Two Dummy Variables Framework

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conditional variance equation in Model 1 (the Dilution-Adjusted GARCH Model) is modified to contain the asymmetric effect as follows:

( )

²

(1/ 2)

D D D

it i it it it

R

= +

r λσ

−

σ

+

u

,

( ¹ )

D D

it it it i it it it

u ≡

σ

Z = −

δ

I

σ

Z

, with

Z

_it

Ω

_t−1

~ (0,1) N

,

(1 ) ,

D

it i it

I

it

σ

= −

δ σ

( )

²

2 2

0 1 ( 1) 2 ( 1) ( 1) .

it i i i t i

u

i t

l u

i i t

σ

=

β

+

β σ

₋ +

β

₋ − ₋ (11)

The parameters

β

_1i,

β

_2i, and

l should satisfy some regularity conditions to ensure

_i that the unconditional volatility of stock return process is finite. In Eq. (11), where

l >0, negative return shocks increase volatility more than positive shocks. Therefore,

i

the conditional volatility function for stock return process accounting for asymmetric and warrant effects is

( )

^{σitD 2}

^{= ⎡} ⎣ ^{(1 −}

^{δi it}

^{I ) /(1 −}

^{δi i t}

^I

^{( 1)−}

^{) ⎤} ⎦ ⎢

²

^⎡ ⎣ ^{(1 −}

^{δi i t}

^I

^{( 1)−}

⁾

^2β0i

⁺

^{β σ1i}

(

^{i tD( 1)−}

)

²

⁺

^β2i

( ^u

^{i tD( 1)−}

^{− l u}

^{i i tD( 1)−}

)

²

^⎤ ⎥ ⎦

.(12)

B. Two Dummy Variables Framework

In general, the dilution effect is due to the possible exercise of warrants; higher stock prices lead to a greater possibility of exercising them. When stock prices are high enough, the dilution effects may already partly or entirely be reflected in the conditional volatility for representing the possibility of wealth transferring from stock holders to warrant holders. This argument is supported by Hauser and Lauterbach (1997). Hence, a threshold dummy variable is added to judge if the stock price is higher than the exercise price, the two dummy variable framework makes the conditional volatility unchanged after warrant introduction until stock price exceeds the exercise price.

Similar to the one dummy variable framework, the original form is presented in

‧

Model 3. Model 4 shows the extended form clarifying the ambiguity of asymmetric effect.

Model 3: The Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices

The threshold dummy variable helps to identify whether the relationship between stock price and exercise price affects the volatility. Since Eq. (10) indicates conditional volatility scales down after warrant introduction, for simplicity we redefine the conditional volatility function with warrant introduction as

2 2 2 2

( 1) 0 1 ( 1) 2 ( 1)

(1 ) ( )

it i i t

I

i i i t i

u

i t

σ

= −

δ ₋ β

+

β σ ₋

+

β ₋ .

Then, by adding a threshold dummy variable

D

_{i t( 1)−} to identify the relation between exercise price and stock price, the conditional volatility can be modified as

ln(

S

_it/

S

_{i t}( 1)₋ )≡

R

_it, conditional volatility is always positive and unconditional volatility is finite. Eq. (13) shows that the warrant introduction leaves the conditional volatility unchanged until the firm’s stock price is higher than the exercise price. We can also express it as

2 2

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23

The conditional expected return of stock i at time t is

(

_{it t} 1

)

_{i it} (1/ 2) _it²

E R

Ω₋ = +

r λσ

−

σ

, and its conditional volatility is

( ) ( )

^{2 2 2 2}

1 1 1 (1 ( 1) ( 1)) ( 0 1 ( 1) 2 ( 1))

t it t t it it i i t i t i i i t i i t

Var

₋

R

Ω₋ =

Var

₋

u

=

σ

= −

δ I

₋

D

₋

β

+

β σ

₋ +

β u

₋ .

Model 4: The Asymmetric Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices

In order to distinguish the dilution effect of warrant introduction from the asymmetric leverage effect, as in Model 2, the conditional volatility function in Model 3 is transformed to

2 2 2 2

( 1) ( 1) 0 1 ( 1) 2 ( 1) ( 1)

(1 ) ( ( ) )

it i i t

I D

i t i i i t i

u

i t

l u

i i t

σ

= −

δ

_{− −}

β

+

β σ

₋ +

β

₋ − ₋ . (14)

With >0

l

_i , negative shocks increase volatility more than positive shocks and

l is

_i the parameter for asymmetric effect.

