---
# System prepended metadata

title: 'Slippage-at-Risk (SaR):'

---

# Slippage-at-Risk (SaR):
## A Forward-Looking Liquidity Risk Framework for Perpetual Futures Exchanges

Otar Sepper -- Draft v.1  
Gauntlet  
`otar@gauntlet.xyz`  

December 2025


## Abstract

We introduce **Slippage-at-Risk (SaR)**, a quantitative framework for measuring liquidity risk in perpetual futures exchanges. Unlike backward-looking metrics such as Value-at-Risk computed on historical returns or realized deficit distributions, SaR provides a *forward-looking* assessment of liquidation execution risk derived from current order book microstructure. The framework comprises three complementary metrics: $\mathrm{SaR}(\alpha)$, the cross-sectional slippage quantile; $\mathrm{ESaR}(\alpha)$, the expected slippage in the distributional tail; and $\mathrm{TSaR}(\alpha)$, the aggregate dollar-denominated tail slippage. We extend the base framework with a *concentration adjustment* that penalizes fragile liquidity structures where a small number of market makers dominate quote provision. Drawing on recent work by [Chitra et al. (2025)](#ref-chitra2025adl) on autodeleveraging mechanisms and insurance fund optimization, we establish a direct mapping from SaR metrics to optimal capital requirements. Empirical analysis using Hyperliquid order book data, including the October 10, 2025 liquidation cascade, demonstrates SaR's predictive validity as a leading indicator of systemic stress. We conclude with practical implementation guidance and discuss philosophical implications for risk management in decentralized financial systems.

**Keywords:** Liquidity risk, perpetual futures, order book microstructure, autodeleveraging, insurance fund, DeFi risk management


## Table of Contents

[TOC]

# 1. Introduction
## 1.1 Motivation

Perpetual futures have emerged as the dominant derivative instrument in cryptocurrency markets, with daily trading volumes regularly exceeding \$100 billion across major venues. Unlike traditional futures contracts with fixed expiration dates, perpetuals use a funding rate mechanism to maintain price alignment with spot markets, enabling indefinite position holding. This innovation has democratized access to leveraged trading but introduced novel systemic risks that existing frameworks inadequately address.

The core vulnerability lies in the interaction between leverage, liquidations, and order book liquidity. When a leveraged position's collateral falls below maintenance requirements, the exchange must liquidate it - selling long positions or buying back short positions at prevailing market prices. During periods of market stress, cascading liquidations can overwhelm available liquidity, resulting in execution prices far worse than theoretical bankruptcy prices. The resulting *bad debt* - the shortfall between bankruptcy price and actual execution price - must be absorbed by the exchange's insurance fund or socialized across profitable traders through autodeleveraging (ADL).

The October 10, 2025 event on Hyperliquid starkly illustrated these dynamics: \$2.1 billion in liquidations over 12 minutes generated \$304.5 million in deficits requiring socialization, with the exchange's queue-based ADL policy expending \$704.6 million in haircuts - an 8$\times$ overutilization relative to the actual deficit [Chitra et al. (2025)](#ref-chitra2025adl). This event was not a black swan but rather the predictable consequence of thin order books under stress conditions that were, in principle, observable before the cascade began.
## 1.2 The Gap in Existing Approaches

Current risk management frameworks for perpetuals exchanges exhibit several critical limitations:

1. **Backward-looking metrics:** Traditional Value-at-Risk (VaR) and Expected Shortfall (ES) are computed on historical return distributions. They capture past volatility but provide no direct measurement of current capacity to absorb liquidations.
2. **Undefined for new assets:** Historical deficit-based metrics cannot be computed for newly listed tokens that have not yet experienced stress events. This creates a dangerous blind spot precisely where risk is highest - illiquid markets with unproven depth.
3. **Static calibration:** Position limits and leverage caps are typically set at listing and adjusted infrequently, failing to adapt to evolving liquidity conditions.
4. **Ignoring microstructure:** The order book - the actual resource that determines execution quality - is rarely incorporated into formal risk metrics.
## 1.3 Contribution and Outline

This paper introduces Slippage-at-Risk (SaR), a framework that directly addresses these limitations by measuring liquidation execution risk from observable order book state. Our contributions are:

1. **Core Framework ([Section 3](#3-the-slippage-at-risk-framework)):** We define the slippage function $S_i(Q)$ and derive three complementary metrics: $\mathrm{SaR}(\alpha)$, $\mathrm{ESaR}(\alpha)$, and $\mathrm{TSaR}(\alpha)$. We establish their interpretation as cross-sectional liquidity quantiles and discuss notional calibration.
2. **Concentration Adjustment ([Section 4](#4-concentration-adjustment)):** We formalize the fragility of concentrated liquidity and derive a haircut formula based on the Herfindahl-Hirschman Index (HHI) that adjusts slippage upward for books dominated by few providers.
3. **Theoretical Connections ([Section 5](#5-theoretical-connections)):** We establish the causal chain from order book depth to deficit distributions and derive the SaR-implied insurance fund formula, connecting our work to [Chitra et al. (2025)](#ref-chitra2025adl).
4. **Extensions ([Section 6](#6-extensions)):** We extend the framework to incorporate cascade effects, time dynamics, cross-token correlation, and spoofing detection.
5. **Empirical Analysis ([Section 7](#7-empirical-analysis)):** We apply the framework to Hyperliquid order book data, validate SaR's predictive properties, and analyze the October 10, 2025 event.
6. **Implementation and Implications ([Sections 8–9](#8-implementation-guide)):** We provide practical guidance for deployment and discuss broader implications for DeFi risk management.


# 2. Preliminaries: Perpetuals Exchange Mechanics

We briefly review the mechanics of perpetual futures exchanges, following the notation of [Chitra et al. (2025)](#ref-chitra2025adl). Readers familiar with these concepts may proceed to [Section 3](#3-the-slippage-at-risk-framework).
## 2.1 Positions and Leverage

A position on a perpetuals exchange is a tuple $p_i = (q_i, c_i, t_i, b_i)$ where:

- $q_i \in \mathbb{R}$ is the signed quantity (positive for long, negative for short)
- $c_i > 0$ is the collateral posted
- $t_i$ is the entry timestamp
- $b_i \in \{-1, +1\}$ indicates direction (long/short)

The *leverage* of position $i$ at time $t$ is:

$$
\ell_{i,t} = \frac{p_t \cdot |q_i|}{c_{i,t}}
$$

where $p_t$ is the mark price and $c_{i,t}$ is current collateral (initial collateral adjusted for realized PnL and funding payments). Exchanges impose a maximum leverage $\ell_{\max}$, typically 10-100$\times$ depending on the asset.
## 2.2 Margin Requirements and Liquidation

Positions are subject to two margin thresholds:

- **Initial Margin (IM):** Minimum collateral required to open a position, typically $\mathrm{IM} = 1/\ell_{\max}$ of notional.
- **Maintenance Margin (MM):** Minimum collateral required to keep a position open, typically $\mathrm{MM} = \mu \cdot \mathrm{IM}$ for some $\mu \in (0,1)$.

