Part VIII · The whole market · Chapter 8D

Two regimes

When hedging calms the market and when it pushes it

Everything in Parts I to VII compresses to a single regime question: is the dealer book net long gamma or net short gamma? The answer sets the sign of the feedback loop between hedging and prices, and with it the whole character of intraday price action.

1 · Two regimes, one mechanism

The two regimes are mirror images of each other, row by row:

A toy example shows the mechanism. Suppose GEX is +$5 billion per 1% move. By Chapter 8A's hedge-flow formula, a 1% rally then has dealers selling about $5 billion of index exposure and a 1% fall has them buying about $5 billion, so each move meets a flow against it. If GEX is −$5 billion instead, the same rally has dealers buying about $5 billion and the same fall has them selling about $5 billion. The news, and the first 1% move it caused, are identical in both cases. Only the sign of the response differs.

The table is the Dealer Mechanics Lab's, and the theory later in this chapter adds two rows to it. The sign of dealer gamma changes liquidity as well as volatility, and neither sign makes prices better informed.

2 · Same shocks, opposite character

The figure below draws six price paths from 100 over 60 steps. Each step's return is a random shock, with a standard deviation of 0.08%, plus a feedback term equal to 0.45 times the previous step's return. The feedback is subtracted when dealers are long gamma (positive GEX) and added when they are short it (negative GEX). Both regimes use the same six runs of random shocks, so the sign of the feedback is the only difference. Switch between the two regimes and compare how far the paths wander.

regime positive GEX negative GEX

Under positive GEX the six paths stay between about 99.5 and 101. Under negative GEX the same shocks spread them from about 98.9 to 102.5, more than twice as wide. In the language of Chapter 3B, each regime makes returns an AR(1) process, with weight $w = -0.45$ (mean reversion) when dealers are long gamma and $w = +0.45$ (momentum) when they are short gamma. One mechanism produces both behaviors.

The same fundamentals can therefore produce qualitatively different price behavior, depending on the dealer-positioning regime. That is why regime-conditional strategy selection, the subject of Part XI, matters. The weight of 0.45 is chosen so the difference shows in six paths. It is an illustration, not a measurement, and the measured effects are far smaller, as a later section shows.

3 · The mathematics of feedback

Published GEX is positive when hedging is stabilizing, but the feedback maths reads more naturally with the opposite sign. So define the destabilizing gamma $G$ as the negation of published GEX, in whatever units the model uses. It is positive when dealers are net short gamma, so that their pro-trend hedging amplifies moves. It is negative when they are net long gamma, so that their contra-trend hedging damps moves. With that sign convention, the Dealer Mechanics Lab writes

in which each symbol has a plain meaning, and the feedback coefficient $\kappa$ carries the state of the market:

Only the product $\kappa G$ matters, so the units of $G$ and $\kappa$ can be anything that makes it a pure number. Taking square roots, and keeping $\kappa G$ below 1, gives $\sigma_\text{eff} = \sigma / (1 - \kappa G)$, and the denominator sorts the regimes:

In the Dealer Mechanics Lab's reading, that divergence is, mathematically, what February 5, 2018 (Volmageddon), March 16, 2020 (the Great Liquidity Crisis) and August 5, 2024 (the yen carry unwind) looked like. Chapter 8H takes the cases in detail and weighs that reading against what regulators and researchers later found.

Where the formula comes from: a feedback multiplier

One simple model produces (8D.1) exactly. It is a toy rather than the Dealer Mechanics Lab's derivation, but it shows what $\kappa$ and $G$ do.

Step 1: the first round. News moves the price by a return $\varepsilon$, with standard deviation $\sigma$. Dealers hedge that move, and their trade moves the price by a further $\kappa G\,\varepsilon$, with the first move if $G > 0$ and against it if $G < 0$.

Step 2: the rounds that follow. The extra move needs hedging too, which adds $\kappa G$ times it, $(\kappa G)^2 \varepsilon$, and that needs hedging in turn. When $\kappa G$ lies between −1 and 1, the rounds shrink and the total is a geometric series:

$$\begin{aligned} \text{move} &= \varepsilon\,\big(1 + \kappa G + (\kappa G)^2 + \cdots\big) \\ &= \frac{\varepsilon}{1 - \kappa G} \end{aligned}$$ (8D.2)

Step 3: the volatility. Dividing a random quantity by a constant divides its standard deviation by the same constant. The total move therefore has standard deviation $\sigma/(1 - \kappa G)$, and squaring it gives (8D.1).

