PART VI 期貨與選擇權

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PART VI 期貨與選擇權

第廿二章選擇權與公司理財：基本概念

22.1 選擇權 22.2 選擇權組合 22.3 選擇權評價

22.4 以選擇權觀點解讀股票與公司債

第廿三章選擇權與公司理財：延伸與應用

23.5 股票選擇權 23.6 擴張選擇權 23.7 延期投資選擇權 23.8 結束營業選擇權

第廿四章認股權證與可轉換公司債

24.9 認股權證 24.10 可轉換公司債

第廿五章衍生性金融商品與避險

25.11 遠期合約 25.12 期貨 25.13 避險

25.14 利率期貨與存續期間 25.15 交換

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22

第二十二章選擇權與公司理財：基本概念

2006 年 7 月 21 日最後修改

22.1 選擇權 22.2 選擇權組合 22.3 選擇權評價

22.4 以選擇權觀點解讀股票與公司債

22.1 選擇權

選擇權（Options）：擁有者擁有某種權力買權（Call Options）：買進的權力賣權（Put Options）：賣出的權力買賣的東西稱為標的物（targets）

買賣的動作稱為履約（exercise）

相約買賣的價格稱為履約價格（striking price, exercise price）

當時標的物的價格稱為現貨價格（spot price）

買賣的最後時間（選擇權有效的截止日）稱為到期日（expiration date）

只有在到期日那天可以履約的選擇權稱為歐式選擇權（European Options）

到期日前任何一天皆可以履約的選擇權稱為美式選擇權（American Options）

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買權的價值

–20

120

20 40 60 80 100

–40 20 40 60

Stock price ($)

Option payoffs ($) Buy a call

Exercise price = $50

50

–20

120

20 40 60 80 100

–40 20 40 60

Stock price ($)

Option payoffs ($)

Exercise price = $50

50

C=^max

{

ST −E^{, 0}

}

其中

C：買權價值，S ：現貨價格，E：履約價格 _T

買進買權（buy a call）或賣出買權（sell a call）

權利金（premium）：賣賣選擇權的價格

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Exercise price = $50;

option premium = $10 Sell a call Buy a call

–20

120

20 40 60 80 100

–40 20 40 60

Stock price ($)

Option payoffs ($)

–20

120

20 40 60 80 100

–40 20 40 60

Stock price ($)

Option payoffs ($)

50 –10

10

價平（At-the Money）：履約價格等於現貨價格的情況

價內（In-the-Money）、價外（Out-of-the-Money）：履約價格不等於現貨價格的情況對選擇權持有者有利者為價內，不利者為價外

Option

Premium = Intrinsic Value

Speculative Value Option +

Premium = Intrinsic Value

Speculative Value +

權利金內含價值投機價值 = + 賣權的價值

–20

20 40 60 80 100

–40 20 40 60

Stock price ($)

Option payoffs ($)

Buy a put

Exercise price = $50; option premium = $10

–10

10 Sell a put

50

P=^max

{

E S− T^{, 0}

}

其中

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P：賣權價值，S ：現貨價格，E：履約價格 _T

例 23-1

買進一履約價格為$50 的賣權，權利金為$10。當現貨價格為$30 時，值得履約，履約價值為：

{ } { }

max E S_T $0 max $30 $40, $0 $20

= − = − =

履約價值 ,

$20 $10 $10

= − = − =

持有者現金流量履約價值權利金

22.2 選擇權組合

基本策略

買進現貨、賣出現貨、買進買權、賣出買權、買進賣權、賣出賣權

Sell a stock Buy a stock

50 60

40 100

–40 40

Stock price ($)

Stock payoffs ($)

–10 10

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Exercise price = $50;

option premium = $10 Sell a call Buy a call

50 60

40 100

–40 40

Stock price ($)

Option payoffs ($)

Buy a put Sell a put

• The seller (or writer) of an option has an obligation.

• The purchaser of an option has an option.

