# Abstract

This thesis proposes a different methodology to measure the speed with which an option’s time value approaches zero as the time to maturity T vanishes other than the Greek letter theta: the slope of a fourth-order polynomial fitted line implied from a term decay plot which plots the log of option price against the log of time to maturity. I hypothesize that when the market state of the underlying asset of options is bullish, the time value of put options decreases faster whereas when the market state of the underlying asset of options is bearish, the time value of call options decreases faster. Three methodologies are used to proxy for the market states of the underlying asset of options including past returns of the underlying asset price, Bry Boschan dating algorithm to classify the market into bull or bear, and Bollinger Bands to classify the market into bull, bear, or mean-reversion. We find international evidence that the hypothesis holds for the index options across countries including Taiwan, Hong Kong, South Korea, Japan, and Malaysia. The empirical results have huge implication for options trading and hedging in that they can construct better trading or hedging strategies by taking advantage of these findings.

# I. Introduction

There are many factors affecting the speed with which an option’s time value approaches zero as the time to maturity T vanishes, including but not limited to the Greek letter theta (θ), underlying asset price of an option (S), the volatility of the underlying asset (σ), and the moneyness of an option. The theta of an option is the rate of change of an option’s premium with respect to the passage of time with all else remaining the same, and theoretical studies show that theta is the lowest for at-the-money call option, other things being equal, as depicted in Figure 1[1]. It means that the market price of an option decreases the fastest when the price of the underlying assets, S, approximates the strike price K. However, empirically, there are few inquiries into the relationship between the time-decaying characteristics of an option’s time value and the patterns of the underlying asset.

Figure 1. Variation of theta of a European call option with stock price. This figure presents the theoretical relationship between theta of a European call option and the underlying asset price implied from the Black–Scholes–Merton formula.

This thesis aims to investigate how the time value of an option interacts with market states of the underlying assets. Specifically, I examine the market prices of at-the-money (ATM) and out-of-the-money (OTM) options since there exists solely time value in these kinds of options without intrinsic value. It is believed that the bull or bear state of underlying asset of an option affects the speed with which an ATM or OTM option’s premium approaches zero as the time-to-maturity vanishes. I hypothesize that when the underlying asset S is bullish, the time value of put options decreases faster, whereas when the underlying asset S is bearish, the time value of call options decreases faster. The hypothesis is intuitive such that when the underlying asset price increases, the market price of put option decreases, in turn pressing the time value of an ATM or OTM put option down and making the time-decaying speed become faster. From the perspective of investors, a higher underlying asset price makes the possibility of an OTM put option to get in the money become lower. As a result, investors lose confidence and want to sell their put option position, fostering the time-decaying characteristics of put options. On the contrary, when the underlying asset price decreases, the market price of call option decreases, in turn pressing the time value of an ATM or OTM call option down and making the time-decaying speed become faster. From the perspective of investors, a lower underlying asset price makes the possibility of an OTM call option to get in the money become lower. As a result, investors lose confidence and want to sell their call option position, fostering the time-decaying characteristics of call options. Figure 2 depicts the illustration of the hypothesis.

Figure 2. Illustration of the hypothesis. This figure presents the hypothesis for the relationship between the time-decaying speed of the time value of options and the market states of the underlying asset.

The motivation for the thesis is to help investors take advantage of the time-decaying characteristics of the time value of options. If investors know under which market state of the underlying asset will the time-decaying speed increases, they can properly implement options strategies like spreads, condor, or short strangle under that market state to gain better return with lower risk.

There are other factors which may affect time decaying characteristics of the time value of options, namely, volatility of the underlying asset σ, option’s implied volatility and moneyness of the option, and we should take them into account in the empirical investigation.

# II. Related Literature

In Carr and Wu (2003), they construct a term decay plot which plots the log of the ratio of option prices to maturity against log maturity in order to measure the speed with which an option’s premium approaches zero as the time-to-maturity vanishes. They study the nature of the price process of an asset underlying an option, finding that while the presence of the jump component varies strongly over time, the presence of the continuous component is constantly felt.

In Figlewski and Freund (1994), convexity of the option pricing function and time decay are considered as factors that lead to risk in an option hedge under discrete rebalancing.

For the classification of the market state, Chen (2009) segregates the stock market into bull and bear whereas Wan-Jung Hsu (2017) segregates the stock market into bull, good bear, and bad bear, and Cooper et al. (2004) defines a bull market as the past three year return to be positive and defines a bear market as the past three year return to be negative.

# III. Methodology

## 3.1 Explanatory variables

Basically, there are three types of explanatory variables, namely, past return of the underlying asset of the option, implied volatility of option, and standard deviation of return of the underlying asset. Let $S_t$ denote daily market price of the underlying asset of the option at time t. Let $R_t$ denote the continuous return of the underlying asset at time t. $R_t$ represents the market states of the underlying assets, where a positive $R_t$ stands for bull market (up-trend market) and a negative $R_t$ stands for bear market (down-trend market). In order to capture different lengths of the market states, I define $R_t$ into different time windows, namely, past month return $R^M_t$, past season return $R_t^Q$, past half year return $R_t^H$, past year return $R_t^Y$, past time-to-maturity return $R_t^{TTM}$. A month is defined as 22 trading days and a year is defined as 252 trading days. It follows that:

$R_t&space;=&space;ln\frac{S_t}{S_{t-1}}$, i = the number of trading days – equation (1)

$R_t^M&space;=&space;ln\frac{S_t}{S_{t-22}}$, monthly return – equation (2)

$R_t^Q&space;=&space;ln\frac{S_t}{S_{t-66}}$, quarterly return – equation (3)

$R_t^H&space;=&space;ln\frac{S_t}{S_{t-132}}$, half year return – equation (4)

$R_t^Y&space;=&space;ln\frac{S_t}{S_{t-252}}$, yearly return – equation (5)

$R_t^TTM&space;=&space;ln\frac{S_t}{S_{t-i}}$, i = time to maturity in the number of trading days. – equation (6)

In Cooper et al. (2004), they define UP (DOWN) markets as the non-negative (negative) returns of the price index over months (t-36) to (t-1), which constitutes past three year return. However, for options their duration mostly last for several months less than a year, so I would like to use monthly return first as the main explanatory variable to proxy for the bull or bear market state.

Another alternative to identify bull and bear market is the Bry-Boschan (1971) dating rule, which has been extensively used in the business and financial cycle literature, including Chen (2009), Harding and Pagan (2003), Nyberg (2013), Chauvet and Potter (2000), Pagan and Sossounov (2003), Candelon et al. (2008, 2012), and Claessens et al. (2012). I apply the Bry-Boschan dating rule, after modification from monthly period to daily period, in order to objectively distinguish between periods of up-trending markets and down-trending markets. The reason I apply Bry-Boschan dating rule with daily period rather than monthly period is because options generally have shorter and limited duration. For the sake of capturing the microstructure information in the underlying asset compatible with limited duration of options, there is a need to segregate market states by shorter time period. For instance, in my Taiwan index options sample from 2010 to 2016, over half of the options (54%) have a duration of only three months.