Step 4: over time. The regime figure spreads the same feedback over successive steps, $r_t = w\,r_{t-1} + \varepsilon_t$. A shock then reaches the price as $\varepsilon_t$ in its own step, $w\,\varepsilon_t$ in the next and $w^2\varepsilon_t$ in the one after, for a total of $\varepsilon_t/(1 - w)$ once the echoes die out. The paths' volatility over many steps follows the same formula, with $w = \kappa G$.

The next figure draws the grind box's rounds one at a time, in units of the news shock $\varepsilon$. The first bar is the news, and each bar after it is one round of hedging, $\kappa G$ times the round before, drawn from where the running total stood. The two sliders set $\kappa G$, from −0.9 to +0.95 and starting at +0.45, and the number of rounds shown, from 0 to 12. Watch the running total approach $\varepsilon/(1 - \kappa G)$, then push $\kappa G$ toward 0.95 and count how many rounds are still large.

feedback κG

+0.45

rounds of hedging

12

total ε/(1 − κG) = 1.82ε

after 12 rounds: 1.82ε (100%)

round 12 adds under 0.01ε

Amplifying. Each round joins the move before it, so the total grows to 1.82 times the news.

At the starting $\kappa G$ of +0.45, every round joins the move before it, and the total reaches 1.82ε, which 12 rounds match to within 0.01ε. At −0.45 the rounds alternate and the total settles at 0.69ε, below the shock itself. At +0.95 the limit is 20ε, but 12 rounds reach only 9.73ε, 49% of it, and the twelfth still adds 0.54ε. At −0.9 the total zigzags, 0.66ε after 12 rounds against a limit of 0.53ε. Near $\kappa G = 1$ the rounds stop dying out: that is the singular point of (8D.1), seen as a process.

Instant or lagged: what the regimes do to autocorrelation

The grind box summed its rounds within one instant, and the regime figure spread them over successive steps. Both give the same multiplier for the volatility of the cumulative move, $\sigma/(1 - w)$. They differ on autocorrelation. If dealers hedge within the instant, each return is larger or smaller than the news, but it carries nothing of the return before: volatility changes and autocorrelation does not. If they hedge one step late, as in the figure, each return carries $w$ times the one before, and the lag-1 autocorrelation is $w$ itself.

The table's rows on mean reversion and momentum are therefore the lagged reading, and two lines of research bear on it. Baltussen, Da, Lammers and Martens (2021) study intraday returns on over 60 futures markets from 1974 to 2020. They find that "the return during the last 30 minutes before the market close is positively predicted by the return during the rest of the day". They link that intraday momentum to the gamma hedging demand of option market makers and leveraged ETFs. The predictive power "reverts over the next days".

Grossman and Zhou (1996) model portfolio insurers in general equilibrium. Like a dealer short gamma, an insurer sells as prices fall and buys as they rise, and a later section returns to the insurers of 1987. Insurers sell to other investors in bad states, and to induce those investors to buy, the risk premium must rise. The authors show that portfolio insurance "increases price volatility, causes mean reversion in asset returns". So the price overshoots within the move and partly comes back afterward. Short gamma means momentum within the move, not necessarily over the days that follow.

The measured sizes are small beside the figure's 0.45. In single stocks, Yang (2015), a job-market paper that was not published, finds that moving from the lowest to the highest quintile of option writers' hedging demand raises daily return autocorrelation from −5.0% to −1.6%. That is a change of 3.4 percentage points of autocorrelation, where the figure's weight of 0.45 is an autocorrelation of 45%. Chapter 8G gives the rest of the measured effects.

