MAXIMIZING FUNCTIONALS OF THE MAXIMUM IN THE SKOROKHOD EMBEDDING PROBLEM AND AN APPLICATION TO VARIANCE SWAPS

• 成果类型: Article
• 署名作者: Hobson, David; Klimmek, Martin
• 署名单位: University of Warwick
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP893
• 发表日期: 2013
• 页码: 2020-2052
• 关键词: martingales; LAW

摘要

The Azema-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-tau, S-tau) depending on the joint law of the stopped process and the maximum. Moreover, for monotonic functions g, they also maximize and minimize E[integral(tau)(0) g (S-t) dt] amongst embeddings of mu, although, perhaps surprisingly, we show that for increasing g the Azema-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 mu(n) converges weakly to mu is a sufficient condition for the associated Azema-Yor and Perkins stopping times to converge. In the case of the Azema-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).