The Bry-Boschan algorithm searches for local maxima (peak) and minima (trough) in the series over a two-sided window of at least 6 days length. Peaks and troughs alternate; i.e., a peak cannot succeed another peak without an intervening trough. In particular, it requires that a complete cycle should have a duration of at least fifteen days from peak to peak or from trough to trough and adjacent pairs between peak and trough should last at least six days. In case of equal values the rule is to choose the last one as the cyclical turn, i.e., the day before the reversal of the cyclical process begins. Assuming yt is the time series data examined, the turning point would be a peak at time t if, $y_{t-6},...,y_{t-1}&space;<&space;y_{t}&space;>&space;y_{t+1},...,y_{t+6}$ , and a trough if, $y_{t-6},...,y_{t-1}&space;>&space;y_{t}&space;<&space;y_{t+1},...,y_{t+6}$. Periods from trough to peak are classified as bull markets, while periods from peak to trough are classified as bear markets, which is depicted in Figure 3[2].

Figure 3. Illustration of Bry-Boschan (1971) dating rule. This figure illustrates how the Bry-Boschan (1971) dating rule objectively distinguishes between periods of up-trending and down-trending markets. A peak or trough is required to have a minimum window of 6 days. A period of up-trending or down-trending market is required to have a minimum phase of six days. Peak to peak or trough to trough is required to have a minimum cycle of 15 days.

Since a certain day after a trough and before a peak classified by the Bry-Boschan dating rule is defined as a bull state whereas a certain day after a peak and before a trough is defined as a bear state, one can know whether the market is bull or bear at certain day given the nearest state in the past, even though one does not know the future market state. Let $B_t$ denote the binary market state at certain time t. It follows that:

$B_t=\begin{Bmatrix}&space;BryBull,&space;\textrm{when&space;the&space;market&space;is&space;bull}\\&space;BryBear,&space;\textrm{when&space;the&space;market&space;is&space;bear}&space;\end{matrix}$ – equation (7)

Figure 4. Illustration of binary classification of market states of the underlying assets by Bry Boschan dating algorithm – subset(last 25%) – Taiwan – TAIWGHT. This figure demonstrates the last 25% of the full sample period of Taiwan options. The white-shaded area is classified as bull market whereas black-shaded area is classified as bear market calculated by Bry Boschan dating algorithm.

Let $IV_t$ denote the implied volatility of the option at time t and $\sigma&space;_t$ denote the rolling window standard deviation of daily return of the underlying asset at time t, e.g., a window of a week or a year. Specifically, $IV_t$ and $\sigma&space;_t$ are divided by their long-term average over a half year period:

$IV^*=\frac{IV_{t}}{IV_{LT}},&space;IV_{LT}&space;=&space;\sum_{t=1}^n\frac{IV_t}{n},n=132$ – equation (8)

$\sigma&space;^*=\frac{\sigma_{t}}{\sigma_{LT}},&space;\sigma_{LT}&space;=&space;\sum_{t=1}^n\frac{\sigma_t}{n},n=132$ – equation (9)

It is well documented that volatility estimates exhibit stationary characteristics such that they usually fall within some certain range. This mean-reverting feature makes a higher volatility converges downward to its long-term average eventually, and vice versa. Given the positive relationship between volatilities and option premiums and the mean-reverting characteristics of volatilities, I use implied volatilities and standard deviations divided by their long-term average to capture more information about the directions premiums would go. For instance, a higher than average volatility leads to a higher option price, and the option price has a higher probability to drop since the volatility will converge downward to its long-term average eventually, making the option premium decrease as well.

## 3.2 Response variable

The response variable is the speed with which an option’s premium approaches zero as the time to maturity T vanishes. The different degree of time-decaying characteristics of option premiums is most easily visualized by a graph. Following Carr and Wu (2003), I focus on a graph which plots the log of option prices against log maturity. I christen such a graph as a term decay plot. In particular, to assess the slope of these term decay plots as a proxy for the time-decaying speed which is defined as SL, I construct a smoothed term decay plot by fitting a simple fourth-order polynomial function:

$ln(P)=a(lnT)^4&space;+&space;b(lnT)^3&space;+&space;c(lnT)^2&space;+&space;d(lnT)&space;+&space;e.$ – equation (10)

Then, the slope of the plot at a certain maturity T is given by $4a(lnT)^3&space;+&space;3b(lnT)^2&space;+&space;2c(lnT)&space;+&space;d$, representing time decaying speed of the option premium, and the curvature is given by 2c, where T is the time to maturity in years and P is the market price of options. In order to investigate whether the time-decaying speed of calls and puts reacts differently across different underlying market states, I classify the time-decaying speed into calls and puts:

$\left\{\begin{matrix}&space;SL_t^C,\textrm{time-decaying&space;speed&space;of&space;calls&space;at&space;time&space;t}\\&space;SL_t^P,\textrm{time-decaying&space;speed&space;of&space;puts&space;at&space;time&space;t}&space;\end{matrix}\right.$ – equation (11)

For each single option, the log of option price is regressed on the log of time to maturity based on the fourth-order polynomial function in equation (10). Then the estimated coefficients of the regression are used to determine the fitted value and the slope which in turn is used as the response variable, the time-decaying speed of the time value of options, as depicted in figure 5.

Figure 5. Illustration of term decay plot. This figure shows the term decay plot of a single option: Taiwan Stock Exchange Capitalization Weighted Stock Index 21-May-2014 9200 Call. The points are the observations of the log of the market price of options with respect to the log of time to maturity. The solid line is the fitted value drawn from the estimated coefficients of the fourth-order polynomial regression in eq. (10). The dashed line is the slope of the fitted line which represents the time-decaying speed of the time value of option.

## 3.3 Model specification of regressions

Different designs of regressions are formed to investigate the relationship between market states of the underlying asset and time-decaying speed of the time value of options, which is the slope of the term decay plot, SL. Two volatility estimates, implied volatility IV and standard deviation σ are used as control variables in all model, so do their value divided by their long-term average, $IV^*$ and $\sigma&space;^*$. Also, moneyness, defined as the natural log of underlying asset price divided by strike price, $ln\frac{S}{K}$, is taken into consideration as a control variable since it affects the speed with which time value shrinks, in light of the Greek letter theta. However, one thing should be aware that whether there is multi-collinearity between moneyness and past return R because strike price K is fixed and both moneyness and past return come from underlying asset price S.

## 3.3.1 Model 1

In model 1, past returns of different time windows are used in order to decide the optimal time window of past return which affects the time-decaying speed the most, namely, past month return RM, past season return RQ, past half year return RH, past year return RY, and past time-to-maturity return RTTM.

$SL_t^C&space;=\beta&space;_0+&space;\beta&space;_1R_t+\beta&space;_2IV_t+\beta&space;_3\sigma&space;_t+\beta&space;_4ln\frac{S}{K}+u$ – equation (12)

$SL_t^P&space;=\beta&space;_0+&space;\beta&space;_1R_t+\beta&space;_2IV_t+\beta&space;_3\sigma&space;_t+\beta&space;_4ln\frac{S}{K}+u$ – equation (13)

## 3.3.2 Model 2

Model 2 is simply model 1 replacing the two volatilities estimates IV and σ by the two volatility estimates divided by their long-term average $IV^*$ and $\sigma&space;^*$, in order to capture the mean-reverting feature of volatilities.

