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Market completion with derivative securities

Schwarz, DC; (2016) Market completion with derivative securities. Finance and Stochastics , 21 (1) pp. 263-284. 10.1007/s00780-016-0317-z. Green open access

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Abstract

Let SF be a ℙ-martingale representing the price of a primitive asset in an incomplete market framework. We present easily verifiable conditions on the model coefficients which guarantee the completeness of the market in which in addition to the primitive asset, one may also trade a derivative contract SB. Both SF and SB are defined in terms of the solution X to a two-dimensional stochastic differential equation: SFt=f(Xt) and SBt:=E[g(X1)|Ft]. From a purely mathematical point of view, we prove that every local martingale under ℙ can be represented as a stochastic integral with respect to the ℙ-martingale S:=(SF,SB). Notably, in contrast to recent results on the endogenous completeness of equilibria markets, our conditions allow the Jacobian matrix of (f,g) to be singular everywhere on R2. Hence they cover as a special case the prominent example of a stochastic volatility model being completed with a European call (or put) option.

Type: Article
Title: Market completion with derivative securities
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s00780-016-0317-z
Publisher version: http://dx.doi.org/10.1007/s00780-016-0317-z
Language: English
Additional information: © The Author(s) 2016 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Keywords: Social Sciences, Science & Technology, Physical Sciences, Business, Finance, Mathematics, Interdisciplinary Applications, Social Sciences, Mathematical Methods, Statistics & Probability, Business & Economics, Mathematics, Mathematical Methods In Social Sciences, Completeness, Derivatives, Integral representation, Diffusion, Martingales, Parabolic equations, Analytic functions, Jacobian determinant, ENDOGENOUS COMPLETENESS, CONTINUOUS-TIME, EQUILIBRIUM
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1534891
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