Reverse-Engineering Jane Street

A constraint-based reconstruction of the model classes, market structure, risk systems, and financial economics that could explain the observed business of a firm reporting $39.6 billion of annual trading revenue.

Summary

Four conclusions I hold, in descending order of confidence:

1. Their revenue growth is volatility and principal risk, not market-share gains. Revenue went from ~$10Bin2023to$39.6B in 2025 without a comparable change in share in any disclosed business line. A bottom-up estimate of conventional market making (§9) reaches roughly 11B at the midpoint, leaving most of the total attributable to volatility convexity, directional risk, and a large venture book that flows through the same revenue line. The July 2026 loss of \~15B came from a concentrated illiquid position while the trading systems kept working, which is the same fact seen from the other side.
2. Latency budget determines model class, not the other way around. A firm with FPGAs on the quote path and thousands of GPUs offline is not running one model. It runs a hierarchy where each tier gets the most capable model that fits its deadline, and complex models reach the quote path only after distillation into something that evaluates in microseconds (§3).
3. Eight model classes are enough to represent the conventional market making identifiable from public evidence, which is about a third of the revenue. Collapsing the thirty-five model instances in sections 4 through 8 gives seven classes for trading plus one for internal allocation (§10). The classes are deletion-irreducible and all twenty-eight pairwise merges fail, though the set is not provably the smallest under a different vocabulary. The same reduction demotes rough volatility and deep hedging, since nothing observed requires either. It explains the ~11B in §9 and not the \~22B to $35B residual, and the principal book is outside it entirely.
4. The methodology transfers to a small account; the resolution does not. Exposure decomposition, markout measurement, hypothesis budgeting, purged validation, and inventory limits are all available to an individual (§11). Their sample sizes, data, costs, and research throughput are not. What a small trader can copy is the machinery for not fooling yourself, which is worth more than any particular signal.

What this document is meant to demonstrate about how I work: every claim is tagged by evidence class, inference is separated from observation, and the corrections appendix records not just what was wrong but the class of reasoning error that produced it. Five such patterns are named there, the most persistent being a habit of writing the strongest available sentence and qualifying it later or not at all. Naming a pattern proved more useful than fixing its instances one at a time, and two of the five were only visible because an earlier fix reproduced the habit in a new place.

---

Contents

• 1. Method
• 2. The Firm as a Constraint Set
• 3. The Latency Hierarchy
• 4. Fair Value and Microstructure
• 5. Volatility Surfaces and Options
• 6. ETFs, Baskets, and Relative Value
• 7. Statistical Alpha and the ML Stack
• 8. Risk, Capital, and the Meta-Model
• 9. The Revenue Bridge
• 10. How Small Can the Set Be?
• 11. What Transfers to a Small Account
• 12. Open Questions
• Appendix A: References
• Appendix B: Sources by Class
• Appendix C: Corrections
• Appendix D: The Full Coarsening Matrix

1. Method

Jane Street does not publish its models. Nobody outside the firm knows what they are.

What exists publicly is a set of constraints, and it is richer than you would expect: financial disclosures from bond offerings, regulatory filings and SEC comment letters, an adversarial regulatory order from India that reconstructs specific trades, a trade secrets complaint describing a strategy’s development lifecycle, two Kaggle competitions built from their production data, an engineering blog, a technical podcast, and job descriptions written by the people doing the work.

A model stack is not free-floating. It is determined by the business it serves, the data it consumes, the latency it must meet, the capital it risks, and the hardware it runs on. All of those are partially observable. So the question here is not “what does Jane Street do.” It is:

Given everything public, what is the smallest set of model classes that could produce the observed behavior?

That is an identification problem under partial observability. The observations are disclosed behavior, hardware, latency budgets, capital and regulatory filings; the unknown is the class of model that generated them. Inverse problems of this shape are rarely uniquely solvable, and this one is not. §10 therefore argues for a weaker and checkable property in place of uniqueness, and says so in its opening lines instead of a footnote.

Reading the document in that frame changes what the middle sections are for. The equations in §§4 through 8 are not demonstrations. They are candidate components in an identification argument, and §10 is where the candidates are reduced, tested and discarded.

Every substantive claim carries a tag.

The [D]/[P] split matters more than it looks. Jane Street is private and does not comment on results, so most of its financial figures reach the public through press reporting sourced to unnamed people. That is not the same epistemic object as a filing, and collapsing the two would undercut the standard the rest of this document holds itself to. Nearly every 2024 through 2026 revenue figure here is [P].

Last verified: September 2026. The recent financial figures should be re-checked against current reporting before being relied on.

Notation

Several symbols are conventional in more than one of the literatures this document draws on. Where a collision was unavoidable I have kept each field’s standard symbol, so the table is the disambiguation.

---

2. The Firm as a Constraint Set

2.1 Scale

Revenue per employee is above $11M. That rules out a large class of explanations immediately. You do not reach that number on agency commissions or on passive spread capture in liquid names, because those businesses compete down to a few basis points and scale with headcount and capital rather than with model quality. Section 9 works the arithmetic.

2.2 Business lines

ETFs and ETPs. Primary market (authorized participant) and secondary. ~24% of primary US-listed ETF activity, ~41% of bond ETF trading, ~17% of European secondary activity [P].

Equities. 200+ venues, 45+ countries [D], average monthly volume ~$2 trillion (2024) [P], plus a retail wholesaling unit that is a named venue in major brokers’ order routing disclosures [P]. I previously carried specific Robinhood order-flow payment figures here and have removed them: the equity-over-options split ran opposite to Robinhood’s usual mix, the implied monthly total was a large fraction of reported industry-wide payments for the same period, and I could not resolve whether the period was monthly, quarterly, or cumulative. The qualitative fact stands; the numbers did not survive checking.

Options. ~8% of OCC transactions **[P]**. For scale, OCC cleared 15.2 billion contracts in 2025, up 24.3%: 8.27B equity, 5.68B ETF, 1.26B index [D].

Fixed income. Pricing more than 16,000 bonds, portfolio trading in baskets of “10 to 1,000 individual bonds in one clip” [D]. Monthly volume ~$230B (2024) [P].

Commodities, index products, digital assets via their JCX platform [D].

A large principal and venture book. Anthropic, CoreWeave, Situational Awareness. Marked through trading revenue, which is why their quarterly numbers swing more than other market makers’ [P].

2.3 Organizational shape

No CEO, no conventional hierarchy, 30 to 40 senior people running it collectively, firm-wide profit sharing with no individual desk P&L, ~16-year average partner tenure [P].

Compensation structure is a model-selection constraint. Pay traders on individual desk P&L and you get siloed, defensively guarded, locally optimized models. A common pool lets you build a shared fair-value layer that every desk consumes, because nobody loses by contributing to it.

The Millennium complaint states that ~25 people worked on the strategy at issue over five years, that it was initially “met with significant skepticism,” and that the firm “risked substantial capital” on an experimental Trading Investigation in 2023 purely to learn how the market worked [D]. That is R&D funded like R&D.

2.4 The technology constraint

• OCaml as the primary language, including a custom compiler fork (OxCaml) [D]
• FPGAs via Hardcaml on the lowest-latency paths [D]
• “Millions of multicast messages per second on a single core” [D]
• Distributed systems built on state machine replication modeled on exchange architecture [D]
• ML inference “at latencies far below those that are typical for ML workflows,” using “heavily optimized CUDA code that stretches the bounds of what GPUs were designed for, to leveraging custom hardware, to writing our own compilers” [D]
• Python for research, OCaml for production, with acknowledged friction between them [D]

The language choice does modeling work as well as latency work. Their stated reason for OCaml is a “powerful type system with good and predictable performance and a low overhead runtime” [D]. The performance half is obvious. The type system half is not.

A firm-wide shared fair-value layer has a specific failure mode: silent unit and identity errors. A price in cents added to a price in dollars. A yield as a decimal in one module and a percentage in another. Shares versus contracts versus notional. The ETF identifier where the basket identifier belongs. None of these throw. They produce quotes wrong by a factor of 100, at wire speed. Phantom types and abstract types make these unrepresentable at compile time. Python cannot, which is why Python is fine for research, where a wrong number gets caught by a human looking at a plot, and dangerous in the quote path, where a wrong number gets lifted by every other market maker inside a millisecond. [R]

The language boundary and the research/production boundary coincide for a reason. The friction is the cost of a deliberate safety property.

---

3. The Latency Hierarchy

Any electronic market maker at this scale runs something structurally like this. The tiering is forced by physics and by the disclosed hardware mix. [S]

The GPU fleet is a Tier 3/4 asset. It is not in the quote path. What crosses from Tier 4 into Tier 1 is distilled: trained offline, then compressed into something that evaluates in microseconds, by explicit distillation, by extracting a piecewise-linear approximation, or by precomputing a response surface into a table. [R]

Supporting evidence is oblique but suggestive. Two blog posts, Visualizing piecewise linear neural networks (2024) and Can you reverse engineer our neural network? (2026) [D]. ReLU networks are the family where you can enumerate exact linear regions and therefore compile, bound, or verify the decision function. That is a niche concern unless you need to put a neural net somewhere a neural net does not fit.

In Young Cho states the inversion directly [D]:

“What is the best model for the job” becomes “given these constraints, what is the best model that can satisfy these constraints?”

A small trader has no Tier 0 or Tier 1. Do not compete where a Tier 1 answer is the edge. Everything available to you is Tier 3 and 4, where returns per unit of speed are lowest and returns per unit of thinking are highest.

---

4. Fair Value and Microstructure

4.1 The central object

Every market making system reduces to one estimate:

$${\hat{S}}_t=\mathbb{E}!\left[S_{t+h}\mid {\mathcal{F}}_t\right]$$

Expected mid at a short horizon, conditional on everything observable. Quote placement, hedging, and inventory are control problems layered on top. [S]

4.2 What the Kaggle competitions reveal

The two competitions are a lightly obfuscated leak of the shape of the internal problem. [S]

2020: 130 anonymized features, targets resp and resp_1 through resp_4, a weight column, date, ts_id. Binary action, Sharpe-like utility over daily P&L [D].

2024: 79 features, nine responders, date_id, time_id, symbol_id, a weight, previous-day responder lags at the first time_id of each day. Target responder_6. Metric is sample-weighted zero-mean $R^2$ [D].

Read as a specification, four things follow.

Multiple simultaneous horizons. Four targets in 2020, nine responders in 2024. They are predicting a vector of forward returns, and the trading decision is a function of the whole vector. A signal strong at 1 second that reverses at 1 minute is a liquidity-provision signal. One weak at 1 second but persistent at 10 minutes is an information signal. Different desks consume different components with different holding periods. [S]

Sample weights. The weight column establishes that unweighted prediction error is not the intended objective. Some observations matter more and the designers knew which. The exact semantics are not established by the column’s existence: tradeable size, dollar risk, liquidity, spread, or a normalization the obfuscation required are all candidates. The defensible claim is the negative one, and it is the useful one. Optimizing unweighted MSE across observations you could never have acted on optimizes the wrong loss. [R]

The metric is zero-mean weighted $R^2$:

$$R^2=1-\frac{{\sum }_i{w}_i(y_i-{\hat{y}}_i{)}^2}{{\sum }_i{w}_i{y}_i^2}$$

Note the denominator. Not $\sum w_i(y_i-\bar{y}{)}^2$. This is correct when you believe the unconditional mean is zero and any apparent drift is noise you must not fit. Top leaderboard scores were on the order of $10^{-2}$: roughly one percent out-of-sample $R^2$, from thousands of competitors with substantial compute. [R] (I could not verify exact final values. The order of magnitude is the claim.)

That number needs its scope stated. One percent is the regime for short-horizon return prediction on liquid instruments. It is not a universal ceiling. Higher $R^2$ is legitimately attainable at longer horizons, in cross-sectional formulations, on volatility targets instead of return targets, on persistent variables, in less efficient markets, and on event-conditioned samples. The defensible version:

For short-horizon return prediction in liquid markets, an out-of-sample $R^2$ materially above the ~1% regime should trigger a leakage and selection-bias audit before it triggers anything else.

symbol_id is a feature and lags cross the day boundary. Cross-sectional structure matters and there is overnight information carry. Whatever they run is not a collection of independent per-symbol models. [S] Which implementation they use is open: entity embeddings, categorical splits in a tree ensemble, target encodings, per-symbol normalization, and symbol-by-feature interactions all encode the same information, and I have no evidence favoring any of them [X]. The schema constrains the problem, not the architecture.

Cho’s framing matches [D]:

“Instead of having, let’s say one unit of data, you have 100 units of data. However, you have one unit of useful data and 99 units of garbage and you do not know what the useful data is.”

4.3 Feature families

Given tick data and this problem shape, a large part of the economically useful feature space is strongly constrained by market microstructure. [S] The constraint is on which information pays. How that information is represented is a separate question. Several encodings of the same information are available for each item below.

