Step 1 of 14

One leveraged position

Start with a single investor who buys an asset priced at 1 using borrowed money. With leverage l, the investor puts in 1/l of their own money (the equity) and borrows the rest (the debt, 1 − 1/l). Leverage 2 means half your own money, half borrowed. Leverage 4 means a quarter of your own money.

Now the price falls by a fraction d (the drawdown). The asset is worth 1 − d. The debt does not shrink. So the equity, which is assets minus debt, absorbs the whole loss:

The lender requires the equity ratio, equity divided by assets, to stay above a maintenance fraction m (say 25%). When it drops below m, the position is in breach and must be reduced. Solving the breach condition for leverage gives a threshold leverage:

Everyone with leverage above l_th(d) is in breach. As the price falls, l_th falls, so the breached set grows. That is the whole balance-sheet mechanism; drag the sliders to see it.

Step 2 of 14

A population of positions

Real markets contain many leveraged positions with different leverage. The paper models leverage as a rescaled normal variable:

Here l is the typical (central) leverage and σ is the relative spread. The floor at 1 (no borrowing) plays no role in what follows, because only the high-leverage tail matters. With l = 2 and σ = 0.16, most positions have leverage between about 1.7 and 2.3. The breached fraction after a drawdown d is the probability that a position’s leverage exceeds the threshold, which is a normal tail probability.

Writing the cutoff in units of Z, the position is breached when Z_i ≥ (l_th/l − 1)/σ, so the fraction of accounts in breach, written F(d) in the paper, is

Step 3 of 14

Count dollars, not accounts

Price impact comes from how much is sold, not from how many people sell. One account holding a hundred million dollars matters more than a thousand accounts holding ten thousand each. So the quantity that drives the feedback is the exposure-weighted breached fraction: the share of total dollars held by breached accounts. The paper calls it the exposure-weighted breach-response function.

Here Q_i is the dollar size of position i and 1{·} is an indicator. If position size is unrelated to leverage, this reduces to the plain account fraction of step 2. If big positions tend to be more leveraged, it does not.

Step 4 of 14

Selling moves the price

The other half of the loop is price impact: selling a fraction B of the vulnerable exposure pushes the price down by some amount. The paper writes the total drawdown as an outside shock plus a feedback term,

where d₀ is the exogenous drawdown (the price would have fallen this much with no forced selling at all), Λ is the feedback amplitude (how much a full liquidation would move the price), and f is a normalized impact function with f(0) = 0, f(1) = 1, and f increasing.

Empirical work on large institutional orders finds that impact grows roughly like the square root of the quantity traded, with a linear regime for very small trades. The paper uses the square root as its main example but checks five different shapes, because it wants to know whether the results depend on the peculiar steepness of the square root at zero (its slope is infinite there), or on concavity at all.

Step 5 of 14

Closing the loop: a fixed-point equation

Now combine the two halves. The breached fraction depends on the drawdown through F (step 3), and the drawdown depends on the liquidated fraction B through the impact law (step 4). A self-consistent state is a value of B that reproduces itself:

The physical version of this iteration is the cascade itself. Round 1: the outside shock d₀ breaches a fraction G(0) = F(d₀). Round 2: those sales deepen the drawdown to d₀ + Λf(G(0)), which breaches G(G(0)). And so on. Each round breaches at least as many as the last because G is increasing. The cascade stops at the lowest fixed point, if there is one.

The key question is what happens when the outside shock d₀ grows. Drag it in the figure below and watch the lower crossing.

Step 6 of 14

Folds, and why only one of them matters

The right-hand panel above is the paper’s central picture. Solve Eq. (6) for the drawdown that breaches exactly a fraction B; call it t(B). It is the exposure-weighted quantile of the individual breach thresholds: the price drop at which the account sitting at cumulative weight B gets margin-called. Then the shock needed to hold the system at state B is

The first term rises with B (you have to push the price further to reach less leveraged accounts). The second term is the help the cascade gives you (sales already made have pushed the price down for free). When the help grows faster than the difficulty, d₀(B) bends backward and the curve becomes S-shaped. The two turning points are folds.

In a reversible system the S-curve gives hysteresis: push the parameter up and the state jumps at the upper fold; pull it back down and the state only jumps back at the lower fold. But liquidation is not reversible. A position that has been sold does not un-sell itself when the price recovers. The paper writes the physical update as

where L_t is the cumulative liquidated fraction. This is an absorbing process: L can only rise. So only the upper fold has physical meaning: it is where the low-liquidation state disappears and the cascade starts. The lower fold and the descending branch describe states the system can never occupy.

