Hobson, D; Tse, ASL; Zhu, Y; (2019) Optimal consumption and investment under transaction costs*. Mathematical Finance , 29 (2) pp. 483-506. 10.1111/mafi.12187.

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Abstract

In this paper, we consider the Merton problem in a market with a single risky asset and proportional transaction costs. We give a complete solution of the problem up to the solution of a first-crossing problem for a first-order differential equation. We find that the characteristics of the solution (e.g., well-posedness) can be related to some simple properties of a univariate quadratic whose coefficients are functions of the parameters of the problem. Our solution to the problem via the value function includes expressions for the boundaries of the no-transaction wedge. Using these expressions, we prove a precise condition for when leverage occurs. One new and unexpected result is that when the solution to the Merton problem (without transaction costs) involves a leveraged position, and when transaction costs are large, the location of the boundary at which sales of the risky asset occur is independent of the transaction cost on purchases.

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