Warrants and CBBC
Bodie, et al. (2021) Ch. 20
Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull
2013
1
Warrant premium


Warrant premium窩輪溢價for call is the difference
between the cost of obtaining one share of stock by
buying and exercising the call warrants($Co x CR
+$X) and buying one share in the stock market at
the current spot price ($So), as a percentage of
current spot price.
Warrant premium for call =
(Co × CR+X)−So
So
× 100%
where Co x CR = cost of CR units of call warrants required to
buy one share of stock
X = exercise price paid to buy one share of stock by exercising
CR units of warrants
So = current market price of one share of stock
2
Gearing


Effective gearing實際槓桿; 有效槓桿 for call is the price
elasticity of a derivative, equal to the percentage
change of the price of call warrants used to buy one
share of stock ∆($Co × CR) / ($Co × CR) in response to
percentage change in current spot price of one share of
stock ∆($S)/$So .
effective gearing for a call =
where
∆(Co× CR)
∆S
×
So
Co×CR
∆(Co× CR)
= Delta 對沖值
∆S
So
= gearing ratio槓桿比率
Co×CR
3
Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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5
Warrant premium
(Co × CR+X)−So
So

Warrant premium for call =

Example: call warrant on Alibaba
($0.023 x 100 +$211.11)−$162.20
×
$162.20
=
$2.30+$211.11−$162.20
×
$162.20
=
$213.41−$162.20
$162.20
=

$51.21
$162.20
× 100%
100%
100%
× 100%
× 100% = 0.315721 x 100% = 31.5721%
If current stock price rises by 31.5721%, to $213.41, the
cost of buying and exercising CR units of stock warrants is
the same as buying one stock in the stock market
Gearing
∆(Co× CR)
∆S

effective gearing for a call =

Example: call warrant on Alibaba
0.1338 ×
×
So
Co×CR
$162.20
$0.023×100
= 0.1338 x 70.5217 = 9.4358


If stock price of Alibaba rises by $1, $(Co x 100) increases
by $0.1338 or $Co increases by $0.1338/100 = $0.001338
or up to $0.023+$0.001338 = $0.024338 per one warrant.
Investors earn $0.001338 x 10,000 warrants per lot =
$13.38.
If Alibaba rises by 1%, $(Co x 100) increases by 9.4358% or
price of one warrant ($Co) rises to $0.023 x (1+9.4358%) =
$0.02517. Investors earn ($0.02517 - $0.023 = $0.00217) x
10,000 warrants per lot = $21.7023 per lot.
7

Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
Warrant premium

Warrant premium for call =

Example: call warrant on HSI
($0.019 x 7,000 +$29,188)−$25,415.69
$25,415.69
=
$133+$29,188−$25,415.69
$25,415.69
(Co × CR+X)−So
So
× 100%
× 100%
× 100% =
$29,321−$25,415.69
$25,415.69
×
100%
=

$3,905.31
$25,415.69
× 100% = 0.153657 x 100% = 15.3657%
When HSI rises by 15.3657%, to 29,321, the cost of
buying and exercising the CR units of the index
warrants is the same as buying the HSI.
Gearing
∆(Co× CR)
∆S

effective gearing for a call =

Example: call warrant on HSI
0.0995 ×
×
So
Co×CR
$25,415.69
$0.019×7,000
= 0.0995 x 191.09541 = 19.01399


If price of HSI rises by 100 points, $(Co x 7,000) increases
by $9.95 or $Co increases by $9.95/7,000 = $0.0014214 or
up to $0.019 + $0.0014214 = $0.0204214 per one warrant.
Investors earn $0.0014214 x 10,000 warrants per lot =
$14.214
If HSI rises by 1%, $(Co x 7,000) increases by 19.01399%
or price of one warrant rises to $0.019 x (1+19.01399%) =
$0.022613. Investors earn ($0.022613 - $0.019 =
$0.003613) x 10,000 warrants per lot = $36.13.
11

Warrant premium


Warrant premium窩輪溢價 for put is the difference
between the amount of selling one share of stock
by selling one share in the stock market at the
current spot price ($So) and buying the put and
exercising the put ($X - $Po x CR), as a
percentage of current spot price.
Warrant premium for put =
So−(X−Po × CR)
So
× 100%
where Po x CR = cost of CR units of put warrants required to
sell one share of stock
X = exercise price received for selling one share of stock
by exercising CR units of warrants
12
So = current market price of one share of stock
Gearing