Order flow imbalance. Cont, Kukanov and Stoikov showed short-horizon price changes are close to linear in order flow imbalance, not trade imbalance. For book events indexed by $n$:

$$e_n=1_{P_n^b\ge P_{n-1}^b}{q}_n^b-1_{P_n^b\le P_{n-1}^b}{q}_{n-1}^b-1_{P_n^a\le P_{n-1}^a}{q}_n^a+1_{P_n^a\ge P_{n-1}^a}{q}_{n-1}^a$$ $${\text{OFI}}_k=\sum _{n\in I_k}{e}_n,\Delta P_k=\beta ,{\text{OFI}}_k+{\epsilon }_k$$

Fitted $\beta$ scales roughly inversely with average book depth, making it the microstructural analogue of Kyle’s $\lambda$. High $R^2$, a handful of arithmetic operations. Tier 1.

Microprice. The naive mid is biased when the book is lopsided:

$$P^{\text{micro}}=\frac{Q^a{P}^b+Q^b{P}^a}{Q^a+Q^b}$$

Note the crossed weighting: bid price weighted by ask size. Stoikov’s refinement fits it as a martingale-adjusted fixed point over imbalance and spread states.

Queue position. In a price-time priority book with a one-tick spread, a dominant component of a passive order’s economics is queue position. Not the only one: spread, fees and rebates, cancellation and replenishment, volatility, hidden liquidity, latency, and adverse selection conditional on fill all enter. The Huang, Lehalle and Rosenbaum queue-reactive model makes arrival intensities functions of book state, which turns fill probability into something computable rather than assumed. [R]

Hawkes intensities. Order flow clusters:

$${\lambda }_i(t)={\mu }_i+\sum _j{\int }_0^t{\varphi }_{ij}(t-s),d{N}_j(s)$$

The branching condition (spectral radius of $\int \varphi$ below one) is what makes the modeled point process stationary rather than explosive. It is a property of the model rather than a claim about the market. The empirical near-instability of fitted kernels matters beyond this layer, but the result usually invoked here needs its hypothesis stated. Jaisson and Rosenbaum (2015) show that nearly unstable Hawkes processes with a light-tailed kernel converge to a classical Brownian CIR process, which is not rough. Roughness requires the heavy-tailed case (Jaisson and Rosenbaum, 2016): with kernel tail $x^{-(1+\nu )}$ and $\nu \in (1/2,1)$, the rescaled limit is an integrated fractional CIR process with Hurst parameter $H=\nu -1/2$. That relation is worth carrying, because it makes the two layers agree on a number. The $H\approx 0.1$ of §5.3 corresponds to $\nu \approx 0.6$, inside the admissible range. So this gives a microstructural mechanism linking order flow to the models in §5.3, conditional on the tail assumption, and one mechanism among possible others. [R]

Adverse selection estimators. Glosten and Milgrom: the spread compensates for informed flow. Kyle: price impact is linear in net order flow with coefficient $\lambda$. A sophisticated market maker will generally estimate conditional adverse selection, plausibly by counterparty, venue, symbol and time of day, though a firm may aggregate those dimensions, learn a latent representation, or use a hierarchical structure instead. [R]

For retail wholesaling, a central problem is separating relatively uninformed retail flow (safe to internalize at tighter prices) from informed flow (wider pricing, hedging, or rejection). Not the only one: order flow payments, spread capture, hedging costs, inventory, segmentation, routing and venue economics, and the fee schedule all bear on P&L. But the classification problem has the richest feature set and the clearest supervised formulation, which is where gradient boosted trees excel.

Cross-asset lead-lag. Hasbrouck (2003) finds the E-mini S&P contract dominates price discovery relative to the cash index and its ETF, so ES leading SPY is a documented empirical result [D]. The rest of this paragraph is inference. The most liquid name in a sector leads the rest. A cointegrated basket leads its ETF. [S] This is where 45 countries and 200 venues becomes a modeling asset rather than an operational cost. They see the leading instrument for nearly everything.

4.4 Quoting as stochastic control

Fair value is not the same as knowing where to quote. The canonical formulation is Avellaneda and Stoikov. With mid $s$, inventory $q$, risk aversion $\gamma$, volatility $\sigma$, horizon $T$, and fill intensity $\lambda (\delta )=A{e}^{-k\delta }$:

$$r(s,q,t)=s-q\gamma {\sigma }^2(T-t)\text{(reservation price)}$$ $${\delta }^a+{\delta }^b=\gamma {\sigma }^2(T-t)+\frac{2}{\gamma }\ln !\left(1+\frac{\gamma }{k}\right)\text{(optimal total spread)}$$

Here $A$ is the fill intensity at zero distance from the mid and $k$ is the rate at which that intensity decays with distance. In words: you do not quote symmetrically around the mid. You quote symmetrically around a price shifted against your inventory. The shift is proportional to position size, volatility, risk aversion, and time remaining.

The width has two separable components, an inventory-risk term scaling with $\gamma {\sigma }^2$ and a market-power term $\frac{2}{\gamma }\ln (1+\gamma /k)$. The second is worth staring at, because $A$ drops out entirely. How often you get filled at the touch does not set your spread. What sets it is how fast fill intensity falls off as you widen, which is a statement about how much pricing power you have.

Guéant, Lehalle and Fernandez-Tapia give closed-form approximations for the infinite-horizon case with inventory limits, which is what a real system needs. Cartea and Jaimungal add short-term alpha, so the reservation price shifts on signal rather than only on inventory, and Hawkes-driven order flow.

A live engine is likely a heavily modified descendant: fair value from §4.3, inventory skew from a control problem, width from an adverse-selection estimate, hard risk overrides at Tier 0. The Avellaneda-Stoikov form is too clean (single asset, constant $\sigma$, exponential fills, no queue, no discrete ticks) but the shape of the solution survives all those relaxations. [R]

4.5 P&L attribution, and why retail market making fails

For fills indexed by $i$, with ${ϵ}_i=+1$ for a buy and $-1$ for a sell:

$$\Pi =\underset{\text{half-spread captured}}{\underbrace{\sum _i{ϵ}_i{q}_i(m_{t_i}-p_i)}};+;\underset{\text{adverse selection (negative)}}{\underbrace{\sum _i{ϵ}_i{q}_i(m_{t_i+\tau }-m_{t_i})}};-;\underset{\text{costs}}{\underbrace{C_{\text{hedge}}+C_{\text{fees}}}};+;\underset{\text{rebates, PFOF}}{\underbrace{R}}$$

The first term is mechanical and positive for fills at favorable prices relative to the contemporaneous mid. (In real data the split is messier: reference mid choice, quote-crossing, and locked or crossed markets all affect it. The economic content is unchanged.)

The second term is the game. Under adverse selection it is negative on average, because fills are not random draws from the arrival process. You get filled disproportionately in the instant before the price moves against you. That conditioning is the content of Glosten-Milgrom.

The value of a passive order is:

$$\mathbb{E}[V]=P(\text{fill})\times (\text{half-spread}+\mathbb{E}[ϵ,\Delta m\mid \text{fill}\left]\right)$$

keeping the sign convention from the line above, so the second term is negative when the market moves against the fill.

The term that kills you is the conditional expectation, not the fill probability. But the two are linked, and the link is the whole mechanism. Queue position drives both: sitting at the back lowers $P(\text{fill})$, since your order only executes once the entire queue ahead is consumed, and full depletion is exactly the state in which price is about to move. So a back-of-queue order fills less often and fills disproportionately in the worst states. The conditioning is not an independent penalty layered on top of a fair fill probability; it is produced by the same queue dynamics that determine the fill probability, which is why §4.3 treats queue position as a first-class modeling object.

Layer latency on top and it compounds. Everyone with a faster fair-value estimate has already cancelled by the time an informed order reaches the back of the book, so what remains is the residual liquidity that survives after everyone with a model has stepped away.

4.6 Markouts

The industry measure of adverse selection. A markout at horizon $\tau$ is post-fill price drift, signed by direction:

$$M(\tau )=ϵ\cdot (m_{t+\tau }-p_{\text{fill}})\cdot q$$

Compute at a ladder of horizons (100 ms, 1 s, 10 s, 60 s, 5 min for equities; longer in illiquid instruments), average over many fills of the same type, plot the curve. [P] as standard industry practice, from vendor and venue methodology documentation; [S] that Jane Street runs it across every venue, counterparty class, symbol and time bucket.

• Rises to a plateau. You captured the spread, flow was uninformed. A healthy wholesaling business.
• Rises then decays. You captured the spread but the flow carried short-lived information. Edge is real, holding period must be shorter than the decay.
• Negative from the start. The counterparty knew something. Widen, reject, or route away.

Markouts determine what a wholesaler pays for a flow stream, whether a venue is toxic, and how the adverse-selection term in the spread gets calibrated. They also transfer completely to a small trader. See §11.

4.7 Execution

When they take risk and must unwind, the machinery is Almgren and Chriss: minimize $\mathbb{E}[\text{cost}]+\lambda \mathrm{Var}[\text{cost}]$ over a liquidation trajectory with temporary impact $h(v)=\eta v$ and permanent impact $g(v)=\gamma v$. The empirically robust version is the square-root law:

$$\Delta P\approx Y,\sigma ,\sqrt{Q/V}$$

impact proportional to the square root of participation, with $Y$ an instrument-dependent constant of order one. The functional forms here are constrained by no-arbitrage arguments, which is a stronger footing than an empirical fit. Huberman and Stanzl (2004) show that permanent impact must be linear to preclude quasi-arbitrage, so the $g(v)=\gamma v$ above is an assumption with a result behind it. Gatheral (2010, Lemma 5.1) constrains the transient part: if impact decays like ${\tau }^{-\alpha }$ and instantaneous impact goes like $v^{\delta }$, absence of dynamic arbitrage requires $\alpha +\delta \ge 1$. (Gatheral writes the decay exponent as $\gamma$. It is reindexed to $\alpha$ here because $\gamma$ is already carrying risk aversion in §4.4 and the permanent-impact coefficient in the line above.) The empirical square root sets $\delta =1/2$, which is admissible only if impact decays at least as fast as ${\tau }^{-1/2}$. Slow decay plus concave instantaneous impact is exactly the combination that lets a sequence of small trades accumulate impact profitably, which is the manipulation the condition rules out. The two exponents are not free to be chosen separately.

Their published intern research topics include “market impact measurement” [D], so this is a documented internal subject.

Candidates contributed. This section supplies candidates for C1, C2, C3 and C5 in §10. It does not claim any particular implementation is identified; the queue-reactive and Hawkes machinery in §4.3 is the least evidenced part of it.

---

5. Volatility Surfaces and Options

Jane Street is ~8% of OCC volume **[P]**, roughly 1.2 billion contracts a year against 2025’s total. The Millennium complaint describes an options strategy generating $50to$100 million per month at peak, developed over five years by ~25 people, using “machine learning models for intra-day predictions” and “heuristic methods to characterize and predict market conditions” [D]. SEBI’s order documents a period where their Indian index options book netted above ₹43,000 crore while cash and futures lost money; I read it through legal analyses rather than the order itself [P].

5.1 What a market maker needs from a vol model

This is where most retail-facing writing goes wrong. A market maker does not need a correctly specified model of volatility dynamics for most pricing tasks, though dynamics do matter for path-dependent exposure and for hedging over horizons where the surface itself moves. What the quoting problem needs is a surface that is:

1. Arbitrage-free. A surface admitting butterfly or calendar arbitrage means someone lifts you on both legs and you have written a free option.
2. Fast to evaluate. Thousands of strikes, continuously.
3. Stable under refit. A surface that jumps between recalibrations generates phantom P&L and spurious hedge trades.
4. Consistent across the book. The same underlying prices identically on the ETF desk, the single-stock desk, and the index desk.

Fit quality to yesterday’s prices is a distant fifth. [S]

5.2 The static parameterization

A framing note covering §5.2 and §5.3. I have no direct evidence of Jane Street’s options model family. These sections answer a different question than the heading implies: what does a technically coherent institutional implementation of these requirements look like? That I can answer from the requirements and from published methods. Whether their implementation matches is a separate claim the evidence does not support. Where I write “almost certainly” below, it attaches to the engineering constraint, not to the firm. Substitute your own preferred parameterization; the argument does not depend on the choice.

The standard is Gatheral’s SVI. In raw form, for total implied variance $w(k)={\sigma }_{BS}^2(k),T$ at log-moneyness $k$:

w(k) = a + b\left{\rho(k - m) + \sqrt{(k-m)^2 + \varsigma^2}\right}

Five parameters per slice, linear asymptotes matching the Lee moment formula, butterfly arbitrage checkable in closed form via Durrleman’s function.

The surface extension is SSVI:

w(k, \theta_t) = \frac{\theta_t}{2}\left{1 + \rho\varphi(\theta_t)k + \sqrt{\left(\varphi(\theta_t)k + \rho\right)^2 + (1-\rho^2)}\right}

with ${\theta }_t$ the ATM total variance and $\phi$ a power law. Its value is that both no-arbitrage conditions reduce to closed-form inequalities:

• Butterfly-free if ${\theta }_t\phi ({\theta }_t)(1+|\rho |)<4$ and ${\theta }_t\phi ({\theta }_t{)}^2(1+|\rho |)\le 4$ (Gatheral and Jacquier, 2014, Theorem 4.2)
• Calendar-free if ${\partial }_t{\theta }_t\ge 0$ and ${\partial }_{\theta }(\theta \phi (\theta ))\ge 0$, together with an upper bound on that same derivative. I omit the full statement of the upper bound rather than paraphrase it; see Gatheral and Jacquier (2014), Theorem 4.1.

Calendar arbitrage is the one that bites a firm quoting across many expiries simultaneously, and it is the reason to use a surface parameterization rather than independent per-expiry slices.

Something in this family, modified for discrete dividends, borrow costs, American exercise, and index expiry term structure, satisfies all four requirements in §5.1. I know of no widely published alternative that satisfies them as cleanly. That is a statement about the available solution space. [R]

5.3 The dynamic layer

A static surface is insufficient for hedging and for path-dependent or forward-starting risk.

Local volatility (Dupire). Exactly reprices the surface by construction:

$${\sigma }_{\text{loc}}^2(K,T)=\frac{{\partial }_{T}C+(r-q)K,{\partial }_{K}C+qC}{\frac{1}{2}{K}^2,{\partial }_{KK}^{2}C}$$

Fast, arbitrage-free if the surface is, wrong about dynamics. It predicts the smile flattens as spot moves, which is not what happens.

Local-stochastic volatility. A Heston-like core plus a leverage function calibrated so vanillas reprice exactly, via the McKean-Vlasov particle method or a forward-PDE fixed point. The most common published institutional approach. The reason to want it is that it prices hedge ratios correctly, and hedge ratios are what a market maker trades on. [R]

Rough volatility. Hurst $H\approx 0.1$, reproducing the observed explosion of ATM skew at short maturity, $\psi (\tau )\sim {\tau }^{H-1/2}$. Given 0DTE volume and rough vol’s selling point being the very short end, I would be mildly surprised if they had no rough-vol machinery. It is expensive to simulate and I have no evidence. [X]

5.4 Hedging under transaction costs

The equation that governs an options book. For a short position continuously delta-hedged at implied vol ${\sigma }_i$ while the underlying realizes ${\sigma }_{r,t}$:

$${\Pi }_T;=;\frac{1}{2}{\int }_0^T{e}^{r(T-t)},S_t^2,{\Gamma }_t^{\text{imp}}\left({\sigma }_i^2-{\sigma }_{r,t}^2\right)dt$$

${\Gamma }_t^{\text{imp}}$ is the option’s gamma computed under the implied volatility used for hedging. The weighting is by the hedging model’s gamma. Flip the bracket for a long position.

The assumptions belong next to the equation. This is a short position, continuously delta-hedged at a constant implied vol ${\sigma }_i$, in a frictionless underlying, with deterministic rates and carry, and no discrete dividends. Relax any of those and additional terms appear: transaction costs (§ below), a stochastic ${\sigma }_i$, financing spreads between long and short stock, and dividend timing risk. The equation decomposes where the P&L comes from under those conditions. It is not a complete accounting of a real book.

Three consequences:

1. The edge is the implied-realized variance spread, not the premium collected. Premium is an accounting artifact.
2. Exposure is dollar-gamma weighted in time. $S_t^2{\Gamma }_t$ is enormous near the strike near expiry and near zero far from it. Realized P&L is dominated by what the underlying did during the few hours when gamma was largest. This is why 0DTE is a structurally different business from 30-day vol selling.
3. Path dependence is total. Two paths with identical realized volatility produce different P&L if they spend different time near the strike.

The degenerate case is a static hedge: fix delta at inception, accept the path. That leaves you short the variance risk premium and short a large amount of path risk, bundled into one premium at trade time, of which only the first is an edge. A market maker separates them: hedge continuously to isolate the variance spread, price the path risk explicitly.

Which runs into continuous hedging being infinitely expensive.

Leland. Discrete rebalancing under proportional costs can be handled approximately by replacing frictionless volatility with an effective volatility differing by a term in $\kappa /(\sigma \sqrt{\delta t})$, where $\kappa$ is the proportional cost and $\delta t$ the rebalancing interval. The direction and exact form depend on position sign, on whether you are adjusting option value, a quoted implied vol, or a hedging vol, and on the bid/ask convention. I am not writing a single formula, because the sign conventions in the literature are inconsistent enough that a compact statement misleads more than it helps. What survives every convention: costs enter as an effective-volatility adjustment, and it grows as you rebalance more often.

Whalley and Wilmott. The more useful result, and worth stating exactly. Under exponential utility with asymptotically small proportional costs, the optimal policy is a no-transaction band around the Black-Scholes delta with half-width

$$H={\left(\frac{3}{2},\frac{e^{-r(T-t)},\kappa ,S,{\Gamma }^2}{a}\right)}^{1/3}$$

$\kappa$ the proportional cost rate, $S$ spot, $\Gamma$ the option’s gamma, $a$ the risk-aversion coefficient of the exponential utility. (Implementations frequently drop the discount factor, giving $H=(3\kappa S{\Gamma }^2/2a{)}^{1/3}$; the two agree at zero rates and differ negligibly at short horizons.)

Note the exponent. Band width scales as the cube root of costs and inversely as the cube root of risk aversion, so tripling costs widens the band by about 44%. Two things follow. Including costs changes the shape of the policy rather than shifting its output: you go from “rehedge continuously” to “rehedge on a state-dependent boundary,” a different kind of object. And the optimal response to higher costs is far less dramatic than intuition suggests.

Deep hedging. Drop the closed form and learn a policy ${\delta }_{\theta }(\cdot )$ by minimizing a convex risk measure of hedged P&L under realistic costs and constraints. Given the GPU fleet, a documented history of deep RL work, and a business whose central difficulty is hedging a large book under costs, something in this family at Tier 3/4 setting policy parameters for Tier 1 execution is likely. [R]

5.5 Settlement design

Scope of this section and the next. What follows analyzes the economics of settlement mechanism design. It does not attempt to resolve the legality of the cited trades or the intent behind them, both of which are contested and neither of which is a modeling question. The trades are useful here because a regulator reconstructed them in public detail, which is rare, and because that detail exposes a general property of how settlement references are constructed.

The India trades are usually discussed as a question of intent. The mechanical question is more interesting: why was this possible there and much harder in US index options? The answer is settlement design.

India. NSE’s index closing value, the derivatives settlement reference, is the weighted average price over the last half hour of trading [D]. A 30-minute VWAP on a single venue, on an index of twelve constituents for Nifty Bank.

United States. SPX monthlies are AM-settled against the Special Opening Quotation, computed from the opening auction prints of all 500 components, which open at staggered times; the value publishes mid-morning once the last component opens. SPXW weeklies and dailies are PM-settled against the official closing index value from the closing auctions [D].

An auction aggregates all interest into a single price-discovery event with published imbalance information. Pushing one means committing size against every participant who sees the imbalance and wants the other side. Pushing a 30-minute VWAP on twelve names requires sustained pressure but no confrontation with concentrated liquidity.

The manipulability of a derivative settlement is a function of the reference price’s construction: number of constituents, length of the averaging window, depth of the venue during that window, and whether the reference is a continuous-trading average or an auction print.

The derivative-to-cash ratio is what makes it pay. On 17 January 2024, BANKNIFTY options notional turnover was ₹103.2 lakh crore against roughly ₹1.05 lakh crore of combined cash and futures [P]. Roughly 100 to 1. When the derivative market is 100x the underlying, the cost of moving the underlying is small relative to the payoff. That is arithmetic about market structure.

5.6 The two documented patterns

SEBI named two [P], both reaching me through legal analyses of the order.

Intra-day Index Manipulation. Two phases. On 17 January 2024: bought ₹4,370 crore of BANKNIFTY constituents and futures between 09:15 and 11:47, aggressively at or above last traded price, lifting the index from 46,573.93 to 47,176.97 against a bearish tape while building short options exposure at 7.3x the cash/futures position. Then sold ₹5,372 crore through the afternoon, aggressively at or below LTP. Net P&L for the day was ₹734.93 crore, with the cash and futures legs losing and the options leg making. (The ₹4,370 and ₹5,372 crore figures are traded notional, not positions or profit; the difference between them is not the P&L.)

Extended Marking the Close. Accumulate cheap out-of-the-money index options before the final period on expiry day, then push the underlying through the settlement window. One published reconstruction: index at 47,420 at 3:15 PM, 47,500 calls at ₹50, aggressive buying in HDFC Bank, ICICI Bank and SBI moved the index roughly 100 points to 47,520, calls went to ₹140, ~₹13.5 crore on that leg against a loss on the equity leg [R], secondary source.

The second pattern only functions because the settlement reference is a short trailing average on a narrow index.

On the index arbitrage defense. Jane Street characterized the activity as standard index arbitrage. The strongest published counter is structural rather than accusatory: for a pure mispricing-capture strategy, per-unit economics should improve as size falls, because smaller size means less impact eroding the convergence. Here impact was the source of profit rather than a cost against it.

Stated as a diagnostic rather than a theorem, since economics can legitimately improve with size: fixed costs and minimum commissions amortize, fee tiers and rebates improve, block liquidity and institutional pricing open up, information acquisition costs amortize, discrete contract sizes stop rounding you off. So the test applies specifically to purported pure mispricing capture, and the conclusion is conditional. If per-unit edge improves with size after netting fixed-cost and fee-tier effects, either your impact model is broken or your edge is not a mispricing.

I use the order for its structural facts (position ratios, notional turnover, timing), which derive from exchange data. I do not rely on its characterization of intent, which Jane Street contests and which is not a modeling question.

5.7 Dealer positioning

The expiry dynamic implies a model of other participants’ hedging demand. Aggregate dealer short gamma means dealers hedge with the move, amplifying it. Long gamma dampens. Open interest by strike is public, so a dealer gamma profile

$$G(S)=\sum _j{\text{OI}}_j\cdot {\Gamma }_j(S)\cdot {\text{sign}}_j\cdot 100\cdot S$$

is constructible by anyone. Units matter here and are often left unstated: as written this is dollars of dealer hedging demand per $1moveintheunderlying.Thewidelyquoted"GEX"conventioninsteadreportsdollarsper1$S/100$, so published figures are not comparable across conventions without checking. The sign assignment is the hard part, and it is where proprietary flow creates asymmetry: a firm with 8% of OCC volume knows the sign for a meaningful fraction of open interest because it wrote it. [R]

5.8 Dispersion and implied correlation

Index variance:

$${\sigma }_I^2=\sum _i\sum _j{w}_i{w}_j{\rho }_{ij}{\sigma }_i{\sigma }_j$$

so implied correlation from index and single-name option prices:

$${\rho }_{\text{imp}}=\frac{{\sigma }_I^2-{\sum }_i{w}_i^2{\sigma }_i^2}{{\sum }_{i\ne j}{w}_i{w}_j{\sigma }_i{\sigma }_j}$$

The dispersion trade is short index vol, long a weighted basket of single-name vol, which is structurally short implied correlation. It exists because index options are persistently richer than constituent-weighted single-name options. Driessen, Maenhout and Vilkov (2009) attribute that gap to a priced correlation risk premium rather than to a mispricing, which changes how you size the trade: the spread is compensation for holding an exposure that hurts when correlation spikes.

Quoting both markets consistently creates this exposure whether or not a firm frames it as a dispersion book. Whether it is then run deliberately, with a target correlation position and a dedicated risk limit, is a separate proposition the evidence does not settle. The risk profile is instructive: correlation goes to one in a crash, so a short-correlation position loses precisely when everything else does. Correlation desks blew up in 2008 and again in early 2018 for this reason. [R]

Candidates contributed. C3 and C4. Rough volatility and deep hedging are raised here as hypotheses and are rejected from the set in §10.4, since nothing observed requires either.

---

6. ETFs, Baskets, and Relative Value

6.1 The structural fact

An ETF is unusual in having a contractually enforced relationship to another observable price. Creation and redemption tie the ETF price $P_t$ to basket value $N_t$ within a band whose width is the round-trip cost.

