Universität zu Köln

Budke, Albrecht (2013). Finite Difference Methods for the Non-Linear Black-Scholes-Barenblatt Equation. PhD thesis, Universität zu Köln.

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Abstract

The Uncertain Volatility model is a non-linear generalisation of the Black-Scholes model in the sense that volatility and correlation can take arbitrary values in given intervals. The value of an option is then given by a non-linear partial differential equation of Hamilton-Jacobi-Bellman type. For this type of equation the concept of viscosity solution has to be considered since in general no smooth solutions in the classical sense exist. To assure the convergence of a discrete scheme it has to be consistent, stable and additionally monotone. Starting from a Finite Difference discretisation we first derive general and structural conditions to adequately price options in the Uncertain Volatility model. Additionally, an optimisation problem has to be solved which can either be done exactly or only approximatively where this choice depends on the discretisation. Finally, sufficient conditions are derived for the different discrete schemes to assure their convergence to the viscosity solution. The obtained theoretical results are finally tested in numerical experiments. The rates of convergence, the effort for excecuting the policy iteration, and the possible gain by using non-uniform grids are analysed.

Item Type:	Thesis (PhD thesis)
Translated title:	Title Language Finite Differenzen Methoden für die nicht-lineare Black-Scholes-Barenblatt Gleichung German
Translated abstract:	Abstract Language Das Uncertain-Volatility-Modell stellt eine nicht-lineare Erweiterung des Black-Scholes-Models insofern dar, dass Volatilität und Korrelation beliebige Werte in vorgegebenen Intervallen annehmen können. Der Wert einer Option ist dann durch eine nicht-lineare partielle Differentialgleichung vom Hamilton-Jacobi-Bellman-Typ bestimmt. Da Lösungen im klassischen Sinne für solche im Allgemeinen nicht existieren, wird das Konzept der Viskositätslösungen, für die die Existenz einer Lösung der PDE garantiert werden kann, verwendet. Um die Konvergenz eines Verfahrens sicherzustellen, muss zusätzlich zu Konsistenz und Stabilität Monotonie nachgewiesen werden. Ausgehend von einem Ansatz mittels finiter Differenzen werden zu erst allgemeine und strukturelle Bedingungen hergeleitet, um Optionen im Uncertain-Volatility-Modell zu bewerten. Abhängig von der Diskretisierung der Gleichung gehört dazu auch die exakte oder approximative Lösung von nicht-linearen Optimierungsproblemen. Anschließend werden hinreichende Bedingungen nachgewiesen, um die Konvergenz der einzelnen Verfahren garantieren zu können. Die entwickelten theoretischen Resultate werden anschließend in numerischen Experimenten praktisch untersucht. Dabei werden insbesondere die Konvergenzraten, der Aufwand für die policy iteration sowie der Nutzen der Verwendung nicht-uniformer Gitter betrachtet. German
Creators:	Creators Email ORCID ORCID Put Code Budke, Albrecht albrecht-budke@gmx.de UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-58479
Date:	7 October 2013
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
Uncontrolled Keywords:	Keywords Language Uncertain Volatility, Black.Scholes-Barenblatt, Black-Scholes, non-linear, option pricing, American options, European options, multi-dimensional, Finite Differences, monotone, monotonicity, viscosity solution, basket options, optimal control English Uncertain Volatility, Black-Scholes-Barenblatt, Black-Scholes, nicht-linear, Optionsbewertung, amerikanische Optionen, europäische Optionen, mehrdimensional, Finite Differenzen, monoton, Monotonie, Viskositätslösung, basket options, optimale Kontrollen German
Date of oral exam:	27 November 2013
Referee:	Name Academic Title Seydel, Rüdiger Prof. Dr. Heider, Pascal Dr. habil.
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Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/5847