–10 10

Buy a call

Sell a put Buy

a put Sell a call

買進賣權與買進現貨（Protective Put Strategy）

Buy a put with an exercise price of $50 Buy the

stock

Protective Put payoffs

$50

$0

$50 Value at

expiry

Value of stock at

expiry

Buy a put with exercise price of $50 for $10

Buy the stock at $40

$40

Protective Put strategy has downside protection and upside potential

$40

$0

-$40

$50 Value at

expiry

Value of stock at expiry -$10

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賣出買權與買進現貨（Coved Call Strategy）

Sell a call with exercise price of $50 for $10 Buy the stock at $40

$40

Covered Call strategy

$0

-$40

$50 Value at

expiry

Value of stock at expiry

-$30 -$30

$10

同時買進買權與賣權

30 40 60 70

30 40

Stock price ($)

Option payoffs ($)

Buy a put with exercise price of $50 for $10

Buy a call with exercise price of $50 for $10

A Long Straddle only makes money if the stock price moves

$20 away from $50.

$50 –20

–20

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同時賣出買權與賣權

–30

30 40 60 70

–40

Stock price ($)

Option payoffs ($)

$50

This Short Straddle only loses money if the stock price moves $20 away from $50.

Sell a put with exercise price of

$50 for $10

Sell a call with an exercise price of $50 for

$10 20

20

Put-Call Parity 買權賣權平價理論

( )

0 0

o 1 T

p S c E + = + r

+

+ = +

賣權價值現貨價值買權價值履約價格現值

bond bond

25 25

Stock price ($)

Option payoffs ($)

Consider the payoffs from holding a portfolio consisting of a call with a strike price of $25 and a bond with a future value of $25.

Call

Portfolio payoff Portfolio value today = c₀+

(1+ r)^T

Portfolio value today = c₀+ E

(1+ r)^T E

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25 25

Stock price ($)

Option payoffs ($)

Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike.

Portfolio value today = p₀+ S₀

Portfolio payoff

22.3 選擇權評價

簡單例子（二項樹評價法）

Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S₀= $25 today and in one year S₁is either $28.75 or $21.25. The risk-free rate is 5%. What is the value of an at-the-money call option?

$25

$21.25 = $25×(1 –.15)

$28.75 = $25×(1.15)

S

₁

$21.25 = $25×(1 –.15)

$28.75 = $25×(1.15)

S

₁

S

₀

Borrow the present value of $21.25 today and buy 1 share.

The net payoff for this levered equity portfolio in one period is either $7.50 or $0.

The levered equity portfolio has twice the option’s payoff so the portfolio is worth twice the call option value.

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$25

$21.25

$28.75

S

₁

S

₀

debt

–

$21.25

portfolio

$7.50

$0

( – ) =

=

C

₁

$3.75

$0

–

$21.25

$2.38

C

₀

$2.38

C

₀

( )

0

1 $21.25 1

$25 $25 20.24 $2.38

2 (1.05) 2

C ⎛ ⎞

= ⎜ − ⎟= − =

⎝ ⎠

有兩個方案：

(1)擁有 2 口買權（E=$25） (2)擁有一張股票，借入$21.25

1.05

則兩者有相同的現金流量，即兩者的價值（現值）相等，故

0 0

$21.25 1 $21.25

2 $25 $25

1.05 2 1.05

C = − ⇒ C = ⎛⎜⎝ − ⎞⎟⎠

避險比例 Delta

_T

T

d C Δ =選擇權價值變動 =dS

股價變動

$3.75 $0 1

$28.75 $21.25 2

Δ = − =

−

買權價格=股價×Delta−借入金額

₀ ₀ 1 $21.25 1

$25 2 1.05 2 C =S × Δ −借入金額= × − ×

風險中立評價法

S(0), V(0)

S(U), V(U)

S(D), V(D) q

1- q

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( ) (1 ) ( )

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

( ) (1 ) ( ) ( ) ( )

(0) (1 )

f f

f

q V U q V D

V r r S S D

q S U q S D q S U S D

S r

× + − ×

⎧ =

⎪ + + × −

⎪ ⇒ =

⎨ × + − × −

⎪ =

⎪ +

⎩

$21.25,C(D)

q

1- q

$25,C(0)

