The Machine Readings · Paper VIII

A sub-series of the Public Doctrine Series

Who Collects the Toll

Why a liquidity provider’s small edge becomes a near-certainty through frequency, why the same arithmetic condemns a taker whose toll exceeds its edge, and why a machine must make its edge positive before it makes it frequent

Dr. Glen Brown · President & CEO, Global Financial Engineering, Inc. and Global Accountancy Institute, Inc.
GFE-PUB-MR-20261103-H-CS · 3 November 2026

I. The Reading

Few firms in modern finance have grown as quickly, or with as few people, as XTX Markets. Founded in London in January 2015 as a spin-out of another quantitative firm, it had become the largest liquidity provider in spot foreign exchange by 2019 and the largest systematic internaliser in European equities in the same year. It now forecasts prices for more than fifty thousand instruments across equities, fixed income, currencies, commodities and digital assets, trades more than two hundred and fifty billion dollars a day, and does so with roughly three hundred employees. In 2022 it reported revenue of £2.5 billion and profit of £1.1 billion. ^[W]

Studying that record raises an observation that is simple to state and easy to underrate. The edge such a firm earns on any one trade is small, a fraction of the spread it quotes; what is large is the number of times it earns it. This paper takes the observation seriously. It asks what frequency does to an edge, for whom it works and against whom, and what it implies for a machine that does not quote prices but takes them, under a fixed bank and a fixed stop.

The answer is the companion to Paper VII of this series, The Toll and the Clock. Paper VII showed that, when the bank and the stop are fixed at birth, the cost of entering, the toll, decides the sign of each position. This paper shows that frequency decides how certainly that sign is expressed in the book. A liquidity provider collects the toll; a directional machine pays it. Both live by the same arithmetic, read from opposite sides.

II. The Mechanism

Frequency multiplies certainty, not edge

Let a book make N independent trades in a period, each with expected result e and dispersion σ. The expected result of the period is N·e. Its dispersion is σ·√N. ^[D] The ratio of the two, which governs how often the period finishes positive, is therefore (e/σ)·√N: it grows with the square root of the number of trades. ^[D] A hundredfold increase in frequency makes the outcome ten times more certain while leaving the edge of each trade exactly where it was. The investment literature has stated the same law for decades as the fundamental law of active management: the information ratio of a strategy is the quality of each independent bet times the square root of its breadth. ^[W]

Figure 1. How frequency turns a small edge into a near-certain period

The chance that a period of N independent trades finishes positive, Φ((e/σ)·√N), for four edges per trade measured in units of the trade’s own dispersion σ. The two upper curves are the case of Section V; the third is an illustrative liquidity provider, not any firm’s actual figure; the fourth shows a negative edge. Derived. Results are in units of risk; money, venue, account and dates are withheld.

Figure 1 draws the consequence. An edge of two thousandths of a standard deviation per trade is invisible on a single trade and nearly invisible over a thousand; over a million it is close to certain. ^[D] This is the sense in which a market maker’s edge is “not big but frequent”: the edge is a sliver, and the frequency converts it into something that looks, day after day, like a salary. When the market maker Virtu Financial filed to list its shares in 2014, it reported a single losing day in 1,238 trading days. ^[W]

The same figure carries the warning. Frequency multiplies the sign of the edge as faithfully as its size. A negative edge of two hundredths of a standard deviation is a coin toss over ten trades and a certainty of loss over ten thousand. ^[D] Frequency does not create an edge; it removes the doubt about which one a book has.

The toll, seen from two sides

A liquidity provider quotes a price at which it will buy and a higher price at which it will sell. Every trader who crosses the spread pays it half the difference. Against that income stands one cost: the trader who crosses may know something, and the provider who traded with a better-informed counterparty loses on the price that follows. The classical models of dealer pricing reduce the provider’s edge to exactly these two terms, the spread collected less the adverse selection suffered. ^[W] A provider’s forecasting machinery exists to make the second term small: to move its quotes before the informed trader can reach them. ^[I]

A directional machine stands on the other side of the same transaction. Paper VII wrote its expected result per birth as E = edge − τ·D/S, where τ is the spread as a share of the bank and D/S the ratio of bank to stop. ^[D] The τ that is a cost to the taker is, in a competitive market, revenue to the maker. ^[I] One firm’s toll is the other’s income, and the same spread that makes the provider’s small edge certain at high frequency makes the taker’s small edge certainly negative at high frequency, whenever the toll exceeds the edge.

III. The Argument

First proposition: frequency is a multiplier of certainty, never of sign

No amount of frequency rescues a negative edge, and any amount of it condemns one. ^[D] The first question an engineer must answer about a trading machine is therefore not how often it trades but whether the edge it multiplies is positive after every cost it pays. Only then is frequency an asset. ^[I]

Second proposition: the order of engineering is sign, then frequency, then size

A machine should first make its edge positive, by refusing the trades whose toll consumes it; then make the edge frequent, by breadth across instruments and lanes; and only then make it large, by sizing. ^[I] Done in any other order, each step amplifies an error made in the step before: size raises the stakes of an edge whose sign is unknown, and frequency hastens a loss that a toll has already made certain. The toll law of Paper VII is the first step taken first.

