Optimal Liquidation in a Defaultable Market

TL;DR

This work studies optimal liquidation of a large position in a defaultable market with price impact. It formulates a stochastic control problem where default is modeled via an intensity-based mechanism tied to a credit index and derives a Hamilton–Jacobi–Bellman equation to characterize the value function. Under a bilinear terminal cost $g(x,y)=Kxy$, the authors obtain an explicit solution with clearly delineated continuation and selling regions, governed by thresholds and exponents $n_0$ and $n_1$ that solve $\tfrac{1}{2}\sigma^2 l(l-1) + \mu l - \delta = 0$ and $\tfrac{1}{2}\sigma^2 l(l-1) + \mu l - (\lambda + \delta) = 0$, and boundaries $F_0$ and $G_{\lambda}(y)$. The analysis reveals how default risk alters the optimal policy, generally making liquidation more aggressive, and the numerical results illustrate sensitivity to $\lambda$ and the limit as $\lambda\to 0$, recovering the default-free benchmark.

Abstract

In this paper we address the problem of optimal liquidation of a large portfolio composed by securities exposed to default risk. The default time is described in terms of a Brownian motion representing the evolution of the value of the firm, whose assets are available in the market for investors. Considering that selling a large number of assets has a significant impact in the price, and hence in the portfolio's value, the control problem involved to describe the optimal strategy to liquidate a large position is analyzed. Under suitable assumptions in the model, an explicit solution is given to the value function and a precise description of the optimal strategy is obtained.

Figures (5)

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Theorems & Definitions (8)

• Remark 2.1
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• proof
• Remark 3.4
• Lemma A.1
• Lemma A.2