The divergence is nonlinear

A linear reading, that at $G = 0$ effective volatility equals $\sigma$, understates what makes the formula matter in practice. Fix $\sigma = 0.15$, a baseline information volatility of 15%, and a small illustrative feedback coefficient $\kappa = 0.0010$, then sweep $G$ across the regimes:

From $G = +800$ to $G = +900$, a roughly 12% increase in destabilizing positioning, $\sigma_\text{eff}$ doubles from 75% to 150%. That is the operational signal: small changes in aggregate positioning produce outsized realized-volatility consequences in the stress regime. The slope of the curve makes this exact: differentiating $\sigma / (1 - \kappa G)$ with respect to $G$ gives

The Dealer Mechanics Lab draws a lesson for the gamma flip, the level at which estimated dealer gamma crosses zero. It reads the curve as the reason the flip matters as a level, even though no single trade is responsible for it. Read the formula closely and it adds a qualification. At $G = 0$ damping turns into amplification smoothly, and nothing in (8D.1) jumps there. The point where the curve turns violent is $\kappa G = 1$, and it depends on $\kappa$, which is liquidity, as much as on positioning. No study found tests the flip as a price level (Chapter 8G).

The value $\kappa = 0.0010$ is only illustrative, since in practice $\kappa$ is fitted to the regime and varies with depth, with dealer aggressiveness and with how concentrated positioning is. The qualitative shape, slow at the wings and explosive near the critical point, is the universal feature. A later section says what $\kappa = 0.0010$ and $G = +800$ mean in dollars.

A note on signs. $G$ is the negation of standard published GEX. Suppose a vendor reports "GEX = +$2.3B" under the JPM-era convention, where positive means stabilizing. The corresponding $G$ in (8D.1) is −$2.3B. Both ways of signing the number coexist in practice, so check which one your model or vendor uses before you read the sign.

Explore the curve

The figure below draws $\sigma_\text{eff} = \sigma/(1 - \kappa G)$ against $G$ from −1 to +1. Its variable is labeled "destabilizing gamma G": the negation of published GEX, positive when dealers are net short gamma. Sliders set the underlying $\sigma$ from 10% to 50%, starting at 20%, and the feedback strength $\kappa$ from 0 to 2, starting at 0.5. Dashed lines mark $\sigma$ and the current $G$, while readouts give $\kappa G$, the effective volatility and the amplification, with a verdict that runs from strongly damped to explosive.

The trigger stress button walks $G$ from −0.6 to +0.6 over five days, which is published GEX flipping from positive to negative. Press it at the defaults, then raise $\kappa$ to 1.5 and press it again.

underlying σ

20%

feedback strength κ

0.50

destabilizing gamma G

+0.00

trigger stress

underlying σ = 20%

κ·G = +0.00

effective σ = 20%

amplification = 1.00×

Neutral. Effective vol equals the underlying.

At the defaults, the stress run lifts effective volatility from 15%, strongly damped at $\kappa G = -0.30$, to 29%, amplifying at $\kappa G = +0.30$: less than double. With $\kappa = 1.5$, the same run reads 24% on day 3 and 43% on day 4, and ends on day 5 at $\kappa G = +0.90$. Effective volatility there is 200%, ten times the underlying, and the verdict reads explosive. The positioning moved the same distance both times, and only the feedback strength differed.

Past $G = 1/\kappa$ the formula has no meaning, so the figure cuts its curve at 200% and holds its readouts at 100 times $\sigma$. To connect the figures, set $\kappa$ to 0.45 and $G$ to +1 and then to −1. The amplification readout shows 1.82× and 0.69×, the regime figure's two multipliers and the cascade's two totals.

4 · The theory behind the formula

The formula is older than the Dealer Mechanics Lab. A line of papers from the 1990s derived it, or close relatives of it, from explicit models of hedgers whose trades move prices. Knowing that literature tells you what the formula assumes and where it stops being true.

Frey and Stremme, and Frey

Frey and Stremme (1997), in Mathematical Finance, split a market into two groups. "Reference traders" are price takers with an ordinary demand for the stock. "Program traders" follow a hedging rule for a book of options they have written, and so must trade whenever the price moves. Setting supply equal to demand gives the price process, and the abstract reports "an explicit expression for the transformation of market volatility" under the hedgers' impact:

In their logarithmic-demand case, as later papers restate it, the volatility becomes $\sigma/(1 - \rho S \phi_S)$. Here $\rho$ measures how much the hedgers' trading moves the price, and $\phi_S$ is how fast their holding changes with the price. That is the book's $\sigma/(1 - \kappa G)$, with $\kappa = \rho$ and $G = S\phi_S$. The last clause of the abstract matters for the rest of this part. A hedged book spread evenly over many strikes changes volatility everywhere a little, while a book concentrated at one strike changes it a lot near that strike and little elsewhere. That is the theoretical root of strike-local effects, pinning among them (Chapter 8E).

