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Valuation and Optimal Strategies in Markets Experiencing Shocks
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.ORCID iD: 0000-0002-6162-906X
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Description
Abstract [en]

This thesis treats a range of stochastic methods with various applications, most notably in finance. It is comprised of five articles, and a summary of the key concepts and results these are built on.

The first two papers consider a jump-to-default model, which is a model where some quantity, e.g. the price of a financial asset, is represented by a stochastic process which has continuous sample paths except for the possibility of a sudden drop to zero. In Paper I prices of European-type options in this model are studied together with the partial integro-differential equation that characterizes the price. In Paper II the price of a perpetual American put option in the same model is found in terms of explicit formulas. Both papers also study the parameter monotonicity and convexity properties of the option prices.

The third and fourth articles both deal with valuation problems in a jump-diffusion model. Paper III concerns the optimal level at which to exercise an American put option with finite time horizon. More specifically, the integral equation that characterizes the optimal boundary is studied. In Paper IV we consider a stochastic game between two players and determine the optimal value and exercise strategy using an iterative technique.

Paper V employs a similar iterative method to solve the statistical problem of determining the unknown drift of a stochastic process, where not only running time but also each observation of the process is costly.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2017. , 30 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 100
Keyword [en]
American options, optimal stopping, game options, jump diffusion, jump to default, free-boundary problems, early exercise premium, integral equation, parabolic pde, convexity, sequential testing, fixed-point approach
National Category
Probability Theory and Statistics
Research subject
Mathematics with specialization in Applied Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-316578ISBN: 978-91-506-2625-4 (print)OAI: oai:DiVA.org:uu-316578DiVA: diva2:1081367
Public defence
2017-05-03, room 80101, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2017-04-11 Created: 2017-03-14 Last updated: 2017-04-11
List of papers
1. Pricing equations in jump-to-default models
Open this publication in new window or tab >>Pricing equations in jump-to-default models
2014 (English)In: Int. J. Theor. Appl. Finance, ISSN 0219-0249, Vol. 17, no 3Article in journal (Refereed) Published
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-313325 (URN)10.1142/S0219024914500198 (DOI)
Available from: 2017-01-19 Created: 2017-01-19 Last updated: 2017-03-14
2. The perpetual American put option in jump-to-default models
Open this publication in new window or tab >>The perpetual American put option in jump-to-default models
2017 (English)In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 89, no 2, 510-520 p.Article in journal (Refereed) Published
Abstract [en]

We study the perpetual American put option in a general jump-to-default model, deriving an explicit expression for the price of the option.

We find that in some cases the optimal stopping boundary vanishes and thus it is not optimal to exercise the option before default occurs. Precise conditions for when this situation arises are given.

Furthermore we present a necessary and sufficient condition for convexity of the option price, and also show that a nonincreasing intensity is sufficient, but not necessary, to have convexity.

From this we also get conditions for when option prices are monotone in the model parameters.

National Category
Probability Theory and Statistics
Research subject
Mathematics with specialization in Applied Mathematics
Identifiers
urn:nbn:se:uu:diva-313326 (URN)10.1080/17442508.2016.1267177 (DOI)000392492800004 ()
Available from: 2017-01-19 Created: 2017-01-19 Last updated: 2017-03-14Bibliographically approved
3. The integral equation for the American put boundary in models with jumps
Open this publication in new window or tab >>The integral equation for the American put boundary in models with jumps
(English)Article in journal (Other academic) Submitted
Abstract [en]

The price of the American put option is frequently studied as the solution to an associated free-boundary problem. This free boundary, the optimal exercise boundary, determines the value of the option. In spectrally negative models the early exercise premium representation for the value of the option gives rise to an integral equation for the boundary. We study this integral equation and prove that the optimal exercise boundary is the unique solution and thus that the equation characterizes the free boundary. In a spectrally positive model, this approach does not give an equation for the boundary. We instead find lower and upper bounds for the true boundary which can be found by solving related equations.

National Category
Probability Theory and Statistics
Research subject
Mathematics with specialization in Applied Mathematics
Identifiers
urn:nbn:se:uu:diva-316576 (URN)
Available from: 2017-03-03 Created: 2017-03-03 Last updated: 2017-03-14
4. Optimal stopping games for a process with jumps
Open this publication in new window or tab >>Optimal stopping games for a process with jumps
(English)Article in journal (Other academic) Submitted
Abstract [en]

This paper presents a study of a general two-player optimal stopping game in a jump-diffusion model. An iterative scheme to find the value of this game is derived, specifically the value is shown to be the limit of a sequence of stopping games for a related diffusion but with a running reward. Furthermore the convergence is uniform and exponential. The special case of a cancellable put option is studied.

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-316577 (URN)
Available from: 2017-03-03 Created: 2017-03-03 Last updated: 2017-03-14
5. Sequential testing of a Wiener process with costly observations
Open this publication in new window or tab >>Sequential testing of a Wiener process with costly observations
(English)Article in journal (Other academic) Submitted
Abstract [en]

We consider the sequential testing of two simple hypotheses for the drift of a Brownian motion when each observation of the underlying process is associated with a positive cost. In this setting where continuous monitoring of the underlying process is not feasible, the question is not only whether to stop or to continue at a given observation time, but also, if continuing,how to distribute the next observation time. Adopting a Bayesian methodology, we show that the value function can be characterized as the unique fixed point of an associated operator, and that it can be constructed using an iterative scheme. Moreover, the optimal sequential distribution of observation times can be described in terms of the fixed point.

National Category
Probability Theory and Statistics
Research subject
Mathematics with specialization in Applied Mathematics
Identifiers
urn:nbn:se:uu:diva-316571 (URN)
Available from: 2017-03-03 Created: 2017-03-03 Last updated: 2017-03-22

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