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Approximation Pricing and the Variance-Optimal Martingale Measure

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  • Schweizer, Martin
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    Abstract

    Let X be a seminmartingale and Teta the space of all predictable X-integrable processes teta such that integral tetat dX is inthe space S square of semimartingales. We consider the problem of approximating a given random variable H element of L square (P) by the sum of a constant c and a stochastic integral of 0 to t with teta s and dXs, with respect to the L square (P)-norm. This problem comes from financial mathematics where the optimal constant V zero can be interpreted as an approximation price for the contingent claim H. An elementary computation yields V zero as the expectation of H under the variance-optimal signed Teta-martingale measure P~, and this leads us to study &Ptilde in more detail. In the case of finite discrete time, we explicitly construct P~ by backward recursion, and we show that P~ is typically not a probability, but only a signed measure. In a continuous-time framework, the situation becomes rather different: We prove that P~ is nonnegative if X has continuous paths and satisfies a very mild no-arbitrage condition. As an application, we show how to obtain the optimal integrand xi element teta in feedback form with the help of a backward stochastic differential equation.

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    Bibliographic Info

    Paper provided by University of Bonn, Germany in its series Discussion Paper Serie B with number 336.

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    Length: pages
    Date of creation: Nov 1995
    Date of revision:
    Handle: RePEc:bon:bonsfb:336

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    Postal: Bonn Graduate School of Economics, University of Bonn, Adenauerallee 24 - 26, 53113 Bonn, Germany
    Fax: +49 228 73 6884
    Web page: http://www.bgse.uni-bonn.de

    Related research

    Keywords: option pricing; variance-optimal martingale measure; backward stochastic differential equations; incomplete markets; adjustment process; mean-variance tradeoff; minimal signed martingale measure;

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    Cited by:
    1. Olivier Gu\'eant & Guillaume Royer, 2013. "VWAP execution and guaranteed VWAP," Papers 1306.2832, arXiv.org, revised Apr 2014.

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