The *equity* of position $i$ at time $t$ is:

$$
e_{i,t} = c_{i,t} + \mathrm{PnL}_{i,t} = c_{i,t} + q_i(p_t - p_{t_i})
$$

A position is liquidated when equity falls below maintenance margin:

$$
e_{i,t} < \mathrm{MM}_i \implies \text{liquidation triggered}
$$

Define the *bankruptcy price* $p_i^{\mathrm{bk}}$ as the price at which equity equals zero:

$$
p_i^{\mathrm{bk}} = p_{t_i} - \frac{c_i}{q_i} \quad \text{(for longs)}
$$

and the *liquidation price* $p_i^{\mathrm{liq}}$ as the price at which equity equals maintenance margin.
## 2.3 Execution and Bad Debt

When liquidating position $i$ with quantity $\Delta q_i = -q_i$, the exchange executes a market order that consumes order book liquidity. The *execution price* $p_i^{\mathrm{exec}}$ depends on available depth. Under a linear market impact model:

$$
p_i^{\mathrm{exec}} = p_t - \frac{\alpha}{2} \Delta q_i
$$

where $\alpha > 0$ is the impact coefficient.

*Bad debt* arises when execution price is worse than bankruptcy price:

$$
D_i = \left( -\tilde{e}(p_i, \Delta q_i) \right)^+ = \left( q_i (p_i^{\mathrm{bk}} - p_i^{\mathrm{exec}}) \right)^+
$$

Total deficit at time $T$ is $D_T = \sum_i D_i$ over all liquidated positions.
## 2.4 Insurance Fund and Autodeleveraging

The insurance fund $\mathrm{IF}_t$ absorbs deficits up to its capacity:

$$
\mathrm{IF}_{t+1} = \mathrm{IF}_t + \text{(liquidation fees)} + \text{(trading fees)} - \min\{\mathrm{IF}_t, D_t\}
$$

When $D_t > \mathrm{IF}_t$, the residual $R_t = (D_t - \mathrm{IF}_t)^+$ triggers autodeleveraging: profitable positions are forcibly closed to cover the shortfall. Different ADL policies (Queue, Pro-Rata, Risk-Aware Pro-Rata) allocate haircuts differently across winners, with significant implications for fairness and revenue [Chitra et al. (2025)](#ref-chitra2025adl).


# 3. The Slippage-at-Risk Framework
## 3.1 Setup and Notation

Consider a perpetuals exchange at time $t$ listing $N$ tokens indexed by $i \in \{1, \ldots, N\}$. For each token $i$, we observe:

- $p_{i,t}$: mid-price
- $\mathrm{OI}_{i,t}$: open interest (total outstanding notional)
- $B_i(\cdot)$: bid-side cumulative depth function
- $A_i(\cdot)$: ask-side cumulative depth function

The order book is represented as a set of limit orders. On the bid side, let $\{(p_k^b, q_k^b)\}_{k=1}^{K_b}$ denote price-quantity pairs with $p_1^b > p_2^b > \cdots > p_{K_b}^b$. The cumulative depth at price $p$ is:

$$
B_i(p) = \sum_{k: p_k^b \geq p} q_k^b
$$

with analogous definition for the ask side.
## 3.2 The Slippage Function

**Definition (Slippage Function)**  
For token $i$, the *slippage function* $S_i: \mathbb{R}_+ \to \mathbb{R}_+$ measures the volume-weighted average execution shortfall when liquidating notional quantity $Q$:

$$
S_i(Q) = \frac{1}{Q} \int_0^Q \left[ p_{\mathrm{mid}} - p(q) \right] dq
$$

where $p(q)$ is the marginal execution price for the $q$-th unit.

For a discrete order book, slippage is computed by walking the book:

**Algorithm: Compute Slippage from Order Book**  

    SLIPPAGE(order_book, Q, mid_price):
    
    Input:  order_book = list of (price, size) sorted best-to-worst
            Q = quantity to liquidate
            mid_price = reference price
    
    Output: slippage as percentage of mid_price
    
    remaining ← Q
    total_cost ← 0
    
    FOR each level (price, size) in order_book:
        
        fill_amount ← min(size, remaining)
        price_impact ← mid_price − price
        total_cost  ← total_cost + fill_amount × price_impact
        remaining   ← remaining − fill_amount
        
        IF remaining = 0: EXIT LOOP
    
    IF remaining > 0:
        RETURN ∞                    # Insufficient liquidity
    
    RETURN total_cost / (Q × mid_price)   # As percentage


**Remark (Directional Slippage)**  
Liquidations are directional: long liquidations sell into bids, short liquidations buy into asks. Define:

$$
\begin{aligned}
S_i^{\mathrm{bid}}(Q) &:\ \text{slippage from selling } Q \text{ (long liquidations)} \\
S_i^{\mathrm{ask}}(Q) &:\ \text{slippage from buying } Q \text{ (short liquidations)}
\end{aligned}
$$

In practice, use the direction matching dominant positioning. If open interest is skewed long, $S_i^{\mathrm{bid}}$ is the relevant measure.
## 3.3 Notional Calibration

The quantity $Q_i$ at which to evaluate slippage should reflect a stress liquidation scenario. We parameterize:
$$
Q_i = \min\left( \beta \cdot \mathrm{OI}_i, \, \mathrm{VaR}_{0.95}(\text{Position Size}_i) \right)
$$

where:

- $\beta \in [0.05, 0.20]$ is the *stress parameter*, representing the fraction of open interest that liquidates simultaneously in a stress scenario.
- $\mathrm{VaR}_{0.95}(\text{Position Size}_i)$ caps $Q_i$ at the 95th percentile of historical position sizes, preventing extreme outliers from dominating.

**Remark (Choosing $\beta$)**  
The appropriate stress parameter depends on market conditions:

- **Normal conditions:** $\beta = 0.05$--$0.10$
- **Elevated volatility:** $\beta = 0.10$--$0.15$
- **Crisis scenarios:** $\beta = 0.15$--$0.20$

During the October 10, 2025 Hyperliquid event, realized liquidations represented approximately 15% of pre-event open interest for affected tokens.
## 3.4 Cross-Sectional SaR

**Definition (Slippage-at-Risk)**  
The *Slippage-at-Risk* at confidence level $\alpha \in (0,1)$ is the $\alpha$-quantile of slippage across all tokens:

$$
\mathrm{SaR}(\alpha) = \inf\left\{ s \in \mathbb{R}_+ : \frac{1}{N} \sum_{i=1}^N \mathbf{1}\{S_i(Q_i) \leq s\} \geq \alpha \right\}
$$

**Interpretation:** $\mathrm{SaR}(0.95) = 3\%$ means that 95% of tokens have slippage at or below 3% when liquidating their calibrated stress notional. The remaining 5% are ``tail'' tokens with critically thin liquidity.
## 3.5 Expected Slippage at Risk (ESaR)

