Developing computational and approximation methods for discrete path dependent options is one of the most important themes of financial derivatives research. In this thesis, various analytical approximation formulas and numerical schemes on discrete path dependent options are investigated under a variety of asset price dynamics. We make a thorough investigation on the analytic tractability of the 3/2 stochastic volatility model and adopt two different approaches (PDE and probabilistic approaches) to derive a closed-form formula for the partial transform of the triple joint transition density (X, I, V) which stand for the log asset price, the quadratic variation and the instantaneous variance, respectively. The closed-form partial transform enables us to deduce a variety of marg...[ Read more ]

Developing computational and approximation methods for discrete path dependent options is one of the most important themes of financial derivatives research. In this thesis, various analytical approximation formulas and numerical schemes on discrete path dependent options are investigated under a variety of asset price dynamics. We make a thorough investigation on the analytic tractability of the 3/2 stochastic volatility model and adopt two different approaches (PDE and probabilistic approaches) to derive a closed-form formula for the partial transform of the triple joint transition density (X, I, V) which stand for the log asset price, the quadratic variation and the instantaneous variance, respectively. The closed-form partial transform enables us to deduce a variety of marginal transition density functions or characteristic functions that are crucial in pricing exotic options and variance derivatives. For pricing discrete arithmetic Asian options, we derive the analytical lower bound and construct an upper bound based on the sharp lower bound under time-changed Lévy processes. The general partially exact and bounded (PEB) approximations are also considered. Finally, numerical algorithms based on Hilbert transform are designed to price barrier options, Bermudan options and finite-maturity discrete timer options.