$28.75,C(D)

) 15 . 1 ( 25

$ 75 . 28

$28.75=$25×(1.15)

$ = ×

) 15 . 1 ( 25

$ 25 . 21

$21.25=$25×(1−.15)

$ = × −

(1.05) $25 $21.25 $5

$28.75 $21.25 $7.50 2 3

q= × − = =

−

$21.25,C(D)

2/3

1/3

$25,C(0)

$28.75,C(D)

2 3 $3.75 (1 3) $0 $2.5

(0) $2.38

(1.05) 1.05

C = × + × = =

影響選擇權價格的因數

Call Put

1. Stock price + –

2. Exercise price – +

3. Interest rate + –

4. Volatility in the stock price + +

5. Expiration date + +

The value of a call option C₀must fall within max (S₀– E, 0) < C₀ < S₀.

The precise position will depend on these factors.

(1)股價（標的物價值）

(2)履約價格

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(3)折現率（利率）

(4)股價變動率（標準差）

(5)到期日

The value of a call option C₀must fall within max (S₀– E, 0) < C₀ < S₀.

25

Option payoffs ($)

Call S_T

loss

E Profit

S_T

Time value Intrinsic value Market Value

Market Value

In-the-money Out-of-the-money

The Black-Scholes Model

0 N( )1 ^rT N( 2)

C = ×S d −Ee⁻ × d

其中

C ：買權價格，S：貨價價格，E：履約價格，0 N d ：標準常態分配累計機率

( )

1

2

1

ln( / )S E (r 2σ T)

d σ T

= + + d₂ =d₁−σ T

r：折現率（無風險利率），σ：股價標準差（volatility）

例 22-3

假設某買權之履約價格E=$150，離到期日T = 個月，標的物現貨價格6 S=$160；若無風險利率r=5%，標的物價格變動率σ =30%，請計算該買權的價格。

【解】

6 6 0.5

T = 個月=12= （年）

2 2

1

1 1

2 2

ln( / ) ( ) ln(160 /150) (0.05 0.30 ) 0.5

0.5282 0.30 0.5

S E r σ T

d σ T

+ + + + ×

= = =

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2 1 0.52815 0.30 0.5 0.3160 d =d −σ T = − =

( )

1

(

0.5282

)

0.7013

N d =N =

( )

2

(

0.3160

)

0.6240

N d =N =

0.05 0.5

0 ( )1 ^rT ( 2) $160 0.7013 $150 0.6240 $20.92 C = ×S N d −Ee⁻ ×N d = × − ×e⁻ ^× × =

請比較

0 1 2

1 2

( ) ( )

( ), ( )

rT

C S N d Ee N d Delta

Delta N d Ee N d

−

⎧ = × − ×

⎪⎨

= × −

⎪⎩

⇒ = = ×

買權價格股價借入金額

借入金額亦即以下兩方案：

(1)持有一口買權

(2)持有Delta=0.7013股標的物

並以無風險利率r=5%借入Ee⁻^rT×N d( ₂)=$91.29 有相同的現金流量。

22.4 以選擇權觀點解讀股票與公司債

假設公司的負債額度為B=$800，令公司的價值為V，則

{ }

max $800, 0 max $800, 0

V

V V

= −

= − −

股東權益價值債權價值

即

股東：擁有履約價格為$800 的買權，標的物為公司

債權人：擁有公司，但賣出履約價格為$800 的買權

依據買權賣權平價理論：

賣權價值現貨價值買權價值履約價格現值 + = +

則債權人價值為

公司價值買權價值履約價格現值賣權價值− = − =$800−賣權

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債權價值= −V max

{

V−$800, 0

}

=$800−max $800

{

−V, 0

}

債權人：擁有$800，但賣出履約價格為$800 的賣權股東價值為

買權價值=賣權價值公司價值履約價格現值賣權+ − = + −V $800 ^{股東權益價值}⁼^{max $800}

{

⁻^V^{, 0}

}

^{+ −}^V ^$800

股東：擁有公司，負債$800，擁有履約價格為$800 的賣權

本章習題

#22-5 #22.6 #22.13 #22.14 #22.19 #22.21 #22.27

1. 22-5(p650)。 Consider a European call option on Stock A that expires on December 21 and has a strike price of $50.