**Definition (Expected Slippage at Risk)**  
The *Expected Slippage at Risk* is the conditional expected slippage among tail tokens:

$$
\mathrm{ESaR}(\alpha) = \mathbb{E}\left[ S_i(Q_i) \,\big|\, S_i(Q_i) > \mathrm{SaR}(\alpha) \right]
$$

$\mathrm{ESaR}$ measures the *severity* of tail liquidity risk, not merely its boundary. It is the slippage analog of Expected Shortfall in the VaR literature.
## 3.6 Total Dollar Slippage at Risk (TSaR)

**Definition (Total Dollar Slippage at Risk)**  
The *Total Dollar Slippage at Risk* aggregates dollar-denominated slippage from tail tokens:

$$
\mathrm{TSaR}(\alpha) = \sum_{i: S_i(Q_i) > \mathrm{SaR}(\alpha)} S_i(Q_i) \cdot Q_i
$$

$\mathrm{TSaR}$ represents the *total execution shortfall* that would materialize if stress liquidations occurred across all tail tokens simultaneously. This quantity maps directly to potential deficit exposure.
## 3.7 Weighted SaR

The basic $\mathrm{SaR}$ treats all tokens equally. For risk management purposes, weighting by exposure is often more appropriate:

**Definition (Open Interest-Weighted SaR)**  
$$
\mathrm{SaR}^w(\alpha) = \inf\left\{ s \in \mathbb{R}_+ : \frac{\sum_{i: S_i(Q_i) \leq s} \mathrm{OI}_i}{\sum_{i=1}^N \mathrm{OI}_i} \geq \alpha \right\}
$$

$\mathrm{SaR}^w(0.95)$ answers: ``What slippage level contains 95% of the open interest (not 95% of tokens)?'' This weights toward tokens with larger aggregate positions.

# 4. Concentration Adjustment
## 4.1 The Fragility of Concentrated Liquidity

Raw slippage $S_i(Q)$ measures depth but ignores its *structure*. Two order books with identical depth profiles can have vastly different fragility:

**Example**  
Consider two order books, each with 1M total bid-side liquidity:

- **Book A (Distributed):** 100 market makers, each providing \$10,000
- **Book B (Concentrated):** 2 market makers, each providing \$500,000

Both yield $S(Q) = s$ for some quantity $Q$. However, if one participant withdraws:

- Book A loses 1% of depth
- Book B loses 50% of depth

Book B's apparent liquidity is fragile - it can vanish suddenly if a dominant provider exits.

This fragility is especially concerning in the context of:

- **Spoofing:** Artificial liquidity placed to manipulate prices, withdrawn before execution
- **Correlated withdrawal:** Market makers pulling quotes simultaneously during stress
- **Single points of failure:** Dominant providers experiencing technical issues or insolvency
## 4.2 Concentration Metrics

For token $i$, let $M_i$ accounts provide liquidity with quantities $\ell_1, \ldots, \ell_{M_i}$. Define shares:

$$
w_m = \frac{\ell_m}{\sum_{j=1}^{M_i} \ell_j}, \quad m = 1, \ldots, M_i
$$

**Definition (Herfindahl-Hirschman Index)**  
$$
\mathrm{HHI}_i = \sum_{m=1}^{M_i} w_m^2
$$

**Properties:**

- $\mathrm{HHI} \in [1/M, 1]$
- $\mathrm{HHI} = 1/M$ when liquidity is uniformly distributed
- $\mathrm{HHI} = 1$ when a single provider supplies all liquidity (monopoly)

**Definition (Effective Number of Providers)**  
$$
N_{\mathrm{eff}} = \frac{1}{\mathrm{HHI}}
$$

$N_{\mathrm{eff}}$ represents the number of equal-sized providers that would produce the observed concentration. In the example in Section 4.1, Book A has $N_{\mathrm{eff}} = 100$ while Book B has $N_{\mathrm{eff}} = 2$.

**Definition (Top-$k$ Concentration Ratio)**  
$$
\mathrm{CR}_k = \sum_{m=1}^k w_{(m)}
$$

where $w_{(m)}$ is the $m$-th largest share. $\mathrm{CR}_1$ measures dominance of the single largest provider.
## 4.3 The Concentration Haircut

We now derive a haircut that adjusts slippage upward based on concentration.

### Withdrawal Scenario Analysis

Consider the scenario where provider $m$ withdraws. Let $S_i^{(-m)}(Q)$ denote slippage computed on the residual book. The *withdrawal-based haircut* is:

$$
h_i^{\mathrm{withdraw}} = \frac{S_i^{(-1)}(Q) - S_i(Q)}{S_i(Q)} = \frac{S_i^{(-1)}(Q)}{S_i(Q)} - 1
$$

where $(-1)$ indicates removal of the largest provider.

### Connection Between HHI and Withdrawal Impact

**Proposition**  
Assume slippage scales inversely with depth: $S \propto 1/L$ where $L$ is total liquidity. If provider $m$ is selected uniformly at random to withdraw, then:

$$
\mathbb{E}\left[ \frac{S^{(-m)}}{S} - 1 \right] = \mathbb{E}\left[ \frac{w_m}{1 - w_m} \right] \approx \frac{\mathrm{HHI}}{1 - 1/M} + O(\mathrm{HHI}^2)
$$

for small concentration.

**Proof**

After provider $m$ withdraws, remaining depth is $(1 - w_m)L$. Under linear slippage:

$$
\frac{S^{(-m)}}{S} = \frac{L}{(1-w_m)L} = \frac{1}{1-w_m}
$$

Thus:

$$
\mathbb{E}\left[ \frac{1}{1-w_m} - 1 \right] = \mathbb{E}\left[ \frac{w_m}{1-w_m} \right]
$$

For small $w_m$, $\frac{w_m}{1-w_m} \approx w_m + w_m^2 + O(w_m^3)$. Taking expectation:

$$
\mathbb{E}\left[ \frac{w_m}{1-w_m} \right] \approx \mathbb{E}[w_m] + \mathbb{E}[w_m^2] = \frac{1}{M} + \mathrm{HHI}
$$

Rearranging yields the result.

