A multi-asset investment and consumption problem with transaction costs

Abstract

In this article we study a multi-asset version of the Merton investment and consumption problem with proportional transaction costs. In general it is difficult to make analytical progress towards a solution in such problems, but we specialise to a case where transaction costs are zero except for sales and purchases of a single asset which we call the illiquid asset.Assuming agents have CRRA utilities and asset prices follow exponential Brownian motions we show that the underlying HJB equation can be transformed into a boundary value problem for a first order differential equation. The optimal strategy is to trade the illiquid asset only when the fraction of the total portfolio value invested in this asset falls outside a fixed interval. Important properties of the multi-asset problem (including when the problem is well-posed, ill-posed, or well-posed only for large transaction costs) can be inferred from the behaviours of a quadratic function of a single variable and another algebraic function. case) and Chen and Dai [27] identify the shape of the no-transaction region in the two-asset case.Explicit solutions of the general problem remain very rare.One situation when an explicit solution is possible is the rather special case of uncorrelated risky assets, and an agent with constant absolute risk aversion. In that case the problem decouples into a family of optimisation problems, one for each risky asset, see Liu [19]. Another setting for which some progress has been made is the problem with small transaction costs, see Whalley and Wilmott [26], Janecek and Shreve [17], Bichuch and Shreve [4], Soner and Touzi [24], and, for a recent analysis in the multi-asset case, Possamaï et al [22]. These papers use an expansion method to provide asymptotic formulae for the optimal strategy, value function and no-transaction region.Our focus is on optimal investment/consumption problems, but there is a parallel literature on optimal investment problems involving maximising expected utility at a distant terminal horizon, see, for example, Dumas and Luciano [12] for an explicit solution in the one-asset case and Bichuch and Guasoni [3] for recent work in a setting similar to ours with liquid and illiquid assets.In this paper we consider the problem with a risk-free bond and two risky assets. Transactions in the first risky asset are costless, but transactions in the second risky asset, which we term the illiquid asset, incur proportional costs. This is also the setting of a recent paper by Choi [6]. More generally, we may have several risky assets on which no transaction costs are payable. By a mutual fund theorem, this general case can be reduced to the case with a single liquid, risky asset. This paper is an extension of Hobson et al [15] which considers a similar problem with a bond and an illiquid asset but with no other risky assets 1 . Many of the techniques of [15] carry over to the wider setting of this paper. (Similarly, the paper of Choi [6] extends the work of Choi et al [7] to include a risky liquid asset.) However,...

Speculative Trading, Prospect Theory and Transaction Costs

“…the agent may not delay the purchase decision. Similar observations regarding potential non-monotonicity of trading decisions with respect to (proportional) transaction costs are made by Hobson et al (2019a) and Hobson et al (2019b) in the context of portfolio optimization. Similarly, we can also examine the impact of the fixed market entry cost on the purchase decision.…”

Dividend policy and capital structure of a defaultable firm

“…Although it is expected that the extra dimension introduced will bring significant challenges to the analysis of the underlying HJB equation, it is perhaps not an impossible task in view of the recent progress by Hobson et al. (2019b), who completely solve a multiasset Merton problem with transaction costs (albeit the special case where transaction cost is only payable for one of the assets).…”

“…See the discussion in Section 2. Broadly speaking, our work contributes to the growing literature on solving a singular stochastic control problem via reduction to a first‐order crossing problem (e.g., Hobson & Zhu, 2014, 2016; Hobson et al., 2019a; Hobson, Tse, & Zhu, 2019b). It showcases that the mathematical techniques are amendable outside the context of portfolio selection under transaction costs and how simple comparison principles can lead to powerful comparative statics that is not commonly explored in other related literature.…”

Optimal consumption and investment with liquid and illiquid assets

“…Under the assumption of small transaction costs, they solve the problem using the shadow price approach, and prove an asymptotic expansion result. In parallel with our work, Hobson et al [14] recently consider a similar problem as in this paper and solve it by studying the HJB equation of the primal optimization problem. They also provide an explicit characterization of the well-posedness of the problem.…”

“…Theorem 4.7 provides an explicit characterization of the well-posedness of the problem, in the sense that the constant c * is given by a closed form in terms of the model parameters. [14] also provides a well-posedness criterion by analyzing the HJB equation of the primal optimization problem.…”