Chen, Chang, Wen and Lin (2013)提出馬可夫調控跳躍過程模型(MMJDM)描述股價指數報酬率，布朗運動項、跳躍項之頻率與市場狀態有關。然而，利率並非常數，本論文以狀態轉換模型配適零息債劵之動態過程，提出狀態轉換下的利率與具跳躍風險的股票報酬之二維模型(MMJDMSI)，並以1999年至2013年的道瓊工業指數與S&P 500指數和同期間之一年期美國國庫劵價格為實證資料，採用EM演算法取得參數估計值。經由概似比檢定結果顯示無論道瓊工業指數還是S&P 500指數，狀態轉換下利率與跳躍風險之股票報酬二維模型更適合描述報酬率。接著，利用Esscher轉換法推導出各模型下的股價指數之歐式買權定價公式，再對MMJDMSI模型進行敏感度分析以評估模型參數發生變動時對於定價公式的影響。最後，以實證資料對各模型進行模型校準及計算隱含波動度，結果顯示MMJDMSI在價內及價外時定價誤差為最小或次小，且此模型亦能呈現出波動度微笑曲線之現象。

To model asset return, Chen, Chang, Wen and Lin (2013) proposed Markov-Modulated Jump Diffusion Model (MMJDM) assuming that the Brownian motion term and jump frequency are all related to market states. In fact, the interest rate is not constant, Regime-Switching Model is taken to fit the process of the zero-coupon bond price, and a bivariate model for interest rate and stock index return with regime-switching and dependent jump risks (MMJDMSI) is proposed. The empirical data are Dow Jones Industrial Average and S&P 500 Index from 1999 to 2013, together with US 1-Year Treasury Bond over the same period. Model parameters are estimated by the Expectation-Maximization (EM) algorithm. The likelihood ratio test (LRT) is performed to compare nested models, and MMJDMSI is better than the others. Then, European call option pricing formula under each model is derived via Esscher transformation, and sensitivity analysis is conducted to evaluate changes resulted from different parameter values under the MMJDMSI pricing formula. Finally, model calibrations are performed and implied volatilities are computed under each model empirically. In cases of in-the-money and out-the-money, MMJDMSI has either the smallest or the second smallest pricing error. Also, the implied volatilities from MMJDMSI display a volatility smile curve.

1.Bailey, W. and Stulz, R. (1989). The pricing of stock index options in a general equilibrium model. Journal of Financial and Quantitative Analysis 24, 1-12.
2.Bakshi, G., Cao, C., and Chen, Z. (1997). Empirical performance of alternative option pricing models. The Journal of Finance 52, 5, 2003-2049.
3.Ball, C. and Torous, W. (1983). A simplified jump process for common stock returns. Journal of Financial and Quantitative Analysis 18, 01, 53-65.
4.Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Eeconomy 81, 3, 637-654.
5.Bo, L., Wang, Y., and Yang, X. (2010). Markov-modulated jump-diffusion for currency option pricing. Insurance: Mathematics and Economics 46, 461-469.
6.Charles, C., Fuh, C.D., and Lin, S.K. (2013). A tale of two regimes: Theory and empirical evidence for a markov-modulated jump diffusion model of equity returns and derivative pricing implications. Working paper.
7.Chen, L.S., Chang, Y.H., Wen, C.H., and Lin, S.K. (2016). Valuation of a defined contribution pension plan: evidence from stock indices under markov-modulated jump diffusion model. Journal of the Chinese Statistical Association 53, 2, 79-106.
8.Duan, J.C., Popova, I., and Ritchken, P. (2002). Option pricing under regime switching. Quantitative Finance 2, 116-132.
9.Elliott, R.J., Chan, L., and Siu, T. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance 1, 4, 423-432.
10.Elliott, R.J., Siu, T.K., and Chan, L. (2007). Pricing options under a generalized markov-modulated jump-diffusion model. Stochastic Analysis and Application 25, 4, 821-843.
11.Gerber, H. and Shiu, E. (1994). Option pricing by Esscher transforms. Transactions of the Society of Actuaries 46, 99, 140.
12.Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357-384.
13.Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6, 2, 327.
14.Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance 42, 2, 281-300.
15.Jarrow, R. and Rosenfeld, E. (1984).Jump risks and the intertemporal capital asset pricing model. The Journal of Business 57, 3, 337-351.
16.Lange, K. A. (1995). Gradient algorithm locally equivalent to the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 57, 2, 425-437.
17.Lin, S.K. Shyu, S.D. and Wang, S.Y. (2013). Option pricing under stock market cycles with jump risks: evidence from Dow Jones industrial average index and S&P 500 index. Working paper.
18.Lin, S. K., Liu, H., and Lee, J.C. (2013). Option pricing under regime-switching jump model with dependent jump sizes: evidence from stock index options. Working paper.
19.Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous Journal of Financial Economics 3, 1-2, 125-144.
20.Mixon, S. (2007). The implied volatility term structure of stock index options. Journal of Empirical Finance 14, 3, 333-354.
21.Scott, L. (1997). Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: Applications of Fourier inversion methods. Mathematical Finance 7, 4, 413-426.
22.Stein, E. and Stein, J. (1991).Stock price distributions with stochastic volatility: an analytic approach. Review of Financial Studies 4, 4, 727.