Reading a Run of One-Sided Flow

The scenario: you're quoting a two-sided market and notice that the last eight trades against your quotes have all been buys — people repeatedly hitting your offer, none hitting your bid. Is this meaningful, or just noise from a small sample? The interview question tests whether you can reason quantitatively about when a run stops looking random.

Framing it as a coin-flip test

If buy and sell orders arrived with equal probability and independently of each other, eight buys in a row is like flipping a fair coin and getting eight heads in a row: probability $(1/2)^8 = 1/256 \approx 0.4\%$. That's unlikely enough that "pure coincidence" is a weak explanation — the more likely explanation is that the assumption of equal, independent probability is wrong right now: either genuine buying interest has picked up (informed flow, or simply a shift in the crowd's collective view) or your quoted price is stale and too low, so buyers keep finding it attractive while sellers ignore it. Both explanations point the same direction: your fair value estimate should move up.

Worked example: updating with a simple Bayesian read

Suppose before the run you believed there was a 50% chance the "true" order flow regime was balanced (buys and sells equally likely) and a 50% chance it had shifted to buy-heavy (say, 80% buys). You observe 8 buys in a row. Under the balanced regime, the probability of 8 straight buys is $(0.5)^8 \approx 0.0039$. Under the buy-heavy regime, the probability of 8 straight buys is $(0.8)^8 \approx 0.168$. Using these as likelihoods with equal 50/50 prior weights:

$$P(\text{buy-heavy} \mid \text{8 buys}) = \frac{0.168 \times 0.5}{0.168 \times 0.5 + 0.0039 \times 0.5} \approx \frac{0.084}{0.0860} \approx 0.977 .$$

A run that started as a coin flip ends with roughly 98% posterior confidence that the regime has genuinely shifted buy-heavy, not that you got unlucky watching a balanced market. That's the quantitative version of "eight in a row is a real signal, not noise" — the exact numbers depend on your assumed regimes and prior, but the mechanism (short unlikely runs update beliefs fast) is the transferable part.

What to actually do about it

Two responses, not mutually exclusive: raise your quoted prices (both bid and offer move up, keeping your spread but re-centering on a higher fair value), and/or widen your offer specifically, since that's the side attracting all the informed-looking flow, while leaving your bid more attractive to bring in some offsetting sell interest. What you should not do is keep quoting exactly where you were on the theory that "it'll revert" — that's betting against the very evidence you just observed, and the coin-flip math above says that evidence is strong.

What this means in practice

Real market-making systems track exactly this kind of run statistic (often called order flow imbalance) continuously and feed it directly into fair-value and quoting logic, precisely because runs this unlikely under a null "balanced" model are common enough in practice to matter and rare enough to be a genuine signal. The interview lesson is to quantify "is this run surprising" rather than answer from gut feel — a specific probability calculation is a much stronger answer than "yes, that seems like a lot of buys."