Effective gearing 實際槓桿; 有效槓桿 for put is the
price elasticity of a derivative, equal to the percentage
change of the price of put warrants used to sell one
share of stock ∆($Po × CR) / ($Po × CR) in response to
percentage change in current spot price of one share
∆($S)/$So .
effective gearing for a put =
where
∆(Po× CR)
∆S
×
So
Po×CR
∆(Po× CR)
= Delta 對沖值
∆S
So
= gearing ratio槓桿比率
Po×CR
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
15
Warrant premium
So−(X−Po × CR)
So

Warrant premium for put =

Example: put warrant on Alibaba
$162.20−($173.88−0.243 x 100)
$162.20
=
=

$162.20−$149.58
×
$162.20
$12.62
$162.20
× 100%
× 100%
100%
× 100%
= 0.077805 x 100% = 7.7805%
When current stock price decreases by 7.7805%, to
$149.58, the amount of buying and exercising the CR
units of put warrants received is the same as selling
one stock in the stock market.
16
Gearing
∆(Po× CR)
∆S

effective gearing for a put =

Example: put warrant on Alibaba
−0.5639 ×


×
So
Po×CR
$162.20
$0.243×100
= −0.5639 x 6.674897 = 3.76397
If stock price of Alibaba falls by $1, $(Po x 100) increases by
$0.5639 or $Po increases by $0.5639/100 = $0.005639 or up
to $0.243+$0.005639 = $0.248639 per one warrant.
Investors earn $0.005639 x 10,000 warrants per lot = $56.39.
If Alibaba falls by 1%, $(Po x 100) increases by 3.76397% or
price of one warrant rises to $0.243 x (1+3.76397%) =
$0.252146. Investors earn ($0.252146 - $0.243 = $0.009146)
x 10,000 warrants per lot = $91.46.
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
19
Warrant premium

Warrant premium for put =

Example: put warrant on HSI
So−(X−Po × CR)
So
$25,415.69−($27,600−0.29 x 10,000)
×
$25,415.69
=
$25,415.69−($27,600−$2,900)
×
$25,415.69
=
$25,415.69−($24,700)
×
$25,415.69
× 100%
100%
100%
100% =
$715.69
$25,415.69
× 100%
= 0.028159 x 100% = 2.8159%

When HSI decreases by 2.8159%, to 24,700, the amount
of buying and exercising the put warrants received is the
same as selling the HSI.
20
Gearing
∆(Po× CR)
∆S

effective gearing for a put =

Example: put warrant on HSI
−0.7536 ×
×
So
Po×CR
$25,415.69
$0.29×10,000
= −0.7536 x 8.7640 = 6.6046


If HSI falls by 100 points, $(Po x 10,000) increases by
$75.36 or $P increases by $75.36/10,000 = $0.007536 or
up to $0.29+$0.007536 = $0.297536 per one warrant.
Investors earn $0.007536 x 10,000 warrants per lot =
$75.36.
If HSI falls by 1%, $(Po x 10,000) increases by 6.6046%
or price of one warrant rises to $0.29 x (1+6.6046%) =
$0.30915. Investors earn ($0.30915 - $0.29 = $0.01915) x
10,000 warrants per lot = $191.50.
21
Callable Bull/Bear contracts




Callable bull/bear contracts (CBBC) 牛熊證, like other
derivatives, track the performance of an underlying
asset without requiring investors to pay the full price
to own the actual asset.
CBBC are issued either as Bull牛 or Bear熊contracts,
allowing investors to take bullish or bearish positions
on the underlying assets基礎資產.
CBBC, like covered warrants, are issued by a third
party, usually an investment bank.
CBBC, like stocks and warrants, are listed on the
Stock Exchange of Hong Kong (SEHK)香港聯合交易
所有限公司, are then traded on the cash market, and
are settled in cash 現金結算.
22
Features of CBBC




The delta of a CBBC is close to one in absolute value.
Thus, if the underlying asset increase in value by $1, a
Bull contract 牛證 increases in value x conversion
ratio 兌換比率 (CR) or entitlement ratio 合約權益比率,
generally by approximately the same amount. A Bear
contract 熊證 generally decreases in value x CR, by
approximately the same amount.
Hence, price movement of CBBC tends to track the
price of underlying asset closely.
Also, unlike warrants, CBBC are issued with the
condition that during their lifespan, they will be called
收回 by the issuers when the price of the underlying
asset reaches 觸及 the Call Price 收回價 and then the
CBBC will expire immediately 打靶.
23
Features of CBBC

Such an event is referred to as a Mandatory Call
Event (MCE) 強制收回事件.