A natural model of the log basis $x_t=\ln P_t-\ln N_t$:

$$d{x}_t=-\kappa (x_t-\mu ),dt+{\sigma }_x,d{W}_t,|x_t|\lesssim c_{\text{create}}$$

with $\kappa$, $\mu$, and effective band width conditional on liquidity regime, time of day, and tradeability of the underlying. [S]

The insight separating a good ETF market maker from a bad one is that $N_t$ is not observed for most interesting ETFs. Jane Street told the SEC directly in a comment letter:

“market participants today typically calculate their own intraday values of an ETF portfolio, and do not rely on the published IIV” [D]

For SPY, computing $N_t$ is trivial and worth nothing. For a high-yield bond ETF, an emerging-market equity ETF trading while its home markets are closed, or a bank loan fund, $N_t$ must be estimated, and that estimate is the product. [S]

6.2 Custom baskets

Before SEC Rule 6c-11, most ETFs operated under exemptive orders requiring creation and redemption baskets to be pro-rata slices. Rule 6c-11 permits custom baskets that differ from a pro-rata slice, subject to written policies and documentation that each is in the fund’s best interest [D].

The economic consequence is large and is why the bond ETF AP business is profitable. [S]

1. The AP can deliver what it has. A market maker holding specific corporate bonds from unrelated client business can create ETF shares by delivering those bonds rather than sourcing a pro-rata slice of 1,200 CUSIPs, most of which did not trade today.
2. The negotiation is over which bonds, not just how many. Composition is negotiated between AP and issuer. Both have views on which bonds are cheap or rich. The AP wants to deliver its least attractive inventory; the fund wants bonds that improve its portfolio. The spread between those views is a recurring negotiated economic quantity, and modeling it well is an unusual edge.
3. It converts inventory risk into a channel. A bond position that would sit on the balance sheet becomes ETF shares sellable in a liquid secondary market. This is the mechanism by which their fixed income market making and ETF business are the same business, and it explains their description of “basket-driven bond inventory from ETFs” [D].

Not a pricing model in the usual sense. An optimization over inventory and a constraint set: choose the basket maximizing the value of what you deliver, subject to the fund’s acceptance criteria, tracking-error tolerances, and your own inventory objectives. [S]

6.3 Estimating the basket

For a bond ETF holding 1,200 bonds of which maybe 150 trade on a given day, fair value requires a price for every bond, every second, from sparse observations. The construction is a factor model in spread space:

$$s_i(t)={\beta }_i^{\top }f(t)+{ϵ}_i(t)$$

with $f(t)$ latent factors (level, slope and curvature of the credit curve; sector; rating bucket; duration; on-the-run status; issue size; liquidity) fitted to whatever is observable, and non-trading bonds priced by projection. Pricing 16,000 bonds is only possible this way. [S]

The closed loop is the interesting part. The ETF’s own traded price is an observation of the factor state. If the ETF is trading and the bonds are not, inference runs backwards, from ETF to bonds. Hasbrouck’s information share and the Gonzalo-Granger permanent-transitory decomposition measure which venue leads price discovery in a cointegrated system, and in credit the ETF leads during stress. Their own paper is titled Credit ETF Trading in Stressed Markets [D].

6.4 Pricing a portfolio trade

$$\text{Quote}=\sum _i{\hat{P}}_i-\underset{\text{ETF, CDX, futures}}{\underbrace{C_{\text{hedge}}}}-\underset{\text{factor-mismatch VaR}}{\underbrace{C_{\text{residual}}}}-\underset{\text{holding period}\times \text{carry}}{\underbrace{C_{\text{inventory}}}}-\underset{\text{is the client informed?}}{\underbrace{C_{\text{adverse}}}}-\text{margin}$$

Every term is a model. The last deserves emphasis: a client asking you to bid on a specific basket has chosen that basket. The selection is informative, and the same adverse-selection logic governing a passive equity quote governs a portfolio bid at a different timescale. [S]

6.5 Hedging

Chosen by cost.

The proxy problem is constrained optimization: minimize tracking-error variance subject to cost and position limits,

$$\underset{h}{\min };(b-h{)}^{\top }\Sigma (b-h)+\lambda ,c(h)$$

Standard portfolio math, but at their scale $\Sigma$ must be estimated well: shrinkage, imposed factor structure, intraday updating. [S]

6.6 Two mechanisms worth naming

The NAV trade. They offer “NAV +/- X bps” execution [D]. Sell a client the ETF at end-of-day NAV plus 5 bps and you have converted “predict the price path” into “replicate the NAV.” NAV comes from official closing prices, closing auctions are the deepest liquidity of the day, so hedge into the close and tracking error against NAV is near zero by construction. The model requirement collapses from a price forecast to an execution model. [S]

The highest-quality edges come from restructuring a trade so that a hard prediction problem becomes an easy replication problem.

Heartbeat trades. Redemptions in kind do not trigger capital gains at the fund level, so an ETF can clear low-basis positions by delivering them in a redemption basket. The heartbeat pattern, a large creation followed shortly by a large redemption around index rebalances, is the operational expression. APs are the counterparty and are compensated for it. A fee-and-inventory business rather than a modeling one, but a real revenue line. [R]

Candidates contributed. C1 and C3, applied to baskets instead of single names. The custom-basket optimization is the clearest case in the document of a friction term reshaping a policy.

---

7. Statistical Alpha and the ML Stack

7.1 What they have said

From the quantitative research page [D]: “analyzing large datasets using a variety of machine learning techniques, building and testing models, creating new trading strategies and writing the code that implements them.”

From the 2017 blog post [D]: Gaussian processes, random forests, adaptive regression splines, genetic algorithms, increasingly deep learning. “Standard techniques almost never work out-of-the-box.” ~2.3 TB/day of market data. “There is no shortage of ways to fool yourself when building ML systems.”

From the 2025 podcast [D]: “tree-based models, or maybe they’re like deep learning models, or maybe they’re just some artisanal regressions that you’ve cooked up.” On cross-asset transfer, Cho asks “how do we transfer the understanding that we’ve developed in the asset classes that we have a lot of experience and competence in?” On multimodal inputs, “how do we interleave things like images or text or market data or what have you to really be able to make use of all the information.” Those two lines are the sourcing for the last two rows of the table in §7.2.

From The Uncertain Art of Accelerating ML Models (Episode 21, October 2024) [D]: Sylvain Gugger, previously a core maintainer of Hugging Face’s Transformers library, is on Jane Street’s ML-infrastructure team helping traders speed up their models. The episode covers learning-rate schedules, PyTorch performance traps, keeping a GPU fed, reproducibility of training runs, and — the load-bearing item — “the unusual size and shape of market data and the need to do inference at very low latencies.”

That last phrase does more work than anything else in the public record for §3. It is a first-party statement that low-latency inference is a live engineering constraint on their neural networks, which is exactly the tension the tier hierarchy predicts and exactly what a firm running nets only in offline research would have no reason to discuss.

From Wrestling the World into Rows (Episode 29, September 2026) [D]: Eric Mannes, a decade at the firm and formerly a commodities trader, helps lead an alternative data team. Named inputs: satellite photographs of parking lots, credit-card panels, weather forecasts. The episode’s stated thesis is “why better ML models raise the value of careful data engineering,” and it covers the day oil futures settled at a negative price and the systems that broke as a result, plus a period when the commodities desk’s risk system was one very large spreadsheet.

Three things follow. Alternative data is a staffed function, not an aspiration, which upgrades the last row of §7.2. The stated thesis is first-party confirmation of the argument in §7.2 that the binding constraint is the data pipeline and not the architecture. And the negative-oil-settlement anecdote is a documented instance of the failure mode this entire document is about: on 20 April 2020 WTI settled at $-37.63, and every model carrying a log-price transform, a lognormal assumption, or a non-negativity constraint became undefined at once. The assumption was invisible until the state space moved outside it.

From Building a Data Warehouse from Scratch (Episode 28, June 2026) [D]: Superstore, a distributed columnar database supporting analytical queries against transactional writes arriving at trading speed, and the Hive, described as a massive compute cluster for research. Jacob Baskin, who built Superstore and studied mechanisms that incentivize people to reveal true preferences, now works on “schemes for bidding on compute time” to allocate the Hive. That is an internal auction, and it is a model class §10 previously missed.

From their published intern research topics [D]: robust low-memory regression on large datasets; statistical methods for non-static, dependent observations; NP-complete optimization in trading systems; market impact measurement; inferring the distribution of asset prices by combining historical data with options data; recognizing “situations where markets behave strangely in a way that is obvious to humans.”

That second-to-last item is the most telling on the list. Combining physical-measure history with risk-neutral option-implied distributions is the pricing kernel and variance risk premium problem. Extracting a good $\mathbb{P}$-measure forecast from $\mathbb{Q}$-measure prices is directly monetizable in an options book, and it is the theoretical bridge between §5 and §7.

7.2 The hierarchy, and why linear models survive

The realistic picture is not “they use deep learning.” It is a layered ensemble where complexity is allocated by signal-to-noise and latency budget. [S]

Artisanal regressions persist alongside the GPU fleet for a reason. At $R^2\approx 0.01$ the signal is small enough that estimation variance and specification risk both become difficult to ignore, and additional model complexity has to earn its keep.

The supporting intuition: OLS with $p$ parameters on $n$ observations has expected excess prediction error scaling like ${\sigma }^{2}p/n$. To keep that well below the $0.01{\sigma }^2$ of signal, you want

$$p\ll n/100$$

That is a statement about the cost of degrees of freedom, not a bias-variance decomposition of the actual problem. Low $R^2$ does not establish that variance dominates bias; misspecification can dominate at any $R^2$, since bias depends on the distance between the true function and the model class rather than on how much variance the signal explains. What low $R^2$ does establish is that every parameter you spend is expensive.

This is why financial ML is hard, and it does not go away when you buy more GPUs. It goes away when you get more independent observations, which markets do not supply on demand. Worse, financial observations are not independent, so effective $n$ is far below row count. A year of 1-minute bars is 98,000 rows and nothing like 98,000 independent observations of anything that matters at a daily horizon.

Cho on the discipline [D]:

“In cases where you have very low amounts of data… you would prefer to not leave that many data points out of sample and just be very careful about the number of hypotheses that you test.”

7.3 Validation

Purged, embargoed cross-validation. Standard K-fold leaks when labels are built over overlapping forward windows. If the label at $t$ depends on returns through $t+h$, any training observation whose own label window overlaps $[t,t+h]$ shares outcome information with the test observation.

Two distinct mechanisms needing distinct remedies:

• Purging removes every training observation whose label window overlaps the test fold. Defined by label geometry, so its extent follows from how the labels were built rather than from a fixed number of days.
• The embargo protects against leakage through temporal proximity: serial correlation in features, slow-moving state variables, and preprocessing that touched neighboring observations. Its width depends on the data-generating process and is not simply the label horizon.

Combinatorial purged CV generates many splits while preserving both.

A hypothesis budget. Cho’s phrase is “not duplicating hypotheses across groups” [D], which is the multiple-testing problem stated organizationally. Fifty researchers each testing 100 specifications is 5,000 trials, and the maximum Sharpe under the null is large. The correction is the deflated Sharpe ratio:

$$\text{DSR}=\Phi !\left(\frac{(\widehat{SR}-S{R}_0)\sqrt{n-1}}{\sqrt{1-{\gamma }_3\widehat{SR}+\frac{{\gamma }_4-1}{4}{\widehat{SR}}^2}}\right)$$

$n$ is the number of return observations, ${\gamma }_3$ and ${\gamma }_4$ are skew and kurtosis, and $S{R}_0$ is the expected maximum Sharpe across $N$ independent trials under a null of zero skill:

$$S{R}_0=\sqrt{\mathrm{Var}[\widehat{SR}]};\left[(1-\gamma ),{\Phi }^{-1}!\left(1-\frac{1}{N}\right)+\gamma ,{\Phi }^{-1}!\left(1-\frac{1}{Ne}\right)\right]$$

with $\gamma \approx 0.5772$, the Euler-Mascheroni constant. Note $n$ and $N$ are different quantities and conflating them is a common error.

Two further implementation traps. $\widehat{SR}$ must be at the same frequency as $n$, so daily returns require a daily Sharpe, not an annualized one. And $\mathrm{Var}[\widehat{SR}]$ in the $S{R}_0$ expression is the variance of Sharpe ratios across the $N$ trials, not the sampling variance of any single strategy’s Sharpe. Both mistakes make $S{R}_0$ too small, which makes the deflation too forgiving, which defeats the point.

Practical consequence: a backtest Sharpe of 2.0 found after 1,000 trials can be statistically less convincing than a Sharpe of 1.0 found after 3. Whether it is depends on sample length, return distribution, and dependence, which is exactly what the formula above accounts for and a raw Sharpe does not.

Reproducibility as a research control. Ron Minsky’s point is that versioned, reproducible pipelines exist to prevent silent contamination of the out-of-sample set across teams rather than for engineering hygiene [D].