Two details from the paper sharpen this. A resting state of the absorbing update only needs L ≥ G(L), not equality, so the equality map is best read as a bifurcation diagram and not as a list of states the market visits. And while the shock is rising, the iteration from zero is already nondecreasing, so the max in Eq. (12) does nothing on the way up. It matters only when the shock is later reduced, where it keeps sold exposure from reappearing. The convergence to the lowest fixed point is the same monotone argument used for clearing payments in interbank networks.

Step 7 of 14

The gain: three factors multiplied

Differentiate the map. The chain rule turns the slope of G into a product of three quantities, each with a clear meaning:

Think of R as a reproduction number. One dollar of forced selling moves the price a little (third factor), that price move lowers the leverage threshold a little (second factor), and that lower threshold catches some density of dollars sitting just below the old threshold (first factor). If the product is below 1, each round of selling triggers less than a full round of new selling and the cascade dies out. The fold is exactly the point where R = 1.

This decomposition is why the paper says the cascade is controlled by “the amount of dollar exposure near the moving maintenance threshold”. A market with the same total leverage can be safe or fragile depending on how the dollars are distributed relative to the threshold.

Step 8 of 14

Where the instability boundary sits

For a power impact law f(B) = B^α (α = ½ is the square root) the fold condition can be solved in closed form. Rewriting the model with leverage as the unknown gives an “equation of state” l(B; Λ), and a fold is where ∂l/∂B = 0. That yields the feedback amplitude at which state B is a fold,

The smallest Λ over all B is the critical feedback Λ_c: below it no fold exists for any leverage and the cascade cannot run away. The B that achieves the minimum is the cusp state B*, and the leverage there is l*. Two facts follow directly from the formula:

• B* does not depend on m or d₀ at all: they enter only through the prefactor b₀.
• Λ_c ∝ b₀/(1 − m) and l* ∝ 1/b₀ (Eq. 17). A larger outside shock (bigger d₀) raises the feedback needed for a cusp but moves the cusp to lower leverage. Fragility at leverage near 1 needs a huge outside shock.

Step 9 of 14

It is not about the square root, or about concavity

A reasonable worry: the square-root law has infinite slope at zero, so maybe the fold is an artifact of that steepness. The paper checks by repeating the analysis for a general impact function. The fold condition becomes

which reads: a fold exists at B if the marginal impact f′ is large enough compared with the slope of the threshold quantile t′ (how much harder it gets to breach the next dollar). The minimum over B is the onset feedback for that law. All five laws from step 4 have a finite onset, including the linear law, whose slope at zero is 1, not infinity, and which is not strictly concave. The paper’s conclusion: neither the square-root singularity nor strict concavity is necessary for a threshold-induced fold. Any increasing impact law can produce one if its marginal response is large enough relative to the weighted threshold density and the balance-sheet slope.

Step 10 of 14

Finite cohorts: the collapse point is random

Everything so far treats the population as infinite: the breached fraction is a smooth function of the drawdown. A real cohort has N accounts with specific leverages and sizes, so the breached fraction is a step function. The question is: by how much does the collapse point of a finite cohort deviate from the fold, and how does that deviation shrink with N?

The fiber-bundle analogy

The paper borrows its finite-size theory from a classic model in materials science. A bundle of N fibers shares a load equally. Each fiber has a random breaking strength. When a fiber breaks, its share of the load moves to the survivors, which can break some of them, and so on. Daniels (1945) worked out the distribution of the load at which the whole bundle fails, and later work by Smith and others refined the finite-size corrections. Replace “load” with “price drop”, “strength” with “breach threshold”, and “load redistribution” with “price impact”, and you have this model, except that fibers here have unequal weights and the impact law is nonlinear.

Collapse as the maximum of a noisy curve

Sort the N accounts by their individual breach thresholds t_i. Let T_N(B) be the threshold of the account at which the cumulative weight first reaches B (the weighted empirical quantile). Once the exposure before it has been sold, the outside shock needed to breach that account is

The low state survives as long as the shock is below every hurdle it has to clear, so the collapse shock is the largest hurdle: the local maximum of a noisy version of the smooth curve d₀(B). Try it below. Every click draws a fresh cohort from the same population.

