Utility Maximization Trading Two Futures with Transaction Costs

Bichuch, Maxim; Shreve, Steven E.

doi:10.1137/110853649

Cited by 44 publications

(59 citation statements)

References 18 publications

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“…Theorem 3 describes the comparative statics in terms of the auxiliary parameters. 4 In general, it is difficult to make categorical statements about the comparative statics with respect to the original market parameters since many of the market parameters enter the definitions of more than one of the auxiliary parameters. However, we have the following results concerning the dependence of p * and p * on the discount rate, and on the drift of the illiquid asset.…”

Section: Comparative Staticsmentioning

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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Section: Comparative Staticsmentioning

confidence: 99%

“…be the value function for a CRRA investor: 4 Since the free boundary value problem does not depend on b 4 , q * and q * are trivially independent of b 4 . We have strong numerical evidence that q * is decreasing in b 2 and q * is increasing in b 2 , but we have not been able to prove this result.…”

Section: Comparative Staticsmentioning

confidence: 99%

A multi-asset investment and consumption problem with transaction costs

2019

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“…The goal then is to obtain explicit asymptotic formulas for optimal trading policies and the associated welfare effect of small transaction costs. Results of this kind were first obtained in simple concrete models that can be solved explicitly in the frictionless case [31,81,85,52,62,12,10,42]. In the last couple of years, there has been a lot of progress in extending these sensitivity analyses to much more general settings [11,14,82,74,2,71,65,55,54,78,17,18,1,38].…”

Section: Introductionmentioning

confidence: 99%

A Primer on Portfolio Choice with Small Transaction Costs

Muhle‐Karbe

Reppen

Soner

2017

Annu. Rev. Financ. Econ.

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This survey is an introduction to asymptotic methods for portfolio-choice problems with small transaction costs. We outline how to derive the corresponding dynamic programming equations and simplify them in the small-cost limit. This allows to obtain explicit solutions in a wide range of settings, which we illustrate for a model with mean-reverting expected returns and proportional transaction costs. For even more complex models, we present a policy iteration scheme that allows to compute the solution numerically.

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“…On models with multiple assets; see also Bichuch and Shreve (), Muthuraman and Kumar (), Dai and Yi (), Dai and Zhong (), and Possamaï, Soner, and Touzi ().…”

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confidence: 99%

“… See Magill and Constantinides (1976),Constantinides (1986),Davis and Norman (1990),Dumas and Luciano (1991),Gerhold et al (2014),and Choi, Sirbu and Zitkovic (2013).3 See Liu(2004),Akian, Menaldi, and Sulem (1996), and Guasoni and Muhle-Karbe (2013a).4 On models with multiple assets; see alsoBichuch and Shreve (2013),Muthuraman and Kumar (2006),Dai and Yi (2009),Dai and Zhong (2010), and Possamaï, Soner, and Touzi(2015).…”

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confidence: 99%

Investing With Liquid and Illiquid Assets

Bichuch

Guasoni

2016

Mathematical Finance

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We find optimal trading policies for long‐term investors with constant relative risk aversion and constant investment opportunities, which include one safe asset, liquid risky assets, and an illiquid risky asset trading with proportional costs. Access to liquid assets creates a diversification motive, which reduces illiquid trading, and a hedging motive, which both reduces illiquid trading and increases liquid trading. A further tempering effect depresses the liquid asset's weight when the illiquid asset's weight is close to ideal, to keep it near that level by reducing its volatility. Multiple liquid assets lead to portfolio separation in four funds: the safe asset, the myopic portfolio, the illiquid asset, and its hedging portfolio.

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Utility Maximization Trading Two Futures with Transaction Costs

Cited by 44 publications

References 18 publications

A multi-asset investment and consumption problem with transaction costs

A multi-asset investment and consumption problem with transaction costs

A Primer on Portfolio Choice with Small Transaction Costs

Investing With Liquid and Illiquid Assets

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