Regime awareness. Cho: “a financial crisis seems to occur roughly every year… the distribution of features or the returns that you might see in your data just kind of changes” [D]. Responses: exponentially weighted refitting, explicit regime variables (realized vol level, cross-sectional dispersion, liquidity), regime-switching models, ensembling across models fitted on different windows. [R]

7.4 The anti-inductive problem

Minsky [D]:

“When you see regularities in the behavior of prices, you’re incentivized to trade against those and that beats them out of the market.”

This is the structural difference from every other ML domain. A cat does not learn to evade your cat classifier. The modeling consequence: signal decay must itself be modeled. Every deployed signal needs a monitored half-life, and the research pipeline needs a replacement rate exceeding aggregate decay.

That is an organizational property as much as a statistical one, and it is the strongest argument that the compensation structure is a competitive advantage rather than a cultural quirk.

Candidates contributed. C6 and C7, both [D]-grade, and C8 via the compute-allocation evidence in §7.1. This is the section where the public record is strongest and the inference is therefore shortest.

---

8. Risk, Capital, and the Meta-Model

8.1 What is documented

15% liquidity buffer against traded capital. $1.22Bbroker-dealernetcapitalagainsta$1M requirement. ~$30Bpartnerequitysupportingadebtprogramofroughly$5.4B in 2025, the April tranche of which was the $1.35Bof6.75$14.6B at ~8% in August 2026. Fitch BB+ positive, S&P outlook raised on business diversification. Fitch attributes risk discipline to “the high level of member ownership” [P], all of it via trade press. I have not seen the rating reports.

I could not find a disclosed VaR figure anywhere public. The risk stack presumably includes per-desk and firm-wide inventory limits with automated Tier 0 enforcement, VaR and expected shortfall at multiple horizons, scenario and stress grids, greeks aggregated across the options book with cross-underlying netting, and a funding model mapping every position to its stressed margin requirement. That is inference from what any firm this size must do. None of it is disclosed. [S]

The 15% buffer is the informative number. Not a VaR figure, a survival figure, sized so a simultaneous margin call can be met without forced liquidation. For a firm whose business is being the buyer when nobody else is, the ability to not be a forced seller is the franchise.

8.2 What July 2026 revealed

They lost approximately $15billioninJuly2026,thefirstdownmonthinaboutadecade,ontheSituationalAwarenessstake(whichtookmargincallsasAIpositionsdeteriorated)andondirectionalAsianequitypositions.PartnerTurnerBatty:"Julywasabadmonth,"andthefirm"closedasignificantportionofourriskinthespecificareaswelostoninJuly,andhavealsoreducedrisk-takinginotherstrategies."Theystillfinishedthefirsteightmonthsabove$40B [P].

The loss did not come from the model stack. It came from a concentrated, illiquid, directional, unhedgeable position and from directional equity bets. Everything in sections 4 through 7 is machinery for extracting small edges at high frequency with tightly controlled inventory, and it appears to have kept working. The first eight months were a record including a $15B month, and the year is not over.

They run at least two businesses with different risk characteristics on one balance sheet:

1. Market making and relative value. Many small, high-Sharpe, rapidly recycled bets. Model-driven, low tail risk per position, managed by inventory limits.
2. Directional and principal investment. Few large, low-Sharpe, long-duration bets. Judgment-driven, enormous tail risk.

The second produced both a reported ~830M gain on Anthropic in Q3 2025 and the \~15B July loss [P]. Not a model. A punt made by people with a lot of capital and reasonable but not superhuman opinions about AI.

Candidates contributed. C7 again, in its risk-governance form. The principal and venture book contributes nothing: §10.5 places it outside the set entirely.

---

9. The Revenue Bridge

Every number in this section is my estimate. None of it is disclosed. The inputs are documented; the capture rates are assumptions, and I give ranges because the honest uncertainty is wide.

Inputs. Primary: OCC 2025 volume of 15.2B contracts, mix 8.27B equity / 5.68B ETF / 1.26B index **[D]**. Reported: 2025 net trading revenue 39.6B; 2023 \~10.0B; equity volume ~2T/month (2024) → \~24T/year; fixed income ~230B/month → \~2.8T/year; ~8% of OCC contracts → ~1.2B contracts; industry-wide US payment for order flow >$4.9B/year across all wholesalers [P]. That last figure feeds no row below; it is here because it is what failed the plausibility check on the order-flow numbers removed from §2.2, and it bounds how large any PFOF contribution to the residual can be.

Note what that mix means for the bridge below. The denominator is primary and the numerator is press. Every conclusion in this section inherits the weaker of the two.

Midpoint roughly $11B.Against$39.6B reported, that leaves a model residual of $22Bto$35B, or 55% to 89% of the total.

One caveat on the equities row specifically. A blended 0.1 to 0.3 bp is a reasonable assumption for high-volume institutional equity and ETF flow, but retail wholesaling captures materially more per unit of notional on a much smaller notional base, and this row does not separate the two. If wholesaling is a meaningful share of that $24T, the row understates. I have left it blended because I cannot split the notional from public data, and I would rather have a visibly crude row than a falsely precise one.

The word residual is doing precise work. This is the gap between a reported revenue figure and a bottom-up estimate built from incomplete public data with assumed capture rates. It is not an unexplained portion of a correctly partitioned income statement, and I have not partitioned anything. A large residual is at least as likely to indict my assumptions as to reveal something about the firm.

Sanity check on the index options row, which is doing the most work. An SPX contract near an index level of 6,500 carries roughly $650,000ofnotional,soonebasispointis$65 per contract. My $20to$80 range corresponds to capturing 0.3 to 1.2 bp, which is plausible for a large index options market maker but is the assumption I would attack first. Index options are about 46% of the estimated subtotal at the midpoint and roughly twice the next largest row, so the bridge is close to a single-assumption model and should be read that way.

Candidates for the residual, in rough order of the weight I give them:

1. 2025 was a high-volatility year and capture rates are convex in volatility. Spreads widen, volumes rise, and the two multiply. A market making business can plausibly earn 2 to 4x calm-market revenue in a genuinely volatile year. Likely the largest single factor, and the reason not to extrapolate 2025.
2. The principal and venture book. Mark-to-market on Anthropic, CoreWeave, Situational Awareness flows through net trading revenue. Not market making revenue in any meaningful sense, and July 2026 shows it flows both ways.
3. Concentrated, capacity-limited opportunities. India contributed $2.3B+ in 2024 from equity derivatives alone [P]. Extremely high margin while they last and invisible in any volume-times-capture calculation.
4. Directional risk-taking inside the market making book. Holding inventory deliberately instead of hedging flat.
5. My capture assumptions are too low, particularly in fixed income and in the ETF primary market, where the custom-basket economics in §6.2 are hard to size from outside.

A note on the correction that produced these numbers. An earlier version of this table had the equities row at $2.4Bto$7.2B, which prices the row at one basis point of $24Twhentherowclaims0.1to0.3bp.Fixingitloweredthesubtotalandraisedtheresidual,whichmakesmyconclusionlookstronger.Acorrectionthatflatterstheauthordeservesmorescrutinythanonethatcostshimsomething,so:theresidualwidenedbecauseIremovedroughly$2B to $6.5Bofestimatedmarketmakingrevenue,andifthetrueequitiescaptureisnearer1bpthan0.1bp,theoriginalfigurewasclosertorightandtheresidualisnearer$15B to $33B. The conclusion below holds under both, which is the only reason I am comfortable stating it.

The strongest point the bridge makes is almost incidental. Revenue went from ~$10Bto$39.6B, roughly 4x, without a corresponding 4x change in market share in any disclosed business line. ETF and options shares were already large in 2023. Volume growth was real but nothing like 4x; OCC contracts grew 24% in 2025. So the growth is predominantly volatility, risk appetite, and the principal book, not displacing competitors in market making.

That is a materially different firm from the one in most popular coverage, and it is the conclusion I hold with the most confidence here.

---

10. How Small Can the Set Be?

The question this document exists to answer asks for the smallest set of model classes consistent with what is observable. A list is not an answer to that.

Neither is what follows, quite, so the standard being met is worth defining before the argument leans on it.

Definitions. A partition of the model instances into classes is:

• Deletion-irreducible if no class can be removed without leaving an observed behavior unexplained.
• Coarsening-irreducible if no two or more classes can be combined without losing a distinction the evidence requires.
• Closed, relative to the candidate set considered in §§4 through 10, if no excluded candidate is required to explain an observed behavior. Unlike the first two, this property is explicitly relative to a universe of candidates I chose.
• Globally minimal if no alternative partition of the same instances, under any vocabulary of primitives, yields fewer classes.

The first three are separate properties and the fourth is not implied by them. This section argues the first three and not the fourth.

One point of logic is worth settling here, because it determines how much work §10.3 has to do. Coarsening-irreducibility appears to require testing merges of every size: pairs, triples, and so on up to all eight at once. It does not. If combining $C_i$ and $C_j$ loses a required distinction $D$, then every partition placing $C_i$ and $C_j$ in the same block also loses $D$, including every multiway merge containing both. Losing a distinction is monotone under coarsening. So pairwise non-mergeability implies non-mergeability at every arity, and checking all $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$ pairs would establish coarsening-irreducibility outright.

That argument is short enough to be checkable, which is the point of stating it. It also has a limit that matters more than the result. It covers coarsenings of this partition, meaning unions of existing blocks. It says nothing about a different partition built from different primitives, which is not a coarsening of this one and is not reachable by any merge test. That is the gap, and it is the same vocabulary gap named above rather than a second independent one.

This section establishes deletion-irreducibility (§10.2), coarsening-irreducibility (§10.3 and Appendix D), and closure (§10.4). It does not establish global minimality, and the reason is worth being exact about, because the obvious explanation is the wrong one. Global minimality does not fail here for want of evidence about Jane Street. More disclosure would not help. Class count is not a property of the firm at all. It is a property of the description language, and the choice of primitives fixes it before any evidence is consulted. Given a fixed vocabulary of primitives, the question is a finite combinatorial check. Absent a canonical vocabulary it is underdetermined, which is a weaker and more accurate claim than the undecidability I asserted in earlier revisions.

A merge test needs a criterion, because any two classes can be merged by describing them loosely enough:

A merge is valid only if it reduces the class count without destroying a distinction the evidence requires.

§10.2 runs the deletion test on all eight classes, so deletion-irreducibility is established. §10.3 argues the three pairs most open to challenge in prose, and Appendix D carries all twenty-eight, so coarsening-irreducibility is established under the stated criterion. Global minimality remains out of reach either way, for the vocabulary reason above and not for want of testing.

10.1 Reduction

The business lines in sections 4 through 8 present roughly thirty-five distinct model instances. Most are the same mathematics pointed at different assets. Collapsed:

Seven classes for the trading business, plus one for internal operations. This reduction is the answer at the level the public evidence and this choice of primitives can support. The model instances developed across §§4 through 8 are represented by these eight objects with different observables substituted in, subject to the scope exclusions in §10.5.

Three of those labels were wrong until this revision, in the same way, and the defect is worth stating because it bears on the whole result. Each named a mechanism where the members share only a role. C3 was “stochastic control with a no-trade region,” but a no-trade region is what you get from proportional or fixed costs (Whalley-Wilmott, the ETF creation band); Almgren-Chriss has linear impact and yields a smooth trading schedule with no region at all. C4 was “constrained interpolation,” which describes SVI and not Dupire, an inversion that takes derivatives of an observed surface. C5 was “Bayesian updating,” but a markout is a measurement and a flow classifier may be discriminative.

The general failure: the merge-validity criterion above was applied to proposed merges and never to the merges performed when the partition was built. Applying it retroactively is what caught these. A class label that names a mechanism not all members use has already destroyed a distinction, which is the thing the criterion forbids.

One collapse survives that scrutiny and is worth stating explicitly. The Avellaneda-Stoikov inventory skew, the Whalley-Wilmott hedging band, the Almgren-Chriss schedule and the ETF creation band appear in four sections as four models. They share one structure: a cost or risk term that makes the optimal policy differ from the frictionless one. The shape of that difference varies with the friction, which is exactly why the class is named for the friction now and not for the shape. And C1 and C5 look like the same regression until you notice that C1 conditions on exogenous state while C5 conditions on an event your own quote caused. That distinction is the entire difference between a fair-value estimate and a fill-conditional one, and it is where retail market making dies.

10.2 Necessity

A class earns its place only if removing it leaves something observed unexplained.

C8 is the honest edge case. It explains nothing about trading and is only necessary if “how the firm operates” includes internal operations. I include it because the question as posed says operates, and flag the scope ambiguity rather than smuggling it past.

10.3 Non-mergeability

The deletion test is not enough. If two classes are really one class described twice, the count is inflated without any class being deletable. Applying the criterion above, three merges are worth testing.