Three scaling laws

Near the fold the smooth curve is a downward parabola. Add to it Gaussian noise that has two parts: a common shift for the whole neighborhood (order N^−1/2) and a Brownian wiggle along B. Maximizing “parabola plus Brownian motion” is a classic problem whose answer scales in a specific way. The exponents that come out are the classical Daniels-Smith fiber-bundle exponents, and the paper says plainly that they are not new. What is new is the variance of the weighted process and the prefactors that follow from it for unequal exposures and nonlinear, price-mediated load sharing. Three predictions result, with computed prefactors and no fitted parameters:

• The spread of the collapse shock across cohorts falls like N^−1/2 (Eq. 27). This comes from the common shift.
• The mean collapse shock sits above the fold by an amount falling like N^−2/3 (Eq. 28). The maximum of noise plus a concave curve is always biased upward, and Brownian scaling sets the exponent.
• The state at takeoff (the B where the jump starts) scatters around B₊ with spread falling like N^−1/3 (Eq. 28). A naive “add one Gaussian error and expand” argument would predict N^−1/4; the paper stresses that the naive argument is wrong because the noise varies along the state axis.

The variance of the common shift is where the weighting enters:

The two terms appear because a weight is a ratio, Q_i/ΣQ_j: both the breached dollars in the numerator and the total in the denominator fluctuate from cohort to cohort. With equal exposures m₂(B) = B and the formula collapses to the familiar B(1 − B) of the Brownian bridge. For the lognormal model m₂(B) = e^s_q² Φ(z + χ), using the 2χ tilt from step 3 (Eq. 25). The spread of the collapse shock is then t′(B₊) √V(B₊) N^−1/2 (Eq. 27).

Step 11 of 14

Heavy tails: one big holder decides

The theory in step 10 needs E[Q²] to be finite. Position sizes in real markets are often heavy-tailed, so the paper repeats the analysis with Pareto sizes, P(Q > x) = x^−μ (Eq. 29), independent of leverage (so χ = 0 and the mean-field map is unchanged). The paper presents this as a less settled regime than step 10: scaling arguments with supporting simulations, not a finished theory.

Both features show up in the collapse point. The typical spread (measured by the interquartile range, which ignores outliers) narrows faster than before, as N^{−(1 − 1/μ)} in place of N^−1/2 (Eq. 30). But the variance is controlled by rare cohorts containing one dominant account. With probability of order N^1−μ a single account holds a fixed fraction ω of all exposure. Such an account usually has a threshold above the fold (only a fraction B₊ ≈ 4% of accounts sit below it). If so, it dilutes everyone else’s weight: the rest of the cohort behaves as if the feedback were Λf((1 − ω)B), the fold moves to a higher shock or disappears, and the collapse comes late. A dominant account below the fold triggers an early collapse instead, but that is rarer by a factor B₊. So the distribution of the collapse shock is skewed to the right, and its standard deviation falls only like N^{−(μ−1)/2} (Eq. 31), slower than the interquartile range.

One big holder, in isolation

The dilution argument can be checked without heavy tails at all. Take a cohort of equal-weight accounts and add a single conservatively levered holder who owns a share ω of all exposure and whose own threshold lies far above the fold. Until that holder is breached, the rest of the cohort sells at most a fraction 1 − ω of the total, so the feedback they generate is Λf((1 − ω)B). For square-root impact this is the same as reducing Λ to Λ√(1 − ω). The fold moves to a larger shock and the jump at the fold shrinks.

The paper tests this with a statistic that does not depend on how “the collapse” is identified: the shock at which the background accounts (everyone except the holder) have first lost 25% of their own exposure. With 100,000 background accounts and holder shares of 0%, 10%, and 20%, that happens at 0.12784, 0.13136, and 0.13561. The mean-field folds of the diluted map are 0.12746, 0.13083, and 0.13509. Simulation and prediction differ by at most 0.00053 in all three cases.

Step 12 of 14

Precursor avalanches before the collapse

Before the macroscopic collapse, each small increase of the outside shock breaches one account, whose sale may breach a few more, and so on: a small avalanche. How big are these avalanches? That is a branching-process question.

An avalanche started at state B has “reproduction number” R(B), the gain from step 7. As the shock ramps up toward the fold, R climbs toward 1, so late avalanches are bigger. Mixing Borel distributions over the ramp, with weight proportional to how many avalanches start at each B (which goes like 1 − R), gives the paper’s Eq. (32). Because 1 − R vanishes linearly at the fold, the integrated distribution steepens from −3/2 to −5/2, the same exponent found for fiber bundles. Sizes here are counted in accounts, and the Borel prediction is derived for equal exposures.

Step 13 of 14

Adding time: partial sales, delays, and fading impact

The binary model says a breached account sells everything instantly and the price move is permanent. The paper’s final section replaces that with a path-resolved simulation in discrete time. Each account keeps a running balance sheet, and:

• A breached account sells only what is needed to restore its maintenance ratio, x* = max(0, A − E/m) (Eq. 34), and only after a random execution delay.
• It can sell at most a fraction r_i of its assets per step, a sale-rate limit.
• Price impact has a memory: the unabsorbed stock of impact decays geometrically with a half-life h_I (Eq. 36), and the price relaxes toward its target at rate γ (Eq. 37). The impact law is the linear-to-square-root crossover from step 4.
• The outside shock is ramped linearly over T steps and then held.

