Pricing American Options on Restricted Underlying Assets


Student thesis: Doctoral Thesis

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Award date29 Oct 2018


The Black-Scholes formula for option pricing is a milestone in modern finance. However, the formulas are only for the ideal settings of the complete market in which any option can be perfectly replicated by a portfolio which consists only of a risk free deposit and the underlying assets. The reality of our financial market unfortunately is incomplete. How to determine the price of options in such an incomplete market has been a very challenging task for decades. Meanwhile, American options add further difficulties to the theoretical studies since free boundaries are involved.

In this thesis, we studied American options in an incomplete market. In particular, we considered the situation in which trades of the underlying risky asset were not allowed by some reasons that, for instance, the transaction cost was extremely high, or that the secondary market was very illiquid. The theoretical approach was based on utility maximization. For the first time, we obtained a closed-form theoretical expression for prices of perpetual American vanilla call and put options in such an incomplete market. We further extended our theoretical study to American exotic options, such as barrier options and lookback options. We also derived the governing equations for American vanilla and exotic options in finite time horizon in this incomplete market. Our theoretical study showed that American options in incomplete markets had many feathers which were dramatically diffrerent form those in a complete market.