快速傅立葉轉換與動差法於台指選擇權之應用

Application of Fast Fourier Transform and Method of Mement Estimation on the Taiwan

李柏翰(Li Bo-Han)

指導教授 ： 白惠明

國立臺北大學/商學院/統計學系/碩士(2014年)

摘要

台灣加權指數選擇權經過這些年來,已發展成為一個成熟的市場,日平均成交量已突破四十萬口。本篇論文主要針對 Black Scholes 、 Variance Gamma 和 Normal Inverse Gaussian 這三種模型,利用快速傅立葉轉換(Fast Fourier Transform)和動差估計法(Method of Moment Estimator),對台指選擇權來做評價。快速傅立葉轉換是離散傅立葉轉換的快速演算法,可用於計算離散傅立葉轉換的逆轉換。快速傅立葉轉換有廣泛的應用,如計算大整數乘法、求解偏微分方程等等。本研究則是利用快速傅立葉轉換來解選擇權的模型價格。
在實證方面,我們將大盤以上漲、下跌、盤整或是整年分別來看,並且利用「均方根誤差」、「平均絕對誤差」和「平均絕對百分比誤差」這三種指標來做為模型的評比依據,並分別於不同價位及模型來觀察其結果。發現在價位方面,三種模型各有優缺點。在深價外是以 Normal Inverse Gaussian 模型的評價能力較佳,在價平以 Black Scholes 模型的評價能力較佳,在價內則以 Variance Gamma 模型的評價能力較佳,至於另外兩種價外時這三種模型結果相差不大。綜合而言,這三種模型在深價外 RMSE 和 MAE 的誤差值均較小,在深價內 MAPE 的誤差值則都最小。

關鍵字

平均絕對誤差 ； 均方根誤差 ； 動差估計法 ； 快速傅利葉轉換 ； 平均絕對百分比誤差

並列摘要

Taiwan weighted index option has evolved into a mature market through the years and the average daily trading volume has exceeded four hundred thousand contracts. In this thesis we apply Fast Fourier Transform and Method of Moment Estimator on three models including Black Scholes, Variance Gamma and Normal Inverse Gaussian to price the Taiwan weighted index option.
Fast Fourier Transformation is a fast algorithm applying discrete Fourier transforms. It can be used to calculate the inverse of discrete Fourier transform. Fast Fourier Transform has a wide range of applications, such as computing large integer multiplication, solving partial differential equations and so on. This study uses Fast Fourier transform to calculate the model prices of options.
In the numerical estimation, we categorize the market into periods of trending up, trending down and consolidation to make a comparison or compare them on the yearly basis. We adopt the "root mean square error", "mean absolute error" and "mean absolute percentage error" as the comparison criteria for different prices and models. When it is deep out-of-the-money, Normal Inverse Gaussian outscores the others. When it is at-the-money, Black Scholes performs better. And when it is deep in-the-money, Variance Gamma is the best. While for the other prices, the three models do not make big difference. Overall, the three modes has small RMSE and MAE values when it is deep out-of-the-money and has small MAPE value when it is deep in-the-money.

並列關鍵字

Root Mean Square Error ； Method of Moment Estimator ； Fast Fourier Transform ； Normal Inverse Gaussian ； Variance Gamma ； Black Scholes ； Mean Absolute Error ； Mean Absolute Percentage Error

參考文獻

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