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Maximizing functionals of the maximum in the Skorokhod embedding problem and an application to variance swaps
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Hobson, David (David G.) and Klimmek, Martin (2013) Maximizing functionals of the maximum in the Skorokhod embedding problem and an application to variance swaps. The Annals of Applied Probability, Volume 23 (Number 5). pp. 2020-2052. doi:10.1214/12-AAP893 ISSN 1050-5164.
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Official URL: http://dx.doi.org/10.1214/12-AAP893
Abstract
The Azéma–Yor solution (resp., the Perkins solution) of the Skorokhod embedding problem has the property that it maximizes (resp., minimizes) the law of the maximum of the stopped process. We show that these constructions have a wider property in that they also maximize (and minimize) expected values for a more general class of bivariate functions F(W τ ,S τ ) depending on the joint law of the stopped process and the maximum. Moreover, for monotonic functions g , they also maximize and minimize E[∫ τ 0 g(S t )dt] amongst embeddings of μ , although, perhaps surprisingly, we show that for increasing g the Azéma–Yor embedding minimizes this quantity, and the Perkins embedding maximizes it.
For g(s)=s −2 we show how these results are useful in calculating model independent bounds on the prices of variance swaps.
Along the way we also consider whether μ n converges weakly to μ is a sufficient condition for the associated Azéma–Yor and Perkins stopping times to converge. In the case of the Azéma–Yor embedding, if the potentials at zero also converge, then the stopping times converge almost surely, but for the Perkins embedding this need not be the case. However, under a further condition on the convergence of atoms at zero, the Perkins stopping times converge in probability (and hence converge almost surely down a subsequence).
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Journal or Publication Title: | The Annals of Applied Probability | ||||
Publisher: | Institute of Mathematical Statistics | ||||
ISSN: | 1050-5164 | ||||
Official Date: | 28 August 2013 | ||||
Dates: |
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Volume: | Volume 23 | ||||
Number: | Number 5 | ||||
Number of Pages: | 32 | ||||
Page Range: | pp. 2020-2052 | ||||
DOI: | 10.1214/12-AAP893 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published |
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