Approximation pricing and the variance-optimal martingale measure

Abstract

Let X be a semimartiangale and let $\Theta$ be the space of all predictable X-integrable process $\vartheta$ such that $\int\vartheta dX$ is in the space $\varsigma^2$ of semimartingales. We consider the problem of approximating a given random variable $H\in L^2(P)$ by the sum of a constant c and a stochastic integral $\int_0^T\vartheta_s dX_s$, with respect to the $L^2(P)$-norm. This problem comes from financial mathematics, where the optimal constant $V_0$ can be interpreted as an approximation price for the contingent clam H. An elementary computation yields $V_0$ as the expectation of H under the variance-optimal signed $\Theta$-martingale measure $\tilda{P}$, and this leads us to study $\tilda{P}$ in more detail. In the case of finite discrete time, we explicitly construct $\tilda{P}$ by backward recursion, and we show that $\tilda{P}$ is typically not a probability, but only a signed measure. In a continuous-time framework, the situation becomes rather different: we prove that $\tilda{P}$ is nonegative is X has continuous paths and satisfies a very mild no-arbitrage condition. As an application, we show how to obtain the optimal integrand $\xi\in\Theta$ in feedback form with the help of a backward stochastic differential equation.