### Parametric Haircut Formula

Based on the above analysis, we propose:

**Definition (Concentration Haircut)**  
$$
h_i^{\mathrm{conc}} = \alpha \cdot \left( \frac{N_{\mathrm{target}}}{N_{\mathrm{eff}}^{(i)}} - 1 \right)^+ + \beta \cdot \left( \mathrm{CR}_1^{(i)} - \mathrm{CR}_1^{\mathrm{thresh}} \right)^+
$$

where:

- $N_{\mathrm{target}}$: target number of effective providers (e.g., 10--20)
- $\mathrm{CR}_1^{\mathrm{thresh}}$: maximum acceptable share for top provider (e.g., 0.25)
- $\alpha, \beta > 0$: sensitivity parameters

The first term penalizes books with too few effective providers. The second term adds an extra penalty when a single account dominates, which may indicate spoofing.
## 4.4 Concentration-Adjusted Slippage and SaR

**Definition (Adjusted Slippage)**  
$$
S_i^{\mathrm{adj}}(Q) = S_i(Q) \cdot \left(1 + h_i^{\mathrm{conc}}\right)
$$

All SaR metrics are then computed using adjusted slippage values:

$$
\begin{aligned}
\mathrm{SaR}^{\mathrm{adj}}(\alpha) &= \text{quantile}_\alpha\left(\{S_i^{\mathrm{adj}}(Q_i)\}_{i=1}^N\right) \\
\mathrm{ESaR}^{\mathrm{adj}}(\alpha) &= \mathbb{E}\left[S_i^{\mathrm{adj}} \,|\, S_i^{\mathrm{adj}} > \mathrm{SaR}^{\mathrm{adj}}(\alpha)\right] \\
\mathrm{TSaR}^{\mathrm{adj}}(\alpha) &= \sum_{i: S_i^{\mathrm{adj}} > \mathrm{SaR}^{\mathrm{adj}}(\alpha)} S_i^{\mathrm{adj}}(Q_i) \cdot Q_i
\end{aligned}
$$


# 5. Theoretical Connections
## 5.1 The Causal Chain: From Order Books to Deficits

Liquidation deficits emerge through a clear causal process:

![Screenshot 2025-12-29 at 12.45.19 PM](https://hackmd.io/_uploads/BknB0EeNWx.png)

SaR measures the *upstream* cause (liquidity conditions), while VaR on historical deficits measures the *downstream* consequence. SaR is a *leading indicator*  - liquidity withdrawal precedes and predicts cascade events.
## 5.2 Forward-Looking vs. Backward-Looking Metrics

**Table: Comparison of Risk Metrics**  
| Metric | Basis | Nature | Information Source | Defined For |
|---|---|---|---|---|
| Historical VaR | Returns | Backward | Past price moves | All assets |
| Deficit VaR | Historical deficits | Backward | Past ADL events | Assets with history |
| **SaR** | Order book depth | **Forward** | Current MM beliefs | **All assets** |
| SaR-implied VaR | SaR distribution | Forward | Current liquidity | All assets |

The forward-looking nature of SaR has critical advantages:

1. **Universally defined:** Computable for any token with an order book, regardless of crisis history.
2. **Encodes market maker beliefs:** Order book depth reflects liquidity providers' assessments of risk.
3. **Leading indicator:** Liquidity withdrawal $\to$ SaR increases $\to$ (lag) $\to$ cascade $\to$ VaR reacts. SaR flags risk at step 2, potentially days before the event.
## 5.3 Optimal Insurance Fund Sizing

[Chitra et al. (2025)](#ref-chitra2025adl) derive the optimal insurance fund by minimizing expected total cost:

$$
\mathrm{IF}^* = \text{argmin}_{\mathrm{IF} \geq 0} \left\{ r \cdot \mathrm{IF} + \kappa \cdot \mathbb{E}[(D - \mathrm{IF})^+] \right\}
$$

where:

- $r$: opportunity cost per unit capital held in IF
- $\kappa$: reputation/social cost per unit deficit socialized
- $D$: random deficit

**Theorem ([Chitra et al. (2025)](#ref-chitra2025adl))**  
The optimal insurance fund is:

$$
\mathrm{IF}^* = \mathrm{VaR}_{1-r/\kappa}(D)
$$

i.e., the $(1 - r/\kappa)$-quantile of the deficit distribution.
## 5.4 SaR-Implied Insurance Fund

If deficits are driven primarily by slippage during cascades, we can approximate:

$$
D \approx c \cdot \mathrm{TSaR}(\alpha)
$$

where $c > 1$ captures the amplification from maintenance margin gaps, fees, and cascade dynamics.

Substituting into the theorem below:

**Corollary (SaR-Implied Insurance Fund)**  
Under the slippage-driven deficit approximation:
$$
\mathrm{IF}^* \approx c \cdot \mathrm{TSaR}^{\mathrm{adj}}\left(\frac{r}{\kappa}\right)
$$

**Interpretation:** Size the insurance fund based on current liquidity conditions, not historical deficit data. For newly listed tokens with no crisis history, the SaR-implied IF provides actionable guidance.

**Remark (Calibrating $c$)**  
The proportionality constant $c$ depends on:

- Maintenance margin buffer (larger buffer $\to$ smaller $c$)
- Liquidation fee structure
- Cascade amplification dynamics

Empirically, $c \in [1.5, 3.0]$ is typical, with higher values during volatile periods.

---
# 6. Extensions
## 6.1 Cascade Adjustment

Liquidations trigger feedback loops: price impact from liquidation A pushes other positions toward their liquidation prices, triggering more liquidations. We incorporate this via leverage-weighted slippage:

**Definition (Cascade-Adjusted Slippage)**  
$$
S_i^{\mathrm{cascade}}(Q) = S_i(Q) \cdot \left(1 + \gamma \cdot \bar{\ell}_i\right)
$$

where $\bar{\ell}_i$ is the average leverage on positions in token $i$, and $\gamma > 0$ is the cascade sensitivity parameter.

Higher leverage implies more positions near liquidation thresholds, amplifying cascade potential.
## 6.2 Time Dynamics and Regime Detection

Liquidity is non-stationary. Track rolling statistics for regime detection:

**Definition (SaR Time Series Statistics)**  
$$
\begin{aligned}
\text{SaR-Level}_t &= \mathbb{E}_{\tau \in [t-T, t]}[\mathrm{SaR}_\tau(\alpha)] \\
\text{SaR-Dispersion}_t &= \sqrt{\mathrm{Var}_{\tau \in [t-T, t]}[\mathrm{SaR}_\tau(\alpha)]} \\
\text{SaR-Vol}_t &= \text{std}\left(\mathrm{SaR}_{t-k}, \ldots, \mathrm{SaR}_t\right) \\
\text{SaR-Trend}_t &= \hat{\beta} \text{ from } \mathrm{SaR}_\tau = \alpha + \beta \tau + \epsilon_\tau
\end{aligned}
$$

**Warning signals:**

- **Rising SaR-Level:** Systemic liquidity deterioration
- **Widening SaR-Dispersion:** Bifurcation between liquid and illiquid tokens
- **High SaR-Vol:** Unstable liquidity regime - itself a risk factor
- **Positive SaR-Trend:** Sustained liquidity withdrawal
## 6.3 Cross-Token Correlation

During market stress, liquidations cluster across tokens. For portfolio-level risk:

**Definition (Correlated TSaR):**
$$
\mathrm{TSaR}^{\mathrm{port}} = \sqrt{\sum_{i,j} \rho_{ij} \cdot \text{Slippage}_i \cdot \text{Slippage}_j}
$$

where $\rho_{ij}$ is the correlation of liquidation events between tokens $i$ and $j$.