Thus, CBBC have a Call Price and a Mandatory call
feature
For Bull contracts, the Call Price must be either equal
to or above the strike price 行 使 價 . For Bear
contracts, the Call Price must be either equal to or
below the strike price.
Investors in CBBC will bear the risk of call because
they may lose all the investments in CBBC.


24
Categories of CBBC




There are two categories of CBBC, namely Category N
CBBC N類牛熊證, and Category R CBBC R類牛熊證.
A Category N CBBC refers to a CBBC where its Call
Price is equal to its strike price. The CBBC holder will
not receive any cash payment once the price of the
underlying asset reaches or goes beyond the call price,
i. e. when the CBBC is called 收回.
For a Category N CBBC, no residual value剩餘價值 will
be paid by the issuers once the price of the underlying
asset reaches or goes beyond the Call Price, i.e. when
the CBBC is called 收回.
.
25
Categories of CBBC


–

–


A Category R CBBC refers to a CBBC where its Call
Price is different from its strike price.
For a Category R Bull contract, the Call Price is above
the strike price.
Price of an underlying asset > Call Price > strike price
For a Category R Bear contract, the Call Price is below
the strike price.
Price of an underlying asset < Call Price < strike price
The Category R CBBC holder may receive a small
amount of cash payment, called residual value 剩餘價值,
once the price of the underlying asset reaches or goes
beyond the Call Price, i.e. when the CBBC is called 收回.
26
.
Theoretical Price of CBBC
Price of a CBBC includes intrinsic value 內 在 值 and
funding cost 財務費用.
 CBBC must be deep in the money 深入價内 and thus has
no time value.
 Like options and warrants, intrinsic value is equal to (S –
X)/CR x contract multiplier for a Bull contract and is equal
to (X – S)/CR x contract multiplier for a Bear contract.
 The funding cost of a CBBC includes the issuer’s
financing or stock borrowing costs, and the issuer’s profit
margin.
 Formula for funding cost per share of CBBC =
(strike price / CR) x funding rate x (days to maturity / 365) x
contract multiplier

27
Theoretical Price of CBBC

–
–
–
Example of a Bull contract:
So = $100; X = $90; Call price= $92; maturity= 3 months,
CR=10 to 1; funding interest rate=2%
Intrinsic value per share of Bull =
(S – X)/CR = ($100- $90)/10 = $1.0
Funding cost per share of Bull =
($90/10) x 2% x (90/365) = $0.0443
Theoretical price per share of the Bull contract at issue =
$1.0 + $0.0443 = $1.0443
28
Theoretical Price of CBBC
Example of a Bear contract:
So = $70; X = $90; Call price= $82; maturity= 3
months,
CR=10 to 1; funding interest rate=2%
– Intrinsic value per share of Bear =
(X – S)/CR = ($90- $70)/10 = $2.0
– Funding cost per share of Bear =
($80/10) x 2% x (90/365) = $0.0394
– Theoretical price per share of the Bear contract
at issue = $2.0 + $0.0394 = $2.0394

29
Value of CBBC before expiry


–
–
–
The value of CBBC before expiry depends upon whether it
is a bull or a bear contract, or it is a Category N or R CBBC.
Example of a Category N Bull contract:
So = $100; X = $85; Call price= $85; maturity= 2 months,
CR=10 to 1; funding interest rate=2%
Intrinsic value per share of Bull =
(S – X)/CR = ($100-85)/10 = $1.5
Funding cost per share of Bull =
($85/10) x 2% x (60/365) = $0.0279
Theoretical price per share of the Bull contract at issue =
$1.5 + $0.0279 = $1.5279
30
Value of CBBC before expiry
–
–
–
–
If St rises to $110 before expiry. the price per share of
the Category N Bull contract is equal to $2.5 + 0.028 =
$2.528.
The stock price rises by 10%, but the price per share of
the bull contract increases by 65.4450%!
If the stock price falls to or even below the Call Price $85
(also = X) before expiry, the Category N Bull contract will
be called, and no residual value is received.
Hence, investors receive nothing.
31
Value of CBBC before expiry

