When Can Markets Crash Without News? A Technical Reading of the Working Paper | ArcaViso

After a sharp market decline, we usually look to the news for an explanation: deteriorating earnings, policy changes, or tighter financing. Yet another set of forces also governs prices: how trading consumes liquidity, how prices recover, and where sale proceeds go. Even with fundamentals unchanged, these mechanisms may leave lasting price losses.

The working paper When Can Markets Crash Without News? formulates this as a reachability problem: given price responses, recovery dynamics, and trading rules, does a trading path exist that leaves sufficiently many assets with sufficiently deep losses at the terminal time?

The argument proceeds through three layers. It first identifies structural barriers that no trading path can cross, then calculates the turnover and inventory needed to reach a declining state, and finally introduces trading-rate limits and daily price bands to construct a two-session path satisfying all constraints. We follow that progression through the mathematics.

1. Combining price dynamics with the trading ledger

Consider a market of $N$ assets, each initially priced at a fixed fundamental benchmark $F>0$ . Asset $i$ has price $S_i(t)$ , change in share holdings $\mathrm d\theta_i$ , and signed dollar trading flow $\mathrm dX_i$ . The model has two basic equations:

\mathrm dS_i=-\rho(S_i-F)\,\mathrm dt+g(S_i)\,\mathrm d\theta_i, \qquad \mathrm dX_i=S_i\,\mathrm d\theta_i.

The first term describes recovery: after departing from $F$ , the price reverts toward the benchmark at rate $\rho>0$ . The second describes trading impact: the positive response function $g(S)$ determines the price change per share traded. The second equation settles trades at the marginal price along the execution path, so price changes and dollar flows must be calculated together.

The model studies a reallocation pool within risky assets. Every dollar received from selling one asset is immediately used to buy another, with paired transfers allowed in both directions between any two assets. Writing $B$ for the incidence matrix of this complete directed trading network gives:

\begin{aligned} \mathrm d\mathbf{X}&=B\,\mathrm d\mathbf{z},\\ \mathrm d\mathbf{z}&\ge0,\qquad \mathbf{1}^\top B=0. \end{aligned}

Each directed edge represents a dollar transfer from the asset sold to the asset bought, so dollar flows across all assets sum to zero at every instant. The paper also requires each asset individually to satisfy the terminal condition:

X_i(T)=\int_0^T S_i\,\mathrm d\theta_i=0.

This sharpens the question: after each asset's net trading dollars have fully offset, how much price displacement can remain?

Dollar closure and share closure have different economic meanings. Define a nonnegative inventory potential:

E(S)=\int_F^S\frac{u-F}{g(u)}\,\mathrm du.

The price equation and trade-settlement rule yield the identity:

\begin{aligned} X_i(T)={}&F[\theta_i(T)-\theta_i(0)]\\ &+E(S_i(T)) +\rho\int_0^T\frac{(S_i-F)^2}{g(S_i)}\,\mathrm dt. \end{aligned}

The last two terms are nonnegative. With the dollar ledger closed, terminal displacement and recovery dissipation must therefore be funded by a reduction in initial share holdings. Requiring the share count to return fully to its starting level would rule out nontrivial dissipative paths through this identity. The paper retains initial inventory and prohibits selling without inventory, allowing it to study residual price risk after dollar closure.

2. A coordinate transformation makes nonlinear trading additive

In the original equation, the price effect of dollar trading depends on the current state. Converting shares to dollars and taking the log price $q_i=\log(S_i/F)$ gives the trading component:

\mathrm dq_i\big|_{\mathrm{trade}} =\frac{g(S_i)}{S_i^2}\,\mathrm dX_i.

The impact of a dollar thus depends on the current price. The paper chooses a different price coordinate:

y_i=\Phi(S_i),\qquad \Phi(S)=\int_F^S\frac{u}{g(u)}\,\mathrm du.

This choice follows from a precise normalization. For a coordinate $\psi(S)$ to satisfy $\mathrm d\psi=\mathrm dX$ during trading, it must obey $\psi'(S)g(S)=S$ . Imposing $\psi(F)=0$ then uniquely determines $\Phi$ .

