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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
Black-Scholes option pricing model.
The value of a call option before its expiration is called premium. The premium is the
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 value and time value, as shown in Equation (1). The intrinsic value of a call option is defined as the difference between the underlying price and strike price or zero, whichever is greatest, as shown in Equation (2).
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
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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.
The other variable that affects the time value is the closeness of the strike price to the money. For an out-of-the-money call option, because the intrinsic value is zero, the call 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 option, as the underlying price increases, the intrinsic value increases, but the time value decreases instead, since the likelihood of the underlying price being beneath the strike 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
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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 variables that affect the premium into account, such as strike price, current underlying price, time until expiration, implied volatility, and risk-free interest rates, Black and
Scholes derive the pricing formula for call options shown in Equation (3).
(3)
where the notations are specified as follows:
C: theoretical premium for a call option S : current underlying price
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)
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(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 used in calculating call option prices represents the total volatilities of the call option prices. At the same time, also indicates the investors’
valuations toward the future uncertainties of the underlying asset.
3-2 Markov-switching Variance Model
Subsection 3-2-1 introduces the Markov Chain and Markov-switching variance
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 denote a vector of an observed variable, and the state or regime index at time t, which is an
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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 is called transition probability, which represents the probability of transitioning from state i at time t-1 to state j at time t. For instance, means the probability of 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 column of P sums to unity: (7)
The two-state Markov-switching variance model is specified as follows. (8)
(9)
otherwise; (10)
(11)
(12)
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(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).
3-2-2 Filtered probability and Smoothed probability
In this subsection we will introduce the filtered probability and smoothed probability
since they help us to infer the state probability at time t.
Though state variable is unobservable, we can still infer the state probability at any time t. Depending on the amount of information used in making inferences on , 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 , refer to inferences about conditional on all the information in the sample: .
To obtain filtered probability and smoothed probability, first, we consider the joint density of and the unobserved :
(14)
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Next, we obtain the marginal density of ,:(15)
With the following transition probabilities:
(17) (18)
we can obtain :
(19)
Lastly, by applying Bayes’ theorem, we obtain filtered probabilities:
(20)
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where , and .
Given an initial probability, , we can obtain filtered probabilities, j by iterating Equation (19) and (20).
Consider the joint probability that and based on full information, ,
(21)
,
and
. (22) Given an initial probability, , the iterations for can be done to derive smoothed probabilities, , .
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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
The Taiwan Futures Exchange is the organization that aims to invigorate futures trading while serving the real economy in Taiwan futures market. It enables traders to 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 TAIEX Futures, Equity Options, and TAIEX Options. In this paper, we choose equity index call options, TAIEX Call Options, as our target. For TAIEX Options, the 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
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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 each month with its strike price closest to the monthly average of the TAIEX of that month, there are still some inaccuracies within this measure, causing the intrinsic values not necessarily equal to zero. But the inaccuracies are minor and acceptable.
4-2 Statistic Characteristics of TAIEX
Table 1 presents the descriptive statistics of TAIEX from 2004/01/02 to 2013/12/31.
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.
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0 100 200 300 400 500 600 700 800 900 1,000
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
Intrinsic Value
Figure 1 Daily Intrinsic Value for TAIEX Call Options Series
from 2004/01/02 to 2013/12/31
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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
Minimum 4089.930
Std. Dev. 1133.813
Skewness -0.394333
Kurtosis 2.728927
Jarque-Bera (Probability) 71.83684 (0.000000)
4,000 5,000 6,000 7,000 8,000 9,000 10,000
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
TAIEX
Figure 2 Daily TAIEX from 2004/01/02 to 2013/12/31
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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 find that the data is right-skewed with Skewness larger than 0 and leptokurtic with Kurtosis larger than 3. For the value of Jarque-Bera statistic, we can reject the hypothesis of normal distribution.
Figure 3 presents the time series of premium for TAIEX Call Options from 2004/01/02 to 2013/12/31. Among the sample period, there exist two extreme values.
One is 900 on 2007/11/01 and the other is 915 on 2011/08/01. The reason for these two 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.
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Table 2 Descriptive Statistics of Premium for TAIEX Call Options
from 2004/01/02 to 2013/12/31
Jarque-Bera (Probability) 2321.805 (0.000000)
0
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
PREMIUM
Figure 3 Daily Premium for TAIEX Call Options Series
from 2004/01/02 to 2013/12/31
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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 Options against Normal and Student-t distribution. We can observe that the premium for TAIEX Call Options is fat-tailed compared to Normal and Student-t distribution with degree 5.
4-3-2 Unit Root Test of Premium for TAIEX Call Options
A unit root test tests whether a sequence of time series data is stationary by using an 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.
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Figure 4 Q-Q plots against Normal and Student-t(5) distribution of Premium for TAIEX Call Options
Table 3 ADF Test and PP Test of Premium for TAIEX Call Options ADF Test Statistic -14.24091 1% Critical value -3.432795
5% Critical value -2.862506 10% Critical value -2.567329 PP Test Statistic -14.19782 1% Critical value -3.432795 5% Critical value -2.862506 10% Critical value -2.567329
-200
0 100 200 300 400 500 600 700 800 900 1,000 1,100
Quantiles of PREMIUM
0 100 200 300 400 500 600 700 800 900 1,000 1,100
Quantiles of PREMIUM
Quantiles of Student's t
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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.
Table 4 presents the descriptive statistics of time value for TAIEX Call Options. We 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 hypothesis of normal distribution.
Figure 5 displays the time series of time value for TAIEX Call Options. The two 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.
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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
Std. Dev. 67.30343
Skewness 1.003969
Kurtosis 4.885771
Jarque-Bera (Probability) 783.7716 (0.000000)
-100 0 100 200 300 400 500
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
Time Value
Figure 5 Daily Time Value for TAIEX Call Options Series from 2004/01/02 to 2013/12/31
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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
Table 5 presents the ADF test and PP test of time value for TAIEX Call Options. We find that both null hypotheses that time value has a unit root can be rejected under 1%
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 process our analysis.
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Figure 6 Q-Q plots against Normal and Student-t(5) distribution of Time Value for TAIEX Call Options
Table 5 ADF Test and PP Test of Time Value for TAIEX Call Options ADF Test Statistic -4.087185 1% Critical value -3.432814
5% Critical value -2.862514 10% Critical value -2.567334 PP Test Statistic -14.011558 1% Critical value -3.432795 5% Critical value -2.862506 10% Critical value -2.567329
-300
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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 results of both premium and time value for TAIEX Call Options. Subsection 5-5 presents the analysis and comparison between our results and results from other related 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.
5-1The characteristics of Time Value Data
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)
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Table 6 Negative time values from 2004/01/02 to 2013/12/31 No. Date TAIEX Premium Intrinsic
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Figure 7-A Variation of Premium and TAIEX from 2004 to 2013
Figure 7-B Corresponding Time values and Variation of TAIEX
from 2004 to 2013
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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
means TAIEX accelerates, time value becomes negative. The correlation coefficient between Δ S and TV is -0.35. To see why time values turn negative when TAIEX rises,
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, 2012/1/31, and 2013/6/28, the time values are all negative. Since the TAIEXs on these days did not change much, the causes of the negative time values have little to do with 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
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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 slight rise in premium by giving negative time values to adjust, and then rising margin of TAIEX is greater than the rising margin of premium. Negative time values will adjust 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, adjusting to positive time values again on 2011/06/10.
For turning points of time values changing from negative to positive, we infer that 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
For turning points of time values changing from negative to positive, we infer that 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