An explicit solution for an optimal stopping/optimal control problem which models an asset sale

TL;DR

This paper derives an explicit solution for an optimal stopping and control problem modeling an asset sale by a risk-averse agent in an incomplete market, revealing complex strategies including potential jumps.

Contribution

It provides a novel explicit solution to a combined stopping and control problem in incomplete markets, with insights into the structure of optimal strategies.

Findings

Explicit solution form for the problem

Optimal strategies may involve jumps

Risk-averse agents may gamble optimally

Abstract

In this article we study an optimal stopping/optimal control problem which models the decision facing a risk-averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corresponds to a feasible wealth process. We formulate this problem as one involving the choice of a stopping time and a martingale. We conjecture the form of the solution and verify that the candidate solution is equal to the value function. The interesting features of the solution are that it is available in a very explicit form, that for some parameter values the optimal strategy is more sophisticated than might originally be expected, and that although the setup is based on continuous diffusions, the optimal martingale may involve a jump process. One interpretation of the solution is that it is optimal for the risk-averse agent to gamble.