Define the recovery response:

\mathcal R(y)=\frac{S(S-F)}{g(S)},\qquad S=\Phi^{-1}(y).

The market equation becomes:

\mathrm d\mathbf{y} =B\,\mathrm d\mathbf{z} -\rho\mathcal R(\mathbf{y})\,\mathrm dt.

Trading dollars now enter additively: the coordinate decrease in one asset equals the increase in another. Nonlinearity is concentrated in recovery and the inverse mapping. Summing the coordinates produces a trading invariant:

H=\sum_i\Phi(S_i),\qquad \mathrm dH=-\rho\sum_i\frac{S_i(S_i-F)}{g(S_i)}\,\mathrm dt.

Trading itself preserves $H$ ; recovery changes $H$ . Whether a broad decline can occur depends on whether recovery can carry the state across a particular boundary. The transformation preserves the original price paths and reachable set while making that boundary visible.

3. Why do elasticities of 1 and 2 create different barriers?

First define the paper's measure of systemic decline. Each asset's terminal log loss is:

L_i(T)=\log\frac{F}{S_i(T)},\qquad \overline L(T)=\frac1N\sum_i L_i(T).

For a coverage fraction $\alpha$ , let $k=\lceil\alpha N\rceil$ and let $Q_\alpha$ denote the $k$ th-largest loss. Define:

\mathcal C_\alpha(T)=\min\{\overline L(T),Q_\alpha(T)\}.

Reaching a threshold $\delta>0$ requires a mean log loss of at least $\delta$ , with at least $k$ assets each losing at least $\delta$ . Coverage and average depth both matter: gains in a few assets offset the mean loss.

The elasticity of the price response is:

e(S)=\frac{Sg'(S)}{g(S)}.

It controls two key curvatures. Setting $G(q)=\Phi(Fe^q)$ gives:

\Phi''(S)=\frac{1-e(S)}{g(S)},\qquad G''(q)=\frac{S^2}{g(S)}[2-e(S)].

If $e(S)\ge1$ at every reachable price, $\Phi$ is concave. Its tangent-line inequality yields:

(S-F)\Phi'(S)\le\Phi(S),\qquad \dot H\ge-\rho H.

Starting from $H(0)=0$ , this implies $H(t)\ge0$ . If every asset fell below $F$ , each term would be negative because $\Phi$ is strictly increasing and $\Phi(F)=0$ , giving $H<0$ . Thus elasticity of at least 1 rules out all assets simultaneously falling below the benchmark.

If $e(S)\ge2$ globally, $G$ is also concave. Jensen's inequality gives:

0\le\frac{H}{N} =\frac1N\sum_iG(q_i) \le G\left(\frac1N\sum_iq_i\right).

Since $G$ is strictly increasing and $G(0)=0$ , the mean log price must be nonnegative and the mean log loss nonpositive. Hence $\mathcal C_\alpha\le0$ at every coverage fraction.

The thresholds therefore address different questions: elasticity 1 governs whether all assets can decline; elasticity 2 additionally governs whether the mean log loss can be positive. Both conditions must hold along the entire permitted path. An elasticity estimate near the initial price supports conclusions only within that price region.

4. A/B/C: how does response shape change the reachable set?

Holding other conditions fixed, the paper compares members of one response family:

g_p(S)=\eta(S/F)^p,\qquad p\in\{2,1,0\}.

All three have the same impact at the initial price $F$ , namely $g_p(F)=\eta$ , constant elasticity $p$ , and normalized log impact per trading dollar $(S/F)^{p-2}$ .

Analytical curves showing normalized log-price impact per trading dollar against relative price for three response functions.

Figure 1 | Three shapes within one response family. Source: V12 Final manuscript and make_figures.py; redrawn by ArcaViso. The vertical axis shows normalized log impact per trading dollar. Analytical model curves; version cutoff: October 17, 2026.