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Chapter 4

Data

Data listed on Hong Kong Exchanges and Clearing Limited (HKEx) is used for demonstration. Hong Kong is one of the world’s three most actively traded warrant markets. Top six exchanges represent almost 90% of the aggregate warrant turnover around the world.⁴ In general, equity warrants have a long expiration period. In the interest of observing total trading period of a warrant, unexpired warrants are excluded; empirical analysis utilizes expired warrant data issued from Jan. 1, 2001, to Dec. 30, 2004. Total observations of each underlying stock return includes its entire warrant trading life, the sample period, and the same amount of time before warrant introduction. The time prior warrant introduction is referred to as the control period.

Observations for each stock include the sample and control periods.

All official daily closing prices of stocks after capital action adjustments are obtained from the Datastream. The data for exercise provisions and other descriptions of the warrants are collected from the annual Fact Book published by HKEx. There were 82 new equity warrants listed on HKEx during 2001-2004. An equal amount of time as warrant lifetime is required for the control group prior its introduction. A

4

In 2005, Hong Kong Exchanges and Clearing Limited (HKEx) published a brief comparison of the

Hong Kong warrant market with oversea counterparts in terms of the number issued and turnover. It

showed that Hong Kong was ranked number two in terms of annual turnover of listed warrants

among world stock exchanges in 2003, just behind Deutsche Börse (DB) of Germany. Clarification

of the double counts problem in Germany, Hong Kong became the world’s most actively traded

warrant market by turnover value in 2003.

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considerable number of stocks are excluded, which include those with another warrant listing during the observation, or those with warrants shortly after an Initial Public Offering (IPO) making the control period too short for comparison.⁵ To avoid the complication of different exchange rates, the warrants traded in currencies other than Hong Kong dollars are also excluded. Lastly, the study also excludes a few coding error warrants or stocks no longer in the public equity market. After elimination, the final sample includes 36 warrants issued from 2001 to 2004 with 37748 observations.

Table summarizes the 36 warrants sorted by listing date. Since almost all Ⅰ subscription periods, except a warrant issued by Regal Hotels Intl. HDG. in 2004, start prior to the listing day of warrants, the starting date of subscription periods is considered the same as the date warrants are introduced. The warrants in Table Ⅰ cover different lengths of time and range from deep-in-the-money to deep-out-of-the-money. Exercise prices of warrants are drawn and plotted with daily returns of the underlying stocks in Appendix A to show the basic patterns of stock returns and to see the relationship between stock and exercise prices.⁶ As indicated in Appendix A, stock returns display smaller volatilities after warrant introduction.

5

Riche Multi-Media HDG. was the only exception, because it went public on Feb. 15, 2000. The control period is briefly unavailable from 19990617-2000214.

6

Each stock is plotted and it is found that most stocks appear to have smaller volatilities after warrant

introduction. Because of page limitation it is only possible to show the stock return process for some

sample companies in Appendix A.

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26 Table Ⅰ

Summary of the 36 warrants issued from Jan. 1, 2001, to Dec. 31, 2004

Sample Sequence Number & Company Listing date Introduction

date Subscription period Date in Control Group Total Date in Sample Number of Valid Observations

Exercise price per unit (HK$) (1) SUN HUNG KAI & CO. 20010115 20010112 20010112-20030111 19990112-20010111 19990112-20030111 1043 3

