Liquidation Equilibrium in Finance

• Liquidation Equilibrium is a quantitative finance framework that defines market states arising from forced asset sales, highlighting endogenous price impacts and strategic agent behavior.
• It employs methods such as stochastic control, Nash equilibrium analysis, and fixed-point theorems to derive optimal liquidation schedules and market-clearing prices.
• This concept underpins systemic risk and market microstructure theory, offering actionable insights for managing execution risk and regulatory constraints.

A liquidation equilibrium is a foundational concept in quantitative finance describing the price, portfolio, and agent behavior arising when one or more large asset positions are compulsorily unwound (“liquidated”) into a market of risk-averse, strategic participants. Unlike classic Arrow–Debreu or Radner equilibria, whose focus is on voluntary, welfare-maximizing portfolios, liquidation equilibrium explicitly models the market-clearing prices, liquidation schedules, and system dynamics induced by forced block sales, fire sales, or regulatory constraints, with endogenous price impacts. This paradigm underpins modern systemic risk, market microstructure, and execution theory. It unites single-agent stochastic optimal control, Nash equilibrium in games of strategic liquidation, and multi-layer networks of assets/liabilities under frictions, and is characterized by equilibrium schedules and prices at which no agent (including liquidators) can improve outcomes unilaterally, given endogenous demand and constraints.

1. Single-Agent Liquidation and Stochastic Control

In the benchmark single-agent setting, liquidation equilibrium is induced by the feedback between the liquidation rate of a large trader and the resulting market price impact, often under stochastic liquidity conditions. For example, in a market with stochastic liquidity captured by a controlled Ornstein-Uhlenbeck volume-impact process $Y_t$, a large investor holds $\Theta_t$ shares and controls a cumulative sales process $A_t$ with dynamics $\Theta_t = \Theta_0 - A_t$ and no short selling ($A_t$ nondecreasing, càdlàg). The uncontrolled (discounted) fundamental price $\bar S$ follows

$$d\bar S_t = \mu \bar S_t\,dt + \sigma \bar S_t\,dW_t,$$

while the market price is subject to transient and multiplicative impact $S_t=f(Y_t)\bar S_t$, with $f$ increasing and $f(0)=1$. The impact process is

$$dY_t = -\beta Y_t\,dt + dB_t - dA_t, \qquad Y_{0-}=y,$$

where $B_t$ can be correlated with $W_t$ ($\rho$ correlation).

The optimal liquidation problem (a two-dimensional singular stochastic control) seeks to maximize the expected discounted liquidation proceeds over infinite time, leading to a variational inequality (HJB) for the value function $V(y, \theta)$: $$\max\{ f(y) - V_y - V_\theta, LV \} = 0, \quad V(y,0) = 0,$$ with $L$ the infinitesimal generator and a “free boundary” curve $y(\theta)$ determining the indifference threshold between “sell” and “wait” regions. The equilibrium is characterized by reflection of the impact process at this boundary, and the solution reduces to a nonlinear ODE for $y(\theta)$ involving a smooth-fit condition and explicit dependence on market resilience ($\beta$), discount/drift ($\delta$), volatility ($\sigma$), and impact function $f$ (Becherer et al., 2016).

At each moment, given current impact $Y_t$ and holdings $\Theta_t$, the agent trades precisely when $Y_t > y(\Theta_t)$. The free boundary $y(\theta)$ encodes the liquidation equilibrium—balancing immediate price impact versus future stochastic resilience—via a trade-off determined entirely endogenously.

2. Multi-Agent Strategic Liquidation and Nash Equilibrium

In multi-agent settings, liquidation equilibrium generalizes to Nash equilibria of linear-quadratic differential games in which $n$ risk-averse participants simultaneously liquidate—affecting both market price (permanent and/or temporary impact) and each other’s proceeds. For instance, in the Almgren–Chriss framework: $$S(t) = S^0(t) + \gamma \sum_{j}(X_j(t) - x_j) + \lambda \sum_j u_j(t),$$ where $X_i(t)$ is agent $i$’s inventory, and $u_i(t) = dX_i/dt$ is the trading rate.

Participants maximize quadratic utilities penalizing both volatility risk and remaining inventory, subject to full liquidation constraints $X_i(0)=x_i$, $X_i(T)=0$. The Nash equilibrium is characterized by a coupled boundary-value problem for $(X_i)$, where, unlike the single-agent case, each agent’s optimal liquidation speed depends on the anticipated aggregate trading rates and inventories of all agents. Existence and uniqueness are guaranteed under quadratic data, and closed-form solutions are available for special cases (e.g., constant parameters, symmetric agents) (Schied et al., 2013).

Qualitative features include phenomena such as predation versus liquidity provision: sufficiently large risk aversion and low permanent impact admit “liquidity provider” Nash equilibria, while low risk aversion or high cross-impact yields “predatory” strategies. This captures endogenous liquidity and strategic feedback central to real-world microstructure.

3. Liquidation Equilibrium in Financial Network and Systemic Risk Models

In systemic risk, networked frameworks model liquidation equilibrium where agents (banks, funds) hold portfolios of illiquid assets and are subject to default/prudential constraints (e.g., leverage limits). Each institution’s sales impact prices endogenously, feeding back through balance sheets and network liabilities. Let $n$ firms have illiquid positions $s_i$, cash $x_i$, liabilities $\bar p_i$, and mark-to-market price $q$. The equilibrium is defined by:

• Clearing payments: $p^* = \bar p \wedge (x + S q^* + A^T p^*)$ (limited liability and mutual exposures).
• Market-clearing prices: $$q^* = F(\textstyle{\sum_i} [s_i \wedge \gamma_i^*])$$, where $F$ is a nonincreasing inverse-demand curve.
• Nash equilibrium liquidation: Each $\gamma_i^*$ solves a constrained game maximizing mark-to-market value given all others' strategies within liquidity/solvency constraints.

