We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox-Ingersoll-Ross (CIR) process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum-Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P 500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.