$SL_t^C&space;=\beta&space;_0+&space;\beta&space;_1R_t+\beta&space;_2IV_t^*+\beta&space;_3\sigma&space;_t^*+\beta&space;_4ln\frac{S}{K}+u$ – equation (14)

$SL_t^P&space;=\beta&space;_0+&space;\beta&space;_1R_t+\beta&space;_2IV_t^*+\beta&space;_3\sigma&space;_t^*+\beta&space;_4ln\frac{S}{K}+u$ – equation (15)

## 3.3.3 Model 3

Bry-Boschan (1971) dating rule is implemented in model 3 as an alternative to decide whether the underlying market state is up-trending or down-trending.

$SL_t^C&space;=\beta&space;_0+&space;\beta&space;_1B_t+\beta&space;_2IV_t+\beta&space;_3\sigma&space;_t+\beta&space;_4ln\frac{S}{K}+u$ – equation (16)

$SL_t^P&space;=\beta&space;_0+&space;\beta&space;_1B_t+\beta&space;_2IV_t+\beta&space;_3\sigma&space;_t+\beta&space;_4ln\frac{S}{K}+u$ – equation (17)

## 3.3.4 Model 4

Model 4 is simply model 3 replacing the two volatilities estimates IV and σ by the two volatility estimates divided by their long-term average $IV^*$ and $\sigma&space;^*$, in order to capture the mean-reverting feature of volatilities.

$SL_t^C&space;=\beta&space;_0+&space;\beta&space;_1B_t+\beta&space;_2IV_t^*+\beta&space;_3\sigma&space;_t^*+\beta&space;_4ln\frac{S}{K}+u$ – equation (18)

$SL_t^P&space;=\beta&space;_0+&space;\beta&space;_1B_t+\beta&space;_2IV_t^*+\beta&space;_3\sigma&space;_t^*+\beta&space;_4ln\frac{S}{K}+u$ – equation (19)

## 3.3.5 Model 5

While Chen (2009) classifies the stock market into binary states, namely, bull and bear, Wan-Jung Hsu (2017) classifies the stock market into three states, namely, bull, good bear and bad bear. The stock market is classified as a bad bear market if a stock bear market is accompanied with a contraction phase of future cash flows, otherwise it is classified as a good bear market. In Wan-Jung Hsu (2017), by applying a multinomial logit model with three alternatives (bull, good bear and bad bear) to predict stock market states, investors can gain much higher returns and decrease unfavorable drawdowns by properly exploiting spreads between growth stocks and value stocks. Unlike Wan-Jung Hsu (2017), I propose different rules in classifying the stock market into three types by making use of Bollinger Bands.

A Bollinger Band, developed by famous technical trader John Bollinger, consists of three lines. Typically, the middle band is a 20-period simple moving average of the price, and the upper and lower bands are 2 standard deviations above and below the moving average. In particular, consider classifying monthly return $R^M$ into 3 states, namely, bull, bear, and mean-reversion. The middle band is defined as a 22-period simple moving average of $R^M$ and the upper and lower bands are one standard deviation above and below the middle band. Let $M_t$ denote market states of the underlying asset. It follows that:

$M_t=&space;\left\{\begin{matrix}&space;\textrm{BollingerBull,&space;if&space;}R^M>&space;\textrm{upper&space;band}\\&space;\textrm{BollingerBear,&space;if&space;}R^M<&space;\textrm{lower&space;band}\\&space;\textrm{BollingerMeanReversion,&space;if&space;lower&space;band}<&space;R^M<&space;\textrm{upper&space;band}&space;\end{matrix}\right.$ – equation (20)

Figure 6. Illustration of triple classification of market states of the underlying assets by Bollinger Bands – subset(last 25%)Taiwan – TAIWGHT. This figure demonstrates the last 25% of the full sample period of Taiwan options. When the monthly return of the underlying asset is higher than the upper band, it is classified as a bull market. When the monthly return of the underlying asset is lower than the lower band, it is classified as a bear market. When the monthly return of the underlying asset is between upper band and lower band, it is classified as a mean-reversion market.

Rather than classifying the market states of the underlying assets of options into solely bull and bear, mean-reversion state is introduced to capture the condition when the past returns, say, monthly return $R_t^M$, is not too far away from its moving average. It is generally believed that trending market, i.e., bull or bear, and mean-reversion market have a huge implication for investors since some hedge funds have a foot in either trend-following model or mean-reversion model.

Then, model 5 is specified using the three market states classification:

$SL_t^C&space;=\beta&space;_0+&space;\beta&space;_1M_t+\beta&space;_2IV_t+\beta&space;_3\sigma&space;_t+\beta&space;_4ln\frac{S}{K}+u$ – equation (21)

$SL_t^P&space;=\beta&space;_0+&space;\beta&space;_1M_t+\beta&space;_2IV_t+\beta&space;_3\sigma&space;_t+\beta&space;_4ln\frac{S}{K}+u$ – equation (22)

## 3.3.6 Model 6

Model 6 is simply model 5 replacing the two volatilities estimates IV and σ by the two volatility estimates divided by their long-term average $IV^*$ and $\sigma&space;^*$, in order to capture the mean-reverting feature of volatilities.

$SL_t^C&space;=\beta&space;_0+&space;\beta&space;_1M_t+\beta&space;_2IV_t^*+\beta&space;_3\sigma&space;_t^*+\beta&space;_4ln\frac{S}{K}+u$ – equation (23)

$SL_t^P&space;=\beta&space;_0+&space;\beta&space;_1M_t+\beta&space;_2IV_t^*+\beta&space;_3\sigma&space;_t^*+\beta&space;_4ln\frac{S}{K}+u$ – equation (24)

# IV. Data and Summary Statistics

For the interests of international evidence across Asia countries near Taiwan, countries under investigation include Taiwan, Hong Kong, South Korea, Japan, and Malaysia. Daily data of index options of these five countries are retrieved since its inception from Thomson Reuters Datastream database. Since the thesis is to dissect the time value of options, ITM options are excluded if there is any day the option is in the money in its entire life, remaining only ATM and ITM options.

Table I

Underlying Assets of Index Options

This table shows the underlying assets of options for the five countries under investigation including Taiwan, Hong Kong, South Korea, Japan, and Malaysia.

As we can see in table II, Taiwan has 6,588 options in total from 2010 to 2017 which constitutes 1,788 trading days. Both the number of calls and puts are 3,294. Of the calls, 37% are classified as OTM options whereas 38% of puts are classified as OTM options. Overall, for every country, 20% to 40% of options are classified as OTM options which are the target of my interest.

Table II

Overview of Index Options

This table gives an overview of the index options under investigation. For each type of index options, number of options and the whole sample period are reported. Number of trading days exclude weekends and holidays. For both calls and puts, the number of ITM and OTM options are reported. Options are classified as ITM if the option is in the money in any date of its entire sample period. On the other hand, options are classified as OTM if the option is either in the money or at the money in its entire sample period. Also, the proportion of ITM and OTM options are reported as a percentage.

Descriptive statistics are structured as follows: For each type of options, visualizations of the three main explanatory variables to measure the market states of the underlying assets are displayed, namely, rolling monthly return of underlying asset, binary classification of market states of the underlying assets by Bry Boschan dating algorithm, and triple classification of market states of the underlying assets by Bollinger Bands. Also, descriptive statistics of dummy variables and numeric variables are reported for each country.

Of the various time windows of returns of the underlying assets, I use the monthly return for consistency to measure the market states of the underlying assets. For the response variable time-decaying speed of the time value of options, SL (slope of the term decay plot), a positive slope means that the market prices of the options decrease as time passes (time value decreases) whereas a negative slope means that the market prices of the options increase as time passes (time value increases). For IV*, the implied volatility divided by its half year long-term average, due to data restrictions that some types of options have few options with maturities longer than half year, monthly long-term average is used rather than half year long-term average. Specifically, in calculating IV*, half year long-term average is replaced by monthly long-term average for two types of options, namely, Japan – JPXNK40 and Malaysia – FBMKLCI.

## 4.1 Descriptive statistics: Taiwan

Figure 7. Rolling monthly return of underlying asset (RM) – Taiwan – TAIWGHT. This figure compares the underlying asset price of the option with its rolling monthly return as a proxy for market states where a positive (negative) monthly return is defined as a bull (bear) market.

As we can see in figure 7, the monthly return of TAIWAN SE WEIGHED TAIEX is relatively stationary compared to the underlying asset price, ranging from -17% to 21%.

Figure 8. Binary classification of market states of the underlying assets by Bry Boschan dating algorithm – Taiwan – TAIWGHT. Parameters for the Bry Boschan dating algorithm are a minimum window of 6 days, a minimum phase of 6 days, and a minimum cycle of 15 days. White-shaded area is classified as bull market whereas black-shaded area is classified as bear market.

Figure 9. Triple classification of market states of the underlying assets by Bollinger BandsTaiwan – TAIWGHT. Parameters of Bollinger Bands applied to monthly returns of the underlying assets are 22 days moving average and one standard deviation. The market is bull (bear) if monthly return is higher (lower) than upper (lower) band whereas the market is mean-reversion if monthly returns falls between upper band and lower band.

Table III

Descriptive Statistics: Dummy Variables – Taiwan – TAIWGHT

This table shows the number of observations of the composition of the binary classification and triple classification of the market states of the underlying assets of options and their proportions. Average duration in days of the bull and bear markets classified by Bry Boschan dating algorithm are also reported, so do their average amplitude which represents the average range of underlying asset price in that market state.

As we can see in table III, for binary classification, the proportion of bull market (57%) is slightly higher than that of bear market (43%). Also, bull market lasts longer (27.7 days) and has greater amplitude (682.6). For triple classification, most of the market states are mean-reversion states (43%) and the proportion of bull market (30%) is slightly higher than that of bear market (27%).

Table IV

Descriptive Statistics: Numeric VariablesTaiwan – TAIWGHT

This table reports the number of observations, mean, standard deviation, minimum, first quartile, median, third quartile, and maximum for each numeric variable. The slope (time-decaying speed of the time value of options), implied volatilities, implied volatility divided by their long-term average and log of moneyness are reported for calls and puts respectively.

As we can see in table IV, the maximum and minimum of the time-decaying speed (SL) of the time value of Taiwan – TAIWGHT options are extreme due to the nature of the fourth-order polynomial regression for the term decay plot and limited observations of some specific options. However, the mean, first quartile, median, and third quartile of time-decaying speed are all positive and reasonably small, representing the general decrease in time value with the passage of time.

## 4.2 Descriptive statistics: Hong Kong – HKHCHIE

Figure 10. Rolling monthly return of underlying asset (RM) – Hong Kong – HKHCHIE. This figure compares the underlying asset price of the option with its rolling monthly return as a proxy for market states where a positive (negative) monthly return is defined as a bull (bear) market.

As we can see in figure 10, the monthly return of HANG SENG CHINA ENTERPRISES is relatively stationary compared to the underlying asset price, ranging from -67% to 32% due to the high volatility during the 2008 financial crisis.

Figure 11. Binary classification of market states of the underlying assets by Bry Boschan dating algorithmHong Kong – HKHCHIE. Parameters for the Bry Boschan dating algorithm are a minimum window of 6 days, a minimum phase of 6 days, and a minimum cycle of 15 days. White-shaded area is classified as bull market whereas black-shaded area is classified as bear market.

Figure 12. Triple classification of market states of the underlying assets by Bollinger Bands – Hong Kong – HKHCHIE. Parameters of Bollinger Bands applied to monthly returns of the underlying assets are 22 days moving average and one standard deviation. The market is bull (bear) if monthly return is higher (lower) than upper (lower) band whereas the market is mean-reversion if monthly returns falls between upper band and lower band.

Table V

Descriptive Statistics: Dummy Variables – Hong Kong – HKHCHIE

This table shows the number of observations of the composition of the binary classification and triple classification of the market states of the underlying assets of options and their proportions. Average duration in days of the bull and bear markets classified by Bry Boschan dating algorithm are also reported, so do their average amplitude which represents the average range of underlying asset price in that market state.

As we can see in table V, for binary classification, the proportion of bull market (50%) is equal to that of bear market (50%). Also, bull market lasts longer (21.2 days) and has greater amplitude (1445.3). For triple classification, most of the market states are mean-reversion states (45%) and the proportion of bull market (26%) is slightly lower than that of bear market (29%).

Table VI

Descriptive Statistics: Numeric Variables – Hong Kong – HKHCHIE

This table reports the number of observations, mean, standard deviation, minimum, first quartile, median, third quartile, and maximum for each numeric variable. The slope (time-decaying speed of the time value of options), implied volatilities, implied volatility divided by their long-term average and log of moneyness are reported for calls and puts respectively.

As we can see in table VI, the maximum and minimum of the time-decaying speed (SL) of the time value of Hong Kong – HKHCHIE options are extreme due to the nature of the fourth-order polynomial regression for the term decay plot and limited observations of some specific options. However, the mean, first quartile, median, and third quartile of time-decaying speed are all positive and reasonably small, representing the general decrease in time value with the passage of time.

## 4.3 Descriptive statistics: South Korea

Figure 13. Rolling monthly return of underlying asset (RM) – South Korea – KOR200I. This figure compares the underlying asset price of the option with its rolling monthly return as a proxy for market states where a positive (negative) monthly return is defined as a bull (bear) market.

As we can see in figure 13, the monthly return of KOREA SE KOSPI 200 is relatively stationary compared to the underlying asset price, ranging from -40% to 20%.

Figure 14. Binary classification of market states of the underlying assets by Bry Boschan dating algorithmSouth Korea – KOR200I. Parameters for the Bry Boschan dating algorithm are a minimum window of 6 days, a minimum phase of 6 days, and a minimum cycle of 15 days. White-shaded area is classified as bull market whereas black-shaded area is classified as bear market.

Figure 15. Triple classification of market states of the underlying assets by Bollinger BandsSouth Korea – KOR200I. Parameters of Bollinger Bands applied to monthly returns of the underlying assets are 22 days moving average and one standard deviation. The market is bull (bear) if monthly return is higher (lower) than upper (lower) band whereas the market is mean-reversion if monthly returns falls between upper band and lower band.

Table VII

Descriptive Statistics: Dummy Variables – South Korea – KOR200I

This table shows the number of observations of the composition of the binary classification and triple classification of the market states of the underlying assets of options and their proportions. Average duration in days of the bull and bear markets classified by Bry Boschan dating algorithm are also reported, so do their average amplitude which represents the average range of underlying asset price in that market state.

As we can see in table VII, for binary classification, the proportion of bull market (58%) is slightly higher than that of bear market (42%). Also, bull market lasts longer (24.3 days) and has greater amplitude (19.8). For triple classification, most of the market states are mean-reversion states (45%) and the proportion of bull market (26%) is slightly lower than that of bear market (29%).

