Pricing Asian Options on Lattices

Path-dependent options are options whose payoff depends nontrivially on the price history of an asset. They play an important role in financial markets. Unfortunately, pricing path-dependent options could be difficult in terms of speed and/or accuracy. The Asian option is one of the most prominent examples. The Asian option is an option whose payoff depends on the arithmetic average price of the asset. How to price such a derivative efficiently and accurately has been a long-standing research and practical problem. Up to now, there is still no simple exact closed form for pricing Asian options. Numerous approximation methods are suggested in the academic literature. However, most of the existing methods are either inefficient or inaccurate or both. Asian options can be priced on the lattice. A lattice divides the time interval between the option initial date and the maturity date into n equal time steps. The pricing results converge to the true option value as n → ∞. Unfortunately, only exponential-time algorithms are currently available if such options are to be priced on a lattice without approximations. Although efficient approximation methods are available, most of them lack convergence guarantees or error controls. A pricing algorithm is said to be exact if no approximations are used in backward induction. This dissertation addresses the Asian option pricing problem with the lattice approach. Two different methods are proposed to meet the efficiency and accuracy requirements. First, a new trinomial lattice for pricing Asian options is suggested. This lattice is designed so the computational time can be dramatically reduced. The resulting exact pricing algorithm is proven to be the first exact lattice algorithm to break the exponential-time barrier. Second, a polynomial time approximation algorithm is developed. This algorithm computes the upper and the lower bounds of the option value of the exact pricing algorithm. When the number of time steps of the lattice becomes larger, this approximation algorithm is proven to converge to the true option value for pricing European-style Asian options. Extensive experiments also reveal that the algorithm works well for American-style Asian options.