We study hedging and pricing of 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 counting process is used to model the switching behavior for the states of the economy. 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 that 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 discuss key properties of our optimal price and optimal hedging strategy.