台指選擇權之波動率-以馬可夫轉換模型分析 - 政大學術集成

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(1)國立政治大學國際經營與貿易研究所碩士論文. 學. ‧ 國. 台指選擇權之波動率-以馬可夫轉換模型分析政治大立 Regime-switched Volatility of TAIEX Options. ‧. Using Markov-switching variance model. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 指導教授：謝淑貞博士研究生：陳宛頤中華民國一 O 四年六月.

(2) 摘要本篇論文使用馬可夫移轉變異數模型探討台指選擇權之買權的波動性。馬可夫移轉變異數模型將條件變異設定為可隨時間變動而改變，甚至移轉到不同區間上。樣本在不同區間下的平滑機率估計值有助於捕捉資料特性，實證結果顯示當樣本落在高波動率區間上時，會對應著重大事件的發生，例如 2004 年台灣 319 槍擊案、2006 年全球股災、2008 年金融海嘯等。當樣本落在低波動率區間上時，會對應著投資人傾向將台股指數的上漲或下跌視為超漲或超跌，而賦予台指選擇權之買權負的時間價值。. 立. 政治大. ‧. ‧ 國. 學. 關鍵字：馬可夫移轉、波動率、台指選擇權、時間價值. n. er. io. sit. y. Nat. al. Ch. engchi. i. i n U. v.

(3) Abstract This paper investigates the volatility of TAIEX Call Options using Markov-switching variance model. The Markov-switching variance model allows the conditional disturbances to change as time passes and even switch between different regimes. The estimation of smoothed probabilities under different regimes facilitates to capture the characteristics of data. The empirical result shows that the high volatility regime is related to extraordinary events, such as 319 shooting incident in 2004, the global stock. 政治大. market crash in 2006, and the Financial Crisis in 2008. When in low volatility regime,. 立. investors tend to treat rise or fall in TAIEX as overreactions and give TAIEX Call. ‧ 國. 學. Options turning points of time values.. ‧. Keywords: Markov-switching variance, volatility, TAIEX Call Options, time value. n. er. io. sit. y. Nat. al. Ch. engchi. ii. i n U. v.

(4) Contents. Abstract…………………………………………………………………….. i Contents……………………………………………………………………. iii List of Figures…………………………………………………………….... iv List of Tables……………………………………………………………….. iv 1. Introduction……………………………………………………………… 1 2. Literature Review……………………………………………………….. 4 3. Methodology…………………………………………………………….. 9 3-1 Pricing of Call Options 3-2 Markov-switching variance model 3-2-1 Markov chain and Markov-switching variance model 3-2-2 Filtered Probability and Smoothed Probability 4. Data……………………………………………………………………….17 4-1 Data Introduction 4-2 Statistics of TAIEX 4-3 Statistics Characteristics of Premium for TAIEX Call Options. 立. 政治大. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat. 4-3-1 Q-Q plot of Premium for TAIEX Call Options 4-3-2 Unit Root Test of Premium for TAIEX Call Options 4-4 Statistics Characteristics of Time Value for TAIEX Call Options 4-4-1 Q-Q plot of Time Value for TAIEX Call Options 4-4-2 Unit Root Test of Time Value for TAIEX Call Options 5. Empirical Results…………………………………………………………29 5-1 The Characteristics of Time Value Data 5-2 The estimation model and probability under different regimes of Premium for TAIEX Call Options 5-3 The estimation model and probability under different regimes of Time Value for TAIEX Call Options 5-4 Comparison of Premium and Time Value for TAIEX Call Options 5-5 Analysis 5-6 Goodness of fit 6. Conclusion……………………………………………………………….. 50. Ch. engchi. i n U. v. 7. References……………………………………………………………….. 52. iii.

(5) List of Figures Figure 1. Daily Intrinsic Value for TAIEX Call Options Series from 2004/01/02 to 2013/12/31…………………………..………. 19 Figure 2 Daily TAIEX from 2004/01/02 to 2013/12/31……………………..20 Figure 3 Daily Premium for TAIEX Call Options Series from 2004/01/02 to 2013/12/31……………………………………22 Figure 4 Q-Q plots against Normal and Student-t(5) distribution of Premium for TAIEX Call Options………………………………24 Figure 5 Daily Time Value for TAIEX Call Options Series from 2004/01/02 to 2013/12/31……………………………………26 Figure 6 Q-Q plots against Normal and Student-t(5) distribution of Time Value for TAIEX Call Options……………………………28 Figure 7-A Variation of Premium and TAIEX from 2004 to 2013……………32 Figure 7-B Corresponding Time values from 2004 to 2013…………………. 32 Figure 8-A Estimation of Smoothed Probabilities of Premium for TAIEX Call Options under high volatility state……………...37. 立. 政治大. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat. Figure 8-B Estimation of Smoothed Probabilities of Premium for TAIEX Call Options under low volatility state……………....37 Figure 9-A Estimation of Smoothed Probabilities of Time value for TAIEX Call Options under high volatility state……….........40 Figure 9-B Estimation of Smoothed Probabilities of Time value for TAIEX Call Options under low volatility state……………..40. Ch. engchi. i n U. v. List of Tables Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8. Descriptive Statistics of TAIEX from 2004/01/02 to 2013/12/31…..20 Descriptive Statistics of Premium for TAIEX Call Options from 2004/01/02 to 2013/12/31……………………………………..22 ADF Test and PP Test of Premium for TAIEX Call Options………..24 Descriptive Statistics of Time Value for TAIEX Call Options from 2004/01/02 to 2013/12/31……………………………………..26 ADF Test and PP Test of Time Value for TAIEX Call Options……..28 Negative time values from 2004/01/02 to 2013/12/31………………30 Economic or Political Events………………………………………..44 Goodness of Fit………………………………………………………49 iv.

(6) 1. Introduction Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) is an important index in viewing Taiwan stock market. The collapse of the stock market can be attributed to extreme events, such as financial crisis, political turmoil, and other economic events. Therefore, the prices of TAIEX Options are a useful source of additional information because they tell people how market participants value extreme. 政治大. events. Recently, in Taiwan, the trading volume of index options takes up the majority. 立. of the whole trading volume in Taiwan futures market. TAIEX Options, which are. ‧ 國. 學. representative of Taiwan stock market, especially account for over 95% in trading. ‧. volume of index options. Consequently, we think TAIEX Options are objectives worth. sit. y. Nat. io. n. al. er. studying and choose them as our research objectives.. v. In financial markets, volatility is one of the most important variables. In terms of. Ch. engchi. i n U. some empirical financial studies, volatility of price change changes over time instead of remaining constant. Researchers even find that the price change exhibits high and low states and the states tend to persist for a while. Generally, historical volatility and implied volatility are two most commonly adopted ways to estimate the option volatility. Most studies indicate that implied volatility contains abundant information contents and has significant predictive power over future volatility. For example, Lamoureux and Lastrapes (1993) find that implied volatility indeed reflects market information 1.

(7) effectively. Poon and Granger (2003) discover that implied volatility best predicts future volatility because compared to other methods; implied volatility contains more relevant information. Knowing the way in which the price volatilities of TAIEX Options change is helpful to one’s understanding of Taiwan stock market since both are closely intertwined. The primary purpose of this paper is to examine the volatility of TAIEX Call Options. 政治大. and analyze the relationships between option premium and time value and see how. 立. investors value future Taiwan stock market. In this paper, we test two variables, option. ‧ 國. 學. premium and time value, for TAIEX Call Options since both are crucial in analyzing. ‧. equity index options. Premium represents the valuation of an option, and it is composed. sit. y. Nat. io. n. al. er. of time value and intrinsic value. Time value is the amount by which the premium of an. v. option exceeds the intrinsic value, and it is also a signal of the implied volatility of the. Ch. engchi. i n U. underlying asset. According to Su et al. (2006), the two-state Markov-switching volatility model outperforms Black-Scholes and CRR as used for pricing index options. Therefore, we choose the two-state Markov-switching variance model with regime hetereskedasticity to examine the volatility of TAIEX Options as well as model regime hetereskedasticity. In this paper, we adopt TAIEX Call Options data from 2004/01/02 to 2013/12/31, and concentrate on two parts. First, we characterize the volatility changes of both premium 2.