Frey (1998), in Finance and Stochastics, gives the cleanest version. One large trader's holding moves the price directly, and the trader replicates a claim by holding its delta. The attempt produces a nonlinear version of the Black-Scholes equation, whose solution Frey proves exists and is unique.

Frey's model in one step

Step 1: the price. Write $\alpha_t$ for the large trader's holding of the stock. Frey's price follows

$$\begin{aligned} dS_t ={}& \mu S_t\,dt + \sigma S_t\,dW_t \\ &+ \rho S_t\,d\alpha_t \end{aligned}$$ (8D.4)

where $dW_t$ is the random news of Chapter 7A and $\rho$ is the "market liquidity parameter": each share the trader buys lifts the price by the fraction $\rho$.

Step 2: the hedge. A trader who replicates a claim worth $V(t, S)$ holds $\alpha_t = V_S$, the claim's delta. When the price moves by $dS$, the holding changes by $V_{SS}\,dS$, plus terms in $dt$ that do not affect volatility.

Step 3: substitute. The random part of the price move becomes $dS = \sigma S\,dW + \rho S V_{SS}\,dS$. Collecting the $dS$ terms gives $dS\,(1 - \rho S V_{SS}) = \sigma S\,dW$.

Step 4: the volatility. Dividing through, the price's volatility is $\sigma/(1 - \rho S V_{SS})$. A claim with a convex payoff has $V_{SS} > 0$, so its replicator buys after rises and sells after falls, and volatility goes up. That is a dealer who has sold the claim, or a portfolio insurer who promised it: short gamma, $G = S V_{SS} > 0$. The equation is singular where $\rho S V_{SS}$ reaches 1.

State Frey's assumptions whenever you quote the formula: impact is linear and permanent, the hedger rebalances continuously to the delta of a known book, and the feedback stays below its critical value. A later section says how each assumption fails.

Other derivations, and a counter-result

Three more papers of the 1990s reach the same shape from different starting points.

• Platen and Schweizer (1998) add hedgers' demand to noise and speculative demand in an equilibrium model. Hedging makes volatility "endogenously determined by agents' trading behavior", and their simulations "quantitatively substantiate the idea that option price distortions can be induced by feedback effects from hedging strategies". The feedback bends the implied volatility smile and skew of Chapters 5E and 5G, not only realized volatility.
• Sircar and Papanicolaou (1998) use reference traders and "programme traders" who hedge by Black-Scholes, and derive a nonlinear pricing equation. Their models "do predict increased implied volatilities with minimal assumptions beyond those of the Black-Scholes theory", and reduce to Black-Scholes when the feedback is removed. When hedgers are a small share of the market, the correction to $\sigma$ is small, which fits a modest effect on typical days.
• Schönbucher and Wilmott (2000) derive the modified price process for an illiquid market and simulate the replication of a put, which is portfolio insurance. The family of models they belong to gives volatility of the form $\sigma/(1 - \rho\lambda(S) S u_{SS})$, rising as the replicating strategy's convexity grows.

Loeper (2018), in The Annals of Applied Probability, gives the rigorous modern version for a hedger whose every trade moves the price linearly. The resulting Black-Scholes equation "is fully non-linear and singular". Loeper shows that it is nonetheless "well posed", and proves the "existence of smooth solutions for a large class of final payoffs". The qualifier "a large class" matters: the equation is still singular, and the results cover payoffs that keep the hedge clear of the singular point.

The sign rule is not a theorem of every model. Basak (1995) models portfolio insurers as optimizing investors in a frictionless general equilibrium. He finds that "the market volatility and risk premium are decreased, and the asset and market price levels increased, by the presence of portfolio insurance". The rule of this chapter comes from models with price impact and limited liquidity. Take those away and the result can reverse.

A newer working paper by Sornette, Ulmann and Wehrli, posted on SSRN, adds the asymmetry the nonlinear table shows. It finds that option hedging creates short-lived directional instabilities. The effect is "strongly asymmetric, with a significantly more pronounced effect on the spot price when the option market maker is short compared to long the option". The damping side is bounded, and the amplifying side is not.

