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  • Pricing of Exchange and Vulnerable Exchange option with Regime Switching
    (2023) Zarban, Ashwaq Ali Y
    Thesis
    This thesis aims to price the exchange option in two different scenarios theoretically. First, we consider a continuous-time, finite-state regime-switching framework to price such an option under double regime-switching jump-diffusion models. We derive the characteristic function as a matrix exponential, which, using matrix notation, allows us to derive the compensators for the synchronous jumps and find the expected values for the price. Using a unit vector representation of the regime-switching process, we can simplify the calculation from a nonlinear function of the value of Markov chain $X_{t}$ into a linear function. Second, as this type of option is being traded in an over-the-counter market, which encounters default risk, we price them considering this risk; these are called defaultable claims, and these options are known as vulnerable options. In the literature, structural and reduced-form models are two models to price such an option. However, we choose the structural model as the reduced form model will never encounter the problem we are addressing in this thesis. The new idea we incorporate in the structural model is the incomplete information setup. We assume that the information about the firm value is only observable at specific times. This approach can be used to price risky debt, deal with the problem of credit spreads approaching zero as the risky bond approaches its maturity, and can be extended nicely to price any derivative; we also study the pricing of these under the regime-switching framework which surprisingly under this incomplete information idea looks like a problem that requires the synchronised jump methodology in deriving the characteristic function.