(8) and time value for TAIEX Call Options by employing the two-state Markov-switching variance model. Second, we analyze the relationships between premium and time value by examining whether their volatilities vary in a similar pattern in the same time period. Our empirical results show that first, when encountering extreme events, such as economic shocks or political controversies, the volatilities of premium and time value for TAIEX Call Options will shift to high volatility state. Second, when there is no. 政治大. significant economic event in the market, investors tend to regard rise or fall in TAIEX. 立. as overreactions and give TAIEX Call Options turning points of time values to adjust.. ‧ 國. 學. In addition to this introduction, the rest of this paper is organized as follows. Section. ‧. 2 gives the literature review. Section 3 introduces the methodology while section 4. sit. y. Nat. io. al. n. conclusion.. er. explains the data. Section 5 performs the empirical results and section 6 presents our. Ch. engchi. 3. i n U. v.

(9) 2. Literature Review A large literature has argued about the specification of disturbances. To further analyze time series data, Engle (1982) develops the AutoRegressive Conditional Hetereskedasticity (ARCH) model, in which the conditional variance is a function of past residuals. Bollerslev (1986) then extends Engle’s ARCH model to Generalized ARCH (GARCH) model by allowing the conditional variance to be a function of the. 政治大. lagged variance. However, there are some drawbacks existing in such models. Diebold. 立. (1986) and Lamoureux and Lastrapes (1990) propose that the high persistence in the. ‧ 國. 學. GARCH model may reflect structural change in the variance process.. ‧. Following this line of thought, Hamilton (1989) introduces the Markov-switching. sit. y. Nat. io. n. al. er. autoregressive model and it becomes the widely employed method in describing time. v. series data. Later on, Turner et al. (1989) examine a variety of models in which the. Ch. engchi. i n U. variance of a portfolio’s excess return depends on a state variable generated by a first-order Markov process. Hamilton and Susmel (1994) apply Markov-switching ARCH (SWARCH) model, which incorporates the features of both Hamilton’s (1988, 1989) switching-regime model and Engle’s (1982) ARCH model, to model the high persistence of variance. Cai (1994) parameterizes a similar model to analyze the volatility of US Treasury-bill yield. Ramchond and Susmel (1998) also estabalish the bivariate SWARCH model to study the relations among major stock market in the 4.

(10) world. While Markov-switching models have been successfully used to model level changes for many economic and financial time series data, many researchers apply the similar models to describe other variables, such as business cycles (Wang, 2007), aggregate output (Huang et al., 1998), exchange rates (Dueker and Neely, 2007), interest rate (Smith, 2002), crude oil market (Zou and Chen, 2013) and stock market (Turner, Startz, and Nelson, 1989).. 立. 政治大. There are some indices and theories concerning the volatility for an equity index. ‧ 國. 學. option. The option pricing theory, Black-Scholes (1973), indicates that volatility plays. ‧. an important role in determining the fair value for an option, or any derivative. sit. y. Nat. io. n. al. er. instrument with option features. The Chicago Board Options Exchange (CBOE). v. proposes the concept of Volatility Index, VXO in 1993 and VIX in 2003. Following the. Ch. engchi. i n U. same concept, the TAIEX Options Volatility Index, which applies the CBOE’s methodology to trading activity in Taiwan option market, reflects current price volatility in the market. Empirical studies discussing volatility forecasting for equity index options are as follows. Figlewski (1997) investigates how best to obtain future volatility forecasts from historical data and from implied volatility in pricing options, and suggests a hypothesis that implied volatilities from different option markets contain relatively more 5.

(11) or less information depending on whether the arbitrage trade in that market is easy or hard. Zhuang, Chang, and Wang (2003) compare the performance of predicting realized volatilities for TAIEX Options between three most commonly adopted estimation models, Historical Volatility (HV), GARCH model, and Implied Volatility (IV). They use TAIEX Options data, including contracts of expiration within one month and contracts of expiration within two months, for both call and put. The sample period is. 政治大. from 2002/3/1 to 2003/2/28 with a total of 6,723 observations. The empirical results. 立. report that first, the predicting performance of Implied Volatility for TAIEX Options is. ‧ 國. 學. better than that of Historical Volatility and GARCH model, especially for current-month. ‧. option contracts. Second, the contained information contents in both Implied Volatility. sit. y. Nat. io. n. al. er. and Historical Volatility are independent in explaining Realized Volatility, therefore,. v. adding Historical Volatility into implied volatility regression model as another regressor. Ch. engchi. i n U. improves the model performance. However, the contained information contents in GARCH model could be explained by Historical Volatility and Implied Volatility, which means adding GARCH(1,1) into implied volatility regression model is useless in enhancing the model performance. Lastly, they found that generally, taking trade volume into consideration does not improve the estimation models’ predicting performances for predicting Realized Volatility. There are other researches concerning the information content implied by option 6.

(12) volatility. For example, Mayhew and Stivers (2003) examine 50 firms with the highest option volume on the CBOE, and indicate that compared to a time-series method, the implied volatility of equity index options provides reliable incremental information about future firm-level volatility. Kuo, Chen, and Chiu (2010) explore whether there are information content in TAIEX VIX and VXO for the future volatility and the return of TAIEX, and the results show that VIX can best describe the future volatility effectively. 政治大. with a positive correlation. Backus, Chernov, and Martin (2011) further quantify the. 立. distributions of consumption growth disasters by using prices of equity index options,. ‧ 國. 學. S&P 500. First, they compare pricing kernels constructed from macro-finance and. ‧. option-pricing models. Second, they compare option prices derived from a macro-based. sit. y. Nat. io. n. al. er. model to those we observe. Lastly, they compare the distribution of consumption growth. v. estimated from international macroeconomic data with one derived from option prices.. Ch. engchi. i n U. The empirical results show that option prices are a reasonably good indicator of the likelihood of disasters in consumption growth, and the probabilities of large negative realizations of consumption growth implied by option prices are smaller than we see in international macroeconomic data. Besides, equity index options are also analyzed in portfolios. Chan, Shih (2005) investigate the correlations among the spot, futures, and options of TAIEX, and find that financial derivatives are more efficient than the underlying asset in conveying 7.

(13) information. Chan, Cheng, and Lung (2005) analyze the impact of option trading activity on implied volatility changes to returns in the S&P 500 Index futures option market. Chang (2006) investigates the lead-lag relationships among the spot, futures, and option markets in TAIEX. Su et al.(2006) apply the TAIEX Options to test the two-state volatility model with Markov process, and show that the two-state volatility model outperforms Black-Scholes and CRR as used for pricing, whether the price. 政治大. conditions are deep in-the-money or out-of-the money, and outperforms Black-Scholes. 立. and CRR as used for hedging, no matter what volatility state of the time periods (high or. ‧ 國. 學. low). Wu, Liao, and Lin (2009) also use TAIEX options to analyze option pricing under. ‧. GARCH-Lévy processes. Kuo, Chen, and Chen (2013) test whether there exists a. sit. y. Nat. io. n. al. sector index options and the TAIEX options.. Ch. engchi. er. common volatility factor and a long-run stable relationship between the electronic. i n U. v. However, recent studies concerning option time value are limited. In this paper, we concentrate on examining the volatilities of TAIEX Call Options and analyzing the relationship between option premium and time value.. 8.

(14) 3. Methodology Subsection 3-1 introduces the pricing of call options, and subsection 3-2 displays the Markov-switching variance model.. 3-1 Pricing of Call Options In this section, we first introduce the characteristics of option pricing and then the. 政治大. Black-Scholes option pricing model.. 立. The value of a call option before its expiration is called premium. The premium is the. ‧ 國. 學. price the buyer pays in order to obtain the right the option might profit in the future.. ‧. Generally, call premium can be represented as the sum of two components, the intrinsic. sit. y. Nat. io. n. al. er. value and time value, as shown in Equation (1). The intrinsic value of a call option is. v. defined as the difference between the underlying price and strike price or zero,. Ch. engchi. whichever is greatest, as shown in Equation (2).. i n U. Call Premium = Intrinsic value + Time value. (1). Intrinsic value = max ( S-K,0 ). (2). where S denotes the underlying price, and K denotes the strike price. The difference between the observed call price and its intrinsic value is called time value. Before expiration, the size of the time value depends on two variables: the time remaining and the intrinsic value. Since an option with a longer time to expiration has 9.

