台灣加權股價指數選擇權(TXO)是目前台灣金融市場中的熱門商品，建構精確可靠的模型來解釋其價格變化的原因或預測其理論價格，不論對各種市場參與者來說皆為十分迫切，也是目前金融研究中的主要議題。
選擇權的評價模型有許多種，目前大多是Black-Scholes模型(1973)和以其為基礎的其他修正模型。本文以另一角度思考台灣加權股價指數選擇權成交價格的變化，將其視為一隨時間變化之動態系統，以微分方程來描述之並建立模型，期望能找出更適用於台灣的模型。
本文使用Li等人（2009）所提出的「二次逼近法」求解，試圖找出最接近實際資料的模型。實證結果發現在本文所使用的三種模型中，三種模型都有不錯的配適結果，而且動態模型對資料整體的趨勢與細部變化都能掌握其變化。針對三種不同動態模型在使用上的選擇，本文提出一些參考的準則。另外，對於降低整體的估計誤差，本文也提出一些可能的方法，供未來研究參考。

TAIEX Index Option (TXO) is the most popular financial derivative in Taiwan option market. Building an accurate model to explain the change of its price or forecast its theoretical price is instant and also a main research subject in Taiwan.
Black-Scholes model (1973) is a most commonly used model in evaluating options. Recently there are many corrections of B-S model in order to reduce error as applying it to empirical data. In this study we treat TXO price as a dynamic system which changes over time and characterize it by differential equations. Our goal is to construct a model more suitable for TXO.
We use“Parabola Approximation” proposed by Li et al. (2009) to solve the differential equations and try to find the model which fits our data the most. Empirical study shows the three models used all produce accurate estimates of TXO prices. We provide some suggestions of model selection and methods which may reduce error further.

[1] 陳威光(2001)，衍生性金融商品：選擇權、期貨與交換，台北市：智盛文化。
[2] 陳松男(2005)，金融工程學（二版），台北市：新陸書局。
[3] 陳松男(2007)，金融數學與隨機微積分，台北市：新陸書局。
[4] 石村貞夫、石村園子(2004)，李詩政、吳文峰譯，廖四郎審定，細說
Black-Scholes微分方程式，台北市：鼎茂圖書。
[5] 石村貞夫、石村園子(2005)，李詩政、吳文峰譯，廖四郎審定，細說
Black-Scholes模型，台北市：添金。
[6] 施能仁、施若竹、施純楨(2008)，衍生性金融工程，台北市：五南。
[7] 倪衍森、吳曼華、鄭亦妏(2005)，在Black-Scholes評價模型下台指選擇權
最適波動性估計方法之研究，管理科學研究，第2卷，第1期，93-109頁。
[8] 林敦舜(2002)，台灣認購權證評價之研究—探討二項式及三項式樹狀模型之
評價差異，國立交通大學經營管理研究所碩士論文。
[9] 葉淵淜(2004)，藉二項式選擇權評價模式預測台灣指數選擇權之決策行為研
究，台中健康暨管理學院經營管理研究所碩士論文。
[10] 林志信(2006)，台指選擇權之三項式評價應用，大同大學事業經營研究所
碩士論文。
[11] 楊佳隆(2006)，台指選擇權時間序列模型與交易分析，國立成功大學會計研
究所碩士論文。
[12] 國立交通大學應用數學系，Dynamical Systems，線上檢索日期：2010年2
月2日，網址：http://www.math.nctu.edu.tw/Dynamics/index.htm。

[13] Black, F., and Scholes, M. (1973), The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81(3), pp. 637-654.
[14] Bollerslev, T. (1986), Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics 31, pp. 307-327.
[15] Bollerslev, T., and Engle, R.F. (1990), Modeling the Coherence in The Short-Run Nominal Exchange Rates: A Multivariate Generalized ARCH Model, Review of Economics and Statistics 72, pp. 498-505.
[16] Box, G.E.P., and Jenkins, G.M. (1970), Time Series Analysis: Forecasting and Control, San Francisco: Holden-Day.
[17] Boyle, P.P., and Vorst, T. (1992), Option Replication in Discrete Time with Transaction Cost, Journal of Finance 47, pp. 271-294.
[18] Brenner, R.J., Harjer, R.H., and Kroner, K.F. (1996), Another Look at Models of The Short-Term Interest Rate, Journal of Financial and Quantitative Analysis 31, pp. 85-107.
[19] Chiras, D.P., and Manaster, S. (1978), The Information Content of Option Prices and a Test of Market Efficiency, Journal of Financial Economics 6, pp. 213-234.
[20] Cox, J.C., and Ross, S.A. (1976), The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics 3, pp. 145-166.
[21] Cox, J.C., Ross, S.A.., and Rubinstein, M. (1979), Option Pricing: A Simplified Approach, Journal of Financial Economics 7, pp. 229-263.
[22] Duan, R., and Li, M.R., and Yang, T. (2008), Propagation of Singularities in the Solutions to the Boltzmann Equation near Equilibrium, Mathematical Models and Methods in Applied Sciences 18(7), pp. 1093-1114.
[23] Engle, R.F. (1982), Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica 50, pp. 987-1008.
[24] Engle, R.F., Ng, V., and Rothschild, M. (1990), Asset Pricing with a Factor-ARCH Covariance Structure: Empirical Estimates for Treasury Bills, Journal of Econometrics 45, pp. 213-237.
[25] Glosten, L.R., Jagannathan, R., and Runkle, D.E. (1993), On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks, Journal of Finance 48, pp. 1779-1801.
[26] Heston, S.L. (1993), A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies 6, pp. 327-343.
[27] Hull, J.C., and White, A. (1987), The Pricing of Options on Assets with Stochastic Volatilities, Journal of Finance 42, pp. 281-300.
[28] Hull, J.C. (2005), Options, Futures, and Other Derivatives, Prentice Hall, New York.
[29] Kremer, J.W., and Roenfeldt, R.L. (1992), Warrant Pricing: Jump-Diffusion vs. Black-Scholes, Journal of Financial and Quantitative Analysis 28, pp. 255-272.
[30] Leland, H.E. (1985), Option Pricing and Replication with Transactions Costs, Journal of Finance 40, pp. 1283-1301.
[31] Lewis, C. D. (1982), Industrial and Business Forecasting Methods, Butterworths, London.
[32] Li, M.R. (2008), Estimates for the Life-Span of the Solutions for Semilinear Wave Equations, Communications on Pure and Applied Analysis 7(2), pp. 417-432.
[33] Li, M.R. (2008), On the Blow-up Time and Blow-up Rate of Positive Solutions of Semi-linear Wave Equations □ u - = 0 in 1-dimensional Space, to appear on Communications on Pure and Applied Analysis (CPAA).
[34] Li, M.R., Shieh, T.H., Yue, C.J., Lee, P., and Li, Y.T. (2009), Parabola Method in Ordinary Differential Equation, submitted to Taiwanese Journal of Mathematics.
[35] Meiss, J. (2007), Dynamical Systems, Scholarpedia 2(2): 1629, Created: 27 May 2006, reviewed: 9 February 2007, accepted: 9 February 2007. Retrieved 2 February 2010 from the World Wide Web: http://www.scholarpedia.org/article/Dynamical_systems
[36] Merton, R.C. (1976), Option Pricing when Underlying Stock Returns Are Discontinuous, Journal of Financial Economics 3, pp. 125-144.
[37] Neftci, S.N. (2000), An Introduction to the Mathematics of Financial Derivatives, Academic Press, San Diego.
[38] Nelson, D. (1991), Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica 59, pp. 347-370.
[39] Poincaré, H. (1892), Les Methodes Nouvelles de la Mecanique Celeste, Paris: Gauthier-Villars et fils. Translated, edited and introduced by Daniel L. Goroff, as New Methods of Celestial Mechanics, (American Institute of Physics, Woodbury, NY, 1993).
[40] Shieh, T.H., and Li, M.R. (2009), Numerical Treatment of Contact Discontinuity with Multi-gases, Journal of Computational and Applied Mathematics 230(2), pp. 656-673.
[41] Shieh, T.H. et al (2009), Analysis on Numerical Results with Different Exhaust Holes, International Communications in Heat and Mass Transfer 36(4), pp. 342-345.
[42] Whalley, A.E., and Wilmott, P. (1997), An Asymptotic Analysis of an Optional Hedging Model for Option Pricing with Transactions Costs, Mathematical Finance 7, pp. 307-324.
[43] Zakoian, J.M. (1994), Threshold Heteroskedasticity Models, Journal of Economic Dynamics and Control 18, pp. 931-955.