These responses yield three trading invariants and three reachability regimes:

Market	Response exponent	Quantity preserved during trading	Main result
A	p=2	Sum of log prices	Rules out positive systemic loss
B	p=1	Sum of prices	Small losses with partial coverage are reachable; fixed initial inventory limits severity
C	p=0	Sum of squared prices	With impulses and less than full coverage, arbitrarily severe log losses can be constructed under uniform finite resource bounds

Market B's limitation follows directly from the inventory potential. For a terminal price $Fe^{-L}$ :

E_1(Fe^{-L})=\frac{F^2}{\eta}(L+e^{-L}-1).

This diverges as $L$ increases. Dollar closure and finite initial inventory require $E_1\le FH_0$ , so fixed inventory rules out arbitrarily large losses.

In market C, $E_0(S)=(S-F)^2/(2\eta)$ remains bounded as the price approaches zero. The paper also constructs paths: for fixed $N$ and $k<N$ , one can choose a sufficiently long but finite $T$ and finite turnover and inventory bounds independent of the loss threshold, such that every finite $\delta$ is reachable by a corresponding path.

Here, “arbitrarily severe” refers to log losses diverging as prices approach zero. The mathematical construction retains at least one receiving asset and permits instantaneous impulses executed along the price-response trajectory. Impulses are settled by integrating marginal prices along execution, with every intermediate price remaining positive.

Changing coverage to all assets, $k=N$ , immediately produces an exact time boundary in the same constant-impact market:

\delta_{\mathrm{crit}}(N,T) =\log\frac{\sqrt N}{1+(\sqrt N-1)e^{-\rho T}}.

Every positive threshold strictly below this boundary is reachable with finite resources; thresholds at or above it are unreachable. The boundary is an unattained supremum. The proof uses the fact that constant-impact trading preserves the squared norm of the price vector, while recovery limits how far that norm can contract. Moving from “most assets” to “all assets” changes the geometry of the problem.

5. From path existence to trading-resource requirements

If $\mathcal C_\alpha(T)\ge\delta$ , at least $k$ assets must fall to or below $s=Fe^{-\delta}$ . Below fundamentals, recovery pushes prices upward, so trading must overcome recovery and cover at least the corresponding coordinate distance. This gives two necessary resource lower bounds:

M\ge k\int_s^F\frac{u}{g(u)}\,\mathrm du,\qquad \sum_iH_i^-\ge k\int_s^F\frac{1}{g(u)}\,\mathrm du.

Here $M$ counts each paired dollar transfer once, and $H_i^-$ is the cumulative number of shares sold in asset $i$ . Dollar distance is measured by $\Phi$ ; share distance by $\Psi(S)=\int_F^S1/g(u)\,\mathrm du$ .

If a qualifying declining asset starts with at most $H_i$ shares, terminal dollar closure also requires:

\int_s^F\frac{F-u}{g(u)}\,\mathrm du\le FH_i.

These inequalities eliminate proposals with insufficient resources. Finding a path also requires handling recovery dissipation, individual ledger closure, and the capacity of other assets to receive the proceeds.

6. Why do finite trading rates constrain losses to second order?

Impulses allow finite trades in zero physical time. A trading-rate cap requires controls to unfold continuously. The paper bounds each asset's gross share-trading rate before offsetting purchases against sales:

r_i^{\mathrm{gross}}(t) =\frac1{S_i(t)}\sum_e|B_{ie}|\dot z_e(t) \le\bar v.

Thus simultaneous trades in opposite directions still count toward the cap. With constant impact $g=\eta$ , price deviations first satisfy a first-order bound:

|S_i(t)-F|\le\frac{\eta\bar v}{\rho}(1-e^{-\rho t}).

Systemic loss faces a stronger restriction. Write $x_i=S_i-F$ and $m=\sum_i x_i$ . Paired dollar flows imply:

\sum_i\dot\theta_i=-\frac1F\sum_ix_i\dot\theta_i, \qquad \dot m=-\rho m-\frac\eta F\sum_ix_i\dot\theta_i.