5 · Portfolio insurance and October 1987

Portfolio insurance was a strategy of the 1980s that gave a stock portfolio a floor without buying listed puts. The insurer replicated a put by trading, selling stock or index futures as prices fell and buying as they rose. That is a synthetic long put, and its hedging is a short-gamma program: it trades with every move, like a dealer short gamma. It is the historical case of the formula's amplifying side.

The Presidential Task Force on Market Mechanisms, known as the Brady Commission, reconstructed the market of October 14 to 20, 1987, from trade records. Its report of January 1988 put the assets under portfolio insurance at about $60 billion to $90 billion, "only 2-3% of equity assets outstanding". On October 19, as summaries of the report give it:

• Sell programs by three portfolio insurers made up just under $2 billion of the New York Stock Exchange's sales of just under $21 billion.
• In futures, three portfolio insurers sold the equivalent of $2.8 billion of stock, and portfolio-insurance sales totaled about $6 billion: $4 billion in futures and $2 billion in stock.
• In the first half hour a few insurers sold futures equivalent to just under $400 million of stock, 28% of public volume. Between 11:40 AM and 2:00 PM they sold about 10,000 futures contracts.
• Roughly 40% of futures sales by traders other than market makers came from portfolio insurers.

The Commission concluded that futures-based strategies were not the "sole cause" of the fall but "a significant factor in accelerating and exacerbating the declines". One account of the report puts insurance selling at "roughly 15% of total stock and futures sold that day". Rubinstein (1988), a co-founder of a portfolio-insurance firm and so an interested party, estimated "perhaps 12 per cent of the dollar change in net sold positions on October 19". He judged a large causal role unlikely. He also noted that the crash undermined two preconditions of every portfolio-insurance program: low transaction costs and price continuity. Replication fails exactly when prices gap, the same weakness a short-gamma dealer has.

Why so large a fall from so small a share? Gennotte and Leland (1990) put the question in their abstract: "traditional models cannot explain how a relatively small amount of selling could cause so large a price drop". In their model some traders are informed, others infer information from prices, and hedging demand is not fully observed. Markets are then "much less liquid in their model than in traditional models". In the authors' words, "Discontinuities (or 'crashes') can occur even with relatively little hedging". The crash needs the hedging to be unobserved or underestimated. If hedging demand is known, its price impact is much smaller.

Two papers of the same period make the information point from other angles. Grossman (1988) argues that without a traded put there is "less information about the future price volatility associated with current dynamic hedging strategies". As a result, "the stocks' future price volatility can rise because of a current lack of information about the extent to which dynamic hedging strategies are in place". Jacklin, Kleidon and Pfleiderer (1992) show that if the market underestimates dynamic hedging, prices first run above fundamentals, and "the price falls when the amount of portfolio insurance is revealed". Black (1988), in an essay on the crash, noted that futures prices went far below their theoretical values just when portfolio insurers wanted to sell.

The lesson for a GEX reader is double-edged. Listed open interest is visible, so the hedging it implies can be anticipated, which is Grossman's case for traded options. The hedging a public map cannot see, in FLEX, over-the-counter and structured-product books, is the kind that did the damage in these models (Chapter 8B).

Kyle and Obizhaeva, in a paper on large bets and stock market crashes in the Review of Finance, measured how badly a linear intuition failed in 1987. Portfolio insurers sold $10.48 billion of index futures and $3.27 billion of NYSE stocks from October 15 to 20, $13.75 billion gross. "Conventional wisdom" predicted declines of 0.51% net and 0.63% gross. Their invariance model, calibrated on over 400,000 portfolio transition orders, predicted 16.77%. The actual decline was 32%.

6 · Two tests of the sign rule

A measured market: currency options

Anderegg, Ulmann and Sornette (2022), in the Journal of International Money and Finance, fit a model of this family to real positions. An earlier version was a Swiss National Bank working paper of 2019. Their model has two aggregate traders, an option market maker and an option market taker, "whose exposures reflect the total outstanding positions of all option traders in the market". The net hedge flow moves the spot rate through "a simple linear permanent impact model". Foreign exchange options trade over the counter, with no public open interest, so the authors rebuilt the market maker's gamma from trade-repository records. Their sample runs from October 21, 2017 to June 30, 2018, in EURUSD and USDJPY.

