A renewal theoretic result in portfolio theory under transaction costs with multiple risky assets

Irle, Albrecht; Prelle, Claas

doi:10.1524/stnd.2009.1049

Cited by 2 publications

(4 citation statements)

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“…If we define ξ n ≡ 0 on {τ n = ∞} in (13), we can see that an admissible proportional strategy can by deduced from an admissible monetary strategy. In fact, there is a one-to-one correspondence between these two kinds of strategies, see Lemma 2.3 and Theorem 2.4 in [15].…”

Section: Description Of the Model And Preliminary Resultsmentioning

confidence: 98%

“…In this subsection we introduce the portfolio model with fixed and proportional transaction costs from [16], [17], and [15]. We consider a financial market model with one bond B and one stock S satisfying…”

Section: Description Of the Model And Preliminary Resultsmentioning

confidence: 99%

“…The article is organized in the following manner. For the sake of completeness, we introduce in Section 2 the model with fixed and proportional costs from [17] and combine the results derived in [15] related to the representation of the maximization problem via risky fraction processes and the subsequent application of coordinate transformation to the whole space R.…”

Section: Introductionmentioning

confidence: 99%

“…A rough guess would see optimal strategies as given by two surfaces in d-dimensional space: As soon as the outer surface is reached by the risky fraction process, trading back occurs to some point at the inner surface. The performance of such strategies was investigated in [15] but the possible optimality of such strategies was not considered here.…”

Section: Introductionmentioning

confidence: 99%

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Optimal portfolio selection under vanishing fixed transaction costs

2017

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In this paper, asymptotic results in a long-term growth rate portfolio optimization model under both fixed and proportional transaction costs are obtained.More precisely, the convergence of the model when the fixed costs tend to zero is investigated. A suitable limit model with purely proportional costs is introduced and the convergence of optimal boundaries, asymptotic growth rates, and optimal risky fraction processes is rigorously proved. The results are based on an in-depth analysis of the convergence of the solutions to the corresponding HJB-equations.(ii) P lim n→∞ τ n = ∞ = 1 and F τn -measurable R d -valued random variables η n , describing the trading volume at τ n . The optimal strategy in the one stock case is to wait until the risky fraction process reaches the boundary of some interval [A, B] ⊂ ]0, 1[ containing the Merton fraction and then trade back to some fixed fraction in ]A, B[ near the Merton fraction and restart the process. This optimal behavior was described for the Kelly criterion in [27]. Bielecki and Pliska then generalized these results in several ways by characterizing the optimal strategies in terms of solutions to quasi-variational inequalities in [7], while existence and uniqueness results for solutions to these HJB-equations in quasi-variational form were established by Nagai in [28] by applying a coordinate transformation to avoid degeneracy.Despite of the feasibility of the optimal trading strategies under fixed transaction costs, the cost structure seems rather unrealistic from the practitioner's point of view.To overcome this problem a combination of fixed and proportional transaction costs was suggested. In some cases, as in [10], [20], and [4], the fixed component of the transaction costs is a constant amount not depending on the wealth. Here, the authors derive solutions for the discounted consumption criterion for the linear utility, asymptotically for the exponential utility, and existence of optimal strategies, resp. Asymptotic results for vanishing fixed costs were recently obtained in [3], [12], [25], where in particular the last paper considers a generalization of our setting. These results are, however, different in nature to the results obtained in this article as the authors do not show convergence of the optimal strategies, but construct asymptotically optimal strategies, which are obviously suboptimal for all fixed positive costs. We also want to mention [14] where a market with price-impact proportional to a power of the order flow is considered and asymptotically explicit formulas are obtained. The precise results and techniques, however, are quite different to ours.Our attention in the present work will be focused on the cases where the fixed component of the transaction costs is a fixed proportion of the investor's wealth, as described above, under the maximization of the asymptotic growth rate. The trading strategies are therefore of impulsive form and the costs paid by the investor at time τ n

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Section: Description Of the Model And Preliminary Resultsmentioning

confidence: 98%