# Continuous-time mean variance portfolio with transaction costs: a proximal approach involving time penalization

M. García-Galicia, A. A. Carsteanu, J. B. Clempner

Research output: Contribution to journalArticleResearchpeer-review

### Abstract

© 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group. This paper proposes a new continuous-time optimization solution that enables the computation of the portfolio problem (based on the utility option pricing and the shortfall risk minimization). We first propose a dynamical stock price process, and then, we transform the solution to a continuous-time discrete-state Markov decision processes. The market behavior is characterized by considering arbitrage-free and assessing transaction costs. To solve the problem, we present a proximal optimization approach, which considers time penalization in the transaction costs and the utility. In order to include the restrictions of the market, as well as those that imposed by the continuous-time space, we employ the Lagrange multipliers approach. As a result, we obtain two different equations: one for computing the portfolio strategies and the other for computing the Lagrange multipliers. Each equation in the portfolio is an optimization problem, for which the necessary condition of a maximum/minimum is solved employing the gradient method approach. At each step of the iterative proximal method, the functional increases and finally converges to a final portfolio. We show the convergence of the method. A numerical example showing the effectiveness of the proposed approach is also developed and presented.
Original language American English 91-111 79 International Journal of General Systems https://doi.org/10.1080/03081079.2018.1522306 Published - 17 Feb 2019

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Transaction Costs
Penalization
Continuous Time
Lagrange multipliers
Proximal Methods
Costs
Optimization
Arbitrage
Computing
Stock Prices
Option Pricing
Markov Decision Process
Discrete-time
Iterative methods
Transform
Optimization Problem
Restriction
Converge

### Cite this

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title = "Continuous-time mean variance portfolio with transaction costs: a proximal approach involving time penalization",
abstract = "{\circledC} 2018, {\circledC} 2018 Informa UK Limited, trading as Taylor & Francis Group. This paper proposes a new continuous-time optimization solution that enables the computation of the portfolio problem (based on the utility option pricing and the shortfall risk minimization). We first propose a dynamical stock price process, and then, we transform the solution to a continuous-time discrete-state Markov decision processes. The market behavior is characterized by considering arbitrage-free and assessing transaction costs. To solve the problem, we present a proximal optimization approach, which considers time penalization in the transaction costs and the utility. In order to include the restrictions of the market, as well as those that imposed by the continuous-time space, we employ the Lagrange multipliers approach. As a result, we obtain two different equations: one for computing the portfolio strategies and the other for computing the Lagrange multipliers. Each equation in the portfolio is an optimization problem, for which the necessary condition of a maximum/minimum is solved employing the gradient method approach. At each step of the iterative proximal method, the functional increases and finally converges to a final portfolio. We show the convergence of the method. A numerical example showing the effectiveness of the proposed approach is also developed and presented.",
author = "M. Garc{\'i}a-Galicia and Carsteanu, {A. A.} and Clempner, {J. B.}",
year = "2019",
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Continuous-time mean variance portfolio with transaction costs: a proximal approach involving time penalization. / García-Galicia, M.; Carsteanu, A. A.; Clempner, J. B.

In: International Journal of General Systems, 17.02.2019, p. 91-111.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Carsteanu, A. A.

AU - Clempner, J. B.

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AB - © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group. This paper proposes a new continuous-time optimization solution that enables the computation of the portfolio problem (based on the utility option pricing and the shortfall risk minimization). We first propose a dynamical stock price process, and then, we transform the solution to a continuous-time discrete-state Markov decision processes. The market behavior is characterized by considering arbitrage-free and assessing transaction costs. To solve the problem, we present a proximal optimization approach, which considers time penalization in the transaction costs and the utility. In order to include the restrictions of the market, as well as those that imposed by the continuous-time space, we employ the Lagrange multipliers approach. As a result, we obtain two different equations: one for computing the portfolio strategies and the other for computing the Lagrange multipliers. Each equation in the portfolio is an optimization problem, for which the necessary condition of a maximum/minimum is solved employing the gradient method approach. At each step of the iterative proximal method, the functional increases and finally converges to a final portfolio. We show the convergence of the method. A numerical example showing the effectiveness of the proposed approach is also developed and presented.

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