Numerical algorithms for exotic financial derivatives

A financial derivative is a financial contract whose value depends upon other underlying variables, which may be the prices of traded securities, stock indices, 3-month interest rates, etc. There has been a phenomenal growth in the number and variety of derivative securities traded in the markets. New exotic derivative products that are tailor-made to meet the individual needs of clients are constantly being invented. The construction of the theoretical framework for the pricing of new derivative securities has been one of the major challenges in this area. The complications of pricing exotic financial derivatives come in two ways. First, the value of a financial derivative may depend on a complex path-dependent structure. Second, some financial contracts come with an early exercise right which permits the holder to exercise the right before the maturity. In this thesis, a general numerical algorithm framework is proposed to value exotic financial derivatives which exhibit path-dependent structures and early exercise rights. First, the forward shooting grid approach, characterized by the augmentation of an auxiliary state vector at each grid node on a lattice tree simulating the stochastic dynamics of underlying variable, is proposed to model the path-dependent structure in a financial contract. The versatility of the approach is demonstrated by applying the method to price some path-dependent options such as Parisian options, reset options and alpha quantile options. Second, the early exercise feature is tackled by a dynamic programming algorithm which compares the early exercise value and continuation value at each grid node in the lattice tree. We propose to combine the forward shooting grid technique and the dynamic programming algorithm to analyze the pricing behavior, optimal calling policy and optimal convert policy of convertible bonds. Furthermore, we propose to extend this approach to value and analyze non-tradable employee stock options with reload provisions. Under the utility maximization approach, the risk aversion, personal wealth of an employee and the reload rights outstanding all enter into the pricing algorithm. The pricing model is then applied to examine the impact of compensation incentive of the reload feature to different types of employees. Finally, we consider the liquidation problem where a trader seeks an optimal trading scheme to unwind a huge portfolio within a time constraint. Subject to the execution time lags and liquidation discount, we propose a numerical algorithm to compute an optimal trading strategy such that the terminal expected value of riskless asset is maximized