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PDE methods for free boundary problems in financial mathematics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

We consider different aspects of free boundary problems that have financial applications. Papers I–III deal with American option pricing, in which case the boundary is called the early exercise boundary and separates the region where to hold the option from the region where to exercise it. In Papers I–II we obtain boundary regularity results by local analysis of the PDEs involved and in Paper III we perform local analysis of the corresponding stochastic representation.

The last paper is different in its character as we are dealing with an optimal switching problem, where a switching of state occurs when the underlying process crosses a free boundary. Here we obtain existence and regularity results of the viscosity solutions to the involved system of variational inequalities.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , viii, 34 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 2008:03
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4777ISBN: 978-91-7178-928-0 (print)OAI: oai:DiVA.org:kth-4777DiVA: diva2:13899
Public defence
2008-06-05, Sal F3, KTH, Lindstedtsvägen 26, Stockholm, 14:00
Opponent
Supervisors
Note
QC 20100630Available from: 2008-05-29 Created: 2008-05-29 Last updated: 2010-07-01Bibliographically approved
List of papers
1. On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing
Open this publication in new window or tab >>On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing
2007 (English)In: Mathematica Scandinavica, ISSN 0025-5521, Vol. 101, no 1, 148-160 p.Article in journal (Refereed) Published
Abstract [en]

The following paper is devoted to the study of the positivity set U = {L phi > 0} arising in parabolic obstacle problems. It is shown that U is contained in the non-coincidence set with a positive distance between the boundaries uniformly in the spatial variable if the boundary of U satisfies an interior C-1 -Dini condition in the space variable and a Lipschitz condition in the time variable. We apply our results to American option pricing and we thus show that the positivity set is strictly contained in the continuation region, which means that the option should not be exercised in U or on the boundary of U.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8543 (URN)000250542100009 ()2-s2.0-35348952197 (Scopus ID)
Note
QC 20100629Available from: 2008-05-29 Created: 2008-05-29 Last updated: 2010-07-01Bibliographically approved
2. Early exercise boundary regularity close to expiry in the indifference setting: A PDE approach
Open this publication in new window or tab >>Early exercise boundary regularity close to expiry in the indifference setting: A PDE approach
(English)Article in journal (Other academic) Submitted
Abstract [en]

The free boundary problem that occurs when pricing American options is studied in a general setting. We investigate the regularity of the free boundary close to initial state using the so called blow-up technique. This problem has been studied extensively and good results are known for the linear, one-dimensional case. The blow-up technique, however, works also for non-linear PDE in higher dimensions. For illustration we apply the technique to the indierence pricing model where the involved PDE is non-linear.

Keyword
American option, obstacle problem, blow-up, indifference pricing
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8544 (URN)
Note
QS 2012Available from: 2008-05-29 Created: 2008-05-29 Last updated: 2012-03-26Bibliographically approved
3. The blow-up technique in terms of stochastics applied to optimal stopping problems in finance
Open this publication in new window or tab >>The blow-up technique in terms of stochastics applied to optimal stopping problems in finance
(English)Manuscript (Other academic)
Abstract [en]

The blow-up technique is a useful tool for local analysis in PDEtheory. It is applicable to non-linear, higher dimensional PDEs. In this paperwe translate the blow-up technique to stochastic terms by considering thestochastic representation of solutions to PDEs. For illustration we apply theblow-up technique to obtain the early exercise boundary regularity close to expiryfor American put and call options in the classic Black-Scholes framework.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8545 (URN)
Note
QC 20100629Available from: 2008-05-29 Created: 2008-05-29 Last updated: 2010-06-30Bibliographically approved
4. A PDE approach to regularity of solutions to finite horizon optimal switching problems
Open this publication in new window or tab >>A PDE approach to regularity of solutions to finite horizon optimal switching problems
2009 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 71, no 12, 6054-6067 p.Article in journal (Refereed) Published
Abstract [en]

We study optimal 2-switching and n-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the n-switching problem we obtain regularity results for the solutions of the variational inequalities. The solutions are C-l,C-l-regular away for the free boundaries of the action sets.

Keyword
Real options; Security design; Backward stochastic differential equation; Default risk; Snell envelope; Stopping time; Stopping and starting; Optimal switching; Viscosity solution of PDEs; Variational inequalities; INVESTMENTS
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8546 (URN)10.1016/j.na.2009.05.063 (DOI)000272606300015 ()2-s2.0-72149106819 (Scopus ID)
Note
QC 20100630Available from: 2008-05-29 Created: 2008-05-29 Last updated: 2010-06-30Bibliographically approved

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Citation style
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