In this thesis we discuss option pricing and hedging under regime switching models. To the standard model we add jumps of various types. In particular, we consider a jump that is synchronous with a change in the regime state. Thus, for example, we can define a process such that the stock price moves to a high volatility state and simultaneously has a large downward jump in returns. This type of model is consistent with market experience. We derive the compensator for our synchronous jumps and price options on such a price process using Fourier transforms. We also test the model on S&P futures options and show that it performs significantly better than a jump diffusion model. Furthermore, we look at the problem of hedging options under finitely many regime states and with finitely many possible jump sizes. We find risk-free hedge portfolios using the risk-free asset, the underlying asset, and finitely many options. Our risk-free trading strategy is consistent with any equivalent martingale measure, and so does not in itself specify which measure should be used to price options.
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