Optimal dividends under a drawdown constraint and a curious square-root rule

In this paper we address the problem of optimal dividend payout strategies from a surplus process governed by Brownian motion with drift under a drawdown constraint, i.e. the dividend rate can never decrease below a given fraction a of its historical maximum. We solve the resulting two-dimensiona...

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Autores principales: Azcue, Pablo, Muler, Nora, Albrecher, Hansjörg
Formato: info:eu-repo/semantics/preprint submittedVersion
Lenguaje:Inglés
Publicado: Finance and Stochastics 2023
Acceso en línea:https://repositorio.utdt.edu/handle/20.500.13098/11842
https://doi.org/10.1007/s00780-023-00500-6
https://doi.org/10.48550/arXiv.2206.12220
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Repositorio Digital Universidad Torcuato Di Tella de Universidad Torcuato Di Tella
id I57-R163-20.500.13098-11842
record_format dspace
spelling I57-R163-20.500.13098-118422023-07-28T14:34:08Z Optimal dividends under a drawdown constraint and a curious square-root rule Azcue, Pablo Muler, Nora Albrecher, Hansjörg In this paper we address the problem of optimal dividend payout strategies from a surplus process governed by Brownian motion with drift under a drawdown constraint, i.e. the dividend rate can never decrease below a given fraction a of its historical maximum. We solve the resulting two-dimensional optimal control problem and identify the value function as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We then derive su cient conditions under which a two-curve strategy is optimal, and show how to determine its concrete form using calculus of variations. We establish a smooth-pasting principle and show how it can be used to prove the optimality of two-curve strategies for su ciently large initial and maximum dividend rate. We also give a number of numerical illustrations in which the optimality of the two-curve strategy can be established for instances with smaller values of the maximum dividend rate, and the concrete form of the curves can be determined. One observes that the resulting drawdown strategies nicely interpolate between the solution for the classical unconstrained dividend problem and the one for a ratcheting constraint as recently studied in [1]. When the maximum allowed dividend rate tends to in nity, we show a surprisingly simple and somewhat intriguing limit result in terms of the parameter a for the surplus level on from which, for su ciently large current dividend rate, a take-the-money-and-run strategy is optimal in the presence of the drawdown constraint. Este documento es una versión del artículo publicado en Finance Stochastics 27, 341–400 (2023) 2023-05-31T15:02:46Z 2023-05-31T15:02:46Z 2023 info:eu-repo/semantics/preprint info:eu-repo/semantics/submittedVersion https://repositorio.utdt.edu/handle/20.500.13098/11842 https://doi.org/10.1007/s00780-023-00500-6 https://doi.org/10.48550/arXiv.2206.12220 eng Albrecher, H., Azcue, P. & Muler, N. Optimal dividends under a drawdown constraint and a curious square-root rule. Finance Stoch 27, 341–400 (2023). https://doi.org/10.1007/s00780-023-00500-6 info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-sa/2.5/ar/ pp. 341 - 400 application/pdf application/pdf Finance and Stochastics
institution Universidad Torcuato Di Tella
institution_str I-57
repository_str R-163
collection Repositorio Digital Universidad Torcuato Di Tella
language Inglés
orig_language_str_mv eng
description In this paper we address the problem of optimal dividend payout strategies from a surplus process governed by Brownian motion with drift under a drawdown constraint, i.e. the dividend rate can never decrease below a given fraction a of its historical maximum. We solve the resulting two-dimensional optimal control problem and identify the value function as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We then derive su cient conditions under which a two-curve strategy is optimal, and show how to determine its concrete form using calculus of variations. We establish a smooth-pasting principle and show how it can be used to prove the optimality of two-curve strategies for su ciently large initial and maximum dividend rate. We also give a number of numerical illustrations in which the optimality of the two-curve strategy can be established for instances with smaller values of the maximum dividend rate, and the concrete form of the curves can be determined. One observes that the resulting drawdown strategies nicely interpolate between the solution for the classical unconstrained dividend problem and the one for a ratcheting constraint as recently studied in [1]. When the maximum allowed dividend rate tends to in nity, we show a surprisingly simple and somewhat intriguing limit result in terms of the parameter a for the surplus level on from which, for su ciently large current dividend rate, a take-the-money-and-run strategy is optimal in the presence of the drawdown constraint.
format info:eu-repo/semantics/preprint
submittedVersion
author Azcue, Pablo
Muler, Nora
Albrecher, Hansjörg
spellingShingle Azcue, Pablo
Muler, Nora
Albrecher, Hansjörg
Optimal dividends under a drawdown constraint and a curious square-root rule
author_facet Azcue, Pablo
Muler, Nora
Albrecher, Hansjörg
author_sort Azcue, Pablo
title Optimal dividends under a drawdown constraint and a curious square-root rule
title_short Optimal dividends under a drawdown constraint and a curious square-root rule
title_full Optimal dividends under a drawdown constraint and a curious square-root rule
title_fullStr Optimal dividends under a drawdown constraint and a curious square-root rule
title_full_unstemmed Optimal dividends under a drawdown constraint and a curious square-root rule
title_sort optimal dividends under a drawdown constraint and a curious square-root rule
publisher Finance and Stochastics
publishDate 2023
url https://repositorio.utdt.edu/handle/20.500.13098/11842
https://doi.org/10.1007/s00780-023-00500-6
https://doi.org/10.48550/arXiv.2206.12220
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AT mulernora optimaldividendsunderadrawdownconstraintandacurioussquarerootrule
AT albrecherhansjorg optimaldividendsunderadrawdownconstraintandacurioussquarerootrule
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