C2 into C3, as “stochastic control under state-dependent transition intensities.” This is the most plausible merge and it fails on two counterexamples already in the document. Whalley-Wilmott is a C3 containing no point process: the state is a diffusion, the cost is proportional, and the no-trade band comes from the cost term with no arrival intensity anywhere in the derivation. The queue-reactive model is a C2 containing no control: it is a generative model of book dynamics with no objective function and no policy. Each survives without the other. A class definition broad enough to hold both would also hold C1 and C6, so the merge does not reduce the count. It dissolves the taxonomy’s discriminating power, which is the property that makes it worth having.

C7 into C6, as “model selection is part of supervised learning.” Regularization controls variance within a single fit. Selection correction addresses the bias introduced by how many fits you looked at. A perfectly cross-validated model that was the survivor of a thousand candidate specifications still needs deflating, and the deflation depends on $N$, a quantity that appears nowhere in the fitting objective. Different input, different failure mode, different remedy.

C4 into C1, as “both fit a function to observed prices.” C1’s output is a point estimate under squared error. C4’s product is a set of inequalities that must hold everywhere, including at strikes where no quote exists. An unconstrained fit through the same option chain is a tradeable arbitrage against you. In C4 the constraint is the deliverable and the fit is incidental, which inverts C1.

C1 and C5 are the fourth candidate merge, and §10.1 handles it: the conditioning event differs.

The remaining twenty-four pairs are in Appendix D, generated from a script that verifies coverage instead of asserting it. Two of them, C1/C6 and C5/C6, are closer calls than any discussed here and are marked as such.

10.4 Rejection

Minimality cuts. These were considered and excluded, and several were in earlier revisions of this document:

Demoting rough volatility and deep hedging is a real cost of taking the question seriously. Both are things I find interesting and both survived six revisions on that basis alone, with no evidence behind either.

10.5 Sufficiency, and the limit of the answer

Two observed facts the set does not explain.

The principal and venture book is not a model, and §8.2 says so. It produced the ~830M Anthropic gain and the \~15B July loss through discretion, with no machinery involved.

More seriously: the eight classes explain the conventional market making business, which §9 estimates at roughly $11Batthemidpointagainst$39.6B of reported revenue. So the honest closing statement of this document is that the eight-class set developed here accounts for something like a third of the observed revenue, and the remainder is volatility convexity, directional risk, and principal investment that no model stack explains.

That is a weaker conclusion than the section heading promises, and it is the correct one. A reconstruction that explained all $39.6B from a model taxonomy would be evidence that the taxonomy had been fitted to the total.

---

11. What Transfers to a Small Account

11.1 The hard no’s

Anyone selling a retail market making bot is selling the losing side of Glosten-Milgrom. The argument is in §4.5.

11.2 Tier A: transfers almost completely

Ideas and measurements. No infrastructure required.

1. Reframe every position by its exposures, not its name.

Retail option structures are marketed by strategy name (the wheel, the collar, the credit spread, the iron condor) and the names obscure what the position is. Two structures with different names and identical exposures are the same trade with different commissions. Decompose everything into the same basis: $\delta$, $\Gamma$, vega, $\theta$, factor betas, financing.

Put-call parity is the canonical demonstration that name and exposure are different objects. For European options on a non-dividend-paying underlying,

$$C-P=S-K{e}^{-rT}$$

so any long-stock-plus-short-call structure is, in exposure terms, a short put plus a long cash position of $K{e}^{-rT}$. The cash leg is not cosmetic. It is why the structure earns financing and why its payoff is bounded above. Rearranging parity for whatever you hold usually reveals a simpler position than the strategy name implies, at higher cost.

Once decomposed, the right questions become askable. Which exposures am I paid to carry, which am I carrying by accident, and is the compensation for each commensurate with its risk?

2. Use the gamma P&L equation as a pre-trade filter.

$$\Pi =\frac{1}{2}\int e^{r(T-t)}{S}_t^2{\Gamma }_t^{\text{imp}}({\sigma }_i^2-{\sigma }_{r,t}^2),dt$$

Before selling any option, estimate both vol terms. Implied you can read. Realized you can estimate. If the spread is not positive and materially larger than transaction costs, you have exposure without edge.

For calibration, and this needs its definitions stated because the sign conventions in this literature are a minefield. Feunou, Jahan-Parvar and Okou define the variance risk premium as risk-neutral minus physical expected realized variance,

$${\text{VRP}}_{t,h}={\mathbb{E}}_t^{\mathbb{Q}}[R{V}_{t,h}]-{\mathbb{E}}_t^{\mathbb{P}}[R{V}_{t,h}],\text{VRP}={\text{VRP}}^U+{\text{VRP}}^D$$

decomposed into upside and downside semivariance components, with skewness risk premium $\text{SRP}={\text{VRP}}^U-{\text{VRP}}^D$. On their S&P 500 sample the downside component averages about +3.4, the upside about −4.4, the skewness premium about −7.8, and the total VRP is close to zero [D]. Consult the paper’s Table 1 for the exact normalization before quoting the magnitudes: variance premia are reported in variance units in some of this literature and in volatility-equivalent units in others, and I have not independently confirmed which convention applies here. The argument below uses only the signs and the relative sizes, which are unaffected.

That last figure is the opposite of what most option-selling commentary assumes. In this decomposition the aggregate premium nearly vanishes, all compensation sits in the downside component, and the upside component is negatively compensated. Selling upside variance, on this evidence, does not pay.

Two caveats. The level of the total VRP is sensitive to how ${\mathbb{E}}^{\mathbb{P}}[RV]$ is forecast, which is an estimation choice rather than an observable; Carr and Wu and Bollerslev, Tauchen and Zhou find a robustly positive total using different physical-expectation estimators, so the disagreement is about the estimator rather than the phenomenon. Treat the level as model-dependent. What is robust across specifications is the composition: the premium is concentrated in, and is compensation for, downside risk.

The practical implication survives both caveats, and most retail option sellers have it backwards. You are being paid for crash insurance, not for time decay. If your position sizing assumes a steady coupon, you have mispriced the trade.

3. Compute markouts on your own fills. (§4.6.)

Highest-value item here, and it costs nothing. For every fill, record fill price, direction, and the mid at 1 minute, 5 minutes, 30 minutes, and 1 day after. Sign and average.

A systematically negative markout curve means you are being adversely selected: limit orders filling precisely when the price is about to move against you, or market orders chasing. Most retail traders have never measured this. It separates “my strategy is wrong” from “my execution is wrong,” which are different problems with different fixes, and almost nobody at small scale distinguishes them.

4. Hypothesis budgeting and deflated Sharpe.

Write down the number of variants tested before looking at the result. Apply the deflation in §7.3. Most retail backtests reporting Sharpe 1.5 after a few hundred parameter sweeps have a deflated Sharpe indistinguishable from zero. This single practice will save more money than any signal you find.

5. Purged, embargoed validation.

Purge every training observation whose label window overlaps the test fold. Add an embargo sized to your feature construction. Nearly every publicly posted retail backtest leaks here and reports a Sharpe 2 to 3x the real one.

6. Inventory-aware position sizing.

The reservation price $r=s-q\gamma {\sigma }^2(T-t)$ is a statement about you, not the market. Willingness to add should fall as position grows, in proportion to ${\sigma }^2$. Most retail blowups come from inventory, not forecasting. Set hard limits per underlying and per factor, and make entry thresholds a function of current inventory.

7. Fractional Kelly, and knowing why.

$f^{\text{*}}=\mu /{\sigma }^2$. Growth rate $g(f)=f\mu -f^2{\sigma }^2/2$ is zero at $f=2f^{\text{*}}$ and negative beyond, so overbetting is not slightly worse than Kelly, it is ruinous. Parameter uncertainty in $\mu$ is severe, so a quarter to a half Kelly is the honest correction.

8. The size test. (§5.6.)

For a strategy you believe captures a mispricing, per-unit economics should improve as size falls. If your backtest looks better per unit at larger size, first net out fixed costs, minimum commissions, fee tiers, and contract rounding. If the effect survives, either your impact model is broken or your edge is not a mispricing.

11.3 Tier B: transfers with real work

9. Restructure prediction problems into replication problems. (§6.6.) Look for trades where “I think X goes up” becomes “X and Y are contractually linked and mispriced by more than the cost of enforcing the link.” Closed-end fund discounts, ETF/underlying pairs in illiquid corners, index-inclusion mechanics, merger spreads, futures/spot basis. Most are competed away in liquid instruments and survive only where capacity is too small for institutions. That capacity constraint is your only structural advantage. Fish where the pond is too small for a $30B balance sheet to bother.

10. Multi-horizon target construction. (§4.2.) Predict at several horizons and read the shape. Positive at 1 day and negative at 5 days is a liquidity-provision signal that reverts. Positive at both is an information signal. Different holding periods, sizing, and exits. Collapsing to a single horizon is why strategies work in backtest and not live.

11. Sample weighting by economic relevance. Weight training observations by the size you could actually have traded.

12. Realized volatility estimation done properly. Close-to-close discards most of the information in the bar. Parkinson uses the high-low range only; Garman-Klass, Rogers-Satchell and Yang-Zhang use full OHLC. All are several times more efficient than close-to-close. Yang-Zhang handles both overnight gaps and drift, which is what you want for item 2.

13. Model transaction costs inside the objective, not after it. Whalley-Wilmott is the archetype: including costs changes the shape of the optimal policy, it does not shift its output. A strategy optimized ignoring costs and then charged costs is a different object from one optimized under costs.

14. Understand your settlement references. (§5.5.) If you hold anything to expiry, know how its settlement value is computed: which window, which venue, auction print or trailing average, how many constituents. This is documented in every exchange rulebook and almost nobody reads it. The AM/PM distinction in SPX alone has produced a lot of unpleasant Friday mornings for people who did not know their contract stopped trading the afternoon before.

11.4 Tier C: useful, mostly aspirational at small scale

15. Arbitrage-free surface fitting. Fitting SVI to a liquid option chain is a tractable weekend project and it immediately shows which quoted options are inconsistent with their neighbors. Whether you can trade the inconsistency after costs is separate, but the diagnostic value is high and the exercise teaches the geometry better than reading does.

16. Order flow imbalance. Computable from affordable Level 2 data. Useful for execution timing even where it is not tradeable alpha.

17. Dealer gamma positioning. Inferable from public OCC open interest. The mechanism is real. Widely disseminated now, so the alpha is thin, but the risk management application, knowing when the market is structurally prone to acceleration, survives dissemination better than the alpha application does.

18. Implied correlation. (§5.8.) Computable from index and constituent chains. Even without trading dispersion, the level tells you how much of the index’s implied vol comes from correlation rather than constituent vol, which is a different regime signal from VIX alone.

11.5 A program, in order

1. Instrument the portfolio in exposure space. Delta, gamma, vega, theta, factor betas. Not P&L by ticker. A weekend of work, and it changes everything downstream.
2. Start recording markouts on every fill. Costs nothing, accumulates immediately, and cannot be recovered retroactively. Do this first because it is the only item with a time cost to delay.
3. Build an implied-versus-realized variance tracker for the underlyings you actually trade. Yang-Zhang realized vol, IV from the chain, spread as a time series, conditioned on regime.
4. Establish a hypothesis log. Every variant, dated, with the result. Deflate before deploying.
5. Build validation infrastructure before signals. Purged CV with embargo, walk-forward, a held-out period you touch once.
6. Set hard inventory limits per underlying and per factor, with entry sizing that decays in current inventory.
7. Only then look for signals, in capacity-constrained corners.

The ordering is the lesson. Jane Street did not win by finding a magic signal. They won by building an environment where signals could be evaluated honestly, deployed safely, and replaced continuously, and by refusing to become a forced seller.

Steps 1 through 6 are available to an individual in kind, which is not the same as available at comparable quality. Their markout samples are larger by many orders of magnitude, their execution and cross-asset data are richer, their costs are lower, their risk limits are more granular, their validation infrastructure is staffed by teams rather than by one person on a weekend, and their ability to spend real capital investigating an anomaly has no small-scale equivalent. What transfers is the methodology, not its resolution.

Step 7 is where a small trader still needs an actual source of edge, and no amount of methodology substitutes for that.

