Hedging portfolio for a market model of degenerate diffusions

Date
2022-11-30
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Source Title
Stochastics
Print ISSN
Electronic ISSN
1744-2516
Publisher
Taylor & Francis
Volume
Issue
Pages
1 - 20
Language
English
Type
Article
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Abstract

We consider a semimartingale market model when the underlying diffusion has a singular volatility matrix and compute the hedging portfolio for a given payoff function. Recently, the representation problem for such degenerate diffusions as a stochastic integral with respect to a martingale has been completely settled. This representation and Malliavin calculus established further for the functionals of a degenerate diffusion process constitute the basis of the present work. Using the Clark–Hausmann–Bismut–Ocone type representation formula derived for these functionals, we prove a version of this formula under an equivalent martingale measure. This allows us to derive the hedging portfolio as a solution of a system of linear equations. The uniqueness of the solution is achieved by a projection idea that lies at the core of the martingale representation at the first place. We demonstrate the hedging strategy as explicitly as possible with some examples of the payoff function such as those used in exotic options, whose value at maturity depends on the prices over the entire time horizon.

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Keywords
Degenerate diffusion, Malliavin calculus, Exotic option, Replicating portfolio, Clark–Ocone formula
Citation
Published Version (Please cite this version)