第一部份
當標的資產價格的變動服從GARCH行程時，Ritchken & Trevor(1999) 提出一以三元樹為基礎的數值演算法，可以對波動性會因時而異的GARCH型美式與歐式選擇權契約提供一評價工具。Cakici & Topyan(2000) 更進一步對Ritchken & Trevor(1999) 演算法的建構過程提出修正，可以提高選擇權的評價結果。然而，在該文獻中，當樹狀圖法在回溯折算選擇權價格時的機率分配上，發生了偏誤，以致於連同Cakici & Topyan 也無法處理GARCH模型評價上的一般化情況。經本文的調整與修正後，除了可使樹狀評價法更完整之外，由此出發，冀望可以進一步針對GARCH族的奇異(exotic)選擇權（例如：障礙(barrier) 選擇權）等在店頭市場交易熱絡的金融商品，利用樹狀法來進行定價與避險的工作。
第二部份
本文利用Ritchken & Trevor(1999)樹形圖演算法，當標的資產價格的變動服從GARCH行程時，成左熙B理障礙選擇權如何定價的問題。研究發現，在波動性會因時而異的GARCH模型下，障礙(barrier)水準與樹狀圖中節點的相對位置，確實會影響評價的偏誤程度，本文也提出了可以有效減少偏誤的改善方法。文中並說明此樹形圖演算法，不僅可以處理歐式GARCH型單邊與雙邊障礙選擇權的定價問題，也可以處理美式GARCH型障礙選擇權的定價問題。
第三部份
本文利用GARCH選擇權評價模型配合馬可夫鏈數值演算法，探討認購權證價格變動的行為。台灣認購權證市場於1997年9月開始進行交易活動，到1999年12月為止，共有16檔個股型認購權證成它a上市交易並期滿下市。本文在標的股票價格服從GARCH行程的條件之下，利用馬可夫鏈矩陣演算法來對認購權證進行評價。另外，權證發行券商常用Black & Scholes與二項式模型來求算認購權證理論價格（例如：元大京華證券商的認購權證公開銷售說明書等）。我們發現在本文所選取的權證樣本之下，三種模型的理論價格皆低估了市場價格，且低估的幅度皆具統計顯著性。並以GARCH模型評價結果最接近市場價格。最後並探討影響GARCH模型價格與理論價格差異的可能因素，研究結果發現：權證距到期日的時間、流動性及權證的價內程度(moneyness)，在解釋價格差異程度上，皆具有統計的顯著性。

Part I
Ritchken and Trevor (1999) propose a lattice approach for pricing American options under discrete time-varying volatility GARCH frameworks. The lattice approach works well for the pricing of the GARCH options, however this approach is inappropriate when the option price is computed on the lattice using standard backward recursion procedures, even the concepts of Cakici & Topyan(2000) is incorporated. This paper shows how to remedy the deficiency and that after our adjustment, the lattice method performs properly for option pricing under the GARCH process.
Part II
In this paper, we show the lattice approach for pricing discretely monitored barrier options in the single and double barrier frameworks under GARCH process. This study extends the Ritchken and Trevor (1999) trinomial method to price barrier option contracts whose volatility process is time varying with the form of GARCH model. The difference between original lattice structure and modified lattice tree for the valuation of barrier options is investigated. We find that, under trinomial lattice of stochastic volatility, the location of barrier affects the option value. This finding is similar to that of Boyle and Lau (1994) based on binomial tree of constant volatility. This article also adopts adjustment parameter, which is a modification of the Ritchken (1995) stretch parameter to compute the option value for single and double barrier option contracts. The results show that the adjusted parameter approach works well for pricing both the European and American GARCH barrier options.
Part III
This paper attempts to employ the GARCH option pricing model proposed by Duan(1995) to empirically examine the pricing of Taiwan stock market related call warrants. We adopt the Markov chain algorithms of Duan and Simonato(2000) for pricing warrants. There exists the deviation between the market price and the theoretical price based on NGARCH process. But the difference between market prices and NGARCH model prices are less than the differences between market prices and BS theoretical prices. We found the NGARCH model performs very well in comparison with the BS model in warrants pricing. As to the difference between market and model prices can be explained by the degree of moneyness, liquidity and time to expiration. These parameters are significant in explaining the difference between market prices and NGARCH model prices in statistical.

Part I:
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4. Bollerslev, T., R. Chou and K. Kroner, 1992, ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence, Journal of Econometrics 52, 1-59.
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6. Boyle, P, 1988, A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis, 23 March, 1-12.
7. Brockhaus Oliver, Michael Farkas, Andrew Ferraris, Douglas Long and Marcus Overhaus, 2000, Equity Derivatives and Market Risk Models, Risk Publications.
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11. Duan,J,C-, Evan Dudley, Genevieve Gauthier and Simonato, 2000, Pricing Discretely Monitored Barrier Option by a Markov Chain, working paper, Hong Kong University of Science and Technology.
12. Duan, J, C- and Simonato, 2000, American Option Pricing under GARCH by a Markov Chain Approximation, Journal of Economic Dynamic and Control.
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14. Engle, R. and V, Ng, 1993, Measuring and Testing of the Impact of News on Volatility, Journal of Finance 48, 1749-1778.