---

12. Open Questions

1. The revenue residual is estimated, not known. §9 narrows it from “unexplained” to roughly $22Bto$35B, or $15Bto$33B under the higher equities-capture assumption, most likely dominated by volatility convexity and the principal book. Every capture rate is my assumption. The index options row is the most load-bearing and least certain.
2. §10 establishes deletion-irreducibility, coarsening-irreducibility and closure, and not global minimality. All twenty-eight pairs are now tested (Appendix D). What remains open is not a gap in the testing but the vocabulary: class count is a property of the chosen primitives and not of the firm, so no evidence of any kind settles whether eight is canonical.
3. No direct evidence of the options model family. SVI/SSVI and LSV are inferred from business requirements and published practice.
4. The distillation hypothesis is inference. The piecewise-linear network posts are suggestive, not dispositive.
5. No public VaR disclosure exists that I could find.
6. The SEBI order is adversarial. I used its structural facts, which derive from exchange data, and avoided its characterization of intent.
7. Most financial figures are [P], not [D]. Jane Street is private and declines to comment on results, so revenue, headcount, capital, and rating figures reach the public through press reporting sourced to unnamed people, much of it a single chain. The [P] tag marks this throughout, but it is worth stating plainly: the quantitative spine of §2.1 and §9 rests on reporting rather than filings. Re-verify near publication.
8. The Extended Marking the Close worked example is a secondary reconstruction. The pattern is documented; the specific numbers are illustrative.
9. The crypto business (JCX) is unmodeled for lack of public detail.
10. No information on their FX business, which for a firm trading in 45 countries must be substantial and must have its own stack.
11. Custom basket economics are structurally clear but unsized. I believe this is a major profit center and cannot put a number on it.

---

Appendix A: References

This list contains both works cited inline and background reading that shaped the analysis without being quoted. One in-text attribution is deliberately absent: I use “Durrleman’s function” for the SVI butterfly check in §5.2 as it appears in Gatheral’s treatment, and have not traced it to a primary source I have read.

Market microstructure and market making

Avellaneda, M., & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3), 217–224.

Bouchaud, J.-P., Bonart, J., Donier, J., & Gould, M. (2018). Trades, quotes and prices: Financial markets under the microscope. Cambridge University Press.

Cartea, Á., Jaimungal, S., & Penalva, J. (2015). Algorithmic and high-frequency trading. Cambridge University Press.

Cont, R., Kukanov, A., & Stoikov, S. (2014). The price impact of order book events. Journal of Financial Econometrics, 12(1), 47–88.

Glosten, L. R., & 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.

Guéant, O., Lehalle, C.-A., & Fernandez-Tapia, J. (2013). Dealing with the inventory risk: A solution to the market making problem. Mathematics and Financial Economics, 7(4), 477–507.

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

Huang, W., Lehalle, C.-A., & Rosenbaum, M. (2015). Simulating and analyzing order book data: The queue-reactive model. Journal of the American Statistical Association, 110(509), 107–122.

Jaisson, T., & Rosenbaum, M. (2016). Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes. The Annals of Applied Probability, 26(5), 2860–2882.

Jaisson, T., & Rosenbaum, M. (2015). Limit theorems for nearly unstable Hawkes processes. Annals of Applied Probability, 25(2), 600–631.

Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica, 53(6), 1315–1335.

Stoikov, S. (2018). The micro-price: A high-frequency estimator of future prices. Quantitative Finance, 18(12), 1959–1966.

Execution and market impact

Almgren, R., & Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3(2), 5–39.

Gatheral, J. (2010). No-dynamic-arbitrage and market impact. Quantitative Finance, 10(7), 749–759.

Huberman, G., & Stanzl, W. (2004). Price manipulation and quasi-arbitrage. Econometrica, 72(4), 1247–1275.

Volatility and options

Bayer, C., Friz, P., & Gatheral, J. (2016). Pricing under rough volatility. Quantitative Finance, 16(6), 887–904.

Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. Review of Financial Studies, 22(11), 4463–4492.

Buehler, H., Gonon, L., Teichmann, J., & Wood, B. (2019). Deep hedging. Quantitative Finance, 19(8), 1271–1291.

Carr, P., & Wu, L. (2009). Variance risk premiums. Review of Financial Studies, 22(3), 1311–1341.

Driessen, J., Maenhout, P. J., & Vilkov, G. (2009). The price of correlation risk: Evidence from equity options. Journal of Finance, 64(3), 1377–1406.

Dupire, B. (1994). Pricing with a smile. Risk, 7(1), 18–20.

Feunou, B., Jahan-Parvar, M. R., & Okou, C. (2015). Downside variance risk premium (Finance and Economics Discussion Series 2015-020). Board of Governors of the Federal Reserve System. Published as Feunou, B., Jahan-Parvar, M. R., & Okou, C. (2018), Journal of Financial Econometrics, 16(3), 341–383. The §11.2 figures are from the 2015 working paper, which is the version I read. Working-paper and published versions of this work differ on the sign and significance of at least one reported relation between the skewness risk premium and the equity premium, so the version is not incidental. Working paper version: Federal Reserve FEDS 2015-020

Gatheral, J. (2006). The volatility surface: A practitioner’s guide. Wiley.

Lee, R. W. (2004). The moment formula for implied volatility at extreme strikes. Mathematical Finance, 14(3), 469–480.

Gatheral, J., & Jacquier, A. (2014). Arbitrage-free SVI volatility surfaces. Quantitative Finance, 14(1), 59–71.

Leland, H. E. (1985). Option pricing and replication with transactions costs. Journal of Finance, 40(5), 1283–1301.

Whalley, A. E., & Wilmott, P. (1997). An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Mathematical Finance, 7(3), 307–324.

Exchange-traded funds

Ben-David, I., Franzoni, F., & Moussawi, R. (2018). Do ETFs increase volatility? Journal of Finance, 73(6), 2471–2535.

Gonzalo, J., & Granger, C. W. J. (1995). Estimation of common long-memory components in cointegrated systems. Journal of Business & Economic Statistics, 13(1), 27–35.

Hasbrouck, J. (2003). Intraday price formation in U.S. equity index markets. The Journal of Finance, 58(6), 2375–2400.

Hasbrouck, J. (1995). One security, many markets: Determining the contributions to price discovery. Journal of Finance, 50(4), 1175–1199.

Pan, K., & Zeng, Y. (2019). ETF arbitrage under liquidity mismatch (Working paper). European Systemic Risk Board.

Petajisto, A. (2017). Inefficiencies in the pricing of exchange-traded funds. Financial Analysts Journal, 73(1), 24–54.

U.S. Securities and Exchange Commission. (2019). Exchange-traded funds (Release No. 33-10695, Rule 6c-11). https://www.sec.gov/files/rules/final/2019/33-10695.pdf

Research methodology

Bailey, D. H., & López de Prado, M. (2014). The deflated Sharpe ratio: Correcting for selection bias, backtest overfitting, and non-normality. Journal of Portfolio Management, 40(5), 94–107.

Harvey, C. R., Liu, Y., & Zhu, H. (2016). …and the cross-section of expected returns. Review of Financial Studies, 29(1), 5–68.

López de Prado, M. (2018). Advances in financial machine learning. Wiley.

Volatility estimation

Parkinson, M. (1980). The extreme value method for estimating the variance of the rate of return. The Journal of Business, 53(1), 61–65.

Rogers, L. C. G., & Satchell, S. E. (1991). Estimating variance from high, low and closing prices. The Annals of Applied Probability, 1(4), 504–512.

Garman, M. B., & Klass, M. J. (1980). On the estimation of security price volatilities from historical data. Journal of Business, 53(1), 67–78.

Yang, D., & Zhang, Q. (2000). Drift-independent volatility estimation based on high, low, open, and close prices. Journal of Business, 73(3), 477–491.

---

Appendix B: Sources by Class

Ordered by evidentiary weight, matching the [D] and [P] tags used throughout.

Primary

First-party statements, regulator and exchange documents, court filings, and original research. Claims resting on these are tagged [D].

Jane Street. Quantitative Research; Performance Engineering; Client Offering; engineering blog archive; Signals and Threads Ep. 22 “Finding Signal in the Noise” (In Young Cho, March 2025), Ep. 21 “The Uncertain Art of Accelerating ML Models” (Sylvain Gugger, October 2024), Ep. 28 “Building a Data Warehouse from Scratch” (Jacob Baskin, June 2026), Ep. 29 “Wrestling the World into Rows” (Eric Mannes, September 2026), and Ep. 1 “Programmable Hardware” (Andy Ray, September 2020), all hosted by Ron Minsky; episode descriptions are first-party, and where I cite episode content I cite the published description; I have not listened to the audio in full; SEC comment letter on ETF rulemaking (S7-15-18); Credit ETF Trading in Stressed Markets; Real-Time Market Data Forecasting competition (Kaggle, 2024) and Market Prediction (Kaggle, 2020).

Exchange and regulatory. NSE Nifty Bank contract specifications; Cboe SPX options specifications; OCC 2025 annual volume; SEC Rule 6c-11.

Court filings. Amended complaint in Jane Street v. Schadewald.

Academic. All entries in Appendix A, and the Feunou, Jahan-Parvar and Okou working paper specifically, which I read directly.

Secondary

Reputable reporting and analysis where the underlying primary evidence is not available to me. Claims resting on these are tagged [P].

2025 full-year results; July 2026 loss; rating agency risk profiles via trade press; payment for order flow totals; SEBI order analysis and strategy reconstruction, both standing in for the order itself.

Tertiary

Used for orientation and for figures I could not source better. Treated as the weakest evidence in the document.

Bond prospectus discussion; index arbitrage critique and worked example; markout methodology; Whalley-Wilmott reference implementation; Jane Street Capital (Wikipedia, for corporate history and dated figures).

---

Appendix C: Corrections

Kept because a document that grades its own confidence should show what it got wrong. The pattern these corrections share is stated in the Summary.

Revision A. Initial analysis.

Revision B. Added the settlement-mechanics analysis (§5.5), which was the largest omission: the India trades were described without explaining why they were possible there and not in US index options. Added markout analysis (§4.6), the revenue bridge (§9) in place of listing the revenue growth as unexplained, dispersion and implied correlation (§5.8), ETF Rule 6c-11 custom baskets and heartbeat trades (§6.2, §6.6), and market making P&L attribution (§4.5).

Revision C. Removed an incorrect put-call parity identity that treated a long-stock-plus-short-call structure as a synthetic short put with no cash leg. Scoped the ~1% $R^2$ claim, which had been generalized from short-horizon liquid-instrument prediction into a near-universal leakage test. Downgraded the sample-weight and symbol_id inferences. Tightened the purged-CV mechanics, which had read as a universal recipe. Made the size test conditional rather than categorical. Corrected the multi-venue data entry in the hard-no list, which cited cost rather than the real constraint. Made the epistemic distinction explicit in §5.2. Removed the covered-call framing throughout.

Revision D. Corrected the variance risk premium discussion (§11.2) after checking the primary source. The prior version stated the S&P 500 premium was persistently positive, then quoted components summing to a negative number. Checking Feunou, Jahan-Parvar and Okou directly showed the total premium on their sample is close to zero, with all compensation in the downside component and the upside component negatively compensated. The section now states definitions explicitly and separates the level of the premium, which is estimator-dependent, from its composition, which is robust. Removed the Leland sign-convention equations as not defensible without specifying the adjusted quantity and bid/ask convention. Verified the Whalley-Wilmott band against an independent implementation and corrected the risk-aversion symbol. Softened remaining absolutes.

Revision E. Fixed a second instance of the “entire economics” overstatement in the retail wholesaling sentence, which survived the same fix applied to the adjacent queue-position paragraph. Worth recording as a pattern: a rhetorical habit corrected in one location tends to survive elsewhere in the same document. Softened the Hawkes-to-rough-volatility claim from “the microstructural origin” to a mechanism.

Revision F. Moved revision history to the end, removed in-text references to prior revisions, added inline source links and an APA reference list, and cut length for tone.

Revision G. Split the [D] tag into [D] primary-documented and [P] reported. The opening had defined [D] as first-party or regulatory evidence, then applied it to press reporting sourced to unnamed people. That was an internal inconsistency of exactly the kind this document criticizes elsewhere, and it mattered most where it was least visible: nearly every financial figure in §2.1 and §9. Reclassified Appendix B by source class to match. Corrected “at $R^2\approx 0.01$, estimator variance dominates bias,” which asserted a bias-variance decomposition the evidence does not support; the $p/n$ calculation is now framed as the cost of degrees of freedom rather than a deduction about the data-generating process. Changed the Hawkes branching condition from market stability to process stationarity, a model property rather than a claim about the world. Stopped half-stating the SSVI calendar condition and cited it out instead. Added the assumptions adjacent to the gamma P&L equation. Softened the deflated Sharpe comparison and the dispersion-book claim. Added an explicit scope statement before the settlement-design examples, separating the economics of the mechanism from the contested legal question.

