Three Essays on Contingent Claims Pricing

This dissertation consists of three research topics in contemporary financial option pricing theories and their applications. The common theme of those topics involves the pricing of financial claims whose value become path-dependent when using the usual lattice approximating schemes. The first essay explores the potential of transformation and other schemes in constructing a sequence of simple binomial processes that weakly converges to the desired diffusion limit. Convergence results are established for the valuation of both European and American contingent claims when the underlying asset prices are approximated by simple binomial processes. It is also demonstrated how to construct reflecting or absorbing binomial processes to approximate diffusions with boundaries. Numerical examples demonstrate that the proposed simple approximations not only converge, but also give more accurate results then existing methods such as Nelson and Ramaswamy (1990), especially for longer maturities. Our purpose in essay 2 is two-fold. First we extend some of the simple lattice-approximation methods for one-dimensional diffusions to higher dimensions and develop special lattices to approximate perfectly correlated diffusions. We then examine current modelling issues of the term structure of interest rates, and demonstrate how to apply the approximation techniques developed here to handle path-dependence and multi-sources of uncertainty in these models. The last essay analyzes the investment decisions of insured banks under fixed-rate deposit insurance. The model takes into account the charter value and allows banks to dynamically revise their asset portfolios. Trade-offs exists between preserving the charter and exploiting deposit insurance. The optimal bank portfolio problem is solved analytically for a constant charter value. In any audit period, banks maximize their risk exposure before some critical time and act cautiously thereafter. The corresponding deposit insurance is shown to be a put option that matures at this critical time rather than at the audit date.