Stable reconstruction of the volatility in a regime-switching local volatility model

Prices of European call options in a regime-switching local volatility model can be computed by solving a parabolic system which generalises the classical Black and Scholes equation, giving these prices as functionals of the local volatilities. We prove Lipschitz stability for the inverse problem of determining the local volatilities from quoted call option prices for a range of strikes, if the calls are indexed by the different states of the continuous Markov chain which governs the regime switches.