(2) GOLD PEAK INDS. 20010214 20010209 20010209-20020208 20000209-20010208 20000209-20020208 522 2.2

(3) COSMOS MACHINERY ENTS. 20010615 20010611 20010611-20030610 19990611-20010610 19990611-20030610 1042 0.4

(4) LUKS GROUP 20010618 20010614 20010614-20040617 19980614-20010613 19980614-20040617 1568 0.9

(5) LEI SHING HONG 20010622 20010619 20010619-20060619 19960619-20010618 19960619-20060619 2608 3

(6) CHINA TRAVEL INTL.INVS. 20010703 20010703 20010703-20030630 19990701-20010702 19990701-20030630 1042 1.22

(7) KITH HOLDINGS 20010711 20010711 20010711-20040630 19980713-20010710 19980713-20040630 1557 2.2

(8) KINGBOARD CHEMICALS HDG. 20010903 20010903 20010903-20031231 19990501-20010902 19990501-20031231 1217 5.8

(9) CITY TELECOM 20011102 20011102 20011102-20041101 19981102-20011101 19981102-20041101 1565 0.11

(10) HAIER ELECTRONICS GP. 20020226 20020226 20020226-20040226 20000226-20020225 20000226-20040226 1043 0.52

(11) PAUL Y ENGR.GP. 20020305 20020301 20020301-20030829 20000901-20020228 20000901-20030829 780 0.4

(12) ASIA ALUMINUM HOLDINGS 20020412 20020410 20020410-20040409 20000410-20020409 20000410-20040409 1044 0.77 (13) FAR EAST PHARM.TECH. 20020507 20020507 20020507-20030506 20010507-20020506 20010507-20030506 521 2.62 (14) HOP HING HOLDINGS 20020603 20020529 20020529-20050430 19990627-20020528 19990627-20050430 1524 0.27 (15) SINOLINK WORLDWIDE HDG. 20020605 20020531 20020531-20031129 20001129-20020530 20001129-20031129 782 1 (16) RICHE MULTI-MEDIA HDG. 20020620 20020617 20020617-20050616 20000214-20020616 20000214-20050616 1392 3.6

(17) HARMONY ASSET 20020625 20020621 20020621-20040630 20000621-20020620 20000621-20040630 1050 0.08

(18) PREMIUM LAND 20020709 20020709 20020709-20030708 20010709-20020708 20010709-20030708 521 0.22

(19) SOUTH CHINA HDG. 20020725 20020723 20020723-20030723 20010723-20020722 20010723-20030723 522 0.42 (20) CHINA STRATEGIC HDG. 20020829 20020829 20020829-20031231 20010401-20020828 20010401-20031231 717 0.16

(21) ALCO HOLDINGS 20020902 20020902 20020902-20050901 19990902-20020901 19990902-20050901 1565 0.98

(22) PACIFIC ANDES INTL.HDG. 20020926 20020926 20020926-20040325 20000926-20020925 20000926-20040325 912 0.85

(23) PEACE MARK HDG. 20030807 20030805 20030805-20050804 20010805-20030804 20010805-20050804 1043 0.65

(24) SOUNDWILL HOLDINGS 20030905 20030903 20030903-20060302 20010302-20030902 20010302-20060302 1304 2 (25) HERITAGE INTL.HDG. 20031016 20031013 20031013-20050412 20020413-20031012 20020413-20050412 781 0.17

(26) EFORCE HOLDINGS 20031210 20031208 20031208-20041207 20021208-20031207 20021208-20041207 521 0.28

(27) ALLIED PROPERTIES 20031205 20031205 20031205-20041206 20021205-20031204 20021205-20041206 522 2.5 (28) KENFAIR INTL.HDG. 20031205 20031202 20031202-20051202 20011202-20031201 20011202-20051202 952 0.69 (29) QUALITY HLTHCR.ASIA 20040114 20040114 20040114-20070113 20010114-20070113 20010114-20070113 1564 2.5 (30) PLAYMATES HOLDINGS 20040524 20040524 20040524-20050523 20030524-20040523 20030524-20050523 520 1.42 (31) CHINA TRAVEL INTL.INVS. 20040602 20040602 20040602-20060531 20020603-2040601 20020603-20060531 1042 1.508 (32) GLOBAL BIO-CHEM TECH.GP. 20040601 20040528 20040528-20070531 20010528-20040527 20010528-20070531 1568 9.8 (33) U-RIGHT INTL.HDG. 20040624 20040618 20040618-20050623 20030618-20040617 20030618-20050623 526 0.2 (34) RONTEX INTL.HDG. 20040630 20040628 20040628-20050627 20030628-20040627 20030628-20050627 520 0.102 (35) REGAL HOTELS INTL.HDG. 20040804 20050202 20050202-20070726 20020726-20050201 20020726-20070726 1304 0.25 (36) MAN YUE INTL.HDG. 20041104 20041104 20041104-20061103 20021104-20041103 20021104-20061103 1044 0.48

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Chapter 5

在文檔中 公司認股權證對股價之影響 - 政大學術集成 (頁 24-30)