a. If Stock A is trading at $55 on December 21. What is the payoff to the owner of the option?

b. If Stock A is trading at $55 on December 21. What is the payoff to the seller of the option?

c. If Stock A is trading at $45 on December 21.What is the payoff to the owner of the option?

d. If Stock A is trading at $45 on December 21. What is the payoff to the seller of the option?

e. Draw the payoff diagram to the owner of this option with respect to the stock price at expiration.

f. Draw the payoff diagram to the seller of this option with respect to the stock price at expiration.

g. If the seller of a call option never receives cash at expiration, why would anyone ever sell a call option?

=$0

2. 22-6(p651)。 Consider a European put option on Stock A that expires on December 21 and has a strike price of $50.

a. If Stock A is trading at $55 on December 21. What is the payoff to the owner of the option?

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b. If Stock A is trading at $55 on December 21. What is the payoff to the seller of the option?

c. If Stock A is trading at $45 on December 21. What is the payoff to the owner of the option?

d. If Stock A is trading at $45 on December 21.What is the payoff to the seller of the option?

e. Draw the payoff diagram to the owner of this option with respect to the stock price at expiration.

f. Draw the payoff diagram to the seller of this option with respect to the stock price at expiration.

= − $5

3. 22-13(p651)。 General Eclectics, Inc., has both European call and put options traded on the Chicago Board of Exchange. Both options have the same exercise price of $40 and both expire in one year. General Eclectic’s stock does not pay dividends. The call and the put are currently selling for $8 and $2, respectively. The risk-free interest rate is 10 percent per annum. What should the stock price of General Eclectics, Inc., be in order to prevent arbitrage?

【解】

T = 、1 E=$40、c₀=$8、p₀ =$2、r=10%

( ) ( )

0 0 0 1 0 0

$8 $40 $2 $42.36

1 ^T 1 10%

c E p S S S

+ r = + ⇒ + = + ⇒ =

+ +

4. 22-14(p651)。 Gimpellian Software, Inc., is a non-dividend-paying common stock that currently trades for $135 per share. Kevin is interested in purchasing a European call option with a strike price of $140 and one year until expiration. Much to his dismay, he discovers that call option are not trade on Gimpellian’s stock .In fact, the only options traded on Gimpellian’s stock are put options with a strike price of $140 and one year until expiration.

These put options are currently trading for $2.The risk-free rate is 25 percent per annum.

a. Compare the market price of the put with the payoff from the immediate exercise of a put option with a strike price of $140. Is this an arbitrage opportunity? Why or why not?

b. Is there a way for Kevin to obtain a synthetic call option with a strike price of $140 and one year until expiration? If there is, what is it? How much would this synthetic call option cost?

c. Drew the payoff diagram at expiration of Kevin’s synthetic call position with respect to the

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stock price.

【解】

T = 、1 S₀ =$135、E=$140、p₀ =$2、r=25%

(a)

這是一個價內的情況：p0 =$2<max

{

E S− 0, 0

}

=$5

然而這是一個歐式賣權，在到期日才能履約，因此沒有套利空間。

(b)

( ) ( ) ( )

0 0 0 0 0 0 1

$2 $135 $140 $25

1 ^T 1 ^T 1 25%

E E

c p S c p S

r r

+ = + ⇒ = + − = + − =

+ + +

5. 22-19(p653)。 Myron Fisher is interested in purchasing a European call option on Meriwether and Associates, Inc., a non-dividend-paying common stock, with a strike price of $50 and 1 year until expiration. Meriwether’s stock is currently trading at $55 per share, and the annual variance of its continuously compounded returns is 0.0625. Treasury bills that mature in one year yield a continuously compounded interest rate of 10 percent per annum. Use the Black-Scholes model to calculate the price of the call option that Myron is interested in buying.

