Approximation pricing and the variance-optimal martingale measure

Let X be a semimartingale and let Theta be the space of all predictable X-integrable processes theta such that integral theta dX is in the space L(2) of semimartingales. We consider the problem of approximating a given random variable H is an element of L(2)(P) by the sum of a constant c and a stochastic integral integral(0)(T) theta(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 claim H. An elementary computation yields V-0 as the expectation of H under the variance-optimal signed Theta-martingale measure (P) over bar, and this leads us to study (P) over bar in more detail. In the case of finite discrete time, we explicitly construct (P) over bar by backward recursion, and we show that (P) over bar is typically not a probability, but only a signed measure. In a continuous-time framework, the situation becomes rather different: we prove that (P) over bar 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 is an element of Theta in feedback form with the help of a backward stochastic differential equation.