Spot volatility "is increased (decreased) by a negative (positive) gamma exposure" of the market maker. The rebuilt gamma was negative, because option takers were net long options. In size, "a negative gamma exposure of the OMM of approximately -1000 billion USD" leads "to an absolute increase in volatility of 0.7% in EURUSD and 0.9% in USDJPY". Here OMM is the option market maker, and about −1,000 billion dollars is roughly what the authors' reconstruction shows. The gamma is in the authors' own measure, whose unit the abstract does not state. The increases are absolute: 0.7 and 0.9 points of annualized volatility, under one point in the two largest currency pairs. The 2019 version explains the larger USDJPY effect by "the higher market impact of a transaction in the USDJPY spot market", EURUSD being the more liquid. Impact is the book's $\kappa$: the same dealer gamma does more in the thinner market.

A simulated market: liquidity and price discovery

Buis, Pieterse-Bloem, Verschoor and Zwinkels (2024), in the Journal of Economic Dynamics and Control, simulate an order book with an agent-based model. That is a simulation in which many traders follow simple rules and prices emerge from their orders. The authors add delta hedgers with a positive or negative net gamma, and compare normal and stressed conditions. Their findings, in the abstract's words:

• "the presence of dynamic hedgers enhances market liquidity under normal conditions";
• "positive gamma helps sustain liquidity in stressed scenarios, while negative gamma depletes it";
• "an increase in the net gamma positioning of dynamic hedgers reduces volatility and increases market stability, whereas a negative gamma positioning increases volatility and makes the market more prone to failure";
• "Price discovery typically worsens when dynamic hedgers become more prevalent, regardless of the sign of their positioning."

These are the two rows the regime table lacked. Negative gamma drains liquidity in stress, which is how $\kappa$ rises in the same step as $G$. Because the two move together, the explosive corner of (8D.1) is reached faster than either alone suggests. And hedging of either sign worsens price discovery, the process by which trading moves prices to what the available information justifies. A damped, pinned market is not a better-informed one. The authors suggest steering hedgers' net gamma as a policy tool, most useful for illiquid instruments. The simulation's background traders place orders by simple random rules, a "zero-intelligence" order book, so the result is a model's, not an estimate.

A market where the dealer's gamma is measured rather than assumed points the same way. Atanasova and co-authors (2025) study the bitcoin options venue Deribit from January 2021 to December 2024. When market makers' aggregate gamma inventory is negative, they "trade in the same direction as BTC price momentum", and option spreads widen. A one-standard-deviation fall in the measure goes with effective spreads 0.12% wider for out-of-the-money calls and 0.19% for puts.

7 · What sets κ: impact, in dollars and points

The units

Only $\kappa G$ matters, but a reader holding a GEX number in dollars needs $\kappa$ in matching units. Take $G$ in billions of dollars per 1% move, the negation of published GEX. Write $\lambda$ for the impact of hedge flow: the index points that $1 billion of buying or selling moves the index.

A 1% move is $0.01S$ points. By Chapter 8A's hedge-flow formula it makes the dealers trade $G$ billion dollars with the move, and that flow moves the index a further $\lambda G$ points. The feedback per unit of move is the second move over the first:

so $\kappa = \lambda/(0.01S)$ per billion dollars. At $S = 6{,}000$, a 1% move is 60 points.

The same rows, restated at the two impacts. $G$ is in billions of dollars per 1% move, so a negative $G$ is a positive published GEX.

Now run the check the other way. Take a constant $\lambda$ of 1 point per $1 billion, and dealers short $20 billion to $100 billion of gamma per 1% move. That gives $\kappa G$ from 0.33 to 1.67: amplifying at the bottom of the range and past the explosion point at the top. Nothing in the evidence shows the index behaving that way on ordinary days (Chapter 8G). Either impact per dollar of hedge flow is smaller and more curved than a constant $\lambda$, or the flow actually hedged is much smaller than gross GEX, or both. What impact research says about $\lambda$ decides which.