【解】

T = 、1 S₀ =$55、E=$50、r=10%、σ = 0.0625=0.25

2 2

1

1 1

2 2

ln( / ) ( ) ln(55 / 50) (0.10 0.25 ) 1

0.9062 0.25 1

S E r σ T

d σ T

+ + + + × ×

= = =

2 1 0.9062 0.25 1 0.6562 d =d −σ T = − =

( )

1

(

0.9062

)

0.8187

N d =N =

( )

2

(

0.6562

)

0.7442

N d =N =

0.10 1

0 ( )1 ^rT ( 2) $55 0.8187 $50 0.7442 $11.30 C = ×S N d −Ee⁻ ×N d = × − ×e⁻ ^× × =

6. 22-21(p653)。 Seth is interested in purchasing a European call option on Brain Enhancers, inc., a non-dividend-paying common stock, with a strike price of $100 and two years until expiration. Brain Enhancer’s stock is currently trading at $100 per share, and the annual variance of its continuously compounded returns is 0.04. Treasury bills that mature in two years yield a continuously compounded interest rate of 5 percent per annum.

a. Use the Black-Scholes model to calculate the price of the call option that Seth is interested in buying.

b. What does put-call parity imply about the price of a put with a strike price of $100 and

(17)

two years until expiration?

【解】

T = 、2 S₀ =$100、E=$100、r=5%、σ = 0.04=0.20 (a)

2 2

1

1 1

2 2

ln( / ) ( ) ln(100 /100) (0.05 0.20 ) 2

0.4950 0.20 2

S E r σ T

d σ T

+ + + + × ×

= = =

d₂ =d₁−σ T =0.4950 0.20 2− =0.2122 N d

( )

1 =N

(

0.4950

)

=0.6897

N d

( )

2 =N

(

0.2122

)

=0.5840

C₀ = ×S N d( )₁ −Ee⁻^rT ×N d( ₂)=$100 0.6897× −$100×e⁻^{0.05 2}^× ×0.5840=$16.13 (b)

5% 2

0 0 ^rT 0 $16.13 $100 $100 $6.61 p =c +Ee⁻ −S = + e⁻ ^× − =

7. 22-27(p654)。 Consider a firm that is financed by both debt and equity. The firm is worth

$1 million today and currently has 700 zero-coupon bonds outstanding that mature in six months. Each bond has a face value of $1000. The firm pays no dividends. The annual variance of the firm’s continuously compounded asset returns is 0.06, and Treasury bills that mature in six months yield a continuously compounded interest rate of 8 percent per annum.

Use the Black-Scholes model to calculate the individual values of the firm’s debt and equity.

【解】

$1, 000 700 $700, 000

B= × = 、T =6個月=0.5年、r=8%、σ = 0.16=0.4 股東權益可以看成擁有履約價格E B= =$700, 000、

現貨價格S V= =$1,000,000的買權：

2 2

1

1 1

2 2

ln( / ) ( ) ln($1,000 / $700) (0.08 0.40 ) 0.5

1.5439 0.40 0.5

S E r σ T

d σ T

+ + + + × ×

= = =

d₂ =d₁−σ T =1.5439 0.40 0.5− =1.2610 N d

( )

1 =N

(

1.5439

)

=0.9387

N d

( )

2 =N

(

1.2610

)

=0.8963

0 1 2

0.08 0.5

( ) ( )

$1,000, 000 0.9387 $700, 000 0.8963 $335,891 C S N d Ee rT N d

e

−

− ×

= × − ×

= × − × × =

$335,891

= 股東權益價值

債權價值可視為擁有公司並賣出買權：

$335,891 $664,109

= −V =

債權價值

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23

第二十三章選擇權與公司理財：延伸與應用

2006 年 7 月 21 日最後修改

23.1 股票選擇權 23.2 擴張選擇權 23.3 延期投資選擇權 23.4 結束營業選擇權

23.1 股票選擇權

股票選擇權（Executive Stock Options）：公司吸引高階主管的福利 例 23-1

某公司總經理獲得 600 萬股股票選擇權，五年後到期。該公司的股價為$39.77，報酬標準差為σ =64.68%。假設無風險利率為r=5%，求該股票選擇權的價值。

