Optimal Control of European Double Barrier Basket Options

  • We consider European double barrier basket call options on two underlyings with an upper and a lower knock-out barrier featuring a finite number of cash settlements at prespecified values of the underlyings between the strike and the upper barrier. The bilaterally constrained cash settlements are considered as controls that have to be chosen such that the Delta of the option is as close as possible to a predefined constant profit/loss. This leads to a control constrained optimal control problem for the two-dimensional Black-Scholes equation with Dirichlet boundary control and finite time control. Based on the variational formulation of the problem in an appropriate Sobolev space setting, we prove the existence of a unique solution and state the first order necessary optimality conditions. A semi-discretization in space by conforming P1 finite elements with respect to a simplicial triangulation of the computational domain gives rise to a semi-discrete control constrained optimal controlWe consider European double barrier basket call options on two underlyings with an upper and a lower knock-out barrier featuring a finite number of cash settlements at prespecified values of the underlyings between the strike and the upper barrier. The bilaterally constrained cash settlements are considered as controls that have to be chosen such that the Delta of the option is as close as possible to a predefined constant profit/loss. This leads to a control constrained optimal control problem for the two-dimensional Black-Scholes equation with Dirichlet boundary control and finite time control. Based on the variational formulation of the problem in an appropriate Sobolev space setting, we prove the existence of a unique solution and state the first order necessary optimality conditions. A semi-discretization in space by conforming P1 finite elements with respect to a simplicial triangulation of the computational domain gives rise to a semi-discrete control constrained optimal control problem for a linear system of first order ordinary differential equations. A further discretization in time by the backward Euler scheme results in a fully discrete optimization problem that is solved numerically by the projected gradient method with Armijo line search. Numerical examples for some selected test cases illustrate the benefits of hedging with European double barrier basket options in case of optimally controlled cash settlements.show moreshow less

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Metadaten
Author:Ronald H. W. HoppeGND, Tobias Lipp
URN:urn:nbn:de:bvb:384-opus4-12059
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/1496
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2011-04)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Contributing Corporation:University of Houston, Universite Pierre et Marie Curie, Paris
Release Date:2011/03/16
Tag:European double barrier basket options; multiple cash settlements; optimal control; Black-Scholes equation
GND-Keyword:Finanzmathematik; Black-Scholes-Modell; Barrier options
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand