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Exchangeability-type properties of asset prices

Part of: Distribution theory - Probability Stochastic processes Mathematical finance

Published online by Cambridge University Press:  01 July 2016

Ilya Molchanov  and
Michael Schmutz
Ilya Molchanov*
Affiliation:
University of Bern
Michael Schmutz*
Affiliation:
University of Bern
*
∗ Postal address: Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland.
∗ Postal address: Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland.
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Abstract

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Let η = (η1,…,ηn) be a positive random vector. If its coordinates ηi and ηj are exchangeable, i.e. the distribution of η is invariant with respect to the swap πij of its ith and jth coordinates, then Ef(η) = Ef(πijη) for all integrable functions f. In this paper we study integrable random vectors that satisfy this identity for a particular family of functions f, namely those which can be written as the positive part of the scalar product 〈u, η〉 with varying weights u. In finance such functions represent payoffs from exchange options with η being the random part of price changes, while from the geometric point of view they determine the support function of the so-called zonoid of η. If the expected values of such payoffs are πij-invariant, we say that η is ij-swap-invariant. A full characterisation of the swap-invariance property and its relationship to the symmetries of expected payoffs of basket options are obtained. The first of these results relies on a characterisation theorem for integrable positive random vectors with equal zonoids. Particular attention is devoted to the case of asset prices driven by Lévy processes. Based on this, concrete semi-static hedging techniques for multi-asset barrier options, such as weighted barrier swap options, weighted barrier quanto-swap options, or certain weighted barrier spread options, are suggested.

Keywords

Barrier optionduality principleexchangeabilityhomogeneous functionLévy processmulti-asset optionpayoffput-call symmetrysemi-static hedgingsymmetryswap invariancezonoid

MSC classification

Primary: 60E05: Distributions 60G51: Processes with independent increments; L'evy processes 91G20: Derivative securities
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 666 - 687
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Supported by the Swiss National Science Foundation, under grants 200021-117606 and 200021-126503.

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