What a Millisecond Is Worth

A resting quote is an option you have written and somebody else holds, and speed is nothing but how fast you can take it back. That makes what a millisecond is worth a calculation rather than a story. On a large exchange-traded fund at 520 dollars and 16 per cent annual volatility, the price moves about 3.4 cents in a second and about a tenth of a cent in a millisecond, and the delay at which a typical move is as large as the half-cent half-spread is 21.3 milliseconds. Below that, being faster wins fractions of a half-cent. Above it, being slower risks the whole spread. That threshold is the entire arena, and it moves enormously: on a ten-dollar small company at 60 per cent volatility quoted fifty cents wide, the same calculation gives 171 minutes, which is another way of saying that speed is worth nothing there at all.

Prerequisites: Lesson 44, for scaling a standard deviation by the square root of time and for the 15.87 that every number below passes through, lesson 53, for the half-spread as the thing a quoter earns and stands to lose, and lesson 54, for the fact that an order in flight is invisible and only its print is public, which is what any race is actually racing against.

The same calculation on five instruments

The threshold is not a property of markets; it is a property of an instrument, and it moves by a factor of three million across ordinary ones. Below is the same arithmetic run five times: annual volatility to a daily figure by 15.87, daily to a second by the square root of 23,400, and then the delay at which that per-second figure reaches the half-spread.

Instrument	Move in one second	Delay at which it equals the half-spread
ETF at $520, 16 per cent, penny spread	3.43 cents	21.3 milliseconds
The same ETF on a 40 per cent day	8.57 cents	3.4 milliseconds
Stock at $400, 40 per cent, five cents wide	6.59 cents	144 milliseconds
Stock at $50, 30 per cent, penny spread	0.62 cents	655 milliseconds
Small company at $10, 60 per cent, fifty cents wide	0.25 cents	171 minutes

Start at the bottom. On a ten-dollar company quoted fifty cents wide, a typical move reaches the half-spread after nearly three hours. Nobody needs to be fast there, and nobody is: the spread is so much larger than anything that can happen inside a second that the entire quoting problem is inventory rather than latency. That row is most of the stock market by name count, and speed is worth nothing in it.

Now the second row against the first. The same instrument on a volatile day moves the threshold from 21.3 milliseconds to 3.4, a factor of six, and nothing about anybody’s hardware changed. The race gets harder on exactly the days when quoting is hardest, which is why spreads widen then: a quoter who cannot get inside the new threshold has to charge more for the option they are writing, and charging more for it is what a wider spread is.

And the middle rows show the two dials working against each other. The four-hundred-dollar stock is more volatile in cents per second than the fund, and its threshold is nonetheless seven times longer, because five cents of spread is ten times the protection. Volatility and spread pull in opposite directions and the ratio is what decides, which is the same shape as lesson 4’s spread-against-stop and lesson 53’s spread-against-informed-edge. The number that matters is never the one printed on the screen by itself.

What none of this tells you is how to trade. It tells you where speed is a factor at all, and the answer is a narrow band of very liquid instruments at very short horizons. If your holding period is measured in hours, the delay you are exposed to is not milliseconds; it is the whole of the position, and the threshold arithmetic has nothing to say about it.

What this does not settle

That the square root of time is the right scaling. Lesson 44 established it and immediately showed where it fails: on the series it measured, scaling one bar up to twenty overstated the actual twenty-bar spread by more than half again, because the bars leaned against each other. At the timescales in this lesson the same objection points the other way, since price changes over milliseconds are more mean-reverting than independent, which makes the moves in the first table an overestimate and pushes the threshold out. The direction is knowable and the size is not, and every figure here should be read as an order of magnitude rather than a measurement.

That a typical move is the right thing to compare. A standard deviation is the middle of a distribution, and what actually costs a quoter money is the tail: the jump on an announcement, the cascade, the seconds when everything moves together. Those are the events the whole business is arranged around, they are not normally distributed, and they do not scale by the square root of anything. The threshold above prices the ordinary case and says nothing about the case that decides the year.