15. Hsieh, K.C. and Ritchken, 2000, An Empirical Comparison of GARCH Option Pricing Models, Working Paper(Case Western Reserve University, USA)
16. Kallsen, J. and M. Taqqu, 1998, Option Pricing in ARCH-Type Models, Mathematical Finance 8, 13-26.
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19. Ritchken, P. and R. Trevor, 1999, Pricing Options Under Generalized GARCH and Stochastic Volatility Process, Journal of Finance 54, 337-402.
Part II:
1. Ahn Dong-Hyun, S. Figlewski and Bin Gao, 1999, Pricing Discrete Barrier
Options with an Adaptive Mesh Model, Journal of Derivatives, Summer,33-43.
2. Black, F. 1976, Studies of Stock Price Volatility Changes, in Proceeding of the
1976 Meetings of the Business and Economic Statistics Section, American
Statistical Association, 177-181.
3. Black, F. and M. Scholes, 1973, The Pricin0g of Options and Corporate Liabilities,
Journal of Political Economy 81, 637-659.
4. Bollerslev, T., 1986,Generalized Autoregressive Conditional Heteroskedasticity,
Journal of Econometrics 31, 307-327.
5. Bollerslev, T., R. Chou and K. Kroner, 1992, ARCH Modeling in Finance: A
Review of the Theory and Empirical Evidence, Journal of Econometrics 52, 1-59.
6. Boyle, P. 1986, Options Valuation Using a Three Jump Process. International
Options Journal, 3, 7-12.
7. Boyle, P, 1988, A Lattice Framework for Option Pricing with Two State Variables. Journal of Financial and Quantitative Analysis, 23 March, 1-12.
8. Boyle, P, and S. H. Lau, 1994, Bumping Up Against the Barrier with the Binomial Method. Journal of Derivatives, Summer, 6-14.
9. Boyle, P, and Y. S. Tian, 1999, Pricing Lookback and Barrier Options under the
CEV Process, Journal of Financial and Quantitative Analysis. Vol. 34, No. 2,
June, 241-264.
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of Derivatives. Fall, 8-22.
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Approach. Journal of Financial Economics, 7, 229-263.
13. Duan, J.C-., 1995, The GARCH Option Pricing Model, Mathematical Finance 5,
13-32.
14.Duan,J,C-, Evan Dudley, Genevieve Gauthier and Simonato, 2001, Pricing
Discretely Monitored Barrier Option by a Markov Chain, working paper, Hong
Kong University of Science and Technology.
15.Duan J.C-, and Jason Z. Wei, 1999, Pricing Foreign Currency and Cross-
Currency Options Under GARCH, Journal of Derivatives, Fall,51-63.
16.Duan, J, C- and Simonato, 2000, American Option Pricing under GARCH by a
Markov Chain Approximation, Journal of Economic Dynamic and Control, 25(11),
1689-1718.
17.Engle, R., 1982, Autoregressive Conditional Heteroskedasticity with Estimates
of the Variance of UK Inflation, Econometrica 50, 987-1108.
18.Engle, R. and V, Ng, 1993, Measuring and Testing of the Impact of News on
Volatility, Journal of Finance 48, 1749-1778.
19.Figlewski Stephen and Bin Gao, 1999, The Adaptive Mesh Model: A New
Approach to Efficient Option Pricing, Journal of Financial Economics 53, 313-351.
20.Gao Bin, J. Z. Huang and M. Subrahmanyam, 1999, The Valuation of American
Barrier Options Using the Decomposition Technique, working paper, New York
University.
21.Hsieh, K.C. and Ritchken, 2000, An Empirical Comparison of GARCH Option
Pricing Models, Working Paper(Case Western Reserve University, USA)
22.Kallsen, J. and M. Taqqu, 1998, Option Pricing in ARCH-Type Models,
Mathematical Finance 8, 13-26.
23.Kamrad, Bardia, and Ritchken, P. 1991, Multinomial Approximating Models for
Options with k State Variables, Management Science 37, 1640-1653.
24.Mandelbrot, B., 1963, The Variation of Certain Speculative Prices, Journal of
Business 36, 394-419.
25.Ritchken, P. 1995, On Pricing Barrier Options, Journal of Derivatives, Winter,
19-28.
26.Ritchken, P. and R. Trevor, 1999, Pricing Options Under Generalized GARCH
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Part III:
1. 李存修(1999),「台灣認購權證個案集」,智勝出版社。
2. Tsun-Siou Lee & Ching Yang(2001), 「An Empirical Analysis of the Market Structure and the Price Behavior of Warrants: The Case of Taiwan」台灣金融財務季刊，第一輯第二期，頁89-101。
3. 李怡宗、劉玉珍、李健瑋(1999),「Black-Scholes 評價模型在台灣認購權證市場之實證」,管理評論，第十八卷第三期，頁83-104。
4. 徐守德、官顯庭和黃玉娟(1998),「台股認購權證定價之研究」,管理評論，第十七卷第二期，頁45-69。
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Discretely Monitored Barrier Option by a Markov Chain”, working paper, Hong
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Asian Financial Crisis-A GARCH Approach,” Journal of Banking and Finance
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Markov Chain Approximation,” Journal of Economic Dynamics and Control.
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