Discontinuities and the American option
This thesis studies the valuation of American options when either prices or dividends or both are discontinuous. Analytical valuation formulas are derived for (1) American call options on stocks with discrete dividends, (2) American options on assets with discontinuous prices, and (3) American put options on stocks with discrete dividends. It is shown that discontinuities in prices or dividends have a similar impact on American options. A general proof technique is developed for the valuation of American options.
Chapter 2 analyzes American call options on stocks paying discrete dividends and obtains results for general price processes. We first prove the existence of finite critical prices for general price processes and with several dividends and show the need for an additional constraint to ensure a positive probability of early exercise. This is in contrast to the usual argument that the present value of the dividends has to be higher than the present value of interest forgone on the strike. The American call option is decomposed as the sum of the corresponding European call and an early exercise premium (EEP). This EEP is shown to have several interesting forms. An analytical formula is derived for the value of the American call option when the stock pays discrete dividends and follows a jump-diffusion process.
In Chapter 3, we derive analytic formulas for the value of American options when the underlying asset follows a jump diffusion process. The early exercise premium has a very different form when compared with the case for diffusion processes, and this can be attributed to the discontinuous nature of the price paths. Analytical formulas are derived for several distributions of the jump amplitude.
Analytical valuation formulas are derived in Chapter 4 for American put options when the underlying stock pays discrete dividends, certain or stochastic, and the stock price evolves as either a geometric Brownian motion or as a jump diffusion process. An innovative and simple proof technique based on “occupation time” rather than first passage time is used to derive these formulas. The approach can be extended easily to general price processes.