  We find the variance-optimal equivalent martingale measure when multivariate
assets are modeled by a regime-switching geometric Brownian motion, and the
regimes are represented by a homogeneous continuous time Markov chain. Under
this new measure, the Markov chain driving the regimes is no longer
homogeneous, which differs from the equivalent martingale measures usually
proposed in the literature.
  We show the solution minimizes the mean-variance hedging error under the
objective measure. As argued by \citet{Schweizer:1996}, the variance-optimal
equivalent measure naturally extends canonical option pricing results to the
case of an incomplete market and the expectation under the proposed measure may
be interpreted as an option price. Solutions for the option value and the
optimal hedging strategy are easily obtained from Monte Carlo simulations. Two
applications are considered.