Each $x_i$ is $O(\bar v)$ , and the share-trading rate is also $O(\bar v)$ , making the term driving the mean price deviation $O(\bar v^2)$ . This is the source of the cancellation of first-order effects in the market average.

Let $d_T=1-e^{-\rho T}$ and $a=\eta\bar v/(F\rho)$ . When $ad_T<1$ , the paper obtains the explicit upper bound:

\mathcal C_\alpha(T)\le a^2\left[d_T-\rho T e^{-\rho T} +\frac{d_T^2}{2(1-ad_T)^2}\right].

For fixed market parameters and time horizon, the maximum positive loss shrinks quadratically as the rate cap becomes small. The paper supplies a continuous construction attaining this order, so the minimum rate needed to reach a small loss threshold satisfies $\bar v_*(\delta)=\Theta(\sqrt\delta)$ .

The construction is instructive. Choose a zero-sum vector $\mathbf{u}$ with every component nonzero. Set $\Delta=T/3$ and $r=e^{-\rho\Delta}$ , and divide dollar trading rates into three equal stages:

\dot X_i=\varepsilon u_i f(t),\qquad f(t)=\begin{cases} 1,&0\le t<\Delta,\\ -(1+r),&\Delta\le t<2\Delta,\\ r,&2\Delta\le t\le T. \end{cases}

The signal's ordinary integral is zero, ensuring terminal dollar closure. Its integral with recovery weights is also zero, canceling the first-order terminal price displacement. When $g$ is three times continuously differentiable near $F$ and local elasticity satisfies $e(F)<1$ , the second-order term leaves a positive loss in every asset. Scaling down $\varepsilon$ lets the path satisfy any prescribed positive rate cap, price band of positive width, and positive initial inventory simultaneously.

Finite rates therefore still permit small declines in all assets. Tighter rate caps reduce the scale of achievable losses. The quadratic law connects these two conclusions.

7. How do daily price bands and receiving assets constrain consecutive declines?

A fixed one-day lower bound $F(1-\beta_-)$ directly gives a loss ceiling of $-\log(1-\beta_-)$ . If the bound resets daily against the previous close, the lower bounds compound over $D$ days:

S_i(T_D)\ge F\prod_{d=1}^D(1-\beta_{-,d}).

The room for cumulative decline expands across sessions, but the sale proceeds must still be absorbed at every instant. Let $J$ be the target asset set and $J^c$ the remaining assets. Instantaneous pairing requires:

-\sum_{i\in J}\dot X_i =\sum_{j\in J^c}\dot X_j.

When the target set is a net seller, the receiving side's capacity is bounded jointly by the remaining assets' prices and share-trading rates:

\sum_{j\in J^c}\dot X_j \le\bar v\sum_{j\in J^c}S_j.

More targets and fewer receiving assets increase the pressure to absorb proceeds. Gains in the receiving assets also raise the market's mean log price, offsetting target losses. A path must therefore satisfy two requirements together: the receiving side must absorb the dollar flow, and the terminal mean loss must remain deep enough.

8. Unpacking the interval-certified ten-asset, two-session construction

The paper combines these constraints in a concrete construction. The parameters are $N=10$ , $F=\eta=1$ , and $\rho=0.4$ , with each trading session lasting 20 model-time units. Nine target assets have daily proportional price bands of ±3%; one receiving asset has ±25% bands. Each resets against the previous close.

The nine targets follow the same continuous piecewise-linear price path $x(t)$ , passing through these nodes:

Model time	0	2.3	17.7	20	22.3	37.7	40
Price of each target asset	1	h	h	0.97	0.995	0.995	0.9409

The first plateau height $h\approx1.020202255079063$ is selected by the exact terminal dollar-closure equation:

101600000000h^2-97138000000h-6646157699=0.

Given $x(t)$ , each target's dollar flow rate follows from the price equation:

u(t)=x(t)[\dot x(t)+0.4(x(t)-1)].

The receiving asset must absorb the opposite flow at nine times this rate at every instant. Its price $y(t)$ therefore solves a scalar nonlinear equation:

\dot y=-0.4(y-1)-\frac{9u(t)}{y},\qquad y(0)=1.

