臺灣博碩士論文加值系統

本研究修正Ritchken and Trevor (1999) 提出的GARCH選擇權樹狀訂價法與Duan, Popova and Ritchken (2002) 提出的Markov-switching 選擇權樹狀訂價法的缺失，以此為基礎，進一步提出更具一般化的Markov-switching GARCH (簡稱MS-GARCH)選擇權樹狀訂價法，以解決標的資產價格動態過程服從MS- GARCH過程下之歐式與美式選擇權評價問題。由於GARCH模型與Markov-switching模型皆可視為MS-GARCH模型的特例，故本文所提之MS-GARCH選擇權樹狀訂價法相當具有一般性。數值分析結果顯示，既有的GARCH選擇權樹狀訂價法與Markov-switching選擇權樹狀訂價法在考慮本文所提出的修正與補強之後，計算上比既有的方法簡單，準確度也較高。除此之外，數值分析結果亦顯示MS-GARCH選擇權樹狀訂價法具有相當良好的收斂性與正確性。

This paper supplements the GARCH lattice algorithm for option pricing developed by Ritchken and Trevor (1999) and proposes possible modifications for Markov-switching lattice algorithm of Duan, Popova and Ritchken (2002), respectively. The numerical analyses show that our modifications for these existent lattice algorithms result in either less complicated procedures or faster rates of convergence. On the other hand, the Markov-switching GARCH models (MS-GARCH) which combine GARCH models with Markov-switching processes are highly flexibility and widely used in financial time-series analysis. Accordingly, this study further develops a new efficient lattice algorithm to price European and American options under discrete time MS-GARCH processes. The new option pricing model considered assumes that the underlying stock price dynamic is modeled by the process in which prices remain in one GARCH process for a random amount of time before switching over into another GARCH process. Since both of GARCH option models and Markov-switching option models are special cases of the MS-GARCH models, the new lattice algorithm is very general. We also conduct numerical analyses to gauge the convergence and accuracy of option prices produced by the lattice to their true values. These results are very satisfactory.

Table of Contents

1. Introduction 1
2. Literature review 3
3. The GARCH option model 6
3.1 The lattice algorithm for the GARCH option model 6
3.2 Possible modifications for the GARCH lattice 12
3.3 Numerical analyses of our modification 16
4. The Markov-switching option model 18
4.1 The lattice algorithm for the Markov-switching option model 19
4.2 Possible modifications for the Markov-switching lattice 22
4.3 Numerical analyses of our modification 24
5. The MS-GARCH model 26
5.1 The lattice algorithm for the MS-GARCH option model 29
5.2 Numerical analyses of MS-GARCH lattice algorithm 36
6. Conclusion 39
References 40

References
Bollen, N.P.B., 1998. “Valuing options in regime-switching models,” Journal of Derivatives 6, 38-49.
Bollen, N.P.B., S.F., Gray, and R.E., Whaley, 2000, “Regime switching in foreign exchange rates: Evidence from currency option prices,” Journal of Econometrics 94, 239-276.
Bollen, N., E., Rasiel, 2003, “The performance of alternative valuation models in the OTC currency options market,” Journal of International Money and Finance 22, 33-64.
Bollerslev, T., R.Y., Chou, and K.F., Kroner, 1992, “ARCH modeling in finance: A review of the theory and empirical evidence,” Journal of Econometrics 52, 5-59.
Cai, J., 1994, “A Markov Model of Switching-Regime ARCH,” Journal of Business and Economic Statistics 12 (3), 309-316.
Cakici, N., K., Tapyan, 2000, “The GARCH option pricing model: a lattice approach,” Journal of Computational Finance 3, 71-85.
Diebold, F.X., 1986, “Modeling the persistence of conditional variances: A comment,” Econometric Reviews 5, 51-56.
Duan, J.C., I., Popova, and P., Ritchken, 2002, “Option pricing under regime switching,” Quantitative Finance 2, 1-17.
Duan, J.C., J.G., Simonato, 2001, “American option pricing under GARCH by a Markov chain approximation,” Journal of Economics Dynamics and Control 25, 1689-1718.
Dueker, M.J., 1999, “Markov switching in GARCH processes and mean- reverting stock-market volatility.” Journal of Business and Economic Statistics 15, 26-34.
Engel, C., 1994, “Can the Markov switching model forecast exchange rates?” Journal of International Economics 36, 151-165.
Engel, C., J.D., Hamilton, 1990, “Long swings in the dollar: Are they in the data and do markets know it?” American Economic Review 80, 689-713.
Gray, S.F., 1996, “Modeling the conditional distribution of interest rates as a regime-switching process,” Journal of Financial Economics 42, 27-62.
Gray, S. F., R. E., Whaley, 2000, “Regime-Switching in Foreign Exchange Rates: Evidence from Currency Option Prices,” Journal of Econometrics 94, 239-276
Hamilton, J.D., 1988, “Rational-expectations econometric analysis of changes in regime. An investigation of the term structure of interest rates,” Journal of Economic Dynamics and Control 12, 385-423.
Hamilton, J.D., 1989, “A new approach to the economic analysis of nonstationary time series and the business cycle,” Econometrics 57, 357-384.
Hamilton, J. D., R., Susmel, 1994, “Autoregressive ConditionalHeteroscedasticity and Changes in Regime,” Journal of Econometrics 64, 307-333.
Hass, M., S., Mittnik, and M.S., Paolella, 2004, “A new approach to Markov- Switching GARCH models,” Journal of Financial Econometrics 2, 493-530.
Klaassen, F., 2002, “Improving GARCH volatility Forecasts with regime- switching GARCH,” Empirical Economics 27, 363-394.
Lamoureux, C.G., W.D., Lastrapes, 1990, “Persistence in variance, structural change, and the GARCH model,” Journal of Business and Economic Statistics 8, 225-234.
Lyuu, Y.D., C.N., Wu, 2005, “On accurate and provably efficient GARCH option pricing algorithms,” Quantitative Finance 5, 181-198.
Maheu, J.M., T.H., McCurdy, 2000, “Identifying bull and bear markets in stock returns,” Journal of Business and Economic Statistics 18, 100-112.
Pagan, A.R., G.W., Schwert, 1990, “Alternative models for conditional stock volatility,” Journal of Econometrics 45, 267-290.
Perez-Quiros, G., A., Timmermann, 2001, “Business cycle asymmetries in stock returns: Evidence from higher order moments and conditional densities,” Journal of Econometrics 103, 259-306.
Ritchken, P., R., Trevor, 1999, “Pricing options under generalized GARCH and stochastic volatility processes,” Journal of Finance 54, 377-402.
Shen, C.H., S.W., Chen, 2004, “Long swing in appreciation and short swing in depreciation and does the market not know it? The case of Taiwan,” International Economic Journal 18, 195-213.
Susmel, R., 2000, “Switching volatility in international equity markets,” International Journal of Finance and Economics 5, 265-283.
Turner, C.M., R., Startz, and C.R., Nelson, 1989, “A Markov model of heteroskedasticity, risk, and learning in the stock market,” Journal of Financial Economics 25, 3-22.
Vigfusson, R., 1997, “Switching between chartists and fundamentalists: A Markov regime-switching approach,” International Journal of Finance and Economics 2, 291-305.
Wu, C.C., 2006, “The GARCH option pricing model: A modification of lattice approach,” Review of Quantitative Finance and Accounting 26, 55-66.