–
–
–
–
Example of a Category N Bear contract:
So = 5,000; X = 5,500; Call price = 5,500; maturity = 2
months, CR = 100 to 1; funding interest rate=2%
Intrinsic value per share of Bear =
(X – S)/CR = (5,500-5,000)/100 x $1 = $5.0
Funding cost per share of Bear =
(5,500/100) x 2% x (60/365) x $1 = $0.1808
Theoretical price per share of the Bear contract at issue =
$5.0 + $0.1808 = $5.1808
If the stock price falls to 4,400 before expiry, the price per
share of the Category N Bear contract is equal to $11.0 +
$0.1808 = $11.1808. The stock index falls by 12%, but the
price per share of the bear contract increases by
32
115.8122%!
Value of CBBC before expiry
–
–
If the stock index rises to or even above the Call
Price 5,500 (also = X) before expiry, the Category
N Bear contract will be called, and no residual
value is received.
Hence, investors receive nothing.
33
Value of CBBC before expiry

–
–
–
–
Example of a Category R Bull contract:
So = $100; X = $85; Call price= $90; maturity= 2 months,
CR=10 to 1; funding interest rate=2%
Intrinsic value per share of Bull =
(S – X)/CR = ($100-85)/10 = $1.5
Funding cost per share of Bull =
($85/10) x 2% x (60/365) = $0.0279
Theoretical price per share of the Bull contract at issue =
$1.5 + $0.0279 = $1.5279
If the stock price rises to $110 before expiry. the price per
share of a Category R Bull contract is equal to $2.5 +
0.028 = $2.528. The stock price rises by 10%, but the
price per share of the bull contract increases by
34
65.4450%!
Value of CBBC before expiry
–
–
–
If the stock price falls to the Call Price $90 before
expiry, the Category R Bull contract will be called, and
the holder may receive the residual value.
The residual value per share is max (St – X, 0)/CR,
where St is the settlement price. The settlement price
of a Bull contract must not be lower than the minimum
trade price of the underlying asset after the Mandatory
Call Event (MCE) and up to and including the next
trading session.
If the stock price fell to Call Price $90 at 10:30am
(morning session), the minimum trade price of the
stock, over the period from 10:30am to 12:00pm, and
from 1:00pm up to 4:00pm (the next trading session),
was $87. $87 is the settlement price St which is higher35
than $85 = X.
Value of CBBC before expiry
–
–
–
–
The residual value is equal to Max(St – X, 0 )/CR
= ($87-85)/10 = $0.2
If the stock price fell to Call Price $90 at 10:30am
(morning session), the minimum trade price of the stock,
over the period from 10:30am to 12:00pm, and from
1:00pm up to 4:00pm (the next trading session), was
also $90.
$90 is the settlement price St which is equal to the Call
Price and higher than $85 = X.
The residual value is the highest equal to Max(St – X, 0)
/CR
= ($90-85)/10 = $0.5
36
Value of CBBC before expiry
–
–
–
If the stock price fell to Call Price $90 at 10:30am (morning
session), the minimum trade price of the stock, over the
period from 10:30am to 12:00pm, and from 1:00pm up to
4:00pm (the next trading session), was $85. $85 is the
settlement price St which is equal to X = $85.
The residual value is the lowest equal to Max(St – X, 0)
/CR
= ($85-85)/10 = $0.0.
If the stock price fell to Call Price $90 at 10:30am (morning
session), the minimum trade price of the stock, over the
period from 10:30am to 12:00pm, and from 1:00pm up to
4:00pm (the next trading session), was $84 < X. The
residual value is equal to Max(St – X, 0) /CR = $0.0
37
Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
38
Value of CBBC before expiry