(15) all the characteristics of an option with a shorter duration but lasts longer, it should carry a higher price, and time value reflects this value. In other words, time value represents the amount of time the call option position has to become more profitable due to a favorable move in the underlying price. In general, the more time to expiration, the greater the time value for an option. As the expiration date nears, time value shrinks to zero, which is known as time decay.. 政治大. The other variable that affects the time value is the closeness of the strike price to the. 立. money. For an out-of-the-money call option, because the intrinsic value is zero, the call. ‧ 國. 學. premium is the time value. Hence, time value is the premium paid to buy the likelihood. ‧. that the underlying price exceeds the strike price in the future. For an in-the-money call. sit. y. Nat. io. n. al. er. option, as the underlying price increases, the intrinsic value increases, but the time value. v. decreases instead, since the likelihood of the underlying price being beneath the strike. Ch. engchi. i n U. price is getting smaller. Therefore, time value can be seen as insurance for an in-the-money call option. As for an at-the-money call option, time value reaches its maximum level and there is no intrinsic value. The implied volatility of the underlying asset is also a factor of time value, though it is hard to quantify. If the underlying asset is highly volatile, a greater degree of price change before expiration is reasonably expected, therefore, the time value will be higher. On the contrary, when the underlying asset exhibits low volatility, the time value will be 10.

(16) lower since the price of the underlying asset is not expected to move much. Consequently, we can view time value as a signal of the implied volatility of the underlying asset. Next, we introduce the Black-Scholes option pricing model. The Black-Scholes option pricing model proposed in 1973 by two economists, Fisher Black and Myron Scholes, is used for calculating the theoretical price for an European option. Taking. 政治大. variables that affect the premium into account, such as strike price, current underlying. 立. price, time until expiration, implied volatility, and risk-free interest rates, Black and. ‧ 國. 學. Scholes derive the pricing formula for call options shown in Equation (3).. ‧. Nat. sit. y. (3). io. n. al. er. where the notations are specified as follows: C: theoretical premium for a call option S : current underlying price. Ch. engchi. i n U. v. K : strike price r : risk-free interest rates T : time until option expiration : standard deviation of the underlying asset’s return ln : natural logarithm (4) 11.

(17) (5) The formula can be divided into two parts. The first part,. , shows the. expected benefit from acquiring the underlying outright. The second part, r) TN(d2), provides the present value of paying the exercise price on the expiration day. The theoretical fair value of a call option is then calculated by taking the difference between these two parts as shown in Equation (3). The standard deviation. 政治大. used in calculating call option prices represents the total. 立. volatilities of the call option prices. At the same time,. also indicates the investors’. ‧ 國. 學. valuations toward the future uncertainties of the underlying asset.. ‧ sit. y. Nat. io. n. al. er. 3-2 Markov-switching Variance Model. v. Subsection 3-2-1 introduces the Markov Chain and Markov-switching variance. Ch. engchi. i n U. model. Subsection 3-2-2 introduces the filtered probability and smoothed probability.. 3-2-1 Markov Chain and Markov-switching Variance Model In this section, we first introduce the Markov chain, and then the Markov-switching variance model as stated in Turner et al. (1989). Markov chain is a stochastic process with Markov property. Let of an observed variable, and. denote a vector. the state or regime index at time t, which is an 12.

(18) unobserved variable. Assume we have a set of states,. , and suppose. can be described by a first-order Markov chain, in that case,. (6) , where K is the number of states, The. .. 政治大. is called transition probability, which represents the probability of transitioning. 立. from state i at time t-1 to state j at time t. For instance,. means the probability of. ‧ 國. 學. going from state 1 at time t-1 to state 2 at time t. Transition probability. can also be. ‧. expressed by the following transition matrix shown in Equation (7). Note that each. sit. y. Nat. io. n. al. er. column of P sums to unity:. Ch. engchi. i n U. v. (7). The two-state Markov-switching variance model is specified as follows. (8) (9) otherwise;. (10) (11) (12). 13.

(19) (13) where. denotes the premium or time value for TAIEX Call Options,. denotes the. unobserved state variable evolving according to a first-order Markov process with transition probabilities in Equation (11), and. denotes the variance of premium or. time value for TAIEX Call Options. We define state 1 as the high volatility state and state 2 as the low volatility state, as represented in Equation (13).. 政治大. 立. 3-2-2 Filtered probability and Smoothed probability. ‧ 國. 學. In this subsection we will introduce the filtered probability and smoothed probability. ‧. io. sit. is unobservable, we can still infer the state probability at. n. al. er. Nat. Though state variable. y. since they help us to infer the state probability at time t.. v ni. any time t. Depending on the amount of information used in making inferences on. Ch. engchi U. ,. we have filtered probabilities and smoothed probabilities. Filtered probabilities, represented as. , refer to inferences about. conditional on information up to. time t:. , and smoothed probabilities, represented as. about. conditional on all the information in the sample:. , refer to inferences .. To obtain filtered probability and smoothed probability, first, we consider the joint density of. and the unobserved. : (14) 14.

(20) where. refers to information up to time. Next, we obtain the marginal density of. .. ,: (15). 政治大. We adopt maximum likelihood estimation method and the log likelihood function is. 立. then given by. ‧ 國. 學 (16). ‧. With the following transition probabilities:. er. io. sit. y. Nat. n. al. we can obtain. :. Ch. engchi U. v ni. (17) (18). (19). Lastly, by applying Bayes’ theorem, we obtain filtered probabilities: (20). 15.

(21) where. , and. .. Given an initial probability, j. , we can obtain filtered probabilities,. by iterating Equation (19) and (20). Consider the joint probability that. and. based on full information,. , (21). 立. 政治大. ‧. ‧ 國. 學. n. al. Given an initial probability,. Ch. er. io. sit. y. Nat. and. ,. v ni. .. (22). e, nthegiterations chi U for. done to derive smoothed probabilities,. ,. 16. can be ..

(22) 4. Data In the following content, subsection 4-1 introduces the data, subsection 4-2, 4-3, and 4-4 describe the statistic characteristics of TAIEX, premium for TAIEX Call Options, and time value for TAIEX Call Options respectively.. 4-1 Data Introduction. 政治大. The Taiwan Futures Exchange is the organization that aims to invigorate futures. 立. trading while serving the real economy in Taiwan futures market. It enables traders to. ‧ 國. 學. develop a solid understanding of the Taifex products and offers hedging vehicles for. ‧. investors. Many different kinds of products are included in Taifex products, such as. sit. y. Nat. io. n. al. er. TAIEX Futures, Equity Options, and TAIEX Options. In this paper, we choose equity. v. index call options, TAIEX Call Options, as our target. For TAIEX Options, the. Ch. engchi. i n U. underlying asset is Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX), and the exercise style is European style. In this study, we choose premium and time value for at-the-money TAIEX Call Options as our research objectives and the data is collected from Taiwan Economic Journal (TEJ) database. In order to employ at-the-money TAIEX Call Options, we choose current-month call option contracts for each month with its strike price closest to the monthly average of the TAIEX of that month. The premium data is directly obtained 17.

(23) from TEJ, and time value data is calculated by subtracting TAIEX from the sum of strike price and premium. The specification of how we obtain time value data will be elaborated later in subsection 4-4. The sample period is from 2004/01/02 to 2013/12/31 with daily frequency, and the total sample number is 2,479 for premium and time value. Figure 1 shows the actual time series for intrinsic value. Though we intend to employ at-the-money TAIEX Call Options by choosing current-month call option contracts for. 政治大. each month with its strike price closest to the monthly average of the TAIEX of that. 立. month, there are still some inaccuracies within this measure, causing the intrinsic values. ‧ 國. 學. not necessarily equal to zero. But the inaccuracies are minor and acceptable.. ‧ sit. y. Nat. io. n. al. er. 4-2 Statistic Characteristics of TAIEX. v. Table 1 presents the descriptive statistics of TAIEX from 2004/01/02 to 2013/12/31.. Ch. engchi. i n U. We find that TAIEX data is left-skewed with Skewness smaller than 0 and platykurtic with Kurtosis smaller than 3. For the value of Jarque-Bera, we can reject the hypothesis of normal distribution. To further understand the historical performance of the Taiwan stock market, Figure 2 shows the time series of TAIEX from 2004/01/02 to 2013/12/31. From Figure 2, we can observe that during 2004 to 2007, the Taiwan stock market was in a bull market trend. In October, 2007, the TAIEX reached its maximum, 9809.88, among the sample period. 18.