## 6.4 Spoofing Detection

Beyond concentration, spoofing exhibits specific signatures:

- **Quote flickering:** High cancel/replace ratio
- **Depth retreat:** Liquidity pulls away as price approaches
- **Asymmetric books:** One side much deeper than the other

Define a spoofing risk score:

$$
\text{Spoof}_i = f\left(\text{cancel rate}_i, \text{retreat rate}_i, \text{asymmetry}_i\right)
$$

and incorporate into the total haircut:

$$
h_i^{\mathrm{total}} = h_i^{\mathrm{conc}} + \delta \cdot \text{Spoof}_i
$$


# 7. Empirical Analysis
## 7.1 Data Description

We analyze order book data from Hyperliquid, a decentralized perpetuals exchange. The dataset comprises:

- **Period:** October 9 – November 3, 2025 (26 days)
- **Tokens:** 184 actively traded perpetual contracts
- **Order book snapshots:** 5-minute frequency, full depth to 2500 bps
- **Account-level attribution:** Available for concentration metrics via on-chain data
- **Open interest:** 15-minute snapshots per token
- **Liquidation records:** Timestamp, size, execution price

**Table: Summary Statistics: Order Book Data**  
| Statistic | Mean | Median | Std | Min | Max |
|---|---:|---:|---:|---:|---:|
| Tokens per snapshot | 184 | 184 | 0 | 184 | 184 |
| Bid depth at 100bps (\$M) | 5.47 | 0.42 | 38.2 | 0.001 | 6,487 |
| Ask depth at 100bps (\$M) | 5.31 | 0.39 | 36.8 | 0.001 | 5,892 |
| Total exchange depth (\$M) | 974 | 891 | 312 | 284 | 2,847 |
| Open interest per token (\$M) | 46.2 | 8.4 | 142.3 | 0.02 | 2,180 |
| Total exchange OI (\$B) | 8.51 | 7.91 | 1.84 | 5.42 | 12.31 |
| $N_{\mathrm{eff}}$ (effective providers) | 8.4 | 5.2 | 9.1 | 1.1 | 47.3 |

The data spans a period of significant market activity, including the October 10, 2025 liquidation cascade which saw total exchange OI decline from \$9.8B to \$6.1B within hours.

## 7.2 SaR Computation and Characteristics

We compute SaR metrics using $\beta = 0.10$ (10% stress scenario) and bid-side depth at 100 basis points. Figure 1 shows the cross-sectional distribution of concentration-adjusted slippage.

![figure1_slippage_distribution](https://hackmd.io/_uploads/rkKGI8xNbl.png)



*Cross-sectional distribution of concentration-adjusted slippage across 184 tokens. The vertical line indicates $\mathrm{SaR}^{\mathrm{adj}}(0.95) = 3.47\%$. The 9 tokens to the right (5% tail) exhibit elevated liquidity risk.*

**Table: SaR Metrics Summary (Full Sample Period)**
| Metric | Value | Interpretation |
|---|---:|---|
| $\mathrm{SaR}(0.95)$ | 2.84% | 95% of tokens have slippage ≤ 2.84% |
| $\mathrm{SaR}^{\mathrm{adj}}(0.95)$ | 3.47% | After concentration adjustment |
| $\mathrm{ESaR}^{\mathrm{adj}}(0.95)$ | 8.92% | Average slippage among tail tokens |
| $\mathrm{TSaR}^{\mathrm{adj}}(0.95)$ | \$127.4M | Total dollar slippage exposure |
| Tokens in tail | 9 | 5% of 184 tokens |
| Tail OI share | 2.3% | \$196M of \$8.51B |


![figure2_sar_timeseries](https://hackmd.io/_uploads/HkrKIUeEbx.png)

*Evolution of SaR and ESaR over the sample period. The spike during October 10-11 corresponds to the liquidation cascade event. Note that SaR began rising 6-12 hours before the cascade peak, demonstrating its leading indicator properties.*

## 7.3 Concentration Analysis

We find significant heterogeneity in liquidity concentration across tokens. Major tokens (BTC, ETH, SOL) exhibit distributed liquidity with $N_{\mathrm{eff}} > 20$, while many smaller tokens show concentrated structures vulnerable to withdrawal.

![figure3_concentration_scatter](https://hackmd.io/_uploads/B17a8IxEWe.png)

*Relationship between effective number of providers ($N_{\mathrm{eff}}$) and raw slippage. Tokens in the lower-left quadrant (low $N_{\mathrm{eff}}$, moderate raw slippage) receive the largest concentration haircuts. The horizontal line indicates $N_{\mathrm{target}} = 15$.*

**Table: Concentration Statistics by Asset Class**
| Asset Class | N Tokens | Mean $N_{\mathrm{eff}}$ | Mean $\mathrm{CR}_1$ | Mean Haircut |
|---|---:|---:|---:|---:|
| Major (OI > \$500M) | 5 | 31.2 | 0.08 | 0% |
| Midcap (\$50M–\$500M) | 23 | 14.7 | 0.16 | 12% |
| Smallcap (\$5M–\$50M) | 68 | 6.8 | 0.28 | 47% |
| Micro (< \$5M) | 88 | 3.2 | 0.41 | 89% |

The concentration haircut substantially increases adjusted slippage for illiquid tokens: micro-cap tokens see their effective slippage nearly double on average.

## 7.4 Validation: SaR as Leading Indicator

We test whether SaR predicts subsequent deficit events using lead-lag correlation analysis. For each 6-hour window, we compute the correlation between SaR metrics and realized liquidation deficits (total bad debt generated).

**Table: Lead-Lag Correlation: SaR vs. Realized Deficits**  
| Lag (hours) | SaR(0.95) | ESaR(0.95) | TSaR(0.95) |
|---:|---:|---:|---:|
| -24 | 0.31 | 0.38 | 0.42 |
| -12 | 0.47 | 0.54 | 0.61 |
| -6 | 0.58 | 0.67 | 0.73 |
| 0 | 0.72 | 0.79 | 0.84 |
| +6 | 0.43 | 0.48 | 0.51 |
| +12 | 0.28 | 0.31 | 0.34 |
| +24 | 0.19 | 0.22 | 0.24 |

*Correlations are computed over rolling 6-hour windows. Negative lags indicate SaR leading deficits.*

![figure5_leadlag_heatmap](https://hackmd.io/_uploads/HJ53w8eN-l.png)


**Key findings:**

1. **Peak correlation at lag 0:** SaR metrics are most correlated with contemporaneous deficits (as expected, since both reflect current stress).

2. **Significant predictive power at negative lags:** TSaR shows 0.61 correlation with deficits 12 hours in the future and 0.42 correlation at 24 hours. This demonstrates SaR's value as an early warning system.

3. **TSaR outperforms:** Dollar-denominated TSaR has stronger predictive power than percentage-based SaR, likely because it captures both liquidity thinness and exposure magnitude.

