Approximation Pricing and the Variance-Optimal Martingale Measure

Let X be a semimartingale and let Θ be the space of all predictable X-integrable processes ϑ such that ∫ϑ dX is in the space J2of semimartingales. We consider the problem of approximating a given random variable H ∈ J2(P) by the sum of a constant c and a stochastic integral ∫T 0ϑsdXs, with respect to the J2(P)-norm. This problem comes from financial mathematics, where the optimal constant V0can be interpreted as an approximation price for the contingent claim H. An elementary computation yields V0as the expectation of H under the variance-optimal signed Θ-martingale measure P̃, and this leads us to study P̃ in more detail. In the case of finite discrete time, we explicitly construct P̃ by backward recursion, and we show that P̃ is typically not a probability, but only a signed measure. In a continuous-time framework, the situation becomes rather different: we prove that P̃ 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 ξ ∈ Θ in feedback form with the help of a backward stochastic differential equation.