Table VIII

Descriptive Statistics: Numeric VariablesSouth Korea – KOR200I

This table reports the number of observations, mean, standard deviation, minimum, first quartile, median, third quartile, and maximum for each numeric variable. The slope (time-decaying speed of the time value of options), implied volatilities, implied volatility divided by their long-term average and log of moneyness are reported for calls and puts respectively.

As we can see in table VIII, the maximum and minimum of the time-decaying speed (SL) of the time value of South Korea – KOR200I options are extreme due to the nature of the fourth-order polynomial regression for the term decay plot and limited observations of some specific options. However, the mean, first quartile, median, and third quartile of time-decaying speed are all positive and reasonably small, representing the general decrease in time value with the passage of time.

## 4.4 Descriptive statistics: Japan – JAPDOWA

Figure 16. Rolling monthly return of underlying asset (RM) – Japan – JAPDOWA. This figure compares the underlying asset price of the option with its rolling monthly return as a proxy for market states where a positive (negative) monthly return is defined as a bull (bear) market.

As we can see in figure 16, the monthly return of NIKKEI 225 STOCK AVERAGE is relatively stationary compared to the underlying asset price, ranging from -51% to 23%.

Figure 17. Binary classification of market states of the underlying assets by Bry Boschan dating algorithmJapan – JAPDOWA. Parameters for the Bry Boschan dating algorithm are a minimum window of 6 days, a minimum phase of 6 days, and a minimum cycle of 15 days. White-shaded area is classified as bull market whereas black-shaded area is classified as bear market.

Figure 18. Triple classification of market states of the underlying assets by Bollinger BandsJapan – JAPDOWA. Parameters of Bollinger Bands applied to monthly returns of the underlying assets are 22 days moving average and one standard deviation. The market is bull (bear) if monthly return is higher (lower) than upper (lower) band whereas the market is mean-reversion if monthly returns fall between the upper band and lower band.

Table IX

Descriptive Statistics: Dummy Variables – Japan – JAPDOWA

This table shows the number of observations of the composition of the binary classification and triple classification of the market states of the underlying assets of options and their proportions. The average duration in days of the bull and bear markets classified by Bry Boschan dating algorithm are also reported, so do their average amplitude which represents the average range of underlying asset price in that market state.

As we can see in table IX, for binary classification, the proportion of bull market (56%) is slightly higher than that of bear market (44%). Also, bull market lasts longer (24.7 days) and has greater amplitude (1445.1). For triple classification, most of the market states are mean-reversion states (44%) and the proportion of bull market (27%) is slightly lower than that of bear market (29%).

Table X

Descriptive Statistics: Numeric Variables – Japan – JAPDOWA

This table reports the number of observations, mean, standard deviation, minimum, first quartile, median, third quartile, and maximum for each numeric variable. The slope (time-decaying speed of the time value of options), implied volatilities, implied volatility divided by their long-term average and log of moneyness are reported for calls and puts respectively.

As we can see in table X, the maximum and minimum of the time-decaying speed (SL) of the time value of Japan – JAPDOWA options are extreme due to the nature of the fourth-order polynomial regression for the term decay plot and limited observations of some specific options. However, the mean, first quartile, median, and third quartile of time-decaying speed are all positive and reasonably small, representing the general decrease in time value with the passage of time.

## 4.5 Descriptive statistics: Japan – JPXNK40

Figure 19. Rolling monthly return of underlying asset (RM) – Japan – JPXNK40. This figure compares the underlying asset price of the option with its rolling monthly return as a proxy for market states where a positive (negative) monthly return is defined as a bull (bear) market.

As we can see in figure 19, the monthly return of JPX-NIKKEI 400 is relatively stationary compared to the underlying asset price, ranging from -40% to 20%.

Figure 20. Binary classification of market states of the underlying assets by Bry Boschan dating algorithmJapan – JPXNK40. Parameters for the Bry Boschan dating algorithm are a minimum window of 6 days, a minimum phase of 6 days, and a minimum cycle of 15 days. White-shaded area is classified as bull market whereas black-shaded area is classified as bear market.

Figure 21. Triple classification of market states of the underlying assets by Bollinger BandsJapan – JPXNK40. Parameters of Bollinger Bands applied to monthly returns of the underlying assets are 22 days moving average and one standard deviation. The market is bull (bear) if monthly return is higher (lower) than upper (lower) band whereas the market is mean-reversion if monthly returns falls between upper band and lower band.

Table XI

Descriptive Statistics: Dummy Variables – Japan – JPXNK40

This table shows the number of observations of the composition of the binary classification and triple classification of the market states of the underlying assets of options and their proportions. Average duration in days of the bull and bear markets classified by Bry Boschan dating algorithm are also reported, so do their average amplitude which represents the average range of underlying asset price in that market state.

As we can see in table XI, for binary classification, the proportion of bull market (63%) is higher than that of bear market (37%). Also, bull market lasts longer (23.7 days) and has smaller amplitude (1026). For triple classification, most of the market states are mean-reversion states (48%) and the proportion of bull market (28%) is slightly higher than that of bear market (24%).

Table XII

Descriptive Statistics: Numeric VariablesJapan – JPXNK40

This table reports the number of observations, mean, standard deviation, minimum, first quartile, median, third quartile, and maximum for each numeric variable. The slope (time-decaying speed of the time value of options), implied volatilities, implied volatility divided by their long-term average and log of moneyness are reported for calls and puts respectively.

As we can see in table XII, the maximum and minimum of the time-decaying speed (SL) of the time value of Japan – JPXNK40 options are extreme due to the nature of the fourth-order polynomial regression for the term decay plot and limited observations of some specific options. However, the mean, median, and third quartile of time-decaying speed are all positive and reasonably small, representing the general decrease in time value with the passage of time, except for the first quartile of the time-decaying speed of calls.

## 4.6 Descriptive statistics: Japan – TOKYOSE

Figure 22. Rolling monthly return of underlying asset (RM) – Japan – TOKYOSE. This figure compares the underlying asset price of the option with its rolling monthly return as a proxy for market states where a positive (negative) monthly return is defined as a bull (bear) market.

As we can see in figure 22, the monthly return of TOPIX is relatively stationary compared to the underlying asset price, ranging from -40% to 20%.

Figure 23. Binary classification of market states of the underlying assets by Bry Boschan dating algorithmJapan – TOKYOSE. Parameters for the Bry Boschan dating algorithm are a minimum window of 6 days, a minimum phase of 6 days, and a minimum cycle of 15 days. White-shaded area is classified as bull market whereas black-shaded area is classified as bear market.

Figure 24. Triple classification of market states of the underlying assets by Bollinger BandsJapan – TOKYOSE. Parameters of Bollinger Bands applied to monthly returns of the underlying assets are 22 days moving average and one standard deviation. The market is bull (bear) if monthly return is higher (lower) than upper (lower) band whereas the market is mean-reversion if monthly returns falls between upper band and lower band.

Table XIII

Descriptive Statistics: Dummy Variables – Japan – TOKYOSE

This table shows the number of observations of the composition of the binary classification and triple classification of the market states of the underlying assets of options and their proportions. Average duration in days of the bull and bear markets classified by Bry Boschan dating algorithm are also reported, so do their average amplitude which represents the average range of underlying asset price in that market state.

As we can see in table XIII, for binary classification, the proportion of bull market (60%) is higher than that of bear market (40%). Also, bull market lasts longer (22.2 days) and has smaller amplitude (115.5). For triple classification, most of the market states are mean-reversion states (46%) and the proportion of bull market (27%) is equal to that of bear market (27%).