Revision H. Corrected an arithmetic error in the §9 equities row, which multiplied $24Tofnotionalby0.1to0.3bpandreported$2.4B to $7.2B.Thatis1to3bp.Therowisnow$0.24B to $0.72B,thesubtotal$4.2B to $17.6B,andtheresidual$22B to $35B.Becausethecorrectionflatterstheconclusion,\S 9nowshowsthearithmeticbothwaysandstatesthattheconclusionhastoholdunderboth.Fixed\S 4.5,whichclaimedaretailparticipanthas"thesame$P(\text{fill})$asanyoneelse"while\S 4.3treatsqueuepositionasadeterminantoffillprobability;thetwoarecausallylinkedthroughqueuedepletion,andthelinkedversionisthebetterargument.Downgraded"theequationin\S 4.5istheproof"toanargument.Addedthepodcastquotesthatsourcethelasttworowsofthe\S 7.2table,whichcarried\ast \ast [D]\ast \ast tagsareadercouldnottrace.Reconciledthe$5.4B debt program in §8.1 with the $1.35B tranche in §2.1. Removed the Robinhood order-flow figures, which failed a plausibility check on direction and magnitude and whose reporting period I could not establish.

Precision: added the sign convention to the passive-order value equation; corrected $k$ from a fill arrival rate to the decay rate of fill intensity, and noted that $A$ drops out of the optimal spread; flagged the variance-premium unit convention as unconfirmed; added two deflated Sharpe implementation traps; corrected Parkinson to a high-low estimator; cited Gatheral (2010) for the impact exponent and Driessen, Maenhout and Vilkov (2009) for the correlation premium; added a notation table for symbols that collide across the source literatures; clarified that the ₹4,370 and ₹5,372 crore figures are traded notional rather than P&L.

Presentation: added a summary, retitled the piece so the revenue figure is not read as a valuation, renamed this appendix, and cut about a third of a “not X, it is Y” construction that had reached roughly thirty instances and read as a tic rather than as emphasis.

Revision I. Corrected the §4.7 impact citation, which attributed to Gatheral (2010) a claim about permanent impact under “a broad class of decay kernels.” Gatheral’s result governs transient impact and constrains the decay and instantaneous-impact exponents jointly ($\alpha +\delta \ge 1$); the linearity-of-permanent-impact result is Huberman and Stanzl (2004), from quasi-arbitrage. The error was introduced by the previous revision’s own precision pass. Fixed “index options are now more than half the estimated subtotal,” which is 46%. Updated §12 and the §9 back-reference, both of which still carried the pre-correction $15Bto$30B residual. Reduced the “rather than” construction from 43 instances to 31, after the previous revision cut the “not X, it is Y” form and the habit reappeared in a different phrase at comparable combined density. (Counts here exclude this appendix’s own quoted mentions of the phrase, which a naive grep includes.)

Revision J. Noted that Gatheral writes the decay exponent as $\gamma$ and that §4.7 reindexes it to $\alpha$, so a reader checking the source does not hit an unexplained mismatch, and added the lemma number. Added ${\delta }_{\theta }$ and option delta to the notation table; $\delta$ carries four meanings in this document and the table had covered two. Corrected “roughly 30” to 32 in the entry above, since a changelog that rounds is inconsistent with its own standard. Moved two generalizations out of the Revision I entry and into the pattern list below, where they belong.

---

Revision K. Full audit pass. Corrected the Hawkes-to-rough-volatility citation: the reference list carried Jaisson and Rosenbaum (2015), whose nearly-unstable limit is a classical CIR process and is not rough. The rough limit is their 2016 heavy-tailed result, and the heavy-tail hypothesis is the load-bearing assumption the earlier text omitted. Added both papers and the relation $H=\nu -1/2$, which ties §4.3 to the $H\approx 0.1$ in §5.3 quantitatively. Fixed a version conflict between the appendices on the variance risk premium paper: Appendix A cited the 2018 published article while Appendix B recorded reading the 2015 working paper, and the two versions disagree on a signed result. Added missing references for Lee, Gonzalo-Granger, Parkinson and Rogers-Satchell, all attributed in text but absent from the list, and stated plainly that the Durrleman attribution is untraced. Corrected “the year was a record” for a year that has four months left. Noted that the industry PFOF figure in §9’s inputs feeds no row. Repositioned the §4.2 [X] tag, whose referent was ambiguous. Added units to the dealer gamma profile in §5.7.

Three of those are the same failure as §4.7 in Revision H: a citation that supports a weaker claim than the sentence attached to it. That is now the second-most-frequent pattern in this appendix and it has its own entry below.

---

Revision L. Restructured §10 from an enumeration into an actual minimality argument, which is what the document’s central question asks for and had never been given. Thirty-five model instances collapse to seven irreducible classes plus one for internal allocation, each with a stated necessity test, and a rejection set naming what was excluded and why. Two consequences worth recording. First, minimality demotes rough volatility and deep hedging out of the set: nothing observed requires either, and both had survived six revisions because I find them interesting. Second, the sufficiency check forces a weaker headline than the section promises, since the set explains the ~11B of conventional market making in §9 and not the \~22B to $35B residual. Added four Signals and Threads episodes never used, two of them from 2026: Gugger on ML infrastructure, whose “need to do inference at very low latencies” is the strongest first-party support the public record offers for the §3 tier hierarchy; Mannes on a staffed alternative-data team, which upgrades the last row of §7.2 from research direction to operating function and independently confirms this document’s data-over-architecture claim; Baskin on internal compute-bidding, which is a model class the map had missed entirely; and the negative WTI settlement of April 2020 as a documented instance of an invisible assumption becoming load-bearing.

That last one is worth a line. A log-price transform is not a modeling choice anyone writes down as an assumption. It became one on 20 April 2020.

Revision M. Retitled §10 and replaced the implicit claim of minimality with the three properties the argument actually establishes: irredundancy, non-mergeability on tested pairs, and rejection closure. The section had been asserting a stronger result than it proved, which §12 already conceded in a place no reader reaches before the claim. Added §10.3, which tests the three merges most open to challenge. The C2-into-C3 merge is the serious one and it fails on two counterexamples already in the document: Whalley-Wilmott is a control problem containing no point process, and the queue-reactive model is a point process containing no control. Corrected the third headline conclusion, which said the eight classes “cover the trading business” while §10.5 said they cover the conventional market making component. Named the problem type in §1 as an identification problem under partial observability, which is what it has been since Revision L without saying so.

The §10 title is the fourth instance of the same pattern and the most structural: the section named for a claim it did not establish, with the qualification filed thirty pages later in §12. A caveat that appears only after the reader has formed the belief is not a caveat.

Revision N. Made the standard in §10 formal: defined irredundancy, closure and global minimality as separate properties, and stated the merge-validity criterion the three tests in §10.3 had been applying without naming. Corrected a claim inside §10’s own limiting paragraph, which said global minimality was “probably not provable at all.” That is a decidability assertion I never established. The accurate statement is narrower and more useful: global minimality is not settleable by evidence about Jane Street, because class count is a property of the description language and is fixed by the choice of primitives before any evidence is consulted. Under a fixed vocabulary it is a finite check. The same overclaim appeared in §12 item 2 and is corrected there too. Added a fifth pattern below.

Revision O. Split the §10 terminology into deletion-irreducibility, coarsening-irreducibility and closure, which are three properties the previous revision had bundled under one word. Added the monotonicity argument: losing a required distinction is preserved under further coarsening, so pairwise non-mergeability implies non-mergeability at every arity, and the merge test does not need to enumerate triples and larger sets. That argument corrects a claim in Revision N, which said checking all pairs was “as close to minimality as this method reaches.” Checking all pairs would in fact establish coarsening-irreducibility outright, which is more than N credited it with and still less than minimality. Softened “the reduction is the answer” in §10.1, which contradicted §10’s own opening two pages earlier.

Retagged the ES/SPY lead-lag claim in §4.3. It carried [S], but Hasbrouck (2003) establishes E-mini price-discovery dominance empirically, so the claim was under-tagged and is now [D] with a citation. That is the first correction in this appendix that moved a tag up, and it is worth noting that a taxonomy of evidence can err toward caution as easily as toward overclaiming; both are misdescriptions. Qualified two sentences outside §10 where the prose asserted more than the surrounding argument: the volatility-dynamics claim in §5.1 and the adverse-selection sign in §4.5.

Revision P. Renamed C3, C4 and C5. All three labels named a mechanism that only some members use, where the members actually share a role: a no-trade region is produced by proportional costs and not by the linear impact in Almgren-Chriss; “constrained interpolation” describes SVI and not Dupire, which is an inversion; “Bayesian updating” describes Glosten-Milgrom and not a markout, which is a measurement. The general failure is that §10’s merge-validity criterion had been applied only to proposed merges and never to the merges performed when I built the partition. Applying it retroactively caught all three at once. This is the most consequential correction since Revision L, because the eight-class count was partly held together by loose labeling, which is the exact thing the criterion forbids.

Corrected §10.5, where “the smallest set of model classes consistent with public evidence” had survived after five revisions of removing that claim everywhere else. It is now “the eight-class set developed here.” A phrase excluded from a section’s argument can persist in the section’s conclusion, which is the hedge-is-an-assertion pattern with the polarity reversed: the qualification was made and then not propagated.

Added one closing line to each of §§4 through 8 naming which classes that section contributes to §10, so the reduction reads as the destination of the preceding analysis.

Revision Q. Completed the coarsening test. Appendix D carries all twenty-eight pairs, generated from a script that verifies coverage instead of asserting it, which moves §10 from arguing coarsening-irreducibility to establishing it under the stated criterion. §12 item 2 no longer lists this as open; what remains open there is the vocabulary, which no amount of testing addresses. The honest limit of the exercise is stated in the appendix: C1/C6 is the weakest cell, and a reader who rejects its witness gets seven classes.

Four terminology and scope repairs. The summary still said “the set is irredundant,” a bundled word §10 retired two revisions earlier. §10.1 said “sections 4 through 7” while §8 contributes C7 and §7.1 motivates C8. Closure was defined as though absolute and is now explicitly relative to the candidate set I chose, which is the property that distinguishes it from the other two. And headline conclusion 3 said the classes “explain” the conventional market making, where the body says public evidence does not identify implementations; “represent” is the word the argument supports.

Five patterns run through these corrections, and the patterns are worth more than the fixes that produced them.

Writing the strongest available sentence, then qualifying it later or not at all. This is the recurring one, present in C, D, E, G, H and I. The §9 arithmetic error is the same failure in a different medium: a number that made the argument cleaner and did not get checked because it pointed the right way.

A correction is an edit, and edits are where errors enter. The §4.7 misattribution was introduced by Revision H’s precision pass, whose entire purpose was fixing errors. Revisions do not monotonically reduce error count, and a freshly corrected passage is the last place to relax scrutiny. It is the first.

A citation is checked for existence, not for scope. Four separate times, a real paper was attached to a claim it does not quite support: Gatheral on permanent impact, Jaisson and Rosenbaum on light-tailed limits, a published article standing in for the working paper actually read, and four in-text attributions with no entry at all. The failure is not fabrication. It is stopping the check once the source is confirmed to exist and to be about roughly the right subject. The fix is to state the hypothesis the result actually requires, which is also what makes the paragraph better.

A hedge is an assertion. §10’s opening paragraph exists to stop the section from overclaiming. It overclaimed. It said global minimality was “probably not provable at all,” which is a decidability claim I had not established and did not need, sitting inside the one passage written specifically to limit what the section claims. The generalization is that qualifying a statement produces a new statement, and the new one arrives unqualified because attention is on the thing being softened. This is close to the edits-introduce-errors pattern above and distinct from it: that one is about the mechanics of revising, this one is about the logical form of hedging.

A surface-form purge relocates a reflex instead of removing it. Cutting the “not X, it is Y” construction in H pushed the same contrastive habit into “rather than” at comparable combined density, so the count moved sideways while the writing did not change. Counting a phrase measures a symptom. The reflex underneath is a preference for defining things by what they are not, and it has to be caught while the sentence is being built.

---

Appendix D: The Full Coarsening Matrix

§10.3 argues three pairs in prose. This is all twenty-eight, generated by a script that encodes each class’s defining primitive and each cell’s witness pair, then verifies that every pair is present and every cell carries two witnesses. The script checks coverage, not correctness: it cannot tell whether a witness is right, only whether I skipped one. A reader checks the witnesses.

A merge is inadmissible when both witnesses exist: a member of the first class lacking the second’s defining primitive, and a member of the second lacking the first’s. One-sided containment would make one class a special case of the other and the merge admissible. By the monotonicity argument in §10, establishing all twenty-eight pairs establishes inadmissibility at every arity.

* Closest calls. The other 23 separate on primitives that share no vocabulary.

Coverage check: 8 classes, 28 pairs required, 28 recorded, none missing, none incomplete, none spurious.

Two caveats. The witnesses are drawn from members already named in §10.1, so this tests the partition as built and not any partition. And C1/C6 is the weakest cell: linear regression is supervised learning, and the separation rests on the microprice being derived, not fitted. If a reader rejects that witness, C1 and C6 merge and the count falls to seven.