Existence (via Kakutani) is generic under continuity and quasi-concavity; uniqueness of the aggregate liquidation (though not its allocation across agents) holds under diagonal strict concavity. The key economic message is that only the system-wide aggregate matters for prices, but individual strategies can be (non-uniquely) distributed (Feinstein, 2015, Feinstein et al., 2015).

The fictitious-default algorithm (a dynamic updating of default sets and clearing payments/prices) computes extremal fixed points and systemically relevant variables.

4. Endogenous Price Impact, Demand Curves, and Microstructure

Liquidation equilibrium provides endogenous price impact functions (“inverse demand”) instead of exogenous curves. In the single-period model, forced sale of external position $Z$ among $n$ risk-averse agents with utility $u_i$ leads to a market-clearing price $v$ solving

$v = \frac{E[Z\,e^{-R(X+Z-v)}]}{E[e^{-R(X+Z-v)}]},$

where $R$ is the “harmonic” aggregate risk aversion. This extends the classical Bühlmann equilibrium to endogenize fire-sale discount, block-trade price impacts, and the full demand curve $s \mapsto v(sq)$. Concavity and monotonicity of $v(s)$ encode liquidity; increased agent risk aversion lowers prices, while agent diversity increases endogenous liquidity (Bichuch et al., 2020).

With exponential utilities, the clearing price $v$ reduces to an Esscher transform, yielding classical linear (or exponential, for Poisson) price-impact formulas for block trades. For CRRA utilities, analogous concave demand curves arise but in nonlinear form.

This approach generalizes to settings with network feedback (systemic risk) and microstructure-aware models (limit order book, VWAP pricing) (Bichuch et al., 2020).

5. Existence, Uniqueness, and Computation

Liquidation equilibrium (in both agent-based and networked forms) typically admits at least one equilibrium under mild conditions (continuity, coercivity, monotonicity). Techniques include:

• Dynamic programming and reflection arguments for stochastic control problems with boundaries (Becherer et al., 2016).
• Fixed-point theorems (Brouwer, Tarski, Kakutani) to show set-valued best-response and payment/price maps admit solutions (Feinstein, 2015, Feinstein et al., 2015, Bichuch et al., 2020).
• Fictitious-default iterative algorithms for multi-layered networks.
• Riccati ODE/PDE analysis and backward stochastic differential equations (BSDEs) for mean-field and stochastic-market settings (Cheng et al., 2024, Fu et al., 2020).

Uniqueness is often aggregate—i.e., the total volume liquidated and systemic prices are uniquely determined, but agent-level allocations may be nonunique absent strict concavity.

6. Liquidation Equilibrium in Portfolio Optimization, Auctions, and Central Clearing

In the context of default management and central clearing, liquidation equilibrium is realized as the unique Radner equilibrium for portfolio reallocation and price formation following exogenous default events. Participants optimize convex risk measures of final P&L (entropic or expected shortfall risk), subject to clearing and balance constraints. The equilibrium is analytically characterized for elliptical (e.g., Gaussian) markets, and allows computation of the “funds transfer price” for competing strategies (pure liquidation, hedging, auction) (Bastide et al., 2023).

The instantaneous price impact of forced liquidation is computed explicitly, as is the decomposition of market risk transfer and counterparty risk transfer. This informs efficient policy decisions (e.g., hedging before auction), rationalizes pre-auction strategies, and measures the welfare/price effect of systemic liquidation.

7. Dynamic Mean-Field, Constrained and Long-Time Regimes

Recent literature extends liquidation equilibrium to mean-field games (major-minor players, self-exciting order flow, no-roundtrip/absorption constraints, trading directionality). In these, a continuum of agents interacts via both execution and price impact, with collective effects (front-running of large traders, stable periodicity) (Chen et al., 2024, Fu et al., 2020, Fu et al., 2023, Fu et al., 2024). Nash equilibria are constructed as the solution to coupled non-linear integral equations (often with endogenous terminal condition).

In infinite-horizon problems, equilibrium can be non-liquidating when exogenous order flow persists—optimal strategies balance ongoing hedging of risk rather than full liquidation, with transition regimes dictated by the underlying BSDE system (Cheng et al., 2024).

Summary Table: Core Forms and Solution Methods

Setting	Main Equilibrium Characterization	Reference
Single-Agent, Stochastic Control	HJB variational inequality, free boundary, reflection	(Becherer et al., 2016)
Multi-Agent, LQ Differential Game	Nash equilibrium, coupled ODE/PDE/BVP	(Schied et al., 2013)
Financial Network/Systemic Risk	Fixed-point (Kakutani) with market-clearing and Nash constraints	(Feinstein, 2015, Feinstein et al., 2015)
One-Period, Aggregator Utility	Endogenous price/quantity curve via risk-sharing fixed point	(Bichuch et al., 2020)
Major/Minor Mean-Field Game	FBSDE system, O(N^{-1/2}) convergence to MFG	(Chen et al., 2024)
Central Clearing Post-Default	Radner equilibrium (convex risk measures, market clearing)	(Bastide et al., 2023)

These frameworks synthesize modern theory and computation of liquidation equilibrium in market microstructure, systemic risk, and optimal execution—where market price, agent strategies, and systemic stability are jointly determined via endogenous strategic interaction, risk, and illiquidity.