Optimal consumption and investment under transaction costs*

Hobson, David; Tse, Alex S. L.; Zhu, Yeqi

doi:10.1111/mafi.12187

Cited by 7 publications

(23 citation statements)

References 17 publications

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“…Our first achievement is to show that the problem of finding the free boundaries and the value function can be reduced to the study of a boundary crossing problem for a family of solutions to a class of first order ordinary differential equations parametrised by the initial values. This allows us to characterise precisely the parameter combinations for which the problem is well-posed (Theorem 1), and in those cases to give an expression for the value function (Theorem 2).These results extend Choi et al [7] and Hobson et al [15] to the case of multiple risky assets.As mentioned above, Choi [6] studies a similar problem. The main difference between this paper and Choi [6] is that we analyse the HJB equation, whereas Choi takes the dual approach and studies shadow prices.…”

supporting

confidence: 59%

“…Essentially then, the investor can ignore the presence of the liquid risky asset, reducing the dimensionality of the problem. This problem is the subject of [15]. If b 2 = 1 then the solution n we define in the next paragraph may pass through singular points.…”

Section: The Problemmentioning

confidence: 99%

“…If b 2 = 1 then the solution n we define in the next paragraph may pass through singular points. See Choi et al [7] or Hobson et al [15] for a discussion of some of the issues.…”

Section: The Problemmentioning

confidence: 99%

“…We adopt the same transformation as [15] to reduce the order of the HJB equation. Recall the relationship between V and G in (2) and the definition p = yθ x+yθ .…”

Section: The Problemmentioning

confidence: 99%

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A multi-asset investment and consumption problem with transaction costs

2019

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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,...

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supporting

confidence: 59%

Section: The Problemmentioning

confidence: 99%

“…If b 2 = 1 then the solution n we define in the next paragraph may pass through singular points. See Choi et al [7] or Hobson et al [15] for a discussion of some of the issues.…”

Section: The Problemmentioning

confidence: 99%

“…We adopt the same transformation as [15] to reduce the order of the HJB equation. Recall the relationship between V and G in (2) and the definition p = yθ x+yθ .…”

Section: The Problemmentioning

confidence: 99%

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A multi-asset investment and consumption problem with transaction costs

2019

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“…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.…”

Section: Resultssupporting

confidence: 56%

Speculative Trading, Prospect Theory and Transaction Costs

Tse

Zheng

2019

SSRN Journal

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A speculative agent with Prospect Theory preference chooses the optimal time to purchase and then to sell an indivisible risky asset as to maximize the expected utility of the round-trip profit net of transaction costs. The optimization problem is formulated as a sequential optimal stopping problem and we provide a complete characterization of the solution. Depending on the preference and market parameters as well as the initial price of the asset, the optimal strategy can be "buy and hold", "buy low sell high", "buy high sell higher" or "no trading". Transaction costs do not necessarily curb speculative trading. For example, while a large proportional transaction cost on sale can unambiguously suppress trading participation, introducing a fixed market entry fee will indeed encourage trading when the asset price level is high.

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Speculative trading, prospect theory and transaction costs

Tse

Zheng

2022

Finance Stoch

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A speculative agent with prospect theory preference chooses the optimal time to purchase and then to sell an indivisible risky asset to maximise the expected utility of the round-trip profit net of transaction costs. The optimisation problem is formulated as a sequential optimal stopping problem, and we provide a complete characterisation of the solution. Depending on the preference and market parameters, the optimal strategy can be “buy and hold”, “buy low, sell high”, “buy high, sell higher” or “no trading”. Behavioural preference and market friction interact in a subtle way which yields surprising implications on the agent’s trading patterns. For example, increasing the market entry fee does not necessarily curb speculative trading, but instead may induce a higher reference point under which the agent becomes more risk-seeking and in turn is more likely to trade.

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Optimal consumption and investment under transaction costs*

Cited by 7 publications

References 17 publications

A multi-asset investment and consumption problem with transaction costs

A multi-asset investment and consumption problem with transaction costs

Speculative Trading, Prospect Theory and Transaction Costs

Speculative trading, prospect theory and transaction costs

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