T = （年） 5 設E S= =$39.77

2 2

1

1 1

2 2

ln( / ) ( ) ln($39.77 / $39.77) (0.05 0.6468 ) 5

0.896 0.6468 5

S E r σ T

d σ T

+ + + + ×

= = =

2 1 0.8960 0.6468 5 0.5503 d =d −σ T = − = −

( )

1

(

0.8960

)

0.8149

N d =N =

( )

2

(

0.5503

)

0.2911 N d =N − =

0.05 5

0 ( )1 ^rT ( 2) $39.77 0.8149 $39.77 0.2911 $23.39 C = ×S N d −Ee⁻ ×N d = × − ×e⁻ ^× × =

$23.39 6,000, 000 $140, 000,000

= × =

股票選擇權價值

23.2 擴張選擇權

擴張選擇權（Option to Expansion）：有擴張經營的權力 例 23-2

期初投資金額$700,000，現金流量如下：（假設折現率為 20%）

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第1年第2年第3年第4年第4年後銷售 $300,000 $600,000 $900,000 $1,000,000 $1,000,000 營業收入 -$100,000 -$50,000 $75,000 $250,000 $250,000 營運資金增額 $50,000 $20,000 $10,000 $10,000 $0

淨現金流量 -$150,000 -$70,000 $65,000 $240,000 $250,000

( ) ( )

( )

$150,000 20%,1 $70,000 20%, 2

$65,000 20%,3 $240,000 20%, 4

$250, 000

20%, 4 20%

$582,561

PVIF PVIF

PVIF

= − × − ×

+ × + ×

+ ×

= 現金流量現值

$700,000 $582,561 $117, 439 $0

NPV = − + = − <

考慮第四年底有擴張 30 家店的權力：

此為期日T = （年），現貨價值4

( )

⁴

$582,561 30

$8, 428, 255 1 20%

S ×

= =

+ ，

履約價格E=$700,000 30× =$21,000,000的買權

假設無風險利率r=3.5%，報酬率標準差σ =50%，則擴張選擇權的價值為：

2

1

ln( / ) ( 2 )

0.27293 S E r σ T

d σ T

= + + = −

d₂ =d₁−σ T = −1.27293 N d

( )

1 =N

(

−0.27293

)

=0.3936 N d

( )

2 =N

(

−1.27293

)

=0.1020

C₀ =$8, 428, 255 0.3936 $21, 000,000× − ×e⁻^{0.035 4}^× ×0.1020=$1, 455,196

付出權利金$117,439 可取得價值$1,455,196，這是一個非常划算的交易！

23.3 延期投資選擇權

23.4 結束營業選擇權

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24

第二十四章認股權證與可轉換公司債

2006 年 7 月 21 日最後修改

24.1 認股權證 24.2 可轉換公司債

24.1 認股權證

認股權證（Warrants）：以某固定價格購買股票的權力 等同於股票買權

綠鞋選擇權（Green Shoe Option）：認購權證的別名

認股權證一定伴隨著新股的發行，因此被視為公司對發行銀行的補貼

影響認購權證價值的因素：

1.

Stock price +

2.

Exercise price –

3.

Interest rate +

4.

Volatility in the stock price +

5.

Expiration date +

6.

Dividends –

認股權證與買權的差異

認股權證買的是新發行股票，因此有股權稀釋（dilution）的問題

認購權證一定是價內（in-the-money）時才會履約，這些履約價值由原有股東承擔，因此原有股票價值會下降（股權稀釋效果）

認購權證的評價

W

n n n

= ×

認股權證價值買權價值 +

其中，n 為原有股數，n 為認股權證股數 _W

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24.2 可轉換公司債

可轉換公司債（Convertible Bonds）：帶有認股權的公司債 例 24-1

A 公司發行 2025 年 12 月到期，年利率 6.75%的可轉換公司債$30 億，轉換比例

（conversion ratio）為 23.53（每張公司債可轉換為 23.53 股普通股），也就是說轉換價格

（conversion price）為$42.5（ $1000 / 23.53 ）；若該公司目前的股價為$22.625，則轉換權 利金（conversion premium）為 88%（$42.25/$22.625）。

