利用適應性有限體積法評價歐式形態的亞式選擇權

Abstract

Numerical methods for pricing Asian options have been researched extensively. The common methods can be classified into three types: lattice methods, PDE methods, and Monte Carlo simulation. The ordinary lattice method needs a large amount of computer memory to keep track of the states on the tree; it is time- and space-consuming. Monte Carlo simulation is straightforward to implement, but its convergence speed is very slow. The most familiar PDE method is the finite difference method. The drawback of traditional finite difference methods is that the accuracy of results depends critically on the spacing of the domain. To achieve high accuracy, the needed number of grid points can be prohibitive. The best numerical method for solving one-dimensional PDEs runs in quadratic time. In this paper, we present a PDE method with O(mn) time complexity and O(m) space complexity based on the adaptive finite volume discretization method and error control techniques, where m and n are the numbers of grid points in the spatial and time dimensions, respectively. We first confirm the practicability and accuracy of our methodology for pricing European calls where closed-form formulas are available. Then we proceed to apply this method to European-style fixed strike Asian options. Our numerical evaluation shows that the number of grid points in the time dimension, n, does not have to be vary large compared to m to get accurate results. Therefore, we only need to increase m for more accurate results. This means the time complexity is basically only linear in m. In our algorithm, we refine areas with higher error variation while leaving others in a coarse discretization. This saves computational time tremendously without sacrificing accuracy. This algorithm also works well for the case with high volatility or high maturity. According to our experiments on Asian options, accuracy of at least 4 digits of precision can be produced in about one second on a personal computer. We also compare our method with other methods in the pricing of European-style Asian options. The results show that it is indeed superior.