Table XIV

Descriptive Statistics: Numeric Variables – Japan – TOKYOSE

This table reports the number of observations, mean, standard deviation, minimum, first quartile, median, third quartile, and maximum for each numeric variable. The slope (time-decaying speed of the time value of options), implied volatilities, implied volatility divided by their long-term average and log of moneyness are reported for calls and puts respectively.

As we can see in table XIV, the maximum and minimum of the time-decaying speed (SL) of the time value of Japan – TOKYOSE options are extreme due to the nature of the fourth-order polynomial regression for the term decay plot and limited observations of some specific options. However, the mean, first quartile, median, and third quartile of time-decaying speed are all positive and reasonably small, representing the general decrease in time value with the passage of time.

## 4.7 Descriptive statistics: Malaysia

Figure 25. Rolling monthly return of underlying asset (RM) – Malaysia – FBMKLCI. This figure compares the underlying asset price of the option with its rolling monthly return as a proxy for market states where a positive (negative) monthly return is defined as a bull (bear) market.

As we can see in figure 25, the monthly return of FTSE BURSA MALAYSIA KLCI is relatively stationary compared to the underlying asset price, ranging from -11% to 9%.

Figure 26. Binary classification of market states of the underlying assets by Bry Boschan dating algorithmMalaysia – FBMKLCI. Parameters for the Bry Boschan dating algorithm are a minimum window of 6 days, a minimum phase of 6 days, and a minimum cycle of 15 days. The white-shaded area is classified as bull market whereas black-shaded area is classified as bear market.

Figure 27. Triple classification of market states of the underlying assets by Bollinger Bands – Malaysia – FBMKLCI. Parameters of Bollinger Bands applied to monthly returns of the underlying assets are 22 days moving average and one standard deviation. The market is bull (bear) if the monthly return is higher (lower) than upper (lower) band whereas the market is mean-reversion if monthly returns fall between the upper band and lower band.

Table XV

Descriptive Statistics: Dummy Variables – Malaysia – FBMKLCI

This table shows the number of observations of the composition of the binary classification and triple classification of the market states of the underlying assets of options and their proportions. The average duration in days of the bull and bear markets classified by Bry Boschan dating algorithm are also reported, so do their average amplitude which represents the average range of underlying asset price in that market state.

As we can see in table XV, for binary classification, the proportion of bull market (57%) is slightly higher than that of bear market (43%). Also, bull market lasts longer (22.7 days) and has greater amplitude (73.1). For triple classification, most of the market states are mean-reversion states (48%) and the proportion of bull market (26%) is equal to that of bear market (26%).

Table XVI

Descriptive Statistics: Numeric Variables – Malaysia – FBMKLCI

This table reports the number of observations, mean, standard deviation, minimum, first quartile, median, third quartile, and maximum for each numeric variable. The slope (time-decaying speed of the time value of options), implied volatilities, implied volatility divided by their long-term average and log of moneyness are reported for calls and puts respectively.

As we can see in table XVI, the maximum and minimum of the time-decaying speed (SL) of the time value of Malaysia – FBMKLCI options are extreme due to the nature of the fourth-order polynomial regression for the term decay plot and limited observations of some specific options. However, the mean, first quartile, median, and third quartile of time-decaying speed are all positive and reasonably small, representing the general decrease in time value with the passage of time.

# V. Empirical Results

For each type of options, linear panel regression model is implemented using R Studio for the six regression models described in section III. For model three and four which use the Bry Boschan dating algorithm to classify the market state of the underlying asset into bull and bear as the main explanatory variable, BryBear is set as the reference category. For model five and six which use the Bollinger Bands to classify the market state of the underlying asset into bull, bear, and mean-reversion as the main explanatory variable, BollingerMeanReversion is set as the reference category since our interest is to see how BollingerBull and BollingerBear affect the time-decaying speed.

Recall the hypothesis that when the underlying asset is bullish, the time value of put options decreases faster, when the underlying asset is bearish, the time value of call options decreases faster. For calls, model 1 and model 2 is consistent with the hypothesis if the estimated coefficient of RM is significantly negative, model 3 and model 4 is consistent with the hypothesis if the estimated coefficient of BryBull is significantly negative, and model 5 and model 6 is consistent with the hypothesis if the estimated coefficient of BollingerBear is significantly positive. For puts, model 1 and model 2 is consistent with the hypothesis if the estimated coefficient of RM is significantly positive, model 3 and model 4 is consistent with the hypothesis if the estimated coefficient of BryBull is significantly positive, and model 5 and model 6 is consistent with the hypothesis if the estimated coefficient of BollingerBull is significantly positive.

Regression summaries below are structured as follows: For each type of options, summaries of the six regression models for calls and puts are reported respectively.

## 5.1 Summaries for the six regression models: Taiwan – TAIWGHT

Table XVII

Regression Summary for Calls – Taiwan – TAIWGHT

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XVII, for Taiwan – TAIWGHT calls, the coefficient of monthly return (RM) is significantly negative for model 1 and 2, meaning that when the market is bearish (RM decreases), the time-decaying speed of the time value of call options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly negative, meaning that the time-decaying speed is higher for bear market compared to bull market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBear is positive though not significant for both models, meaning that the time-decaying speed is higher for bear market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

Table XVIII

Regression Summary for Puts – Taiwan – TAIWGHT

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XVIII, for Taiwan – TAIWGHT puts, the coefficient of monthly return (RM) is significantly positive for model 1 and 2, meaning that when the market is bullish (RM increases), the time-decaying speed of the time value of put options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to bear market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5, the coefficient of BollingerBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis. However, for model 6, the coefficient of BollingerBull is significantly negative, meaning that the time-decaying speed is lower for bull market compared to mean-reversion market, which is not consistent with the hypothesis. Possible reasons may due to the different model specifications between model 5 and model 6 in that model 6 uses volatilities divided by their long-term average rather than the original volatilities used by model 5.

## 5.2 Summaries for the six regression models: Hong Kong – HKHCHIE

Table XIX

Regression Summary for Calls – Hong Kong – HKHCHIE

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XIX, for Hong Kong – HKHCHIE calls, the coefficient of monthly return (RM) is significantly negative for model 1 and 2, meaning that when the market is bearish (RM decreases), the time-decaying speed of the time value of call options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly negative, meaning that the time-decaying speed is higher for bear market compared to bull market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBear is significantly positive, meaning that the time-decaying speed is higher for bear market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

Table XX

Regression Summary for Puts – Hong Kong – HKHCHIE

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XX, for Hong Kong – HKHCHIE puts, the coefficient of monthly return (RM) is significantly positive for model 1 and 2, meaning that when the market is bullish (RM increases), the time-decaying speed of the time value of put options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to bear market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5, the coefficient of BollingerBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis. However, for model 6, the coefficient of BollingerBull is significantly negative, meaning that the time-decaying speed is lower for bull market compared to mean-reversion market, which is not consistent with the hypothesis. Possible reasons may due to the different model specifications between model 5 and model 6 in that model 6 uses volatilities divided by their long-term average rather than the original volatilities used by model 5.