How far from the binary fold?

The paper adds a reference point. Take the same population and the same crossover impact law, but with complete, instantaneous liquidation and permanent impact, as in steps 5 to 9. That binary model has its upper fold at an outside shock of 0.10469 (total drawdown 0.13683, breached exposure 0.01891). The path-resolved simulations with permanent impact have large-cohort median transition shocks between 0.19537 and 0.19565. With an 80-step half-life the range is 0.23210 to 0.23523. So partial sales, delays, rate limits, and finite price relaxation account for the larger first shift away from the binary fold, and impact decay adds a further delay. The paper calls this split descriptive and not additive, because the mechanisms interact nonlinearly.

Three results

The cascade has an intrinsic duration. With slow ramps, the time between 10% and 90% of the final selling converges to about τ_c = 32 steps regardless of ramp length. Measured in units of the shock, the width of the transition is τ_c times the ramp rate (Eq. 38), so it shrinks like 1/T. The narrowing is consistent with a discontinuous transition in the slow-driving limit; it only looks smooth if you drive fast. For fast ramps (T ≤ 1000) the final sold fraction is larger, 0.915 at T = 250 against 0.84 for slow ramps, because the shock keeps rising while the cascade runs.

Transient impact makes ignition unreliable. With permanent impact the cohort-to-cohort spread of the transition shock falls with N as expected, from 0.0043 at N = 1000 to 0.0007 at N = 64,000 (exponent about −0.45). With a half-life of 80 steps it does not fall: 0.0064 at N = 2000, 0.0070 at 8000, 0.0051 at 64,000. The reason is stalled ignition: near the fold the cascade grows slowly, so the first wave of sales competes with the fading of its own price effect. In a subset of cohorts the first wave dies after selling a few percent, and the macroscopic collapse ignites later at a larger shock. The distribution of transition shocks becomes bimodal, and its width is set by the distance between the two modes, not by ordinary sampling noise. The paper is careful about what this implies: over the cohort sizes studied, no convergence to a unique transition shock is established under transient impact, so it asserts no deterministic large-cohort limit.

The paper tests this reading directly on a matched set of 128 cohorts. Cohorts whose transition shock lay more than 0.008 above the median (24 of them, 18.8%) sold only 9.6% of exposure in their first recruitment wave, against 36.9% for the rest. Between that first wave and the eventual collapse, their unabsorbed impact stock fell by a median 69%; a decay of more than 25% occurred in 79% of the late cohorts and in 8.7% of the others. The median quiet interval after the first wave was 243 steps for the late group and zero for the others. So the late cohorts are exactly the ones whose first wave was weak and whose price impact faded before the wave could recruit the full cascade.

Fading impact delays and fragments the cascade but does not remove it. Extending the ramp to d₀ = 0.45 shows that even a 3-step half-life produces a largest selling event five times the no-feedback baseline. As memory shortens, the shock at which the largest event occurs rises from 0.194 to 0.39. When the half-life drops below the cascade duration, the deleveraging fragments: the number of separate events per cohort is exactly one for h_I ≥ 28 and approaches two for h_I ≤ 4. An apparent suppression can be an artifact of stopping the ramp too early: with the ramp ended at 0.30 and h_I = 20, some cohorts show only partial deleveraging, and the same cohorts collapse when the ramp continues.

Step 14 of 14

What to take away, and what the model does not say

The whole argument in five sentences:

1. Leveraged positions are margin-called when the price falls past a threshold that depends on their leverage (step 1).
2. Their forced sales move the price, which moves the threshold for everyone else, so the breached dollar fraction obeys a fixed-point equation (steps 2 to 5).
3. When the product of exposure density near the threshold, balance-sheet sensitivity, and marginal impact reaches 1, the low-liquidation fixed point disappears and the state jumps: an absorbing cascade. This does not depend on the square-root impact law or on strict concavity, and it shifts predictably with the shock size, the maintenance rule, and how dollars are distributed across leverage (steps 6 to 9).
4. For a finite cohort, the collapse point is the maximum of a noisy curve. With light-tailed sizes its spread, upward bias, and takeoff scatter shrink like N^−1/2, N^−2/3, and N^−1/3, the classical fiber-bundle exponents, with prefactors computed from a new weighted variance; with heavy-tailed sizes a rare large, conservatively levered holder can delay the collapse (steps 10 to 12).
5. Once time is resolved the cascade has a finite duration; fading impact delays it, shrinks it, can fragment it, and can stall a weak first wave so that the transition shock becomes bimodal across cohorts (step 13).

