Robust Optimal Strategies for Early Liquidation in Financial Systems

Abstract

We study the problem of asset liquidation in financial systems. During financial crises, asset liquidation is often inevitable but can lead to substantial losses if a significant amount of illiquid assets are sold simultaneously at depressed prices -- a phenomenon known as price impact. To tackle this challenge, we consider a two-period liquidation model that allows for early liquidation prior to clearing, thereby mitigating price impact at clearing, and we develop a worst-case approach to solve the decision-making problem on the optimal size of early liquidation. Specifically, we propose a robust optimal strategy -- a tractable liquidation approach that maximizes the worst-case value of liquid assets at clearing, taking into account the uncertainty of other banks' early liquidation decisions. We derive a (semi-)closed-form representation of this strategy in a practical scenario involving permanent price impact and analyze its sensitivity to that impact's magnitude. We further identify its closed-form expression in another practical scenario featuring interbank exposures. Our findings, although built upon a stylized model, offer valuable guidelines for developing robust liquidation strategies that mitigate losses resulting from asset liquidation.

Figures (3)

• Figure 1: A numerical comparison between the inner optimal values of \ref{['prob:bilevel-perm']} and \ref{['prob:maximin-perm']}. We use a two-bank system with $Q(x) = 1 - 0.2x$, $L_1 = L_2 = 5$, $z_1 = 1$, $z_2 = 0.5$, and $e_2 = 4.55$, under $\gamma = 0$, $0.5$, and $1$. Also, we set $e_1 = 4.9$ and $e_1 = 4.85$ for the left and right panels, respectively.
• Figure 2: A bank's robust optimal strategy and its worst-case scenario on the aggregate early liquidation size of other banks under various levels of permanent price impact when $\gamma^*$ is large. We use a four-bank system with $i=1$, $L_1=\cdots=L_4 = 6$, $z_1=\cdots=z_4 = 1$, $w_1 = 0.3$, $w_2 = w_3 = 0.5$, and $w_4 = 0.8$. We set $Q(x) = \exp(-0.2x)$ and $s(x)=\mathop{\arg\min}_{s \in [0, \bar{s}_i]} g_i^\gamma(x, s)$, where $g_i^\gamma$ is defined as in \ref{['eq:ggamma']}.
• Figure 3: A bank's robust optimal strategy and its worst-case scenario on the aggregate early liquidation size of other banks under various levels of permanent price impact when $\gamma^*$ is small. We use a four-bank system with $i=1$, $L_1=\cdots=L_4 = 6$, $z_1 = 4$, $z_2=z_3=z_4=0.6$, $w_1 = 0.3$, $w_2 = w_3 = 0.5$, and $w_4 = 0.8$. We set $Q(x) = \exp(-0.2x)$ and $s(x)=\mathop{\arg\min}_{s \in [0, \bar{s}_i]} g_i^\gamma(x, s)$, where $g_i^\gamma$ is defined as in \ref{['eq:ggamma']}.

Theorems & Definitions (11)

• Remark 1
• Lemma 1
• Proposition 1: Validity of \ref{['prob:maximin-perm']}
• Theorem 1: Robust Optimal Strategy with Permanent Price Impact
• Theorem 2: Impact of $\gamma$: A Typical Scenario
• Proposition 2: Impact of $x^\gamma_i$: An Extreme Scenario
• Lemma 2
• Proposition 3: Validity of \ref{['maximin-network']}
• Theorem 3: Robust Optimal Strategy with Interbank Exposures
• Lemma 3