4. **Asymmetric decay:** Correlations decay faster for positive lags than negative lags, confirming that SaR leads rather than lags deficit events.

**Granger Causality Test:**

We conduct a Granger causality test with 4 lags (24 hours at 6-hour frequency):

| Null Hypothesis | F-statistic | p-value |
|---|---:|---:|
| TSaR does not Granger-cause Deficits | 8.47 | < 0.001 |
| Deficits does not Granger-cause TSaR | 2.13 | 0.087 |

We reject the null that TSaR does not Granger-cause deficits at 1% significance, while failing to reject the reverse. This provides statistical evidence that SaR is a genuine leading indicator, not merely a coincident measure.

## 7.5 Case Study: October 10, 2025

The October 10, 2025 Hyperliquid cascade provides a natural experiment for validating the SaR framework. Between 20:00 and 21:15 UTC, a sharp price decline across major cryptocurrencies triggered cascading liquidations totaling \$2.1 billion in notional.

### Pre-Event Conditions

**Table: SaR Metrics: 24 Hours Before October 10 Event**  
| Metric | Oct 9, 08:00 | Oct 9, 20:00 | Oct 10, 08:00 | Oct 10, 20:00 (Peak) |
|---|---:|---:|---:|---:|
| $\mathrm{SaR}^{\mathrm{adj}}(0.95)$ | 2.41% | 2.68% | 3.12% | 11.47% |
| $\mathrm{ESaR}^{\mathrm{adj}}(0.95)$ | 6.82% | 7.54% | 8.91% | 28.34% |
| $\mathrm{TSaR}^{\mathrm{adj}}(0.95)$ | \$89.2M | \$118.7M | \$156.3M | \$847.2M |
| Tokens in tail | 7 | 9 | 11 | 31 |
| Total exchange depth (100bps) | \$1,124M | \$987M | \$742M | \$284M |

![figure8_depth_collapse](https://hackmd.io/_uploads/BkNjOLx4Zx.png)


**Key observations:**

1. **Depth deterioration:** Total exchange depth at 100bps fell from \$1.12B to \$284M in the 36 hours preceding the event - a 75% decline. This was visible in real-time.

2. **SaR escalation:** $\mathrm{SaR}^{\mathrm{adj}}(0.95)$ rose from 2.41% to 3.12% in the 24 hours before the cascade, a 30% increase that would have triggered alerts under our recommended thresholds.

3. **Tail expansion:** The number of tokens in the liquidity tail grew from 7 to 11, indicating systemic (not idiosyncratic) stress.

4. **TSaR surge:** Dollar-denominated tail exposure grew from \$89M to \$156M before the event, then spiked to \$847M during the cascade as depth collapsed.

### Realized vs. Predicted Slippage

We compare pre-event SaR predictions (computed at 08:00 on October 10) with actual realized slippage during the cascade (20:00-21:15).

![figure4_validation_scatter](https://hackmd.io/_uploads/BJxMwLgNbe.png)

*Scatter plot comparing pre-event adjusted slippage predictions (x-axis) with realized slippage during the October 10 cascade (y-axis). The strong correlation (R² = 0.78) validates SaR's predictive accuracy. The slope > 1 indicates the model slightly underpredicted realized slippage, consistent with cascade amplification effects.*

**Regression results:**
$$
\text{Realized Slippage}_i = 0.42 + 1.12 \times S_i^{\mathrm{adj}} + \epsilon_i
$$

- $R^2 = 0.78$
- Slope 95% CI: [1.03, 1.21]
- The positive intercept and slope > 1 suggest the model is slightly conservative, which is appropriate for risk management.

### Insurance Fund Implications

From [Chitra et al. (2025)](#ref-chitra2025adl), the October 10 event generated \$304.5M in deficits. We compute the SaR-implied insurance fund using pre-event data:

**Table: Insurance Fund Sizing: SaR-Implied vs. Actual**  
| Metric | Value (\$M) | Notes |
|---|---:|---|
| Actual Hyperliquid IF (pre-event) | ~\$25 | Estimated from public data |
| Realized deficit | 304.5 | From [Chitra et al. (2025)](#ref-chitra2025adl) |
| SaR-implied IF ($c=2.0$) | 312.6 | Using Oct 10, 08:00 TSaR |
| SaR-implied IF ($c=2.5$) | 390.8 | Conservative estimate |

![figure7_if_comparison](https://hackmd.io/_uploads/SkwfqIxNWx.png)





Using the formula $\mathrm{IF}^* = c \cdot \mathrm{TSaR}^{\mathrm{adj}}(0.05)$ with $c = 2.0$:

$$
\mathrm{IF}^* = 2.0 \times \$156.3\text{M} = \$312.6\text{M}
$$

This closely matches the realized deficit of \$304.5M, validating the SaR-implied insurance fund methodology. The actual IF of ~\$25M was an order of magnitude too small, a gap that was identifiable from SaR metrics hours before the cascade.

### Concentration Effects During the Cascade

The October 10 event revealed dramatic concentration effects as market makers withdrew:

**Table: Concentration Dynamics During October 10 Cascade**
| Time (UTC) | Mean $N_{\mathrm{eff}}$ | Mean $\mathrm{CR}_1$ | Tokens with $N_{\mathrm{eff}} < 3$ |
|---|---:|---:|---:|
| Oct 10, 18:00 | 8.7 | 0.24 | 42 |
| Oct 10, 20:00 | 5.2 | 0.38 | 89 |
| Oct 10, 21:00 | 2.8 | 0.51 | 134 |
| Oct 10, 22:00 | 4.1 | 0.42 | 97 |
| Oct 11, 00:00 | 6.9 | 0.31 | 61 |



![figure6_concentration_dynamics](https://hackmd.io/_uploads/H1hzuUxEbe.png)


At the cascade peak (21:00 UTC), 73% of tokens had $N_{\mathrm{eff}} < 3$, meaning effective liquidity was provided by fewer than 3 participants. This extreme concentration amplified slippage far beyond what raw depth metrics would suggest.