What the paper claims as new

The paper is explicit about overlap with earlier work. Kavya (2026) independently developed a threshold-density model of stop-loss and margin-liquidation cascades with the same local instability criterion, shock-driven discontinuous transitions, and a Galton-Watson description of precursors. The finite-size exponents of step 10 are the classical Daniels-Smith results. So the paper does not rest its novelty on the fold, the R = 1 criterion, the branching mechanism, or the exponents. It rests on four things:

• unequal dollar exposures and their correlation with leverage (steps 3 and 8);
• the weighted finite-cohort variance, including the fluctuation of the normalization, and the parameter-free prefactors for nonlinear price-mediated load sharing (step 10);
• the heavy-tail regime in which a dominant holder delays the loss of stability (step 11);
• the competition between the cascade’s own duration and the memory of price impact (step 13).

Limits the paper states

• Nothing is calibrated to a real market. The parameter sets are theoretical examples, and Λ in particular is chosen so that a transition lies inside the numerical window.
• Only plain margined cash positions obey Eq. (1). Options, swaps, portfolio-margin accounts, and leveraged exchange-traded products would need their own state variables and sale weights.
• There is one asset and no network. The paper deliberately removes portfolio overlap to isolate the common-price channel.
• The population is long-only and static. Nobody enters or leaves, nobody reduces risk voluntarily before a margin breach, liquidity providers do not react to the order flow, and the impact law does not change with market state. Pre-emptive selling matters in real episodes; it removes exposure from a later mechanical cascade but adds selling pressure earlier, and the paper says the net effect cannot be signed without a behavioral rule. The transition studied is the mechanically triggered component of a broader deleveraging process.
• The impact function is fixed in advance. The paper points to a different mechanism it leaves out: a finite carrying capacity, in which the remaining ability of buyers to absorb sales is itself a state variable, depleted by past forced sales and replenished over time.
• The heavy-tail standard-deviation scaling is conditional (step 11), and the precursor statistics count accounts, not dollars (step 12).
• A credible empirical test would need the exposure distribution, its joint distribution with leverage, the maintenance rules, the impact function on a stated horizon, and the initiating shock, all measured before the event, and would then classify collapses out of sample instead of inferring a hidden parameter from the fact that a crash happened. If sizes are heavy-tailed, the largest holders must be measured individually. The paper names markets with mechanical liquidation rules and visible positions, such as perpetual futures and on-chain collateralized lending, as direct settings for such a test.

Check yourself

Glossary

Leverage l

Assets divided by equity. Leverage 2 means half of the position is borrowed.

Maintenance fraction m

Minimum allowed equity-to-assets ratio. Falling below it triggers a margin call.

Drawdown d; outside shock d₀

Fractional price decline. d₀ is the part not caused by forced selling; it is the control parameter, also called the external field.

Breach-response function F(d)

The exposure-weighted share of positions in breach after a total drawdown d.

Exposure Q; tilt χ

Dollar size of a position. χ = ρ_qz s_q summarizes how size correlates with leverage in the lognormal-normal model.

Impact function f; feedback amplitude Λ

Normalized price response to selling a fraction of vulnerable exposure, and its overall scale.

Equality map G; gain R

The self-consistency map B = G(B) and its slope. R = 1 marks a fold.

Fold; cusp

A point where two fixed points merge and vanish; a point where two folds merge as a second parameter varies.

Absorbing process

A process with a state it can enter but never leave; here, sold exposure stays sold.

Collapse field d_0,+^(N); takeoff state

The shock at which a finite cohort’s macroscopic cascade starts, and the sold fraction just before it starts.

Weighted Daniels theory

The Daniels-Smith fiber-bundle finite-size theory with its classical exponents, extended here to unequal fiber weights and nonlinear load transfer through a weighted variance.

Borel distribution

Total progeny of a Galton-Watson process with Poisson offspring.

Dominant holder; dilution

One account with a fixed share ω of all exposure. If it is conservatively levered, the rest of the cohort generates feedback Λf((1 − ω)B) and loses stability later.

Stalled ignition

A first wave of forced sales that dies out because its price impact fades before it recruits the full cascade.

Impact half-life h_I; cascade duration τ_c

How fast a price move from selling fades, and how long the cascade takes to run once it ignites.

The paper uses no empirical data. Its Python source, fixed random seeds, and generated tables are at github.com/bonabeau-gif/Absorbing_Margin_Liquidation_Cascades.