## 5.3 Summaries for the six regression models: South Korea

Table XXI

Regression Summary for Calls – South Korea – KOR200I

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXI, for South Korea – KOR200I calls, the coefficient of monthly return (RM) is significantly negative for model 1 and 2, meaning that when the market is bearish (RM decreases), the time-decaying speed of the time value of call options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly negative, meaning that the time-decaying speed is higher for bear market compared to bull market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBear is significantly positive, meaning that the time-decaying speed is higher for bear market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

Table XXII

Regression Summary for Puts – South Korea – KOR200I

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXII, for South Korea – KOR200I puts, the coefficient of monthly return (RM) is significantly positive for model 1 and 2, meaning that when the market is bullish (RM increases), the time-decaying speed of the time value of put options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to bear market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5, the coefficient of BollingerBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis. However, for model 6, the coefficient of BollingerBull is negative, meaning that the time-decaying speed is lower for bull market compared to mean-reversion market, which is not consistent with the hypothesis. Possible reasons may due to the different model specifications between model 5 and model 6 in that model 6 uses volatilities divided by their long-term average rather than the original volatilities used by model 5.

## 5.4 Summaries for the six regression models: Japan – JAPDOWA

Table XXIII

Regression Summary for Calls – Japan – JAPDOWA

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXIII, for Japan – JAPDOWA calls, the coefficient of monthly return (RM) is significantly negative for model 1 and 2, meaning that when the market is bearish (RM decreases), the time-decaying speed of the time value of call options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly negative, meaning that the time-decaying speed is higher for bear market compared to bull market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBear is significantly positive, meaning that the time-decaying speed is higher for bear market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

Table XXIV

Regression Summary for Puts – Japan – JAPDOWA

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXIV, for Japan – JAPDOWA puts, the coefficient of monthly return (RM) is significantly positive for model 1 and 2, meaning that when the market is bullish (RM increases), the time-decaying speed of the time value of put options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to bear market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

## 5.5 Summaries for the six regression models: Japan – JPXNK40

Table XXV

Regression Summary for Calls – Japan – JPXNK40

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXV, for Japan – JPXNK40 calls, the coefficient of monthly return (RM) is significantly negative for model 1 and 2, meaning that when the market is bearish (RM decreases), the time-decaying speed of the time value of call options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly negative, meaning that the time-decaying speed is higher for bear market compared to bull market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBear is significantly positive, meaning that the time-decaying speed is higher for bear market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

Table XXVI

Regression Summary for Puts – Japan – JPXNK40

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXVI, for Japan – JPXNK40 puts, the coefficient of monthly return (RM) is significantly positive for model 1 and 2, meaning that when the market is bullish (RM increases), the time-decaying speed of the time value of put options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to bear market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

## 5.6 Summaries for the six regression models: Japan – TOKYOSE

Table XXVII

Regression Summary for Calls – Japan – TOKYOSE

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXVII, for Japan – TOKYOSE calls, the coefficient of monthly return (RM) is significantly negative for model 1 and 2, meaning that when the market is bearish (RM decreases), the time-decaying speed of the time value of call options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly negative, meaning that the time-decaying speed is higher for bear market compared to bull market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBear is significantly positive, meaning that the time-decaying speed is higher for bear market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

Table XXVIII

Regression Summary for Puts – Japan – TOKYOSE

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXVIII, for Japan – TOKYOSE puts, the coefficient of monthly return (RM) is significantly positive for model 1 and 2, meaning that when the market is bullish (RM increases), the time-decaying speed of the time value of put options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to bear market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBull is positive though not significant for both models, meaning that the time-decaying speed is higher for bull market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

## 5.7 Summaries for the six regression models: Malaysia

Table XXIX

Regression Summary for Calls – Malaysia – FBMKLCI

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of call options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXIX, for Malaysia – FBMKLCI calls, the coefficient of monthly return (RM) is significantly negative for model 1 and 2, meaning that when the market is bearish (RM decreases), the time-decaying speed of the time value of call options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly negative, meaning that the time-decaying speed is higher for bear market compared to bull market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBear is significantly positive, meaning that the time-decaying speed is higher for bear market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

Table XXX

Regression Summary for Puts – Malaysia – FBMKLCI

In this table, each column represents one model. All models have the same response variable which is the time-decaying speed of the time value of options (SL). Estimated coefficients and standard errors in parentheses are reported.

In table XXX, for Malaysia – FBMKLCI puts, the coefficient of monthly return (RM) is significantly positive for model 1 and 2, meaning that when the market is bullish (RM increases), the time-decaying speed of the time value of put options increases, which is consistent with the hypothesis. For model 3 and 4, the coefficient of BryBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to bear market classified by the Bry Boschan dating algorithm, which is consistent with the hypothesis. For model 5 and 6, the coefficient of BollingerBull is significantly positive, meaning that the time-decaying speed is higher for bull market compared to mean-reversion market classified by the Bollinger Bands, which is consistent with the hypothesis.

# VI. Robustness Tests

Rather than using the monthly return of the underlying asset of options, I use a shorter time window, namely, weekly return, to test the Taiwan – TAIWGHT index options to see if the hypothesis sill holds. As we can see in table XXXI, for Taiwan – TAIWGHT calls, the coefficient of weekly return (RW) is significantly negative for model 1 and 2, meaning that when the market is bearish (RW decreases), the time-decaying speed of the time value of call options increases, which is consistent with the hypothesis. Also, for Taiwan – TAIWGHT puts,  the coefficient of weekly return (RW) is significantly positive for model 1 and 2, meaning that when the market is bullish (RW increases), the time-decaying speed of the time value of put options increases, which is consistent with the hypothesis. As a result, the hypothesis is robust in using different time window of past returns of the underlying asset of options as a measurement of the market states, namely, monthly or weekly returns.

Table XXXI

Robustness tests

This table tests the hypothesis using weekly return of the underlying asset of options rather than monthly return as a measure of market state of the underlying asset for Taiwan – TAIWGHT index options.

# VII. Conclusion

First, we summarize whether the hypothesis holds for each type of index options in table XXXII. As we can see in table XXXII, almost all of the empirical results (81 out of 84 linear panel regression models) supports the hypothesis that when the market state of the underlying asset of options is bullish, the time value of put options decreases faster whereas when the market state of the underlying asset of options is bearish, the time value of call options decreases faster. The empirical results are internationally consistent with the hypothesis across countries including Taiwan, Hong Kong, South Korea, Japan, and Malaysia. And the results are robust in using different methodologies to classify the market states of the underlying assets of options, namely, monthly or weekly returns of the underlying asset price, Bry Boschan dating algorithm to segregate into bull or bear market, and Bollinger Bands to segregate into bull, bear, or mean-reversion market.

Second, the findings have some implications for options traders in that they can capitalize on the different time-decaying speed of out of the money options with respect to different market states of the underlying assets. Further construction of trading or hedging strategies is encouraged to take advantage of the empirical findings in this thesis.

Table XXXII

Consistency of Hypothesis

This table summarizes whether the empirical result is consistent with the hypothesis that when the market is bullish (bearish), the time-decaying speed of the time value of put (call) options increases for each type of index options. T represents the hypothesis holds whereas F represents the hypothesis does not hold.

