  We study hedging and pricing of unattainable contingent claims in a
non-Markovian regime-switching financial model. Our financial market consists
of a bank account and a risky asset whose dynamics are driven by a Brownian
motion and a multivariate counting process with stochastic intensities. The
interest rate, drift, volatility and intensities fluctuate over time and, in
particular, they depend on the state (regime) of the economy which is modelled
by the multivariate counting process. Hence, we can allow for stressed market
conditions. We assume that the trajectory of the risky asset is continuous
between the transition times for the states of the economy and that the value
of the risky asset jumps at the time of the transition. We find the hedging
strategy which minimizes the instantaneous mean-variance risk of the hedger's
surplus and we set the price so that the instantaneous Sharpe ratio of the
hedger's surplus equals a predefined target. We use Backward Stochastic
Differential Equations. Interestingly, the instantaneous mean-variance hedging
and instantaneous Sharpe ratio pricing can be related to no-good-deal pricing
and robust pricing and hedging under model ambiguity. We discuss key properties
of the optimal price and the optimal hedging strategy. We also use our results
to price and hedge mortality-contingent claims with financial components
(equity-linked insurance claims) in a combined insurance and regime-switching
financial model.