Symmetry reduces the ten-asset system to two price trajectories. Exact dollar closure for the targets, combined with instantaneous pairing, also closes the receiving asset's terminal dollar ledger.

Figure 2 | Floating-point numerical reconstruction of the interval-certified path. Source: improved_multiday_witness.py from V12 Final; redrawn by ArcaViso. Prices are normalized to their initial values; the horizontal axis measures model trading time. The panels use different vertical ranges; dashed lines mark daily price bands. Version cutoff: October 17, 2026.

The nine targets close at their lower limits twice, giving a cumulative price loss of $1-0.97^2=5.91\%$ . High-precision numerical reconstruction gives $y(40)\approx1.405160822$ , so:

Q_{0.9}=-\log(0.9409)\approx0.06092,

\overline L=-\frac1{10}\log[(0.9409)^9y(40)] \approx0.0208114.

The mean loss is the smaller term, making systemic loss approximately 0.02081. This corresponds to a decline of about 2.06% in the cross-sectional geometric mean of terminal prices. The receiving asset's approximately 40.5% gain is included in this calculation.

The rigorous certification uses fixed-point interval arithmetic, with outward rounding for multiplication and division. It computes enclosures across 80,000 time steps over the six target-path segments, checking prices between steps, trading rates, and resource integrals. The certificate establishes:

\mathcal C_{0.9}(40)>0.020,\qquad \bar v_{\mathrm{used}}<0.285,\qquad M<3,\qquad H_i^-<2.

Thus two initial shares per asset suffice for the path, total paired dollar turnover is less than 3, and all execution prices remain inside their respective rolling price bands. The certificate also encloses the receiving asset's terminal price within $1.4047874<y(40)<1.4055543$ and uses rational bounds for the exponential function to verify that mean loss strictly exceeds 0.020.

For this fixed two-session, ten-asset problem, consider all feasible paths that bring the same nine assets to their lower closing limits on both days while satisfying $\mathcal C_{0.9}(40)\ge0.020$ . Let $\bar v_*$ be the infimum of the required cap on each asset's gross share-trading rate. The quadratic barrier excludes rates that are too low, while the constructed path supplies a feasible upper bound. Together they give:

0.044<\bar v_*<0.285.

This interval leaves a clear next question: among all feasible paths, what is the minimum trading rate needed to reach the same target? The existing construction proves that the feasible set is nonempty. Finding the exact optimum requires solving an optimal-control problem with rolling price bands, inventory constraints, and ledger closure.

9. Research contribution and next steps

The paper separates market stability into three layers that can be checked individually: response geometry determines structural reachability; trading dollars and inventory determine resource sufficiency; rates and daily price bands determine whether a path can be executed. The response coordinate $\Phi$ connects all three, generating elasticity barriers and expressing resource lower bounds as coordinate distances.

The framework suggests studying broad declines alongside the receiving side's capacity, the recovery timescale, and changes in impact per dollar as prices fall. The symmetric ten-asset construction supplies a complete existence example. Heterogeneous assets, restricted trading networks, direct cross-asset impact, and mechanisms governing behavioral choices offer directions for extending the model.

The scope here is fixed fundamentals, exogenous impact functions, a common recovery rate, and paired reallocation within risky assets. Daily proportional price bands are a model rule. Applying the results to a particular market requires estimating response and recovery parameters and matching the model to actual execution, cash accounts, and exchange rules. The paper is currently a working paper. The figures show analytical curves and constructed paths; numerical curves come from floating-point reconstruction, while rigorous feasibility follows from an independent interval certificate for the specified path.

Paper and reproducibility materials

This article follows the V12 Final mathematical baseline and the SIFIN typeset manuscript. The model and coordinates appear in manuscript Section 2; elasticity barriers in Section 3; A/B/C and the full-coverage boundary in Section 4; resource lower bounds in Section 5; finite rates in Section 6; and the multiday construction and certificate in Section 7 and the appendices. Figures have been redrawn to ArcaViso standards, preserving the original model and calculation conventions.