What impact research says

The idea of one number for impact per unit of flow goes back to Kyle (1985). In his model of an informed trader, the market makers' price rises in proportion to net order flow. The slope, known as Kyle's lambda, is the price move per unit of flow, and market depth is its inverse. In that model the slope is set by how much information the flow might carry. Hedge flow is mechanical, and a market that can recognize it has less reason to move far on it, so its impact tends to be smaller and to fade.

Measured impact is not linear in size. The price impact of a metaorder, a large order worked in pieces, grows roughly as the square root of its size, the law of Chapter 2D, (2D.5). In the form the research uses,

where $I$ is the move, $Q$ the size of the order, $V_D$ the daily volume and $\sigma_D$ the daily volatility, with $Y$ a constant of order one. Sato and a co-author survey all liquid Tokyo Stock Exchange stocks over eight years. They find the exponent "equal to 1/2 within statistical errors at both the individual stock level and the individual trader level". Bucci and co-authors (2019) find "a clear crossover between a linear market impact regime and a square-root regime" in 8 million US institutional trades. Zarinelli and co-authors (2015) find that the square root fits "only across about two orders of magnitude in order size", a logarithm fitting better across five. They also find that impact relaxes on average to "approximately 2/3 of the peak impact". Tóth and co-authors (2011) trace the law to a latent order book that "vanishes around the current price".

Four more findings bracket $\lambda$ for the index:

• The permanent anchor. Gabaix and Koijen (2021), in an NBER working paper, estimate that "investing $1 in the stock market increases the market's aggregate value by about $5", a price elasticity of demand of 0.2. It is an estimate for persistent flows over a quarter, and it is contested. At an assumed US market value of $55 trillion, $1 billion of persistent flow would add about $5 billion, roughly 1 bp or half an index point at 6,000. Hedge flow that later reverses leaves less than that behind, while its transient impact is larger.
• Predictable flows. Brøgger (2021) finds that the daily rebalancing of leveraged volatility products is predictable in size, sign and timing. Yet "larger and more predictable flows have smaller price impact coefficients, and ... there are no excess profits from trading ahead of rebalancing flows". A flow everyone can see coming is partly absorbed in advance, which is Gennotte and Leland's point from the other side.
• Stress. Kyle and Obizhaeva's crash comparisons show how far impact can rise. On May 6, 2010, a seller's program of 75,000 E-mini contracts, about $4.1 billion, coincided with a 5.12% drop in five minutes, from 1,113 to 1,056, against an invariance prediction of 0.61%. In 1987 the actual fall was about twice the prediction, and in 2010 about eight times. The authors read the larger-than-predicted drops as a sign that "rapid selling exacerbates transitory price impact".
• Depth over time. A working paper by Boyarchenko, Larsen, Venter and Whelan is described in Larsen's 2024 doctoral thesis. It finds that "the depth and dealer number of the S&P 500 futures contract have decreased by 80% over a ten-year period from 2012 to 2022". It is unpublished, and the figure depends on how it defines depth. If it holds, the same GEX moves the index more than it did a decade before, so GEX numbers from different years are not comparable without a depth adjustment.

No published estimate gives the impact of dealer hedge flow in SPX in points per $1 billion. The pinning figure's range of 0 to 3 points per $1 billion spans the calm-day bracket for flows worked over a day. Stress would need values several times higher.

The figure below puts the three views of $\lambda$ on one chart. The horizontal axis is $G$, dealers' short gamma in billions of dollars per 1% move, the negative of published GEX, from −$100 billion to +$100 billion and starting at +$20 billion. The vertical axis is the loop gain $\kappa G$ for a 1% move. The straight line is constant impact at the $\lambda$ you set, from 0 to 3 points per $1 billion and starting at 1, by (8D.5), with spot starting at 6,000. The band is square-root impact on the worked example's assumptions, with $Y$ from 0.5 to 1. A green strip marks the gains of 0.18 to 0.30 read from the studies, and a dashed line the explosive gain of 1.

dealers short, G

short $20B

impact λ

1.0 pt per $1B

spot

6,000

constant: κG = +0.33, ×1.50

square root: +0.10 to +0.20

constant λ explodes at G = $60B

Amplifying. Dealers short $20 billion per 1% move multiply each move by 1.50 with constant impact, and by 1.11 to 1.25 with square-root impact.

