\gamma$ alone.
2. The skew vanishes at $T$. Because the inventory penalty is $\propto (T-t)$, the model stops protecting inventory at the close - precisely when it is most exposed. The paper flags this; the fix is an explicit terminal penalty $\alpha q_T^2$ or a rolling horizon (never let $T-t\to 0$).
3. $\gamma$ and $k$ are not separately identifiable in the spread. The spread $\frac{2}{\gamma }\ln (1+\gamma /k)$ couples them; a mis-estimated $k$ masquerades as a wrong $\gamma$ (or vice versa). Fit them jointly to fill data, not by eye.
4. Risk-neutral ≠ profitable. At $\gamma \to 0$ the model degenerates to symmetric quoting, which has higher mean P&L here but unbounded inventory risk. “Maximize P&L” and “control variance” are different objectives; AS optimizes the CARA utility, not the raw mean.

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5. References

• Avellaneda & Stoikov (2008), Quantitative Finance 8(3)
• Cartea & Jaimungal (2015), Risk metrics and fine tuning of high-frequency trading strategies, Mathematical Finance 25(3), 576–611
• Menkveld (2013), High frequency trading and the new market makers, J. Financial Markets 16(4)

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6. Connected Graph Bridges

• Back: 03 · The AS Model
• Forward: 05 · Failure Modes · Index Hub
• Sibling: Inventory Management & Quote Skewing · VaR & Expected Shortfall