(24) Intrinsic Value 1,000 900 800 700 600 500 400 300 200 100 0. 2006. 2007. 立. 政治大 2008. 2009. 2010. 2012. ‧ 國. 學. Daily Intrinsic Value for TAIEX Call Options Series. ‧. Nat. from 2004/01/02 to 2013/12/31. io. sit. Figure 1. 2011. y. 2005. n. al. er. 2004. Ch. engchi. 19. i n U. v. 2013.

(25) Table 1. Descriptive Statistics of TAIEX from 2004/01/02 to 2013/12/31. Observations. 2479. Mean. 7265.653. Median. 7439.960. Maximum. 9809.880. 治政 4089.930 大. Minimum. 立. -0.394333 2.728927. Jarque-Bera (Probability). ‧. Kurtosis. 學. Skewness. 1133.813. ‧ 國. Std. Dev.. 71.83684 (0.000000). n. er. io. sit. y. Nat. al. Ch. 10,000. e n TAIEX gchi. i n U. v. 9,000. 8,000. 7,000. 6,000. 5,000. 4,000. 2004. 2005. 2006. 2007. 2008. 2009. 2010. 2011. Figure 2 Daily TAIEX from 2004/01/02 to 2013/12/31. 20. 2012. 2013.

(26) After the Global Financial Crisis aroused in 2008, the Taiwan stock market deteriorated and changed to a bear market. In November, 2008, the TAIEX plunged and dropped to its minimum level, 4089.93, among the sample period.. 4-3 Statistic Characteristics of Premium for TAIEX Call Options Table 2 presents the descriptive statistics of premium for TAIEX Call Options. We. 政治大. find that the data is right-skewed with Skewness larger than 0 and leptokurtic with. 立. Kurtosis larger than 3. For the value of Jarque-Bera statistic, we can reject the. ‧ 國. 學. hypothesis of normal distribution.. ‧. Figure 3 presents the time series of premium for TAIEX Call Options from. sit. y. Nat. io. n. al. er. 2004/01/02 to 2013/12/31. Among the sample period, there exist two extreme values.. v. One is 900 on 2007/11/01 and the other is 915 on 2011/08/01. The reason for these two. Ch. engchi. i n U. extreme values is due to the way we select the data. From October, 2007 to November, 2007, the TAIEX plunged greatly because of the US subprime mortgage crisis. Hence, the strike price we choose for the call option contracts falls from 9600 in October, 2007 to 8800 in November, 2007, and the sudden decline in the strike price makes the premium soar from 380 in 2007/10/31 to 900 in 2007/11/01. The second extreme value, 915 on 2011/08/01, is due to the same reason as well, and this time period corresponds to the global stock market crash led by the European sovereign debt crisis. 21.

(27) Table 2. Descriptive Statistics of Premium for TAIEX Call Options from 2004/01/02 to 2013/12/31. Observations. 2479. Mean. 180.4041. Median. 158.0000. Maximum. 915.000. Minimum. 9.0000. 治政 112.0199 大. Std. Dev.. 立. 學. Kurtosis. 1.493150. ‧ 國. Skewness. 6.682412. Jarque-Bera (Probability). 2321.805 (0.000000). ‧. PREMIUM. n. al. 900. Ch. 800. engchi. er. io. sit. y. Nat. 1,000. i n U. v. 700 600 500 400 300 200 100 0. 2004. 2005. 2006. 2007. 2008. 2009. 2010. 2011. 2012. Figure 3 Daily Premium for TAIEX Call Options Series from 2004/01/02 to 2013/12/31. 22. 2013.

(28) 4-3-1 Q-Q plot of Premium for TAIEX Call Options Q-Q plot is a graphical method to compare two distributions by plotting their quantiles against each other. If the two distributions are identical, the points in the Q-Q plot will approximately lie on the 45° line y=x. Here we use Q-Q plot to compare the distribution of the sample data to two theoretical distributions, Normal distribution and Student-t distribution. Figure 4 shows the Q-Q plots of premium for TAIEX Call. 政治大. Options against Normal and Student-t distribution. We can observe that the premium for. 立. TAIEX Call Options is fat-tailed compared to Normal and Student-t distribution with. ‧. ‧ 國. 學 y. Nat. io. n. al. er. 4-3-2 Unit Root Test of Premium for TAIEX Call Options. sit. degree 5.. v. A unit root test tests whether a sequence of time series data is stationary by using an. Ch. engchi. i n U. autoregressive model. Here we use Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test to examine whether premium and time value data for TAIEX Call Options are stationary. Table 3 presents the ADF test and PP test of premium for TAIEX Call Options. We find that both null hypotheses that premium has a unit root can be rejected under 1% significance level. In other words, there is no unit root in the series of premium data.. 23.

(29) 600. Quantiles of Normal. 500. 400. 300. 200. 100. 0. -100. -200. 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1,000. 1,100. 900. 1,000. 1,100. Quantiles of PREMIUM 1,000. 立. 600. 400. 政治大. -400. 0. 100. 200. 300. 400. 500. 600. 700. 800. Quantiles of PREMIUM. io. Q-Q plots against Normal and Student-t(5) distribution of Premium for TAIEX Call Options. Table 3. al. n. Figure 4. Nat. -600. y. -200. sit. 0. er. 200. ‧. ‧ 國. 學. Quantiles of Student's t. 800. Ch. engchi. i n U. v. ADF Test and PP Test of Premium for TAIEX Call Options. ADF Test Statistic. PP Test Statistic. -14.24091. 1% Critical value. -3.432795. 5% Critical value. -2.862506. 10% Critical value. -2.567329. 1% Critical value. -3.432795. 5% Critical value. -2.862506. 10% Critical value. -2.567329. -14.19782. 24.

(30) 4-4 Statistic Characteristics of Time Value for TAIEX Call Options Since TEJ doesn’t provide time value data for TAIEX Call Options directly, we obtain time value data by applying Equation (24) derived from Equation (23). P= max(S-K,0) + TV. (23). TV=P–max(S-K,0). (24). where P denotes premium for TAIEX Call Options, S the TAIEX, K the strike price, and TV the time value.. 政治大. 立. Table 4 presents the descriptive statistics of time value for TAIEX Call Options. We. ‧ 國. 學. find that the data is also right-skewed with Skewness larger than 0 and leptokurtic with. ‧. Kurtosis larger than 3. About the value of Jarque-Bera statistic, we can reject the. sit. y. Nat. io. n. al. er. hypothesis of normal distribution.. v. Figure 5 displays the time series of time value for TAIEX Call Options. The two. Ch. engchi. i n U. highest values among the sample period are 441 on 2009/05/04 and 423 on 2008/03/20. The minimum value among the sample period is -84.36 on 2012/07/03.. 25.

(31) Table 4. Descriptive Statistics of Time Value for TAIEX Call Options from 2004/01/02 to 2013/12/31. Observations. 2479. Mean. 116.6487. Median. 108.5400. Maximum. 441. Minimum. -84.36000. 治政 67.30343 大 1.003969. Std. Dev.. 立. Skewness. ‧ 國. 4.885771. Jarque-Bera (Probability). 783.7716 (0.000000). ‧. Nat. al. n. 300. y. sit. io. 400. Time Value. er. 500. 學. Kurtosis. Ch. 200. engchi. i n U. v. 100. 0. -100. 2004. 2005. Figure 5. 2006. 2007. 2008. 2009. 2010. 2011. 2012. Daily Time Value for TAIEX Call Options Series from 2004/01/02 to 2013/12/31. 26. 2013.

(32) 4-4-1 Q-Q plot of Time Value for TAIEX Call Options Figure 6 shows the Q-Q plots of time value for TAIEX Call Options against Normal and Student-t distribution. We can perceive that the time value for TAIEX Call Options is also fat-tailed compared to Normal and Student-t distribution with degree 5.. 4-4-2 Unit Root Test of Time Value for TAIEX Call Options. 政治大. Table 5 presents the ADF test and PP test of time value for TAIEX Call Options. We. 立. find that both null hypotheses that time value has a unit root can be rejected under 1%. ‧ 國. 學. significance level, indicating that there is no unit root in the time value series. Since the. ‧. series for premium data and time value data are both stationary, we can continue to. sit. y. Nat. io. n. al. er. process our analysis.. Ch. engchi. 27. i n U. v.

