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Convexity of the optimal stopping boundary for the American put option
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2004 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 299, no 1, 147-156 p.Article in journal (Refereed) Published
Abstract [en]

We show that the optimal stopping boundary for the American put option is convex in the standard Black–Scholes model. The methods are adapted from ice-melting problems and rely upon studying the behavior of level curves of solutions to certain parabolic differential equations.

Place, publisher, year, edition, pages
2004. Vol. 299, no 1, 147-156 p.
National Category
Natural Sciences
URN: urn:nbn:se:uu:diva-92195DOI: 10.1016/j.jmaa.2004.06.018OAI: oai:DiVA.org:uu-92195DiVA: diva2:165180
Available from: 2004-10-08 Created: 2004-10-08 Last updated: 2013-06-20Bibliographically approved
In thesis
1. Selected Problems in Financial Mathematics
Open this publication in new window or tab >>Selected Problems in Financial Mathematics
2004 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing.

In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility.

In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied.

Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary.

A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility.

In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion.

Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2004. 17 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 38
Mathematical analysis, American options, convexity, monotonicity in the volatility, robustness, optimal stopping, parabolic equations, free boundary problems, volatility, Russian options, game options, excessive functions, superreplication, smooth fit, Matematisk analys
National Category
Mathematical Analysis
urn:nbn:se:uu:diva-4574 (URN)91-506-1774-5 (ISBN)
Public defence
2004-10-29, 2146, Buliding 2, Polacksbacken, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 13:15
Available from: 2004-10-08 Created: 2004-10-08Bibliographically approved

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Ekström, Erik
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