Third proposition: the toll decides how long the truth takes to appear

Solve the √N law for the number of trades needed before an edge stands two standard errors clear of zero, and it becomes N* = (2σ/e)². ^[D] Because the toll subtracts from e, it does not merely lower the edge; it lengthens, quadratically, the time a book must wait before its own record can show whether the edge is there at all. ^[D]

Figure 2. The births a book must wait for before its edge can be seen, against the toll

N* = (2σ/e)² with e = 0.029 R − τ·0.044 (the common edge and median bank-to-stop ratio of Paper VII) and σ = 0.20 R. Navy: independent births. Gold: births as they actually arrive, inflated by the measured variance ratio of 2.07 (Section V). Derived. Results are in units of risk; money, venue, account and dates are withheld.

With the edge and geometry measured in Paper VII, a book born at no toll can see its edge in about 190 independent births; at a toll of one-fifth of the bank, about 390; at two-fifths, about 1,230; at one-half, about 3,270; and as the toll approaches the edge itself, never. ^[D] A machine that admits dear births is not only earning less; it is choosing to be unable to know, for months, whether it is earning anything.

Fourth proposition: frequency is counted in independent trades

The √N law assumes that the trades are independent. Positions born together on related instruments are not: a session move, a factor, a shock reaches them all at once, and the dispersion of their sum grows faster than √N. ^[D] If the average correlation between trades in a period is ρ, a book of N trades behaves like one of N/(1+(N−1)ρ). ^[D] A correlation of a few hundredths, invisible in any single pair, halves the effective frequency of a book that makes two dozen trades a day. ^[D] Breadth is real only where it is independent; a hundred positions on one factor are, for this arithmetic, a few. ^[I]

Frequency does not create an edge. It removes the doubt about which one a book has.

Fifth proposition: the edge lives in the measurement

The firm this paper began with does what it does with about three hundred people, and its public record is one of research infrastructure: computing at scale, storage built for research data, engineers whose work is the speed of its models. ^[W] The lesson for a smaller house is not to imitate its business, which is a different business, but its ratio: very few people, and the whole of the edge in the machine and in the instruments that measure it. ^[I] A machine that cannot measure its own toll, its own independence and its own edge to the standard error cannot know which of the three to engineer next.

IV. What Would Falsify This Reading

The square-root test. Over periods of growing length, the dispersion of the book’s result must grow as the square root of its effective number of births. If it grows faster, the births are more correlated than measured and the effective frequency is lower than this paper claims; if it grows as N itself, the book is effectively one trade. ^[I]

The variance-ratio test. The ratio of the day-to-day dispersion of the book to the dispersion predicted from independent births was measured at 2.07. If, under the toll law and the narrower portfolio, the ratio rises, breadth is being bought on a common factor and the gain in frequency is illusory. ^[I]

The waiting test. Under the toll law the births kept carry an edge of about 0.19 standard deviations each. Their mean must stand two standard errors clear of zero within roughly two hundred to four hundred births, depending on how they cluster. If it has not after four hundred, either the edge is smaller than measured or the births are less independent than measured, and the paper must be revised. ^[I]

The sign test. If raising the frequency of the book, by adding instruments or lanes, lowers the mean result per birth, the new births are paying a toll the edge cannot carry, and the second proposition’s order has been broken. ^[I]

V. The Case

The case is a governed multi-asset book over six weeks, the same case as Paper VII: about a thousand births, some twenty-five a day. ^[W]

Table 1. The case in the arithmetic of frequency

Resolved births only, which favours the quick banks and therefore flatters e; Paper VII sets out why. Derived from the birth ledger and the one-minute price record. Results are in units of risk; money, venue, account and dates are withheld.

Births per day were about 25. Across the 32 days with three or more resolved births, the day-to-day dispersion of their summed result was 2.07 times what independent births would give, an average correlation of about 0.04 between births of the same day, and therefore about 12 independent births a day where the ledger shows 25. ^[D] Three days in four finished positive. ^[D] A normal approximation to the same numbers predicts fewer, about three in five; the difference is the shape of the distribution, not an error: most days carry no death at all and finish positive, while the mean is carried down by the few days that do. ^{[D] [I]}

Table 1 is the paper’s argument in six rows. The toll law did not only raise the edge of the births it keeps; by raising e and lowering σ together, it cut the number of births the book must wait for before its edge can be seen from about 950 to about 110, or, as births actually cluster, from about 1,970 to about 230. ^[D] The book has since adopted the law, and its first births under it are too few, and too early, to be read. ^[W] The test of this paper is the one set in Section IV: the kept births must show their edge within a few hundred births, not a few thousand.

References

Grinold, R. C. (1989). The fundamental law of active management. Journal of Portfolio Management 15(3), 30–37.

Glosten, L. R. and Milgrom, P. R. (1985). Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. Journal of Financial Economics 14(1), 71–100.

Ho, T. and Stoll, H. R. (1981). Optimal dealer pricing under transactions and return uncertainty. Journal of Financial Economics 9(1), 47–73.

Virtu Financial, Inc. (2014). Form S-1 Registration Statement, U.S. Securities and Exchange Commission: one losing trading day in 1,238.

XTX Markets: company history, scale, headcount and reported 2022 results, as summarised in the public record (Wikipedia, “XTX Markets”, accessed 3 November 2026), and the firm’s published research-infrastructure work.

Dr. Glen Brown (2026). The Toll and the Clock. The Machine Readings, Paper VII. Public Doctrine Series.