Inverse Optimal Stopping and Optimal Closure of Illiquid Positions

Abstract

Many economic situations are modeled as stopping problems. Examples include job search, pricing of American options, timing of market entry and irreversible investment decisions. The first part of the thesis analyzes optimal stopping in a dynamic mechanism design framework. It deals with a principal-agent problem where the principal and the agent have different preferences over stopping rules. The agent privately observes a one-dimensional Markov process that influences her payoff. Based on her observation the agent decides when to stop. In order to induce the agent to employ a different stopping rule the principal commits to a transfer that depends only on the time the agent stopped. The goal is to characterize the set of stopping rules that can be implemented using such a transfer. <br /> To this end the well-known single crossing condition from static mechanism design is transferred to optimal stopping problems. In a discrete-time framework it is shown that under this condition a stopping rule is implementable if and only if it is of cut-off type. If time is continuous, a cut-off rule is implementable provided that the associated threshold satisfies certain regularity assumptions. The transfer admits a closed form representation based on the reflected version of the underlying Markov process. A uniqueness result for the transfer is provided. As a consequence one obtains a new nonlinear integral equation characterizing the optimal stopping boundary in one-dimensional stopping problems. <br /> The second part of this thesis analyzes the problem of how to close a large asset position in an illiquid market. The first goal is to characterize trading strategies that make very high liquidation costs unlikely. To this end a model that allows for a price-sensitive closure of the position is set-up. It provides a simple device for designing and controlling the distribution of the revenues from unwinding the position. The risk inherent in the open position is modeled by a functional that can be interpreted as the time-average of the squared value-at-risk of the open position. Market illiquidity is reflected by a linear, temporary price impact. The stochastic control problem consists of minimizing a weighted sum of the execution costs and the risk functional.<br /> By appealing to dynamic programming, semi-explicit formulas for the optimal execution strategies are derived in a discrete-time framework. Within the continuous-time version of the model the optimal trading rates can be characterized in terms of a partial differential equation (PDE) describing by how much they differ from the optimal risk-neutral trading rate. The PDE possesses a singularity and does not, in general, have a closed-form solution. A uniqueness result for solutions in the viscosity sense is provided, allowing in the following to identify the value function and optimal trading rates. It is shown that optimal strategies from the discrete model converge to the continuous-time optimal trading rates.<br /> In the next step this model is generalized by incorporating a stochastic price impact. The liquidation constraint is relaxed by introducing a set of scenarios where the position does not have to be closed. A purely probabilistic solution of this not necessarily Markovian control problem is provided by means of a backward stochastic differential equation (BSDE). The BSDE in this problem possesses a singular terminal condition. It is shown that a minimal supersolution of the BSDE exists. Special cases for which the control problem has explicit solutions are discussed.<br /> Finally, the impact of a cross-hedging opportunity on liquidation strategies is analyzed. Suppose there is an open position to be closed in an illiquid forward market (e.g. a commodity market) before delivery. The liquidity of the asset increases as the delivery date approaches. Therefore, an early closure eliminates the risk inherent in the open position but also omits the opportunity of reducing execution costs. Assume further that there is a proxy market where forwards of a correlated asset are traded. Liquidity in the proxy market is high and thus performing a cross-hedge reduces execution costs. However, since the prices are not perfectly correlated, this hedging strategy entails basis risk. Using techniques from singular stochastic control theory allows to obtain an optimal trade-off between execution costs and basis risk. Explicit optimal hedging strategies for simple liquidity dynamics are derived.