–
–
–
–
Example of a Category R Bear contract:
So = 5,000; X = 5,500; Call price= 5,300; maturity= 2 months,
CR=100 to 1; funding interest rate=2%
Intrinsic value per share of Bear =
(X – S)/CR x contract multiplier = (5,500-5,000)/100 x $1
= $5.0
Funding cost per share of Bull =
(5,500/100) x 2% x (60/365) x $1 = $0.1808
Theoretical price per share of the Bear contract at issue
= $5.0 + $0.1808 = $5.1808
If the stock index falls to 4,400 before expiry. the price per
share of the Category R Bear contract is equal to $11+
$0.1808 = $11.1808. The stock index falls by 12%, but the
price per share of the bear contract increases by 115.8122%!
Value of CBBC before expiry
–
–
–
If the stock price rises to the Call Price 5,300 before
expiry, the Category R Bear contract will be called, and
the holder may receive the residual value.
The residual value per share is Max(X – St, 0)/CR,
where St is the settlement price. The settlement price of
a Bear contract must not be higher than the maximum
trade price of the underlying asset after the Mandatory
Call Event (MCE) and up to and including the next
trading session.
If the stock index increased to Call Price 5,300 at
3:30pm (afternoon session), the maximum trade price of
the stock index, over the period from 3:30pm to 4:00pm,
and from 9:30pm up to 12:00pm the next day (the next
trading session), was 5,400. 5,400 is the settlement
40
price St which is lower than 5,500 = X.
Value of CBBC before expiry
–
–
–
–
The residual value is equal to Max(X – St, 0)/CR x
contract multiplier
= (5,500-5,400)/100 x $1 = $1.0
If the stock index increased to Call Price 5,300 at
3:30pm (afternoon session), the maximum trade price of
the stock index, over the period from 3:30pm to 4:00pm,
and from 9:30pm up to 12:00pm the next day (the next
trading session), was 5,300.
5,300 is the settlement price St which is lower than
5,500 = X.
The residual value is the highest equal to Max(X – St,
0)/CR x contract multiplier
41
= (5,500-5,300)/100 x $1 = $2.0
Value of CBBC before expiry
–
–
–
If the stock index increased to Call Price 5,300 at 3:30pm
(afternoon session), the maximum trade price of the stock
index, over the period from 3:30pm to 4:00pm, and from
9:30pm up to 12:00pm the next day (the next trading
session), was 5,500. 5,500 is the settlement price St which
is equal to X = 5,500.
The residual value is the lowest equal to Max(X – St, 0)/CR
x contract multiplier
= (5,500-5,500)/100 x $1 = $0.0
If the stock index increased to Call Price 5,300 at 3:30pm
(afternoon session), the maximum trade price of the stock
index, over the period from 3:30pm to 4:00pm, and from
9:30pm up to 12:00pm the next day (the next trading
session), was 5,600 > X. The residual value is equal to
Max(X – St, 0) / CR x contract multiplier = 0.0 x $1 = $0.
Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
43
Valuation of CBBC at expiry




CBBC can be held until maturity if not called before
expiry, or sold on the Stock Exchange before expiry.
For a Bull contract, the cash settlement at normal
expiry T, will be Max(ST – X, 0)/CR where ST is the
settlement price as determined on the valuation day, X
is the strike price.
For a Bear contract, the cash settlement at normal
expiry T, will be Max(X – ST, 0)/CR where ST is the
settlement price as determined on the valuation day, X
is the strike price.
For CBBC with a Hong Kong stock as underlying, the
settlement price of the CBBC will be the closing price of
the underlying on the last trading day.
44
Valuation of CBBC at expiry

For CBBC with a Hong Kong stock index (Hang Seng
Index HSI or Hang Seng China Enterprises Index
HSCEI) as underlying, the settlement level of the CBBC
will be the same as the index level for settling the
relevant expiring index futures on the settlement day of
the index futures contract
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
46
Value of CBBC at expiry

–

–
Example of a Category R Bull contract:
So = $100; X = $85; Call price= $90; maturity= 2 months,
CR=10 to 1; funding interest rate=2%
If the stock price closes at $120 on the last trading day, the
cash settlement amount per share of Bull contract is equal to
Max(ST – X, 0)/CR = ($120-85)/10 = $3.5
Example of a Category R Bear contract:
So = 5,000; X = 5,500; Call price= 5,300; maturity= 2 months,
CR=100 to 1; funding interest rate=2%
If the stock index closes at 4,400 on the settlement day of
the CBBC and the index futures contract, the cash
settlement amount per share of Bear contract is equal to
Max(X – ST, 0)/CR x contract multiplier = (5,500-4,400)/100 x
47
$1 = $11
Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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Examples of CBBC
Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
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