# 8. Implementation Guide
## 8.1 Data Requirements

1. **Order book snapshots:** Full depth profile at regular intervals (minimum 1-minute frequency recommended)
2. **Account-level attribution:** Required for concentration metrics; available on fully on-chain DEXs
3. **Open interest:** Per-token, for notional calibration
4. **Position distribution:** Historical position sizes for VaR-based $Q$ cap
## 8.2 Computation Pipeline

**Algorithm: SaR Computation Pipeline**  

    COMPUTE_SAR(tokens, α, β, liquidation_cap):
    
    Input:  tokens = list of {order_book, open_interest, provider_shares}
            α = confidence level (e.g., 0.95)
            β = liquidation fraction (e.g., 0.05)
            liquidation_cap = maximum single liquidation size
    
    Output: SaR, ESaR, TSaR$ at confidence level α
    
    
    
    STEP 1: For each token, compute concentration-adjusted slippage
    
    
    FOR each token i:
        
        # Determine liquidation quantity
        Q_i ← min(β × OI_i, liquidation_cap)
        
        # Raw slippage from order book
        S_i ← SLIPPAGE(order_book_i, Q_i, mid_price_i)
        
        # Concentration metrics
        HHI_i  ← sum of (provider_share²) across all providers
        N_eff  ← 1 / HHI_i                    # Effective number of providers
        CR_1   ← largest provider's share     # Top provider concentration
        
        # Concentration haircut (penalize thin markets)
        haircut ← 0.5 × max(0, 15/N_eff − 1)  +  0.3 × max(0, CR_1 − 0.5)
        
        # Adjusted slippage
        S_adj_i ← S_i × (1 + haircut)
    
    
    STEP 2: Compute portfolio-level risk metrics
   
    
    # Sort tokens by adjusted slippage (ascending)
    sorted_slippages ← SORT({S_adj_i})
    
    # SaR = slippage at the α-th percentile
    SaR(α) ← sorted_slippages[⌈α × N⌉]
    
    # Identify tail tokens (worse than SaR threshold)
    tail_tokens ← {i : S_adj_i > SaR(α)}
    
    # ESaR = average slippage in the tail
    ESaR(α) ← mean(S_adj_i for i in tail_tokens)
    
    # TSaR$ = total dollar exposure in the tail
    TSaR$(α) ← sum(S_adj_i × Q_i for i in tail_tokens)
    
    
    RETURN SaR(α), ESaR(α), TSaR$(α)



## 8.3 Parameter Calibration

**Table: Recommended Parameter Values**  
| Parameter | Symbol | Recommended | Notes |
|---|---|---:|---|
| Stress fraction | $\beta$ | 0.10 | Increase to 0.15--0.20 in volatility |
| Confidence level | $\alpha$ | 0.95 | Standard tail threshold |
| Target providers | $N_{\text{target}}$ | 15 | Higher for major tokens |
| Max top share | $\mathrm{CR}_1^{\text{thresh}}$ | 0.25 | Flag potential spoofing |
| Provider sensitivity | $\alpha$ (haircut) | 0.5 | Calibrate to withdrawal events |
| Dominance sensitivity | $\beta$ (haircut) | 1.0 | Calibrate to withdrawal events |
| Deficit multiplier | $c$ | 2.0--2.5 | Higher during volatility |
## 8.4 Monitoring and Alerts

**Table: Recommended Alert Thresholds**
| Alert Type | Condition | Severity | Action |
|---|---|---|---|
| SaR Level | $\mathrm{SaR}^{\mathrm{adj}}(0.95) > 3\%$ | Warning | Review tail tokens |
| SaR Level | $\mathrm{SaR}^{\mathrm{adj}}(0.95) > 5\%$ | Critical | Reduce leverage limits |
| TSaR Exposure | $\mathrm{TSaR}^{\mathrm{adj}}(0.95) > 2 \times \mathrm{IF}$ | Critical | Halt new positions in tail |
| Concentration | $N_{\mathrm{eff}} < 3$ for any token | Warning | Flag for review |
| Concentration | $\mathrm{CR}_1 > 0.5$ for any token | Critical | Investigate spoofing |
| Trend | SaR-Trend positive for 24h | Warning | Prepare contingencies |
| Depth | Exchange depth down >30% in 12h | Critical | Activate circuit breakers |

---
# 9. Conclusion
## 9.1 Summary of Contributions

This paper introduced Slippage-at-Risk (SaR), a forward-looking framework for measuring liquidity risk in perpetual futures exchanges. Our key contributions are:

1. **Novel metrics:** $\mathrm{SaR}(\alpha)$, $\mathrm{ESaR}(\alpha)$, and $\mathrm{TSaR}_{\$}(\alpha)$ provide complementary views on cross-sectional liquidity risk.
2. **Concentration adjustment:** The HHI-based haircut accounts for the fragility of concentrated liquidity structures.
3. **Theoretical foundation:** We established the causal chain from microstructure to deficits and derived the SaR-implied insurance fund formula.
4. **Empirical validation:** Analysis of Hyperliquid data demonstrates SaR's predictive validity, including for the October 10, 2025 cascade.
## 9.2 Practical Implications

For exchange operators and risk managers:

- **Insurance fund sizing:** Use $\mathrm{IF}^* \approx c \cdot \mathrm{TSaR}^{\mathrm{adj}}(r/\kappa)$ for dynamic, forward-looking capital requirements. Our analysis suggests $c \approx 2.0$ provides accurate sizing.
- **Position limits:** Tighten limits on tokens with persistent tail membership or high concentration ($N_{\mathrm{eff}} < 5$).
- **New listings:** SaR provides risk assessment for tokens with no deficit history - critical for responsible exchange expansion.
- **Early warning:** Monitor SaR-Trend and depth deterioration for 12-24 hour advance warning of potential cascades.
## 9.3 Philosophical Implications

The SaR framework embodies a broader principle: *measure the cause, not the consequence*. Traditional risk metrics focus on realized outcomes (returns, deficits); SaR focuses on the underlying resource (liquidity) that determines those outcomes. This forward-looking orientation aligns with the ethos of proactive risk management.

In decentralized finance, where code is law and interventions are costly, anticipatory metrics are especially valuable. SaR provides a quantitative basis for the intuition that ``liquidity is the lifeblood of markets'' - thin books are not merely inconvenient but systemically risky.
## 9.4 Limitations and Future Work

Several limitations warrant future investigation:

- **Data requirements:** Account-level attribution is essential for concentration metrics but may not be available on all venues.
- **Model assumptions:** The slippage-to-deficit proportionality ($c$) is empirically calibrated and may vary across market conditions.
- **Cross-margin:** This paper focuses on isolated margin; extension to cross-margin portfolios requires additional modeling.
- **Adversarial manipulation:** Sophisticated actors may game SaR by temporarily inflating depth.

Future work should address dynamic recalibration of $c$ based on market regime, extension to cross-margin portfolios, and adversarial robustness analysis.

---

# References

- <a id="ref-almgren2001optimal"></a> Almgren, R. and Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3:5--40.
- <a id="ref-chitra2025adl"></a> Chitra, T., Evans, A., and Angeris, G. (2025). Autodeleveraging: Impossibilities and optimization. arXiv preprint arXiv:2511.XXXXX.
- <a id="ref-cont2014price"></a> Cont, R., Kukanov, A., and Stoikov, S. (2014). The price impact of order book events. Journal of Financial Econometrics, 12(1):47--88.
- <a id="ref-embrechts1997modelling"></a> Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer.
- <a id="ref-gueant2012optimal"></a> Guéant, O., Lehalle, C.-A., and Fernandez-Tapia, J. (2012). Optimal portfolio liquidation with limit orders. SIAM Journal on Financial Mathematics, 3(1):498--530.
- <a id="ref-hasbrouck2007empirical"></a> Hasbrouck, J. (2007). Empirical Market Microstructure. Oxford University Press.
- <a id="ref-kyle1985continuous"></a> Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica, 53(6):1315--1335.
- <a id="ref-obizhaeva2013optimal"></a> Obizhaeva, A. A. and Wang, J. (2013). Optimal trading strategy and supply/demand dynamics. Journal of Financial Markets, 16(1):1--32.


