Approximation Pricing and the Variance-Optimal Martingale Measure
AbstractLet 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 InfoPaper provided by University of Bonn, Germany in its series Discussion Paper Serie B with number 336.
Date of creation: Nov 1995
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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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- G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
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- Olivier Gu\'eant & Guillaume Royer, 2013. "VWAP execution and guaranteed VWAP," Papers 1306.2832, arXiv.org, revised Apr 2014.
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