Multi-Dimensional Portfolio Optimization with Proportional Transaction Costs

Multi-Dimensional Portfolio Optimization with Proportional Transaction Costs
Author: Kumar Muthuraman
Publisher:
Total Pages: 32
Release: 2004
Genre:
ISBN:
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We provide a computational study of the problem of optimally allocating wealth among multiple stocks and a bank account, in order to maximize the infinite horizon discounted utility of consumption. We consider the situation where the transfer of wealth from one asset to another involves transaction costs that are proportional to the amount of wealth transferred. Our model allows for correlation between the price processes, which in turn gives rise to interesting hedging strategies. This results in a stochastic control problem with both drift-rate and singular controls, that can be recast as a free boundary problem in partial differential equations. Adapting the finite element method and using an iterative procedure that converts the free-boundary problem into a sequence of fixed boundary problems, we provide an efficient numerical method for solving this problem. We present computational results that describe the impact of volatility, risk aversion of the investor, level of transaction costs and correlation among the risky assets on the structure of the optimal policy. Finally we suggest and quantify some heuristic approximations.

Worst-Case Portfolio Optimization with Proportional Transaction Costs

Worst-Case Portfolio Optimization with Proportional Transaction Costs
Author: Christoph Belak
Publisher:
Total Pages:
Release: 2016
Genre:
ISBN:
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We study optimal asset allocation in a crash-threatened financial market with proportional transaction costs. The market is assumed to be in either a normal state, in which the risky asset follows a geometric Brownian motion, or in a crash state, in which the price of the risky asset can suddenly drop by a certain relative amount. We only assume the maximum number and the maximum relative size of the crashes to be given and do not make any assumptions about their distributions. For every investment strategy, we identify the worst-case scenario in the sense that the expected utility of terminal wealth is minimized. The objective is then to determine the investment strategy which yields the highest expected utility in its worst-case scenario.We solve the problem for utility functions with constant relative risk aversion using a stochastic control approach. We characterize the value function as the unique viscosity solution of a second-order nonlinear partial differential equation. The optimal strategies are characterized by time-dependent free boundaries which we compute numerically. The numerical examples suggest that it is not optimal to invest any wealth in the risky asset close to the investment horizon, while a long position in the risky asset is optimal if the remaining investment period is sufficiently large.

Optimal Investment with Transaction Costs and Stochastic Volatility Part I

Optimal Investment with Transaction Costs and Stochastic Volatility Part I
Author: Maxim Bichuch
Publisher:
Total Pages: 29
Release: 2015
Genre:
ISBN:
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Two major financial market complexities are transaction costs and uncertain volatility, and we analyze their joint impact on the problem of portfolio optimization. When volatility is constant, the transaction costs optimal investment problem has a long history, especially in the use of asymptotic approximations when the cost is small. Under stochastic volatility, but with no transaction costs, the Merton problem under general utility functions can also be analyzed with asymptotic methods. Here, we look at the long-run growth rate problem when both complexities are present, using separation of time scales approximations. This leads to perturbation analysis of an eigenvalue problem. We find the first term in the asymptotic expansion in the time scale parameter, of the optimal long-term growth rate, and of the optimal strategy, for fixed small transaction costs.The Companion piece for this paper are available at the following URL: "http://ssrn.com/abstract=2659918" http://ssrn.com/abstract=2659918.

Stochastic Optimization Models in Finance

Stochastic Optimization Models in Finance
Author: W. T. Ziemba
Publisher: World Scientific
Total Pages: 756
Release: 2006
Genre: Business & Economics
ISBN: 9812773657
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A reprint of one of the classic volumes on portfolio theory and investment, this book has been used by the leading professors at universities such as Stanford, Berkeley, and Carnegie-Mellon. It contains five parts, each with a review of the literature and about 150 pages of computational and review exercises and further in-depth, challenging problems. Frequently referenced and highly usable, the material remains as fresh and relevant for a portfolio theory course as ever. Sample Chapter(s). Chapter 1: Expected Utility Theory (373 KB). Contents: Mathematical Tools: Expected Utility Theory; Convexity and the Kuhn-Tucker Conditions; Dynamic Programming; Qualitative Economic Results: Stochastic Dominance; Measures of Risk Aversion; Separation Theorems; Static Portfolio Selection Models: Mean-Variance and Safety First Approaches and Their Extensions; Existence and Diversification of Optimal Portfolio Policies: Effects of Taxes on Risk Taking; Dynamic Models Reducible to Static Models: Models That Have a Single Decision Point; Risk Aversion over Time Implies Static Risk Aversion; Myopic Portfolio Policies; Dynamic Models: Two-Period Consumption Models and Portfolio Revision; Models of Optimal Capital Accumulation and Portfolio Selection; Models of Option Strategy; The Capital Growth Criterion and Continuous-Time Models. Readership: Postdoctoral and graduate students, researchers, academics, and professionals interested in portfolio theory and stochastic optimization.