At the start, constant impact gives $\kappa G = 0.33$ and multiplies every move by 1.50, while square-root impact gives 0.10 to 0.20, a multiplier of 1.11 to 1.25. Drag $G$ to +$100 billion: the straight line crosses the explosive gain at $60 billion and reads 1.67, while the band stays between 0.22 and 0.45. Across the Lab's range of $20 billion to $100 billion the band stays within reach of the studies' strip, and the straight line has left it before $20 billion. Drag $G$ below zero and long gamma damps, with the two laws apart in the same way: at −$50 billion, constant impact shrinks each move to ×0.55 and square-root impact to ×0.76 to ×0.86.

8 · What the formula leaves out

Each of the formula's assumptions fails in a known way, and each failure has a literature:

1. Linear, permanent impact. Impact is concave and partly transient, as above.
2. Continuous hedging of a known book. Almgren and Li (2016) find that an optimal hedger "trades smoothly towards the classical Black-Scholes delta, with trading intensity proportional to instantaneous mishedge and inversely proportional to illiquidity". Aubert, Chevalier and Ly Vath (2025) model a market maker whose hedging has transient as well as permanent impact, and who hedges in discrete bursts rather than continuously. Either way the flow is spread out and lumpy, not delivered at each tick.
3. Everyone else's demand is unchanged. Other traders anticipate predictable flow and trade against it, as Brøgger's result shows, so gross mechanical flow is not net price pressure.
4. Feedback below its critical value. Near $\kappa G = 1$ the smooth model gives way to gaps and crashes, as in Gennotte and Leland's model and Kyle and Obizhaeva's comparisons.

Two further omissions matter for anyone reading a map of open-interest gamma. The first is new trades. Egebjerg and Kokholm (2024), in an SSRN working paper, split the impact of market makers' hedging in two. The "gamma effect" is rebalancing an unchanged book as spot moves, and the "inventory effect" is hedging new option trades. Studying the gamma effect alone "can underestimate the total effect when the OMM's inventory adjustments reinforce the gamma impact and overestimate it when the adjustments move in the opposite direction". In SPX data, both components "are significant and with impact in line with the model predictions". A burst of customer call buying is hedged at once and moves the index even when the stock of gamma has not changed. A map of yesterday's open interest shows only the first effect.

The second omission is other hedgers. $G$ in the formula is everyone who trades with the move, not only option dealers. A leveraged or inverse ETF with assets $A$ and multiple $L$ trades $L(L-1)xA$ after a day's return $x$, always with the move (Chapter 8H). Tuzun (2013), in a Federal Reserve working paper, estimated for the fund assets of that time that "A 1% increase in broad stock-market indexes induces LETFs to originate rebalancing flows equivalent to $1.04 billion worth of stock". Volatility-targeting funds and trend followers also trade with moves, on longer lags. Target-date funds trade against them: Parker, Schoar and Sun (2023) find that they move retirement investors into "macrocontrarian strategies that sell stocks after relatively good stock market performance". Dealer GEX is one term in the market's net $G$, and a map of option gamma shows none of the others.

9 · How large the feedback is in practice

The dealer-feedback story is real and quantifiable, but its practical effect is modest on a typical day. Chapter 8G reviews the studies that measure it and sets them side by side in their own units. They run from Barbon and Buraschi's intraday momentum and reversal in stocks to the 0DTE studies of Dim, Eraker and Vilkov and of Adams and co-authors.

In the Dealer Mechanics Lab's reading, the episodes that everyone remembers are February 2018, March 2020, the GME squeeze and the August 5, 2024 unwind. They are the right tail of this distribution, where feedback briefly approached the unstable threshold (Chapters 8H and 8I). On most days, GEX is a probabilistic prior you can use to bias your strategy selection, not a deterministic forecast.

What to do with a regime reading is Chapter 8J's subject, and the book's two sources weigh it differently. The Dealer Mechanics Lab pairs the sign of GEX with the shape of the VIX term structure as a decision boundary for choosing strategies. The Edge Book keeps the sign as a low-confidence filter, worth 10% to 15% of one filter's weight, and ignores it when it disagrees with the term structure (Chapter 10H). Chapter 8J sets out both, and turns the regime into a multiplier on size rather than a trigger for trades.

10 · Check yourself