Modelling Latency as a Distribution

A liquidity-taking strategy backtests using a flat 300-microsecond latency: whenever the signal fires, the simulator assumes the order arrives 300 μs later, checks whether the target price was still available at that instant in the historical book, and fills if so. On that assumption the backtest reports an 88% fill rate. Live, the fill rate is 61%. The strategy's real latency averages almost exactly 300 μs — it's the flat assumption that's wrong, not the average. Real latency is right-skewed, and the difference between "usually 300 μs, occasionally 900 μs" and "always 300 μs" is invisible to any backtest using only the mean.

Why the mean is the wrong summary

A liquidity-taking order only fills at the target price if it arrives before that price disappears. The order book moves on its own clock — sometimes the price sits available for 2 milliseconds, sometimes for 100 microseconds. Fill probability is not a function of average latency; it's the probability that a random latency draw lands under a random, and often correlated, price-availability window:

$$P(\text{fill}) = P(L < W),$$

where $L$ is your latency (a random variable) and $W$ is how long the price stays available (also random, and typically shorter exactly when the market is moving fast — which is also when latency tends to spike, because busy periods stress the same systems). Replacing $L$ with its mean answers a different, easier question and gets a different, wrong answer whenever $L$'s distribution has meaningful spread — which real latency, being bounded below by physics but unbounded above by queueing and contention, always does.

Worked example: log-normal latency versus a flat assumption

Suppose measured live latency fits a log-normal distribution with median 280 μs and 95th percentile 850 μs (mostly fast, with a fat slow tail from OS jitter and network contention). The price-availability window $W$, from historical book data, is exponential with mean 500 μs. Using the flat-latency backtest ($L=300$ μs always): fill rate is $P(W>300) = e^{-300/500} \approx 0.549$, roughly 55%.

Now draw $L$ from the log-normal and compute $P(L<W)$ by Monte Carlo across 100,000 paired draws. That fraction comes out near 46% — lower still, because the log-normal's fat right tail produces slow draws the flat 300 μs assumption never generates, and those are exactly the draws most likely to miss a short-lived $W$.

Worked example: why correlation makes it worse

Let $W$'s mean shrink to 200 μs specifically during the 10% of ticks flagged high-volatility, and $L$'s median simultaneously rise to 450 μs during those same ticks (contention on the same busy systems). Fill rate during the volatile 10% drops to roughly 18%, staying near 52% the rest of the time — a blended rate near 49%, barely different from the uncorrelated case. But the strategy's P&L concentrates in exactly the volatile ticks where fill rate is worst, so the realized outcome is closer to the 18% regime than the blended average suggests.

Widen the standard deviation on the plot above and picture the right tail stretching out: that stretch is exactly what a flat-latency backtest deletes, and it decides whether a fast-moving order gets filled or missed.

What this means in practice

Fit a distribution to measured live latency, let it vary by regime rather than using one global fit, and simulate fill probability as $P(L<W)$ per event rather than a single pass/fail check. See Wire-to-Wire Latency Decomposition for where the tail in $L$ comes from, and Modelling Fill Probability for a Resting Order for extending this into a full simulator.