That the volatility numbers are yours. Sixteen per cent for a broad fund and sixty for a small company are round figures chosen to make the arithmetic legible, not measurements of anything. Lesson 44’s four steps produce the number for your instrument in about ten minutes, and the threshold moves as the square of the ratio, so a volatility estimate that is off by a factor of two moves the answer by a factor of four.

That the half-spread is what a stale quote loses. It is what the quoter earns, which is why it is the natural comparison, but the loss on being picked off is the size of the move, not the size of the spread, and the two are only equal at the threshold by construction. Above the threshold the loss grows with the square root of the delay while the premium stays fixed, which is the whole reason the threshold is worth computing and also the reason it is not a break-even.

That any of this bears on a retail decision. It does not. Nothing in this lesson changes an order type, a size, a stop or an instrument, and the honest use of it is to stop attributing your fills to a race you are not in. Lesson 54 already located the costs that are yours, and none of them were latency.

And the concession that costs this lesson its frame: the option analogy is exact for the quoter and only a metaphor for anyone else. A quoter genuinely has written something and genuinely can cancel it, so the arithmetic above is a real valuation of a real position. A trader sending a marketable order has written nothing, holds nothing, and is exposed to latency in a completely different way — through the print their own execution makes, which lesson 54 described and which this page has not priced at all. So the number in the claim is what a millisecond is worth to a market maker. What a millisecond is worth to you is a question this lesson has replaced rather than answered, and the replacement is easier to compute, which is precisely why it should be treated with suspicion.

Problems

1. Compute the threshold for your own instrument. Take lesson 44’s four steps to an annual volatility, divide by 15.87 for a day, divide by the square root of the seconds in your session for a second, then square the ratio of your half-spread to that figure. That is the delay at which a typical move reaches your half-spread. Ten minutes, and if the answer is longer than a second you can stop reading anything about speed.
2. Watch the threshold move with the day. Compute it on a quiet day and on a day with a scheduled release, using the same instrument and the realised volatility of each. The threshold falls with the square of the volatility ratio, so a doubling cuts it by four. Then check what the spread did on the same two days. If the spread widened by roughly the factor the threshold shrank, you have watched a quoter reprice an option, which is what a widening spread is. Half an hour.
3. Find the horizon at which your own exposure stops being about milliseconds. Take your median holding period and compute the typical move over it with the same scaling. Compare it with your average winner and with the half-spread. On a two-hour hold the first number will be tens or hundreds of times the third, and that ratio is the honest measure of how much latency can matter to you. Do it for the three instruments you trade most, because the answer differs by more than you expect. An evening, and it ends the subject.

Sources. Eric Budish, Peter Cramton and John Shim, “The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response” (The Quarterly Journal of Economics, 2015), for the finding this lesson turns on: the duration of a typical arbitrage opportunity collapsed over six years while the profit per opportunity did not fall, which is what makes speed a race for share rather than for size. Matteo Aquilina, Eric Budish and Peter O’Neill, “Quantifying the High-Frequency Trading Arms Race” (The Quarterly Journal of Economics, 2022), for races decided in single-digit microseconds measured from exchange message data, and for a total prize far smaller as a share of volume than the popular accounts imply. Joel Hasbrouck and Gideon Saar, “Low-Latency Trading” (Journal of Financial Markets, 2013), for what low latency does to spreads and depth, which is the counterweight to reading any of this as pure extraction. Michael Lewis, Flash Boys (W. W. Norton, 2014), for the popular account and for the label lesson 54 corrected, which is worth reading precisely because so much of what people believe about this comes from it.

Speed buys one thing: a shorter life for the option you wrote when you posted a quote. That makes it computable. On a large fund at 16 per cent volatility a second is worth 3.4 cents of exposure and a millisecond a tenth of a cent, and the delay at which the typical move reaches the half-cent half-spread is 21.3 milliseconds. On a small company quoted fifty cents wide the same calculation gives 171 minutes, and speed is worth nothing there. The race is real and it is a race for share of a fixed stream of half-cents, not for larger ones, which is why opportunities that got ten times shorter did not get any less valuable each. None of it is a market you can enter and none of it explains your fills. Lesson 56 goes looking for the trades that never appear on any of these venues at all, and asks what the tape can and cannot tell you about them.

Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.