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Essays in Finance Related Problems with PDE Approach
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is comprised of three scientific articles, all in PDE problems related to finance. The first two papers address problems concerning bonds and the third one treat s asymmetry problem in the mean field game theory.

We investigate the free boundary for the problem of valuation of American type convertible bond with extra call feature, in paper one. In this case the free boundary behaviour is of interest since it shows the optimal conversion strategy. We study the regularity of the free boundary and also the solution to the pricing of this special financial derivative. The touching point of the free boundary and fixed boundary is discussed, as well.

The second paper is devoted to the study of bonds with call feature and conversion. Further, it is shown that the price of a Bermudan type bond tends to that of an American one on condition that the conversion time goes to the whole life-span of the bond.

Our main goal in the third paper is to investigate a symmetry case regarding the stationary problem in mean field game of Lasry-Lions. Starting with a priori unknown ring-'shaped region, taking the boundaries as level sets and admitting harmonic solutions, it is implied that the region has spherical symmetry. Moreover, therein we attempt to state the model to an higher dimensional case.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. , vii, 14 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 13:04
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-134786ISBN: 978-91-7501-959-8 (print)OAI: oai:DiVA.org:kth-134786DiVA: diva2:668058
Public defence
2013-12-16, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20131129

Available from: 2013-11-29 Created: 2013-11-28 Last updated: 2013-12-12Bibliographically approved
List of papers
1. Analysis of free boundaries for convertible bonds, with a call feature
Open this publication in new window or tab >>Analysis of free boundaries for convertible bonds, with a call feature
2014 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 59, no 7, 912-928 p.Article in journal (Refereed) Published
Abstract [en]

Convertible bonds give rise to the so-called free boundary; i.e. an unknown boundary between continuation and conversion regions of the bond. The characteristic feature of such a bond, with an extra call feature, is that the free boundary may reach all the way to the fixed boundary. Our intention in this paper is to study the behaviour of the free boundary in the vicinity of a touching point with the fixed boundary. Along the lines of our analysis, we will also produce some results on regularity of solutions (value of the bond) up to the fixed boundary. Our methods are robust and of general nature, and can be applied to fully nonlinear equations. In particular, we shall obtain uniform results for the regularity of both solutions and their free boundaries.

Keyword
convertible bonds, free boundary, regularity, blow up
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-134796 (URN)10.1080/17476933.2013.795954 (DOI)000334162800002 ()2-s2.0-84898918435 (Scopus ID)
Note

QC 20140515

Available from: 2013-11-28 Created: 2013-11-28 Last updated: 2017-12-06Bibliographically approved
2. Analysis for Bonds with conversion/call/Bermudan feature
Open this publication in new window or tab >>Analysis for Bonds with conversion/call/Bermudan feature
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-134809 (URN)
Note

QS 2013

Available from: 2013-11-29 Created: 2013-11-29 Last updated: 2013-11-29Bibliographically approved
3. A symmetry problem with applications to finance
Open this publication in new window or tab >>A symmetry problem with applications to finance
2014 (English)In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 174, no 4, 617-626 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we will consider an overdetermined problem in an unknown ring-shaped domain, and we prove that the domain has to be spherical ring. We will apply this problem to an n-dimensional toy model in the mean field game theory introduced by Lasry-Lions too. We show that if in Lasry-Lions model, the additional extra data ”the boundary is a level set” is assumed, then the region, where the solution is harmonic, has to have spherical symmetry.

Keyword
Mean field game, Symmetry
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-134810 (URN)10.1007/s00605-013-0544-8 (DOI)000339875900007 ()2-s2.0-84905013340 (Scopus ID)
Note

QC 20140901

Available from: 2013-11-29 Created: 2013-11-29 Last updated: 2017-12-06Bibliographically approved

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CiteExportLink to record
Permanent link

Direct link
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Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
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