# SFB 303 Discussion Paper No. B - 336

**Author**: Schweizer, Martin

**Title**: Approximation Pricing and the Variance-Optimal Martingale Measure

**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.

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

**JEL-Classification-Number**: G13

**Creation-Date**: November 1995

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