(33) 600 500. Quantiles of Normal. 400 300 200 100 0 -100 -200 -300 -100. -50. 0. 50. 100. 150. 200. 250. 300. 350. 400. 450. 500. 550. 600. 650. 700. 750. 800. 650. 700. 750. 800. Quantiles of Time Value 1,000. 政治大. 立. 600. 400. 0. Nat 50. 100. 150. 200. 250. 300. 350. 400. 450. 500. Quantiles of Time Value. io. al. 550. 600. Q-Q plots against Normal and Student-t(5) distribution of Time Value. Table 5. n. Figure 6. -50. y. -400. sit. -200. er. 0. ‧. ‧ 國. 200. -600 -100. 學. Quantiles of Student's t. 800. i n C for TAIEX Options h e n Call gchi U. v. ADF Test and PP Test of Time Value for TAIEX Call Options. ADF Test Statistic. PP Test Statistic. -4.087185. 1% Critical value. -3.432814. 5% Critical value. -2.862514. 10% Critical value. -2.567334. 1% Critical value. -3.432795. 5% Critical value. -2.862506. 10% Critical value. -2.567329. -14.011558. 28.

(34) 5. Empirical Results Section 5 first presents the characteristics of time value observed from the time series data, then presents our empirical results. Subsection 5-1 shows the characteristics of time value for TAIEX Call Options. Subsection 5-2 and 5-3 show the estimation models and estimated smoothed probabilities under different regimes of premium and time value for TAIEX Call Options respectively. Subsection 5-4 compares the empirical. 政治大. results of both premium and time value for TAIEX Call Options. Subsection 5-5. 立. presents the analysis and comparison between our results and results from other related. ‧ 國. 學. literature. Subsection 5-6 shows the goodness of fit. The estimation of parameters and. ‧. smoothed probabilities under different regimes are conducted by software Eviews 8.. n. Ch. engchi. er. io. sit. y. Nat. al. 5-1The characteristics of Time Value Data. i n U. v. Before discussing the empirical results of time value, a closer look at the time series data of time value was taken to find out the characteristics of time value for TAIEX Call Options by observing Figure 5. Referring to Figure 5, most of the time value data are positive, while some are negative in certain time periods. Since negative time values are different from ordinary, we wonder their meaning behind. The times when negative time values occur are shown in Table 6. Figure 7-A shows the corresponding rising or falling margin of TAIEX(Δ S) 29.

(35) Table 6. 3. 160.97. 33.03. 2004/8/26. 5,813.39. 311. 313.39. -2.39. 2004/8/27. 5,797.71. 340. 297.71. 42.29. 2005/5/27. 5,991.55. 109. 91.55. 17.45. 2005/5/30. 6,009.52. 109. 109.52. -0.52. 2005/5/31. 6,011.56. 100. 111.56. -11.56. 2005/6/1. 5,971.62. 20.5. 0.00. 20.50. 2005/6/17. 6,293.56. 97. 93.56. 3.44. 6,296.89 政治95 大 96.89 6,278.46 93 78.46. 立 8,892.83 2007/6/28. -1.89 14.54. 304. 292.83. 11.17. 273. 283.21. 2007/7/2. 8,939.49. 49. 學. -10.21. 0.00. 49.00. 2010/6/23. 7,582.15. 188. 182.15. 5.85. 2010/6/24. 7,589.89. 181. 189.89. -8.89. 2010/6/25. 7,474.71. 122. 74.71. 47.29. 2010/7/28. 7,784.81. 206. 184.81. 21.19. 2010/7/29. 7,798.99. 192. 198.99. -6.99 10.37. 2011/6/1. 160.63 v i n C 8,988.84 h e n g c h171i U 88.84 9,062.35 335 362.35. 2011/6/2. 8,991.36. 293. 291.36. 1.64. 2011/6/3. 9,046.28. 322. 346.28. -24.28. 2011/6/7. 9,057.10. 340. 357.10. -17.10. 2011/6/8. 9,007.53. 289. 307.53. -18.53. 2011/6/9. 9,000.94. 298. 300.94. -2.94. 2011/6/10. 8,837.82. 181. 137.82. 43.18. 2011/6/13. 8,712.95. 126. 12.95. 113.05. 2011/6/14. 8,829.21. 157. 129.21. 27.79. 2011/6/15. 8,831.45. 131. 131.45. -0.45. 2011/6/16. 8,654.43. 85. 0.00. 85.00. 2011/7/5. 8,784.44. 96. 84.44. 11.56. ‧ 國. 8,883.21. n. al. 2011/5/31. 7,760.63. 171. 30. ‧. 2007/6/29. io. 8. Value. 194. 2010/7/30 7. Value. Nat. 6. Time. 5,660.97. 2005/6/21. 5. Intrinsic. 2004/8/23. 2005/6/20 4. Premium. y. 2. TAIEX. sit. 1. Date. er. No.. Negative time values from 2004/01/02 to 2013/12/31. 82.16 -27.35.

(36) 10. 8,824.44. 110. 124.44. -14.44. 2011/7/7. 8,773.42. 110. 73.42. 36.58. 2012/1/30. 7,407.41. 241. 207.41. 33.59. 2012/1/31. 7,517.08. 309. 317.08. -8.08. 2012/2/1. 7,549.21. 52. 0.00. 52.00. 2012/6/19. 7,273.13. 174. 173.13. 0.87. 2012/6/20. 7,334.63. 197. 234.63. -37.63. 2012/6/21. 7,279.05. 154. 179.05. -25.05. 2012/6/22. 7,222.05. 111. 122.05. -11.05. 2012/6/25. 7,166.38. 73. 66.38. 6.62. 2012/6/26. 7,137.93. 67. 37.93. 29.07. 2012/6/27. 7,183.01. 79 83.01 治政 7,169.61 82 69.61 大 7,296.28 128 196.28. -4.01. 2012/6/28. 立 7,345.16 2012/7/2. 2012/6/29. 145.16. -45.16. 218.36. 2012/7/4. 7,422.59. 159. 學. -84.36. 222.59. -63.59. 2012/7/5. 7,387.78. 162. 187.78. -25.78. 2012/7/6. 7,368.59. 146. 168.59. -22.59. 2012/7/9. 7,309.96. 126. 109.96. 2013/5/30. 8,243.29. 61. 8,201.02. 199. y. 16.04. sit. ‧ 國. 134. ‧. 7,418.36. 0.00. 61.00. 201.02. -2.02 -16.22. 2013/6/6. 191.22 v i n C 8,181.91 h e n g c h168i U 181.91 8,096.14. 143. 96.14. 46.86. 2013/6/7. 8,095.20. 135. 95.20. 39.80. 2013/6/10. 8,160.55. 155. 160.55. -5.55. 2013/6/11. 8,116.15. 151. 116.15. 34.85. 2013/6/27. 7,883.90. 18. 0.00. 18.00. 2013/6/28. 8,062.21. 29. 62.21. -33.21. 2013/7/1. 8,036.00. 47. 0.00. 47.00. io. 2013/6/3. n. al. 2013/6/4 2013/6/5. 12. -68.28. 2012/7/3. Nat. 11. 12.39. 100. 8,191.22. 175. 31. er. 9. 2011/7/6. -13.91.

(37) ΔP and ΔS 250 200 150 100 50 0 -50 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 -100 -150 -200 -250 -300. 立. ΔP ΔS. 政治大. Figure 7-A Variation of Premium and TAIEX from 2004 to 2013. ‧ 國. y. al. n. 0. sit. io. 50. er. 100. -100. ‧. Nat. 150. -50. 學. 200. ΔS and TV. 1. 4. i n U. v. 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61. Ch. engchi. -150 -200. Figure 7-B Corresponding Time values and Variation of TAIEX from 2004 to 2013. 32. TV ΔS.

(38) and premium(Δ P) in Table 6. Figure 7-B shows the corresponding time values and Δ S in Table 6. From Figure 7-A, we find that basically, Δ P andΔ S move in same directions. There is a positive relationship between TAIEX and premium because the higher the TAIEX, the more advantages the call buyers possess, thus the higher the call premium. From Figure 7-B, we perceive that roughly speaking, changing margin of TAIEX (Δ S) and time value (TV) move in opposite directions. When Δ S is up, which. 政治大. means TAIEX accelerates, time value becomes negative. The correlation coefficient. 立. between Δ S and TV is -0.35. To see why time values turn negative when TAIEX rises,. ‧ 國. 學. a closer look at the data in Table 6 was taken. From Table 6, we find that some negative. ‧. time values appear at the end of the month. For example, on 2005/5/31, 2007/6/29,. sit. y. Nat. io. n. al. er. 2012/1/31, and 2013/6/28, the time values are all negative. Since the TAIEXs on these. v. days did not change much, the causes of the negative time values have little to do with. Ch. engchi. i n U. TAIEX. Therefore, we presume the causes of the negative time values are related to MSCI Taiwan Index Futures. Because the settling day of MSCI Taiwan Index Futures lies at the end of the month, investors expect TAIEX to be depressed when approaching the end of the month. Therefore, investors are willing to give only small rises to premiums for TAIEX Call Options by giving negative time values to adjust. For other negative time values, we further examine their turning points, whether changing from positive to negative or from negative to positive, to discuss the 33.

