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2014 | OriginalPaper | Buchkapitel

A Note on Market Completeness with American Put Options

verfasst von : Luciano Campi

Erschienen in: Inspired by Finance

Verlag: Springer International Publishing

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Abstract

We consider a non necessarily complete financial market with one bond and one risky asset, whose price process is modeled by a suitably integrable, strictly positive, càdlàg process S on [0,T]. Every option price is defined as the conditional expectation under a given equivalent (true) martingale measure \(\mathbb{P}\), the same for all options. We show that every positive contingent claim on S can be approximately replicated in L ²-sense by investing dynamically in the underlying and statically in all American put options (of every strike price k and with the same maturity T). We also provide a counterexample to static hedging with European call options of all strike prices and all maturities t≤T.

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Vorheriges Kapitel Dynamic Hedging of Counterparty Exposure

Nächstes Kapitel An f-Divergence Approach for Optimal Portfolios in Exponential Lévy Models

We recall that, in this

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setting, weak orthogonality between two martingales M and N in

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is equivalent to strong orthogonality, i.e.

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. See, e.g., Lemma 2, Sect. IV, in Protter’s book [16].

Indeed, the support of \(dL_{u} ^{k} (S)\) is {u:S _u=S _u−=k} (see, e.g., Protter’s book [16], Theorem 69, p. 217) while that of \(\mathbf{1}_{\{S_{u}\leq k\}}du\) is {u:S _u≤k}. Thus, their intersection is contained in {u:S _u=k} which is at most countable and so it has zero Lebesgue measure.

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Titel: A Note on Market Completeness with American Put Options
verfasst von: Luciano Campi
Verlag: Springer International Publishing
Buch: Inspired by Finance
Print ISBN: 978-3-319-02068-6

Electronic ISBN: 978-3-319-02069-3

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-02069-3_4