I 本論文應用數值積分方法來迅速且正確地評價選擇權的價值。吾人所建議的數 值積分方法為高斯數值積分，因為其能達到數值積分的最高階次，所以可以非 常逼近真實的選擇權價格。高斯數值積分的理念在於它不僅能夠選擇積分點的 權重同時也能自由地決定積分點的位置，因此在同樣的積分點數之下，高斯數 值積分的收斂階次將會是辛普森法的兩倍。數值結果顯示，本方法可以應用在 非常廣泛的選擇權類型上同時也能應用在不同的標的資產演化過程上。利用本 方法，我們將能進一步萃取市場上美式選擇權或其他新奇選擇權的隱含波動度以從事更進一步的研究。 This paper develops an efficient and accurate method for numerical evaluation of the integral equations in option pricing problems. We suggest using the Gaussian quadratures, the highest order method in numerical integration, to approximate the option values. The idea of Gaussian quadratures is to give ourselves the freedom to choose not only the weight coefficients, but also the location of the abscissas at which the function is evaluated. It turns out that we can achieve Gaussian quadrature formulas whose convergence order is, essentially, twice that of Newton-Cotes formula (such as the Simpson's rule) with the same number of points. The numerical results are extremely well for a broa d range of options and underlying asset price processes. With this powerful tool, it would be possible to extract information such as implied volatility from the market prices of American options and other exotic options.