(39) motivation and information behind. After observing the data in Table 6 closely, we find that most of the time, the reason for the time values to change from positive to negative is because the investors are pessimistic in the rise of TAIEX. For example, referring to Table 6, from 2004/8/23 to 2004/8/26, TAIEX was up 153 points, while premium was up only 117 points. We infer that when TAIEX goes up, investors pessimistically do not think the rise of TAIEX would continue, hence, investors are willing to give only a. 政治大. slight rise in premium by giving negative time values to adjust, and then rising margin. 立. of TAIEX is greater than the rising margin of premium. Negative time values will adjust. ‧ 國. 學. to positive time values as TAIEX goes down later. For example, referring to Table 6, as. ‧. TAIEX went down, time values from 2011/06/03 to 2011/06/09 increased gradually,. sit. y. Nat. io. n. al. er. adjusting to positive time values again on 2011/06/10.. v. For turning points of time values changing from negative to positive, we infer that. Ch. engchi. i n U. when TAIEX goes down, investors do not believe that the fall of TAIEX would continue, and do not want premium to fall too much, so investors raise time values to keep premium from slipping, resulting in less fall margins in premium than in TAIEX. For example, referring to Table 6, from 2010/6/24 to 2010/6/25, TAIEX dropped 115 points while premium dropped only 59 points, with time value changing from negative to positive. Since time values represent the investors’ expectations and attitudes toward the future 34.

(40) underlying price, together with the above mentioned, we can draw a conclusion that except for the ones that occurred at the end of the month, the occurrence of turning points of time values means that the investors’ expectations toward the future TAIEX are not matching to the current trend of TAIEX. Instead, investors hold expectations which are contrary to the current TAIEX trend toward the future TAIEX. This also explains the negative correlation between Δ S and time value in Figure 7-B.. 政治大. 立. 5-2The estimation model and probability under different regimes of Premium for. ‧ 國. 學. TAIEX Call Options. ‧. For premium for TAIEX Call Options, the result of the estimation model is shown. sit. y. Nat. io. n. al. er. below, with standard errors in parentheses.. Ch. engchi U. v ni. (25) (26). (27) otherwise; where. denotes the premium for TAIEX Call Options,. (28) the unobserved state. variable which evolves according to a first order Markov process with transition probabilities in Equation (27), and. the variance of the premium for TAIEX Call. Options. 35.

(41) From the results of the transition probabilities shown in Equation (27), we find that is obviously greater than. , and. is also greater than. , implying that the. series in one state at time t will tend to stay in the same state at time t+1. Next, we examine the estimated smoothed probabilities under different regimes from the following figures, Figure 8-A and Figure 8-B. Figure 8-A and 8-B show the estimation of smoothed probabilities under different regimes of premium for TAIEX. 政治大. Call Options. In Figure 8-A which shows the smoothed probabilities under state 1, the. 立. high volatility state, we can observe that the data is of high volatility during some. ‧. ‧ 國. 學. periods.. The first high volatility period is the first half of year 2004. After checking the series. sit. y. Nat. io. n. al. er. data from Figure 3, we find that the premium for TAIEX Call Options increased. v. significantly and then dropped markedly during this period. The fluctuations were. Ch. engchi. i n U. affected by the 2004 Presidential Election in Taiwan and the tense relationship between Taiwan and China. Referring to Figure 2, the political events led to a huge drop in TAIEX, therefore affecting the premium for TAIEX Call Options. The second high volatility period is about from May, 2006 to June, 2006. The period corresponds to the global stock market crash triggered by the uncertain interest rate policy announced by Fed. After the inflation rate increased, the investors’ worry whether Fed will raise the interest rate or not raised, together with other pressures and 36.

(42) Smoothed Regime Probabilities P(S(t)= 1) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0. 2004. 2005. 2006. 2007. 立. P(S(t)= 2). io. 0.4. al. n. 0.3 0.2 0.1 0.0. 2004. 2005. 2006. 2013. sit. 0.5. 2012. er. 0.6. 2011. y. Nat. 0.7. 2010. Smoothed Regime Probabilities. ‧ 國. 0.8. 2009. ‧. 0.9. 2008. 學. 1.0. 政治大 Figure 8-A. Ch. 2007. engchi 2008. 2009. i n U. v. 2010. 2011. 2012. Figure 8-B Figure 8-A, 8-B Estimation of Smoothed Probabilities of Premium for TAIEX Call Options. 37. 2013.

(43) factors, the confidence crisis bursted, hence resulting in the global stock market crash. The third high volatility period starts from the second half of year 2007 to 2010, also known as 2008 Financial Crisis. The US housing bubble which emerged in 2007 and Lehman Brothers’ bankruptcy in September, 2008 led to severe Financial Crisis. According to Figure 8-A, the smoothed regime probabilities under state 1 seem to be highly volatile from 2008 to 2010, indicating that the states the premiums for TAIEX. 政治大. Call Options are in switch frequently between high volatility state and low volatility. 立. state. Referring to Figure 3, we observe that during 2008 Financial Crisis, the premiums. ‧ 國. 學. for TAIEX Call Options fluctuate and sometimes reach a relatively higher degree.. ‧. The fourth high volatility period is from August, 2011 to December, 2011. This period. sit. y. Nat. io. n. al. er. corresponds to the global stock market crash, which was caused by the fear of contagion. v. of European sovereign debt crisis and the downgrading of the US credit rating. Aside. Ch. engchi. i n U. from the periods the above mentioned, the rest of the periods remain low volatility according to Figure 8-A. Generally, we can conclude that when an economic shock or political controversy happens, the states the premiums for TAIEX Call Options are in will shift to state 1, the high volatility state.. 38.

(44) 5-3 The estimation model and probability under different regimes of Time value for TAIEX Call Options For time value of TAIEX Call Options, the result of the estimation model is shown below, with standard errors in parentheses. (29) (30). 立. (31). otherwise;. (32). ‧ 國. 學. where. 政治大. denotes the time value for TAIEX Call Options,. the unobserved state. ‧. variable which evolves according to a first order Markov process with transition. sit. y. Nat. io. al. n. Options.. the variance of the time value for TAIEX Call. er. probabilities in Equation (31), and. Ch. engchi. i n U. v. From the results of transition probabilities shown in Equation (31), we find that is substantially greater than. , and. is also greater than. , implying that the. series in one state at time t will tend to stay in the same state at time t+1. Next, we examine the estimated smoothed probabilities under different regimes from the following figures, Figure 9-A and Figure 9-B. Figure 9-A and 9-B show the estimated smoothed probabilities under different regimes of time value for TAIEX Call Options. From Figure 9-A, we can observe that 39.

(45) Smoothed Regime Probabilities P(S(t)= 1) 1.0. 0.8. 0.6. 0.4. 0.2. 0.0. 2004. 2005. 2006. 2007. 立. 2008. P(S(t)= 2). 0.0. 2004. 2005. 2012. 2013. y. sit. n. al. er. io. 0.2. 2013. Smoothed Regime Probabilities. Nat. 0.4. 2012. ‧. 0.6. 2011. 政Figure治 9-A 大. ‧ 國. 0.8. 2010. 學. 1.0. 2009. 2006. 2007. Ch. 2008. 2009. eFigure n g c9-Bh i. i n U 2010. v. 2011. Figure 9-A, 9-B Estimation of Smoothed Probabilities of Time value for TAIEX Call Options. 40.

