Expected P&L of a Quote

Computing a market maker's expected profit means weighting spread capture against the probability and cost of adverse fills.

1. The idea

To evaluate a quote, decompose expected P&L by conditioning on who trades against it: informed flow that moves the price against the maker, and uninformed (noise) flow that does not.

If a fraction $π$ of flow is informed and moves fair value by $δ$ against the maker after the trade, while the rest is uninformed and fair value is unchanged, the expected P&L per fill is a weighted average.

This framework — expected value as a mixture over trader types — is the core building block for pricing bid-ask spreads, sizing quotes, and later for adverse-selection and inventory-risk reasoning.

2. Why it matters

Interviewers often give you fill probabilities and information probabilities and ask you to compute break-even spread — this is a direct application of conditioning and linearity of expectation.

3. Core mathematics

Expected P&L per fill

E [P&L] = \frac{S}{2} - π δ

$S /2$ = half-spread captured, $π$ = probability counterparty is informed, $δ$ = expected adverse price move given informed

Break-even half-spread

\frac{S}{2} = π δ

4. How to recognize it

Think of this when…

Given a probability that a trade is 'informed' vs 'noise'
Asked to find the minimum spread for a market maker to break even
Involves an expected loss conditional on adverse information

5. Worked example

Worked example

A market maker quotes a half-spread of $0.05. With probability 0.2 a trade is informed and fair value moves $0.30 against the maker after the fill; otherwise fair value is unchanged. Is the quote profitable in expectation, and what is the minimum half-spread to break even?

1
Expected loss from informed flow per fill.
$0.2 \times 0.30 = 0.06$
2
Compare to the half-spread captured.
$E [P&L] = 0.05 - 0.06 = - 0.01$
3
Solve for break-even half-spread.
$\frac{S}{2} = 0.2 \times 0.30 = 0.06$
Answer: The quote loses $0.01 in expectation; the maker needs at least a $0.06 half-spread ($0.12 total spread) to break even.

6. Common mistakes

Forgetting to weight the adverse move by its probability rather than assuming it always happens
Double-counting the spread by adding it on both the buy and sell side incorrectly
Assuming the maker can always offset immediately at the current midpoint

7. Quick check

If informed probability rises, the break-even spread should:

Expected P&L per fill is $S /2 - π δ$ . This is an application of:

8. Practice

1 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Conditional Expectation Bayes' Theorem

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