| Summary: | 碩士 === 國立臺北教育大學 === 資訊科學系碩士班 === 101 === Derivatives are popular financial instruments and play essential roles in financial markets, so it is critical to evaluate them efficiently and accurately. In 1973, Black and Scholes arrived at their ground-breaking analytical pricing formula. However, some empirical studies show that the lognormal diffusion process, which has been widely used to model the underlying asset's price dynamics, does not capture the empirical findings satisfactorily. Therefore, many alternative processes have been proposed, and a very popular one is the jump-diffusion process. On the other side, many stochastic interest rate models are put forward.
Since most derivatives have no analytical formulas, they must be priced by numerical methods like lattices. The pricing results converge to the theoretical values when the number of time steps increases. Unfortunately, the nonlinearity error introduced by the nonlinearity of the value function may cause the pricing results to converge slowly or even oscil1ate significantly.
This study proposes an accurate and efficient lattice for the jump-diffusion process. This lattice has successfully solved the oscillation problem. In addition, the time complexity of our lattice is lower than those of existing lattice methods by at least half an order. Numerous numerical calculations confirm the superior performance of our lattice to existing methods in terms of accuracy, speed, and generality.
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