6.9.2 The Market-Maker P&L Decomposition

Basic Prerequisites: 01 · From Zero and Adverse Selection - 04 · Spread Decomposition.

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1. Intuition & Practical Objective

The five terms of Page 01 are a mental model. To run a desk, a risk system, or an academic study, you need them as a measurable decomposition: for every fill, attribute the dollars to spread, to the mid-price move that followed, to the inventory carried, and to the exchange’s ledger. This is the same accounting in three places - Menkveld (2013) for one HFT firm, Hasbrouck (2007) Ch 11 for NYSE specialists, and any modern desk’s TCA (transaction-cost analysis) report.

The objective of this page is precise: given a maker’s quote, fill, and the mid-price path, write down P&L as a sum of terms each with a distinct economic owner - the maker (spread), the informed trader (adverse selection), the financing desk (inventory), the exchange (fees/rebates). If you can name the owner of each dollar, you can name the lever to move it.

The one-sentence essence. “Decompose every fill into spread earned + rebate received − mid-price move against you − inventory carry − access fees, and attribute each term to the party who earned it.”

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2. Mathematical Ground Truth & Derivations

2.1 The per-fill decomposition

Fix a maker and a fill index $i$. Let ${\text{side}}_i=+1$ if the maker sold (a buyer lifted his ask) and $-1$ if he bought. Let $m_i$ be the pre-trade mid, and let the maker’s fill price be $m_i+{\text{side}}_{i}h$. Over the maker’s holding horizon the mid moves by $\Delta m_i=m_{i+\Delta }-m_i$.

The maker’s position after the fill is $-{\text{side}}_i$ (he sold $\Rightarrow$ short). Marking the position to the terminal mid:

$$\underset{\text{half-spread earned}}{\underbrace{{\text{spread}}_i=h}},\underset{\text{mid P\&L on the inventory}}{\underbrace{{\text{move}}_i=(-{\text{side}}_i)\Delta m_i}}.$$

So the per-fill P&L is

$${\pi }_i=h+{\text{side}}_i\text{-dependent rebate}-{\text{side}}_i\Delta m_i-c_{\text{inv}}-f_{\text{take}}\cdot 1[\text{taker}].$$

Averaging over fills, and writing $\lambda \equiv \mathbb{E}[{\text{side}}_i\Delta m_i]$, the systematic part of the mid move is the adverse-selection cost (the maker’s fills are selected): for a maker who sells, the mid tends to rise; for one who buys, to fall - so $\mathbb{E}[{\text{side}}_i\Delta m_i]>0$ and it enters as a cost:

$$\boxed{\mathbb{E}[\pi ]=\underset{\text{spread}}{\underbrace{h}}+\underset{\text{rebate}}{\underbrace{r}}-\underset{\text{adverse sel.}}{\underbrace{\lambda }}-\underset{\text{inventory}}{\underbrace{c_{\text{inv}}}}-\underset{\text{access fees}}{\underbrace{\mathbb{E}[f_{\text{take}}]}}}$$

2.2 The two “inextricable” terms, separated

The decomposition’s whole power is separating $\lambda$ (permanent, information) from $c_{\text{inv}}$ (transient, inventory). Hasbrouck (2007) Ch 11: information effects are permanent, inventory effects are temporary - the mid keeps the permanent part, the transitory part reverts. Operationally:

• $\lambda$ = $\mathbb{E}[\Delta m]$ that does not revert - estimate with the trade-price response, or via the “realized cost” $p_t-m_{t+5}$ in Hasbrouck Ch 14.
• $c_{\text{inv}}$ = the reverting component times the carrying cost $-\rho \sigma$ (see Inventory Management & Quote Skewing for the $\mathbb{E}[\Delta m]=-\rho \sigma z$ form).

2.3 The rebate as a signed fee, and additive neutrality

Redefine the maker’s fee as $f_m=-r$. Then $\pi =h-\lambda -c_{\text{inv}}-f_m-f_{\text{take}}1[\text{taker}]$. In this form the rebate is just a negative fee, and the decomposition is symmetric: maker and taker are two sides of one exchange-fee schedule. Page 03 shows when that symmetry is neutral (fine ticks) and when it breaks (tick friction).

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3. Computational Implementation - a Monte Carlo MM desk

Standard library only. Each simulated fill earns a half-spread, receives a rebate, and then the mid moves with a systematic adverse component $\delta$ (conditioned on the maker’s side) plus zero-mean noise. We then reconcile the simulated P&L against the analytic decomposition, with and without the rebate.

Three solid numbers to carry forward:

• Net P&L $=+$ $0.0050/share with the rebate, + $0.0030 without - the rebate is 40% of net.
• The simulated adverse-selection term is $-0.005979$ vs the analytic $\lambda =\delta =0.006$ - a 0.4% sampling gap over $5\times 10^5$ draws, i.e. the decomposition is tight.
• The headline spread is $2h=$ $0.020; the true take is $0.0050 - the spread overstates the maker’s margin by 4×.

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4. Failure Modes & First-Principles Breakdowns

1. The adverse-selection term is a conditional mean, not the raw move. $\lambda =\mathbb{E}[\text{side}\cdot \Delta m]$ requires conditioning on the maker’s own side. Measuring $\mathbb{E}[\Delta m]$ unconditionally gives ≈0 and makes the maker look cost-free. This is the winner’s curse in estimation form (see GM - 05 · Failure Modes).
2. Inventory carry is non-linear in size. $c_{\text{inv}}$ above is a linearised per-share cost. The true cost is convex in inventory ($\frac{1}{2}\gamma {\sigma }^2{q}^2$), so a “flat” per-share number hides the blow-up risk when $|q|$ grows - the mechanism of 05 · Failure Modes.
3. Holding-horizon mismatch. $\Delta m$ is measured over some horizon; a different horizon moves dollars between “adverse selection” and “inventory.” Standardise one horizon (e.g. 5 trades / 5 minutes) and hold it fixed, or the decomposition is not comparable across days.
4. Fee timing is contractual, P&L is accrual. Rebates arrive monthly and are tier-dependent; a desk that books rebate revenue per fill but pays a flat monthly schedule will mis-state its true marginal economics on a tier-change day.
5. $h$ is per-fill, but fills are autocorrelated. Adverse selection arrives in bursts (informed flows cluster); treating fills as i.i.d. understates the variance of $\pi$ and the drawdown risk even when the mean is right.

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5. References

• Menkveld, Albert J. (2013), High frequency trading and the new market makers, JFM 16(4), 712–740
• Hasbrouck (2007)
• Stoll (1978) and Ho & Stoll (1981)
• Copeland & Galai (1983), Information effects on the bid-ask spread, JF 38

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · Maker-Taker Fees & Rebates - where the $+r$ and $-f_t$ terms come from.
• Inputs: Spread Decomposition (estimating $\lambda$) · Inventory Management (the $c_{\text{inv}}$ curve) · Roll Model (the transitory/reverting part)