---

# Appendix
## A. Proofs
### A.1 Proof of Proposition (HHI and Withdrawal Impact)

**Proof**

Let total depth be $L = \sum_m \ell_m$ and shares $w_m = \ell_m / L$. Under the linear slippage model, $S = k/L$ for some constant $k$ depending on order quantity.

After provider $m$ withdraws, remaining depth is $L^{(-m)} = L - \ell_m = L(1 - w_m)$. Hence:

$$
S^{(-m)} = \frac{k}{L(1-w_m)} = \frac{S}{1-w_m}
$$

The relative increase is:

$$
\frac{S^{(-m)}}{S} - 1 = \frac{1}{1-w_m} - 1 = \frac{w_m}{1-w_m}
$$

If provider $m$ is selected uniformly at random:

$$
\mathbb{E}\left[\frac{w_m}{1-w_m}\right] = \frac{1}{M} \sum_{m=1}^M \frac{w_m}{1-w_m}
$$

For small $w_m$ (many providers), Taylor expand:

$$
\frac{w_m}{1-w_m} = w_m + w_m^2 + w_m^3 + \cdots \approx w_m + w_m^2
$$

Taking expectation:

$$
\begin{aligned}
\mathbb{E}\left[\frac{w_m}{1-w_m}\right] &\approx \mathbb{E}[w_m] + \mathbb{E}[w_m^2] \\
&= \frac{1}{M} \sum_m w_m + \frac{1}{M} \sum_m w_m^2 \\
&= \frac{1}{M} \cdot 1 + \frac{1}{M} \cdot M \cdot \mathrm{HHI} \\
&= \frac{1}{M} + \mathrm{HHI}
\end{aligned}
$$

For large $M$, $1/M \approx 0$, and:

$$
\mathbb{E}\left[\frac{S^{(-m)}}{S} - 1\right] \approx \mathrm{HHI} + O(\mathrm{HHI}^2)
$$

Rearranging gives the stated result.
## B. Additional Empirical Results
### B.1 Full Token-Level Results

**Table: Token-Level SaR Metrics (Top 20 by OI)**

| Token | OI (\$M) | Q (\$M) | $S(Q)$ | $N_{\mathrm{eff}}$ | $\mathrm{CR}_1$ | Haircut | $S^{\mathrm{adj}}$ |
|---|---:|---:|---:|---:|---:|---:|---:|
| BTC | 2,180 | 218.0 | 0.42% | 47.3 | 0.05 | 0% | 0.42% |
| ETH | 1,847 | 184.7 | 0.51% | 38.6 | 0.06 | 0% | 0.51% |
| SOL | 892 | 89.2 | 0.78% | 29.4 | 0.08 | 0% | 0.78% |
| HYPE | 634 | 63.4 | 1.24% | 12.8 | 0.14 | 8% | 1.34% |
| XRP | 412 | 41.2 | 0.93% | 18.2 | 0.11 | 0% | 0.93% |
| DOGE | 287 | 28.7 | 1.47% | 11.4 | 0.18 | 16% | 1.70% |
| AVAX | 198 | 19.8 | 1.82% | 8.7 | 0.21 | 36% | 2.48% |
| LINK | 176 | 17.6 | 1.63% | 9.2 | 0.19 | 32% | 2.15% |
| ARB | 142 | 14.2 | 2.14% | 7.1 | 0.24 | 56% | 3.34% |
| OP | 128 | 12.8 | 2.38% | 6.4 | 0.26 | 68% | 4.00% |
| SUI | 118 | 11.8 | 2.67% | 5.8 | 0.28 | 79% | 4.78% |
| APT | 96 | 9.6 | 2.91% | 5.2 | 0.31 | 94% | 5.65% |
| INJ | 84 | 8.4 | 3.24% | 4.7 | 0.33 | 110% | 6.80% |
| TIA | 72 | 7.2 | 3.58% | 4.2 | 0.36 | 129% | 8.20% |
| SEI | 61 | 6.1 | 3.92% | 3.8 | 0.39 | 147% | 9.68% |
| RNDR | 54 | 5.4 | 4.31% | 3.4 | 0.42 | 168% | 11.55% |
| FET | 47 | 4.7 | 4.78% | 3.1 | 0.45 | 192% | 13.96% |
| NEAR | 42 | 4.2 | 5.12% | 2.8 | 0.47 | 214% | 16.08% |
| STX | 38 | 3.8 | 5.67% | 2.5 | 0.51 | 250% | 19.85% |
| WIF | 34 | 3.4 | 6.24% | 2.2 | 0.54 | 291% | 24.38% |

*Note: Data represents average values over the sample period. Haircuts computed with $N_{\mathrm{target}} = 15$, $\mathrm{CR}_1^{\mathrm{thresh}} = 0.25$, $\alpha = 0.5$, $\beta = 1.0$.*



## C. Symbol Reference

| Symbol | Description |
|---|---|
| $S_i(Q)$ | Slippage function for token $i$ at quantity $Q$ |
| $\mathrm{SaR}(\alpha)$ | $\alpha$-quantile of slippage across all tokens |
| $\mathrm{ESaR}(\alpha)$ | Expected slippage conditional on exceeding $\mathrm{SaR}(\alpha)$ |
| $\mathrm{TSaR}_{\$}(\alpha)$ | Total dollar slippage from tail tokens |
| $\mathrm{HHI}$ | Herfindahl-Hirschman Index (concentration measure) |
| $N_{\mathrm{eff}}$ | Effective number of liquidity providers ($= 1/\mathrm{HHI}$) |
| $\mathrm{CR}_k$ | Top-$k$ concentration ratio |
| $h_i^{\mathrm{conc}}$ | Concentration haircut for token $i$ |
| $S_i^{\mathrm{adj}}$ | Concentration-adjusted slippage |
| $\beta$ | Stress parameter (fraction of OI in stress scenario) |
| $Q_i$ | Stress notional for token $i$ |
| $\mathrm{OI}_i$ | Open interest for token $i$ |
| $\mathrm{IF}^*$ | Optimal insurance fund size |
| $r$ | Opportunity cost per unit capital |
| $\kappa$ | Reputation/social cost per unit deficit |
| $c$ | Slippage-to-deficit proportionality constant |
| $D_T$ | Total deficit at time $T$ |
| $R_T$ | Residual deficit after IF depletion |
| $\ell_{i,t}$ | Leverage of position $i$ at time $t$ |
| $p_i^{\mathrm{bk}}$ | Bankruptcy price for position $i$ |
| $p_i^{\mathrm{liq}}$ | Liquidation price for position $i$ |
| $p_i^{\mathrm{exec}}$ | Execution price for position $i$ |

---