(46) the time values for TAIEX Call Options are of high volatility during three main periods. The first high volatility period is the first half of year 2004, which was influenced by the 2004 Presidential Election in Taiwan. Referring to Figure 5, the time values for TAIEX Call Options from April, 2004 to August, 2004 fluctuate in a descending trend, implying that the unstable political factors affect the investors’ confidence in the securities market.. 政治大. The second high volatility period is from the second half of year 2007 to the end of. 立. 2009. This period corresponds to the 2008 Financial Crisis. After checking Figure 5, we. ‧ 國. 學. find that around 2008 Financial Crisis, the time values for TAIEX Call Options fluctuate. ‧. greatly, which implies that during an economic crisis, investors’ confidence in securities. n. al. er. io. sit. y. Nat. market diminish.. v. The third high volatility period is from August, 2011 to April, 2012. This period. Ch. engchi. i n U. corresponds to the global stock market crash attributed to European Sovereign Debt Crisis. Referring to Figure 5, the time values for TAIEX Call Options rise during the period. Excluding the periods the above mentioned, the rest of the time values for TAIEX Call Options in the sample period remain low volatility according to Figure 9-A. In general, we conclude that when an economic shock or political controversy happens, time values for TAIEX Call Options rise and the states the time values for TAIEX Call Options are in shift to high volatility state, suggesting that market participants’ attitude 41.

(47) toward securities market become conservative and pessimistic due to the rise of the underlying assets’ future uncertainties.. 5-4 Comparison of Premium and Time Value for TAIEX Call Options In this section, we compare the results of both premium and time value for TAIEX Call Options by observing the figures of the estimated smoothed regime probabilities. 政治大. for the high volatility state, Figure 8-A and Figure 9-A.. 立. From Figure 8-A and Figure 9-A, it is obvious to see that the estimated smoothed. ‧ 國. 學. regime probabilities for premiums are much more volatile than that for time values for. ‧. TAIEX Call Options. The estimated smoothed regime probabilities for time values seem. sit. y. Nat. io. n. al. er. to be steadier and smoother. This outcome also signifies that the two-state. v. Markov-switching variance model is very suitable for explaining time values for TAIEX. Ch. engchi. i n U. Call Options. By comparing the volatility states in Figure 8-A and Figure 9-A, we perceive that the figures of the estimated smoothed regime probabilities of both premium and time value for TAIEX Call Options are almost consistent. The time periods that are of high volatility are nearly the same, which means that when encountering a certain economic or political shock, the volatilities of both premium and time value for TAIEX Call Options react in the same way, shifting from low volatility state to high volatility state. The only noticeable difference between Figure 8-A and 9-A 42.

(48) Smoothed Regime Probabilities P(S(t)= 1) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0. 2004. 治 of Premium for TAIEX Call 政 Probabilities Estimation of Smoothed 大立 under high volatility state Options. 2005. 2007. 1.0. 2009. 2010. Smoothed Regime Probabilities P(S(t)= 1). sit. io. al. n. 0.2. 0.0. 2004. 2005. 2013. er. 0.4. 2012. y. Nat. 0.6. 2011. ‧. 0.8. 2008. 學. ‧ 國. Figure 8-A. 2006. 2006. Ch. 2007. engchi 2008. 2009. i n U. v. 2010. 2011. 2012. 2013. Figure 9-A Estimation of Smoothed Probabilities of Time Value for TAIEX Call Options under high volatility state. 43.

(49) is the time period when the 2006 global market crash happened. During 2006 global stock market crash, the estimated smoothed regime probabilities of premium for TAIEX Call Options shift to high volatility state, while that of time value for TAIEX Call Options still stay in the low volatility state. Therefore, we list out the major economic and political events which influence the volatilities of premium and time value for TAIEX Call Options chronologically in the following table, Table 7.. 立. Table 7. 政治大 Economic or Political Events. ‧ 國. 學. Year. Events Taiwan’s Presidential Election. n. al. Financial Crisis. Ch. engchi. er. io 2011. Global Stock Market Crash. sit. y. Nat. 2006 2008. ‧. 2004. i n U. v. Global Stock Market Crash. Taiwan stock market is closely related with the election events. On March 19, 2004, the day before Taiwan’s Presidential Election, President Chen Shui-bian was assassinated while he was campaigning, known as the 319 shooting incident. The incident made the market participants panic, and caused a substantial fall in TAIEX. Since Taiwan stock market is closely linked with the US stock market, TAIEX will be 44.

(50) affected whenever there is a stock market crash in the US. The global stock market crash in 2006 was due to Japan because Bank of Japan imposed a tight money policy in an attempt to control the inflation in May, 2006, and this policy made the hot money in the world decrease substantially. Later on, the unexpected policy of raising the interest rates announced by Fed triggered the global stock market crash, and the crash started in emerging markets. The Financial Crisis in 2008 started when Lehman Brothers. 政治大. collapsed, and caused a sharp decline in TAIEX. In August, 2011, the global stock. 立. market crash was caused by the fear of contagion of the European sovereign debt crisis,. ‧ 國. 學. and the concerns for the downgrading of the US credit rating.. ‧. Referring to Table 7, we see that when an economic or political shock happens,. sit. y. Nat. io. al. n. volatility state.. er. volatilities of both premium and time value for TAIEX Call Options change to high. Ch. engchi. i n U. v. 5-5 Analysis In this section, we analyze our empirical results and compare our results with results in other related literature. In this research, we particularly derive the time value data by applying Equation (24), observe the characteristics of time value from the time series data in Figure 5, and examine not only the volatilities of equity index option premium, but also the 45.

(51) volatilities of time value since time value matters in analyzing options. The consistency of Figure 8-A and Figure 9-A, the estimated smoothed regime probabilities of both premium and time value for TAIEX Call Options, also indicates that the implied information for both premium and time value are consistent. In subsection 5-1, we conclude that the occurrence of turning points of time values mean that the investors’ expectations toward the future TAIEX are contrary to the. 政治大. current trend of TAIEX. After comparing time values’ time series data and the estimated. 立. smoothed regime probabilities, Figure 5 and Figure 9-A, we find that negative time. ‧ 國. 學. values and turning points mostly occur in low volatility periods. Hence, we can reach a. ‧. conclusion that when in low volatility period; investors’ expectations toward the future. sit. y. Nat. io. n. al. er. TAIEX are against the current TAIEX trend by giving turning points of time values. In. v. other words, when there is no significant economic event in the market, investors tend. Ch. engchi. i n U. to regard rise or fall in TAIEX as overreactions and give TAIEX Call Options turning points of time values to adjust. On the contrary, when there are major economic events happening, investors raise the time values since the uncertainties of the future TAIEX increase, as shown in Figure 5. When examining the volatilities of premium and time value respectively, eyeballing the time series data obtains the same result as the result of applying the two-state Markov-switching variance model. The time periods which are in high volatility state 46.

(52) TV 500. 400. 300. 200. 100. 0. -100. 2004. 2005. 2006. 2007. 2008. 2009. 2010. 2011. 2012. 2013. 2012. 2013. 政治大 Daily Time Value for TAIEX Call Options Series 立. Figure 5. Smoothed Regime Probabilities P(S(t)= 1). ‧. 1.0. 學. ‧ 國. from 2004/01/02 to 2013/12/31. 0.8. sit. n. al. er. io. 0.4. y. Nat. 0.6. 0.2. 0.0. 2004. 2005. 2006. Ch 2007. engchi 2008. 2009. i n U. v. 2010. 2011. Figure 9-A Estimation of Smoothed Probabilities of Time Value for TAIEX Call Options under high volatility state. 47.

(53) are consistent. For example, the high volatility periods estimated by observing Figure 3 are consistent to the results estimated by the two-state Markov-switching variance model, as shown in Figure 8-A. This fact proves that it is adequate to apply the two-state Markov-switching variance model to premiums and time values of TAIEX Call Options. Moreover, our results suggest that there would be loss information if we do not classify the volatilities into two states and assume the volatilities exhibit only one state.. 立. 政治大. In Section 2, Literature Review, we introduced Zhuang, Chang, and Wang (2003),. ‧ 國. 學. and they report that the predicting performance of Implied Volatility is better than. ‧. Historical Volatility and GARCH model, especially Implied Volatility derived from. sit. y. Nat. io. n. al. er. current-month TAIEX Options. In this paper, we also adopt current-month TAIEX. v. Options and implied volatility because the premium data input in Markov-switching. Ch. engchi. i n U. variance model is derived from Black-Scholes option pricing model. This similarity enhances the credibility of our results as well. In this study, we differentiate different economic states through prices of TAIEX Call Options, and classify the prices into two states, high and low, by employing the Markov process with a two-state volatility regime. We regard the high volatility state as the state with strong or fast information arrival, and regard the low volatility state as the state with weak or slow information arrival in our paper. Similarly, Backus, Chernov, and 48.