可轉換公司債的評價

{ }

max max

= +

+

可轉換公司債價值純公司債價值, 轉換價值選擇權價值

純公司債價值, 隱含價值投機價值

純公司債價值認購權證價值 例 24-2

A 公司離到期還有 37 期利息（半年支息一次，利息為 $1,000 6.75% 0.5× × =$33.75），

若折現率為r=8%（一期折現率為 4%），則純公司債價值為

( ) ( )

$33.75 PVIFA 4%,37 $1,000 PVIF 4%,37 $880.36

= × + × =

純公司債價值

轉換價值為

{ }

max S 23.53, $880.36

= ×

轉換價值

其中 S 為股價，得知，股價 $880.36

$37.41 23.53

S> = 時轉換價值會高於純公司債價值。

Convertible Bond Value

Stock Price Straight bond

value Straight bond

value Conversion

Value

= conversion ratio

floor value

Convertible bond values Convertible bond

values

Option value Option value

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例 24-3

B 公司有 1,000 股普通股，以及 100 張面額$1,000 的（不付息）可轉換公司債流通在外，轉換比例為 10 股，請討論轉換時機。

{ }

^{10 100}

max min $1,000 100, ,

1,000 10 100 2

V × V V

⎧ ⎫

= ⎨⎩ × + × × = ⎬⎭

債權人價值

亦即

(1)當V <$10,000時，公司全部價值都需支付給債權人債權人價值= <V $10,000

(2)當 $10,000≤ ≤V $20,000時債權人價值=$10,000 V<

(2)當V >$20, 000時（值得轉換），

$10, 000

2

=V >

債權人價值

24.3 發行認股權證與可轉換公司債的理由

利息考量現金流量考量代理成本風險承擔

24.4 轉換策略

買回（callable）：發行公司可以以某事先設定價格買回公司債

理論上，公司債價值等於或大於買回價格時為適當買回時機，但是因為有買回宣告期，實務上公司總是在債券價格低於買回價格時買回。Why?

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本章習題

#24.12 #24.14

1. 24-12(p694)。 Sportime Fitness Center, Inc., issued convertible bonds, each with a par value of $1000 and a conversion price of $25 per share. The bonds are available for

immediate conversion. The current price of Sportime’s common stock is $24 per share. The current market price of each of the firm’s convertible bonds is $990.The convertible bond’s straight value is not known.

a. what is the minimum price that each of Spotime’s convertible bonds should sell for?

b. Explain the difference between the current market price of each convertible bond and the value of the common stock into which it can be immediately converted.

【解】

$1, 000

$25 40

=公司債面額= =

轉換比例轉換價格

0 $24 40 $960

=S × = × =

(現在)轉換價值轉換比例

(a) ^{可轉換公司債價值}⁼^max

{

純公司債價值轉換價值^,

}

⁺投機價值^≥轉換價值因此，該可轉換公司債至少值$960。

(b)

可轉換公司債值高於(現在)轉換價值$960 的部分為股價變動的投機價值：

(1)若未來股價走低，可轉換公司債至少保有純公司債的價值 (2)若未來股價走高，轉換價值也會跟著增加

2. 24-14(p694)。 Hannon Home Products, Inc., recently issued $430,000 worth of 8 percent convertible debentures. Each convertible bond has a face value of $1,000.Each convertible bond can be converted into 28 shares of the firm’s common stock anytime before maturity.

The current price of Hannon’s common stock is $31.25 per share, and the market value of each of Hannon’s convertible bonds is $1,180. Answer the following questions related to Hannon’s convertible bonds:

a. What is the conversion ratio?

b. What is the conversion price?

c. What is the conversion premium?

d. What is the conversion value?

e. If the value of Hannon’s common stock increases by $2, what will the conversion value be?

