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Dynamic Hedging: CVaR Minimization and Path-Wise Comparison Open Access


Other title
conditional value-at-risk
dynamic hedging
stochastic modelling
path-wise comparison theorem
Neyman-Pearson lemma
Type of item
Degree grantor
University of Alberta
Author or creator
Smirnov, Ivan
Supervisor and department
Melnikov, Alexander (Mathematical and Statistical Sciences)
Examining committee member and department
Cadenillas, Abel (Mathematical and Statistical Sciences)
Choulli, Tahir (Mathematical and Statistical Sciences)
Melnikov, Alexander (Mathematical and Statistical Sciences)
Frei, Christoph (Mathematical and Statistical Sciences)
Schmuland, Byron (Mathematical and Statistical Sciences)
Swishchuk, Anatoliy (Mathematics and Statistics, University of Calgary)
Department of Mathematical and Statistical Sciences
Mathematical Finance
Date accepted
Graduation date
Doctor of Philosophy
Degree level
Imposing constraints on the class of the available self-financing strategies may eliminate the possibility of using replicating or superhedging strategies, which leads to the problem of partial hedging. In the present work, the partial hedging problem is investigated from the viewpoint of the contingent claim seller who aims to minimize the shortfall risk through dynamic hedging un- der the constraint on the initial capital. The shortfall risk is measured via conditional value-at-risk, a coherent quantile risk measure. Another problem consists in finding a strategy that minimizes hedging costs under a constraint on conditional value-at-risk of the hedging portfolio. In a complete market, an explicit algorithm for constructing the optimal hedging strategy in both problems is presented, along with a number of detailed illustrations. In the incomplete case, the optimal solution is no longer explicit, however a cer- tain generalization of the Neyman-Pearson lemma may be used to deduce the general structure of the optimal strategy. Some of such generalizations as- sume weak compactness of the set of densities of equivalent sigma-martingale measures. We show that this requirement is in fact never satisfied in the in- complete market setting and provide detailed discussion of the matter in both the discrete- and continuous-time cases. Finally, we demonstrate how path- wise comparison can be used in the problem of approximate option hedging and pricing, and we illustrate the approach in the framework of the constant elasticity of variance model.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
Melnikov, A. and Smirnov, I. (2012). Dynamic hedging of conditional value-at-risk. Insurance: Mathematics and Economics, 51(1):182-190.

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File title: Introduction
File title: Dynamic hedging: CVaR minimization and path-wise comparison
File author: Ivan Smirnov
Page count: 140
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