(54) Martin (2011) also use prices of equity index options to identify extreme events (disasters) in consumptions. What we have in common is that we both try to distinguish different macroeconomic states through the prices of equity index options, and the result of Backus, Chernov, and Martin (2011) is valid. This fact suggests that prices for equity index options are an important and effective indicator in valuing financial markets.. 5-6 Goodness of Fit. 立. 政治大. In this section, we employ AIC and SBC to examine the goodness of fit of the fitting. ‧ 國. 學. model, the two-state Markov-Switching variance model, for premium and time value for. ‧. TAIEX Call Options, as shown in Table 8.. sit. y. Nat. io. n. al. er. From Table 8, we see that AIC and SBC are about the same for both premium and. v. time value for TAIEX Call Options. Both AIC and SBC are calculated by software Eviews 8.. Ch. engchi. Table 8. i n U. Goodness of Fit AIC. SBC. Premium. 13.38125. 13.39063. Time Value. 13.39924. 13.40862. 49.

(55) 6. Conclusion In this paper, we employ a two-state Markov-switching variance model with regime hetereskedasticity to analyze the volatility of premium and time value for TAIEX Call Options. The two-state Markov-switching variance model adequately characterizes the behavior of premium and time value for TAIEX Call Options. Our empirical analysis has demonstrated that first, when an economic shock, a political turmoil or other. 政治大. extraordinary event occurs, the premium and time value for TAIEX Call Options will. 立. shift to high volatility regime. Second, when there is no significant economic event in. ‧ 國. 學. the market, investors tend to regard rise or fall in TAIEX as overreactions and give. ‧. TAIEX Call Options turning points of time values to adjust.. sit. y. Nat. io. n. al. er. However, there is still something left to be further studied. First, there are other kinds. v. of Markov-switching models, and we are also interested in whether they fit TAIEX. Ch. engchi. i n U. Options data. Second, while a higher VIX indicates that traders expect higher volatility, and a lower VIX indicates that they expect lower volatility in the equities market, we can further compare TAIEX Options Volatility Index provided by Taiwan Futures Exchange to our empirical results. Lastly, theoretically, time value should not appear to be negative. However, through our empirical process, there are negative time values in the sample period. We infer the reason for negative time value is because we adopt current S instead of expected S in section 4. Hence, premium reflects the current value 50.

(56) of option instead of future value of option, and this is an issue to be discussed in future studies.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 51. i n U. v.

(57) 7. Reference I. Chinese References 王祝三，莊益源，張鐘霖(2003)，「波動率模型預測能力的比較-以臺指選擇權為例」，臺灣金融財務季刊，4(2)，41-63。吳仰哲，廖四郎，林士貴(2009)，「Lévy 與 GARCH- Lévy 過程之選擇權評價與實證分析：臺灣加權股價指數選擇權為例」，管理與系統，17(1)，49-74。徐正憲(2014)，「馬可夫轉換模型在黃金現貨、石油價格之實證研究」，政大統計系. 政治大張志向(2006)，「台指選擇權推出對領先落後關係的影響：內含價值與權利類型」，立碩士論文。. ‧ 國. 學. 亞太經濟管理評論，10(1)，1-26。. 郭玟秀，陳仁龍，邱永金(2010)，「台指選擇權隱含波動率指標對真實波動率與指. ‧. 數報酬的資訊內涵之研究」，創新與管理，7(2)，127-146。. sit. y. Nat. 粘瑞益(1999/2006?)，建構臺灣股市之隱含波動度避險模型—以馬可夫轉換模型為. io. al. er. 例，第六屆證券暨期貨金椽獎，市場組佳作。. n. 郭維裕，陳鴻隆，陳威光(2013)，「選擇權市場效率性檢定；隱含波動率成對交易. Ch. 檢定法」，管理與系統，23(3)，425-458。. engchi. i n U. v. 詹錦宏，施介人 (2005)，「台股指數現貨、期貨與選擇權價格發現之研究」，臺灣金融財務季刊，6(1)，31-51。 Ⅱ. English References. Backus, D., Chernov, M., Martin, I. (2011), “Disasters Implied by Equity Index Options,” The Journal of Finance, 66(6), 1969-2012. Black, F., Scholes, M. (1973), “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637-659. 52.

(58) Bollerslev, T. (1986), “Generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, 307-327. Cai, J. (1994), “A Markov Model of Switching-Regime ARCH,” Journal of Business and Economic Statistics, 12(3), 309-316. Cont, R., Deguest, R. (2013), “Equity Correlations Implied by Index Options: Estimation and Model Uncertainty Analysis,” Mathematical Finance, 23(3), 496-530. Chan, K.C., Cheng, L.T.W., Lung, P.P. (2005), “Asymmetric Volatility and Trading Activity in Index Futures Options,” The Financial Review, 40, 381-407.. 政治大. Chen, J., Zou, W. (2013), “A Markov regime-switching model for crude-oil markets:. 立. Comparison of composite likelihood and full likelihood,” The Canadian Journal of. ‧ 國. 學. Statistics, 41(2), 353-367.. Daouk, H., Guo, J.Q. (2004), “Switching Asymmetric GARCH and Options on a. ‧. Volatility Index,” The Journal of Futures Markets, 24(3), 251-282.. Nat. sit. y. Diebold, F.X. (1986), “Modelling the persistence of conditional variance: A comment,”. n. al. er. io. Econometric Reviews, 5(1), 51-56.. i n U. v. Dueker, M., Neely, C. (2007), “Can Markov switching models predict excess foreign. Ch. engchi. exchange returns?” Journal of Banking and Finance, 31, 279–296. Engle, R.F. (1982), “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 50(4), 987-1008. Figlewski, S. (1997), “Forecasting Volatility,” Financial Markets, Institutions & Instruments, 6(1). Hamilton, J.D. (1989), “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle,” Econometrica, 57(2), 357-384. Hamilton, J.D., Susmel, R. (1994), “Autoregressive conditional heteroskedasticity and changes in regime,” Journal of Econometrics, 64, 307-333. 53.

(59) Huang, Y.L., Kuan, C.M., Lin, K.S. (1998), “Identifying the turning points and business cycles and forecasting real GNP growth rate in Taiwan,” Taiwan Economic Review, 26, 431–457. Kim, C.J., Nelson, C.R., Startz, R. (1998), “Testing for mean reversion in heteroskedastic data based on Gibbs-sampling-augmented randomization,” Journal of Empirical Finances, 5, 131-154. Kim, C.J., Nelson, C.R. (1999), “State-space models with regime switching : classical and Gibbs-sampling approaches with applications,” 1st edition, 59-93, England, The MIT Press.. 立. 政治大. Lamourex, C.G., Lastrapes, W.D. (1990), “Heteroskedasticity in Stock Return Data:. ‧ 國. 學. Volume versus GARCH Effects,” The Journal of Finance, 45(1), 221-229. Lamourex, C.G., Lastrapes, W.D. (1993), “Forecasting Stock-Return Variance: Toward. ‧. an understanding of Stochastic Implied Volatilities,” The Review of Financial Studies,. sit. y. Nat. 6(2), 293-326.. n. al. er. io. Mayhew, S., Stivers, C. (2003), “Stock Return Dynamics, Option Volume, and the. i n U. v. Information Content of Implied Volatility,” The Journal of Futures Markets, 23(7), 615-646.. Ch. engchi. Poon, S.H., C.W.J. Granger (2003), “Forecasting Volatility in Financial Market,” Journal of Economic Literature, 41, 475-539. Ramchond, L., Susmel, R. (1998), “ Volatility and Cross Correlation Across Major Stock Markets,” Journal of Empirical Finance, 5(4), 397-416. Smith, D.R. (2002), “Markov-switching and stochastic volatility diffusion models of short-term interest rates,” Journal of Business and Economic Statistics, 20, 183–197. Su et al. (2006), “Pricing and Hedging performance of Taiwan Stock Index Options under two-state volatility condition,” Proceedings of the 11th annual conference of 54.

(60) Asia Pacific Decision Sciences Institute, June 14-18, 707-711. Turner, C.M., Startz, R., Nelson, C.R. (1989), “A Markov Model of Hetereskedasticity, Risk, and Learning in the Stock Market,” Journal of Financial Economics, 25, 3-22. Wang, J. (2007), “Research on the Markov switching model,” Quantitative and Technical Economics, 3, 39–48.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 55. i n U. v.

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