【解】

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(a)

=28 轉換比例 (b)

$1,000

$35.71

= 面額 = 28 =

轉換價格轉換比例

(c)

$35.71

1 1 14.27%

$31.25

=轉換價格− = − =

轉換權利金

股票價格 (d)

0 $31.25 28 $875

=S × = × =

轉換價值轉換比例

(e) 轉換價值=S0×轉換比例=

(

$31.25 $2+

)

×28=$931

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25

第二十五章衍生性金融商品與避險

2006 年 8 月 5 日最後修改

25.1 遠期合約 25.2 期貨 25.3 避險

25.4 利率期貨與存續期間 25.5 交換

25.1 遠期合約

衍生性金融商品（derivatives）

現金流量取決於其他金融商品價格或指數的金融契約

避險（hedge, hedging risk）

以衍生性金融商品降低財務風險

遠期合約（forward contracts）

承諾在未來某特定時間完成約定交易的合約

25.2 期貨

期貨（Future Contracts）

遠期合約的制式化版本

期貨與遠期合約的差異：

(1)到期月的交割日期由賣方決定

(2)在期貨交易所交易：標準商品、標準數量、共同到期日 (3)隨時做市價重估（marking to market）

逐日結算交易商品價格變動而產生的盈虧

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Currency per U.S. $ equivalent U.S. $

Wed Tue Wed Tue

Japan (yen) 0.007142857 0.007194245 140 139

1-month forward 0.006993007 0.007042254 143 142 3-months forward 0.006666667 0.006711409 150 149 6-months forward 0.00625 0.006289308 160 159

Currently $1 = ¥140.

The 3-month forward price is $1=¥150.

交易商品、合約數量與交易所：

Contract Contract Size Exchange Agricultural

Corn 5,000 bushels Chicago BOT Wheat 5,000 bushels Chicago & KC

Cocoa 10 metric tons CSCE OJ 15,000 lbs. CTN Metals & Petroleum

Copper 25,000 lbs. CMX Gold 100 troy oz. CMX Unleaded gasoline 42,000 gal. NYM Financial

British Pound £62,500 IMM Japanese Yen ¥12.5 million IMM Eurodollar $1 million LIFFE

25.3 避險

避險者（hedgers）與投機者（speculators）

交易商品的買、賣雙方為避險者，投機者不擁有商品也不需要該商品三者接會受交易商品價格變動影響

套利（arbitrage）與價格發現（price discovering）

利用多種商品間定價偏誤來獲取（無風險）報酬的行為沒有套利空間的價格即為合理的商品價格

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25.4 利率期貨與存續期間

存續期間（duration）

用以衡量利率的風險

1 2

1

( ) 1 ( ) 2 ( )

(1 ) (1 )

T

N t

t t

N t

t t

PV C PV C PV C T

D PV

C t D r

C r

=

× + × + + ×

=

×

= + +

∑

"

Discount Present Years x PV Years Cash flow factor value / Bond price

0.5 $40.00 0.96154 $38.46 0.0192 1 $40.00 0.92456 $36.98 0.0370 1.5 $40.00 0.88900 $35.56 0.0533 2 $40.00 0.85480 $34.19 0.0684 2.5 $40.00 0.82193 $32.88 0.0822 3 $1,040.00 0.79031 $821.93 2.4658

$1,000.00 2.7259 years Bond price Bond duration Duration is expressed in units of time; usually years.

Duration is expressed in units of time; usually years.

25.5 交換

交換（swaps）

簽約雙方同意交換未來的現金流量

利率交換（interest-rate swaps）：浮動利率與固定利率交換 貨幣交換（current swaps）：不同貨幣間交換

交換銀行（the swap bank）：撮合與仲介角色

兩公司原資金成本：

A 公司：美元 8%、英鎊 11.6%

B 公司：美元 10%、英鎊 12%

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$9.4%

Firm B

$8%

$8% £12%£12%

Swap Bank

Firm A

£11%

$8%

£12%

結果：

A 公司以 8%借入美元交給交換銀行，並取得利率 11%之英鎊（省 0.6%）

B 公司以 12%借入英鎊交給交換銀行，並取得利率 9.4%之美元（省 0.6%）

銀行美元部位賺 1.4%，英鎊部位虧 1%，合計賺 0.4%

PART VI 期貨與選擇權