Asymptotic analysis for Merton's problem with transaction costs in power utility case

Choi, Jin Hyuk

doi:10.1080/17442508.2013.879142

Cited by 8 publications

(18 citation statements)

References 10 publications

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“…() to calculate the next order term. Based on Theorem , we give a short and simple argument that yields the results of Janeček and Shreve () and Choi (), and extend to a boundary case in which the width of the no‐transaction wedge is exceptionally of square root order.…”

Section: Introductionmentioning

confidence: 74%

“…() to show that (when

m_{M} > 0

and

q_{m} \notin false{0, 1 false}

)

p_{*} = q_{M} - {normalΔ}_{1}^{1 / 3} ξ^{1 / 3} - {normalΔ}_{2} {normalΔ}_{1}^{2 / 3} ξ^{2 / 3} + O false(ξ false); p^{*} = q_{M} + {normalΔ}_{1}^{1 / 3} ξ^{1 / 3} - {normalΔ}_{2} {normalΔ}_{1}^{2 / 3} ξ^{2 / 3} + O false(ξ false),

where

{normalΔ}_{2} = \frac{2 m_{M}}{3 q_{M} {(1 - q_{M})}^{2} δ^{2} R} = \frac{σ^{2} R false(2 σ^{2} R β - μ^{2} (1 - R) false)}{3 μ {(σ^{2} R - μ)}^{2}} .

In principle, Choi's results can be extended to give an expansion to any order. (In fact, Janeček & Shreve, , assume

λ = γ

, and Choi, , assumes

γ = 0

but it is straightforward to translate their results to a more general setting. )…”

Section: Discussion and Implications Of The Main Resultsmentioning

confidence: 99%

“…() and this paper make great strides toward a complete solution of the Merton problem for general levels of the transaction cost parameters. We can also connect to the literature on the Merton problem in the small transaction cost regime (Janeček & Shreve, ; Choi, ). Janeček and Shreve () show that, typically, for small transaction costs, the no‐transaction wedge is centered around the Merton line and has width of the order of the one‐third power of the size of the transaction cost.…”

Section: Introductionmentioning

confidence: 83%

“…Janeček and Shreve () show that, typically, for small transaction costs, the no‐transaction wedge is centered around the Merton line and has width of the order of the one‐third power of the size of the transaction cost. Choi () applies results from Choi et al. () to calculate the next order term.…”

Section: Introductionmentioning

confidence: 99%

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Optimal consumption and investment under transaction costs*

Hobson

Tse

Zhu

2018

Mathematical Finance

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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. K E Y W O R D SMerton problem, proportional transaction costs, no transaction region, well-posedness, leverage Mathematical Finance. 2019;29:483-506.wileyonlinelibrary.com/journal/mafi

show abstract

Section: Introductionmentioning

confidence: 74%

“…() to show that (when

m_{M} > 0

and

q_{m} \notin false{0, 1 false}

)

p_{*} = q_{M} - {normalΔ}_{1}^{1 / 3} ξ^{1 / 3} - {normalΔ}_{2} {normalΔ}_{1}^{2 / 3} ξ^{2 / 3} + O false(ξ false); p^{*} = q_{M} + {normalΔ}_{1}^{1 / 3} ξ^{1 / 3} - {normalΔ}_{2} {normalΔ}_{1}^{2 / 3} ξ^{2 / 3} + O false(ξ false),

where

{normalΔ}_{2} = \frac{2 m_{M}}{3 q_{M} {(1 - q_{M})}^{2} δ^{2} R} = \frac{σ^{2} R false(2 σ^{2} R β - μ^{2} (1 - R) false)}{3 μ {(σ^{2} R - μ)}^{2}} .

In principle, Choi's results can be extended to give an expansion to any order. (In fact, Janeček & Shreve, , assume

λ = γ

, and Choi, , assumes

γ = 0

but it is straightforward to translate their results to a more general setting. )…”

Section: Discussion and Implications Of The Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal consumption and investment under transaction costs*

Hobson

Tse

Zhu

2018

Mathematical Finance

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show abstract

“…These aspects compete and determine the size of the effect of the transaction costs on consumption when the liquid risky asset becomes tradable in the market. 6 The inequalities (5.14) and (5.15) are followed by the following observation.…”

Section: 3mentioning

confidence: 94%

Optimal consumption and investment with liquid and illiquid assets

Choi

2019

Mathematical Finance

Self Cite

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I consider an optimal consumption/investment problem to maximize expected utility from consumption. In this market model, the investor is allowed to choose a portfolio that consists of one bond, one liquid risky asset (no transaction costs), and one illiquid risky asset (proportional transaction costs). I fully characterize the optimal consumption and trading strategies in terms of the solution of the free boundary ordinary differential equation (ODE) with an integral constraint. I find an explicit characterization of model parameters for the well‐posedness of the problem, and show that the problem is well posed if and only if there exists a shadow price process. Finally, I describe how the investor's optimal strategy is affected by the additional opportunity of trading the liquid risky asset, compared to the simpler model with one bond and one illiquid risky asset.

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Optimal Investment in an Illiquid Market with Search Frictions and Transaction Costs

Choi

Gang

2023

Appl Math Optim

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Asymptotic analysis for Merton's problem with transaction costs in power utility case

Cited by 8 publications

References 10 publications

Optimal consumption and investment under transaction costs*

Optimal consumption and investment under transaction costs*

Optimal consumption and investment with liquid and illiquid assets

Optimal Investment in an Illiquid Market with Search Frictions and Transaction Costs

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