Optimal portfolio selection under vanishing fixed transaction costs

Abstract

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

Moment-constrained optimal dividends: precommitment and consistent planning

“…(18)) that H • ; c, x c , xc satisfies the second item above; moreover, H • ; c, x c , xc satisfies the first item above (with x c = y and xc = ȳ) within Case (i) (of Proposition 3.1). Hence, (21) holds in Case (i). It can be similarly shown that (21) holds in Case (ii).…”

“…Remark 3.5. The properties (21) and (22) are established in [41, p. 675] in a setting similar to that of the present paper. The property (23) is established in [33,Remark 2] for a model based on a Wiener process with drift.…”

“…To the best of our knowledge, such an analysis has not been carried out for the exact optimal dividend problem (17). However, slightly different problems without absorption are studied in [38] and the references therein, and it is investigated what happens when the fixed cost c is sent to zero; see also [3,21,39]. One main finding is that while the value of the problem converges to the one without fixed costs as c 0 (as in Proposition 3.2), the derivative with respect to c converges to −∞.…”

Moment constrained optimal dividends: precommitment \& consistent planning

“…To the best of our knowledge, such an analysis has not been carried out for exactly the optimal dividend problem (3.2). However, slightly different problems without absorption are studied in [34], and the references therein, and it is investigated what happens when the fixed cost c is sent to zero, see also [2,18,35]. One main finding is that while the value of the problem converges to the one without fixed costs as c ց 0 (as in Proposition 3.2) the derivative with respect to c converges to −∞.…”

On an approximation of average cost per unit time impulse control of Markov processes

“…This kind of cost functional is also important in mathematics of finance under the name of Kelly criterion -see [16]. In particular impulse control appear in the case of growth optimal portfolio under proportional transaction costs in [10] and [7] and references there in. Average cost per unit time impulse control in a general (locally compact separable) state space was considered in [21].…”