# List of Tables

Table I Underlying Assets of Index Options

Table II Overview of Index Options

Table III Descriptive Statistics: Dummy Variables – Taiwan – TAIWGHT

Table IV Descriptive Statistics: Numeric Variables – Taiwan – TAIWGHT

Table V Descriptive Statistics: Dummy Variables – Hong Kong – HKHCHIE

Table VI Descriptive Statistics: Numeric Variables – Hong Kong – HKHCHIE

Table VII Descriptive Statistics: Dummy Variables – South Korea – KOR200I

Table VIII Descriptive Statistics: Numeric Variables – South Korea – KOR200I

Table IX Descriptive Statistics: Dummy Variables – Japan – JAPDOWA

Table X Descriptive Statistics: Numeric Variables – Japan – JAPDOWA

Table XI Descriptive Statistics: Dummy Variables – Japan – JPXNK40

Table XII Descriptive Statistics: Numeric Variables – Japan – JPXNK40

Table XIII Descriptive Statistics: Dummy Variables – Japan – TOKYOSE

Table XIV Descriptive Statistics: Numeric Variables – Japan – TOKYOSE

Table XV Descriptive Statistics: Dummy Variables – Malaysia – FBMKLCI

Table XVI Descriptive Statistics: Numeric Variables – Malaysia – FBMKLCI

Table XVII Regression Summary for Calls – Taiwan – TAIWGHT

Table XVIII Regression Summary for Puts – Taiwan – TAIWGHT

Table XIX Regression Summary for Calls – Hong Kong – HKHCHIE

Table XX Regression Summary for Puts – Hong Kong – HKHCHIE

Table XXI Regression Summary for Calls – South Korea – KOR200I

Table XXII Regression Summary for Puts – South Korea – KOR200I

Table XXIII Regression Summary for Calls – Japan – JAPDOWA

Table XXIV Regression Summary for Puts – Japan – JAPDOWA

Table XXV Regression Summary for Calls – Japan – JPXNK40

Table XXVI Regression Summary for Puts – Japan – JPXNK40

Table XXVII Regression Summary for Calls – Japan – TOKYOSE

Table XXVIII Regression Summary for Puts – Japan – TOKYOSE

Table XXIX Regression Summary for Calls – Malaysia – FBMKLCI

Table XXX Regression Summary for Puts – Malaysia – FBMKLCI

Table XXXI Robustness tests

Table XXXII Consistency of Hypothesis

# List of Figures

Figure 1. Variation of theta of a European call option with stock price

Figure 2. Illustration of hypothesis

Figure 3. Illustration of Bry-Boschan (1971) dating rule

Figure 4. Illustration of binary classification of market states of the underlying assets by Bry     Boschan dating algorithm – subset(last 25%) – Taiwan – TAIWGHT

Figure 5. Illustration of term decay plot

Figure 6. Illustration of triple classification of market states of the underlying assets by Bollinger Bands – subset(last 25%) – Taiwan – TAIWGHT

Figure 7. Rolling monthly return of underlying asset (RM) – Taiwan – TAIWGHT

Figure 8. Binary classification of market states of the underlying assets by Bry Boschan dating algorithm – Taiwan – TAIWGHT

Figure 9. Triple classification of market states of the underlying assets by Bollinger Bands – Taiwan – TAIWGHT

Figure 10. Rolling monthly return of underlying asset (RM) – Hong Kong – HKHCHIE

Figure 11. Binary classification of market states of the underlying assets by Bry Boschan dating algorithm – Hong Kong – HKHCHIE

Figure 12. Triple classification of market states of the underlying assets by Bollinger Bands – Hong Kong – HKHCHIE

Figure 13. Rolling monthly return of underlying asset (RM) – South Korea – KOR200I

Figure 14. Binary classification of market states of the underlying assets by Bry Boschan dating algorithm – South Korea – KOR200I

Figure 15. Triple classification of market states of the underlying assets by Bollinger Bands – South Korea – KOR200I

Figure 16. Rolling monthly return of underlying asset (RM) – Japan – JAPDOWA

Figure 17. Binary classification of market states of the underlying assets by Bry Boschan dating algorithm – Japan – JAPDOWA

Figure 18. Triple classification of market states of the underlying assets by Bollinger Bands – Japan – JAPDOWA

Figure 19. Rolling monthly return of underlying asset (RM) – Japan – JPXNK40

Figure 20. Binary classification of market states of the underlying assets by Bry Boschan dating algorithm – Japan – JPXNK40

Figure 21. Triple classification of market states of the underlying assets by Bollinger Bands – Japan – JPXNK40

Figure 22. Rolling monthly return of underlying asset (RM) – Japan – TOKYOSE

Figure 23. Binary classification of market states of the underlying assets by Bry Boschan dating algorithm – Japan – TOKYOSE

Figure 24. Triple classification of market states of the underlying assets by Bollinger Bands – Japan – TOKYOSE

Figure 25. Rolling monthly return of underlying asset (RM) – Malaysia – FBMKLCI

Figure 26. Binary classification of market states of the underlying assets by Bry Boschan dating algorithm – Malaysia – FBMKLCI

Figure 27. Triple classification of market states of the underlying assets by Bollinger Bands – Malaysia – FBMKLCI

# References

Candelon, Bertrand, Jameed Ahmed, and Stefan Theo Marcella Straetmans. “Predicting and capitalizing on stock market bears in the us.” METEOR, Maastricht research school of Economics of TEchnology and ORganizations, 2012.

Candelon, Bertrand, Jan Piplack, and Stefan Straetmans. “On measuring synchronization of bulls and bears: The case of East Asia.” Journal of banking & finance 32.6 (2008): 1022-1035.

Chauvet, Marcelle, and Simon Potter. “Coincident and leading indicators of the stock market.” Journal of Empirical Finance 7.1 (2000): 87-111.

Chen, Shiu-Sheng. “Predicting the bear stock market: Macroeconomic variables as leading indicators.” Journal of Banking & Finance 33.2 (2009): 211-223.

Claessens, Stijn, M. Ayhan Kose, and Marco E. Terrones. “How do business and financial cycles interact?” Journal of International Economics 87.1 (2012): 178-190.

Cooper, Michael J., Roberto C. Gutierrez, and Allaudeen Hameed. “Market states and momentum.” The Journal of Finance 59.3 (2004): 1345-1365.

Figlewski, Stephen, and Steven Freund. “The pricing of convexity risk and time decay in options markets.” Journal of banking & finance 18.1 (1994): 73-91.

Harding, Don, and Adrian Pagan. “A comparison of two business cycle dating methods.” Journal of Economic Dynamics and Control 27.9 (2003): 1681-1690.

Hlavac, Marek (2015). stargazer: Well-Formatted Regression and Summary Statistics Tables.

John C. Hull, Options, Futures, and Other Derivatives. Boston: Pearson Education, 2011, p.387-388

Nyberg, Henri. “Predicting bear and bull stock markets with dynamic binary time series models.” Journal of Banking & Finance 37.9 (2013): 3351-3363.

Pagan, Adrian R., and Kirill A. Sossounov. “A simple framework for analyzing bull and bear markets.” Journal of Applied Econometrics 18.1 (2003): 23-46.

Peter Carr and Liuren Wu, “What type of process underlies options? A simple robust test,”      the Journal of Finance, vol. LVIII, no. 6 • December 2003

Wan-Jung Hsu, “Predicting and Capitalizing on Two Types of Stock Bear Markets in the U.S,” Ph.D. dissertation, University of Washington, 2017.

[1] John C. Hull, Options, Futures, and Other Derivatives. Boston: Pearson Education, 2011, p. 387-388

[2] Wan-Jung Hsu (2017)

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