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Applications of option Applications of option pricing analysispricing analysis
CLIFFORD W. SMITH JR.CLIFFORD W. SMITH JR.The University of RochesterThe University of Rochester
19791979
Presentation:
892625 應子健
892627 廖尉呈
892642 李承儒
Presentation:
892625 應子健
892627 廖尉呈
892642 李承儒
1. Introduction1. Introduction
歷史背景歷史背景 :: 19731973 年，年， Black-Scholes Black-Scholes 模型理論剛剛發模型理論剛剛發
表，尚未廣泛應用於市場上。這篇論文的表，尚未廣泛應用於市場上。這篇論文的發表在發表在 19791979 年，利用 年，利用 Black-Scholes Black-Scholes 模模型定價公司資產、負債與其他衍生性金融型定價公司資產、負債與其他衍生性金融商品。這篇商品。這篇 paperpaper 在當時算是很創新的概在當時算是很創新的概念，不過在今天看來卻是屬於必備的知識。念，不過在今天看來卻是屬於必備的知識。
2. The pricing of European 2. The pricing of European optionsoptions
THE PRICING OF EUROPEAN PUT AND CALL OPTIONTHE PRICING OF EUROPEAN PUT AND CALL OPTION AssumptionsAssumptions 1.1. There are no penalties for short sales ;There are no penalties for short sales ;2.2. Transactions costs and taxes are zero ;Transactions costs and taxes are zero ;3.3. The market operates continuously ;The market operates continuously ;4.4. The riskless rate is known and constant ;The riskless rate is known and constant ;5.5. The stock price follows a continuous Ito process ;The stock price follows a continuous Ito process ;6.6. The stock pays no dividends ;The stock pays no dividends ;7.7. The option can only be exercised at the terminal date The option can only be exercised at the terminal date
of the contract .of the contract .
European call option pricing European call option pricing The price of call optionThe price of call option ：：
The solution can be written in general form asThe solution can be written in general form as
where ; ; ; ; . where ; ; ; ; .
T
TrXSrT
T
TrXS XNeSNc
2//ln)2/()/ln( 22
rTXScc ,,,, 20
Sc 02
c 0
Xc 0
rc 0
Tc
European put option pricing European put option pricing Consider two portfoliosConsider two portfolios ：： Portfolio IPortfolio I ：： one European call + one shone European call + one sh
are of stock sold short + X pure discount are of stock sold short + X pure discount B(T) and face value of one dollarB(T) and face value of one dollar
Portfolio IIPortfolio II ：： one European put with saone European put with same terms as the callme terms as the call
Portfolio Current value Stock price at Portfolio Current value Stock price at T=0T=0
I I IIII 0 0
Relationship Relationship
between the between the
terminal value of terminal value of
portfolios portfolios
I and III and II
0* S
*SX )(;, TXBSXTSc XS *0 XSXS **
XTSWP ;, *SX
**III VV **
III VV
This solution can be written in general forThis solution can be written in general form asm as
)(,,;, TXBSXTScXTSp
rT
T
TrXSrT
T
TrXS XeSNXeSNp
)2/()/ln()2/()/ln( 22
T
TrXSrT
T
TrXS NXeSN
)2/()/ln()2/()/ln( 22
),,,,( 2 rTXSpp
3. The pricing of corporate 3. The pricing of corporate liabilitiesliabilities
Pricing of the debt and equity of a firmPricing of the debt and equity of a firm
公司發行債券，實質上相當於股票持有者將所持有的資產賣公司發行債券，實質上相當於股票持有者將所持有的資產賣給債券持有者，並且再買一個可以在債券到期日時以債券票給債券持有者，並且再買一個可以在債券到期日時以債券票面價買回資產的買權。面價買回資產的買權。
若票面價為 若票面價為 X X ，執行日當天公司價值為 ，執行日當天公司價值為 VV* * ，則在執行日時，則在執行日時股票持有者的收入為 股票持有者的收入為 max [ 0, Vmax [ 0, V** - X] - X] ，取決於公司價值 ，取決於公司價值 VV** 是否大於 是否大於 XX ，與選擇權相當類似。，與選擇權相當類似。
AssumptionsAssumptions1.1. 公司發行還本時付息債券，並且在到期日之前限制任公司發行還本時付息債券，並且在到期日之前限制任
何股利發放。何股利發放。2.2. 公司總價值不會受到資本結構改變而影響。（符合 公司總價值不會受到資本結構改變而影響。（符合 MM
odigliani-Miller worldodigliani-Miller world ））3.3. 公司價值的動態變化有相同的期望值；並且在任何有公司價值的動態變化有相同的期望值；並且在任何有
限時間之中是一個有著固定變動報酬率的指數常態分限時間之中是一個有著固定變動報酬率的指數常態分佈。佈。
4.4. 存在已知的固定無風險利率。存在已知的固定無風險利率。
By Black-Scholes call option solution yieldsBy Black-Scholes call option solution yields}{}{ )2/()/ln()2/()/ln( 22
T
TrXVrT
T
TrXV XNeVNE
By payoff graphBy payoff graph
In risk neutral world, They should have equal price, or arIn risk neutral world, They should have equal price, or arbitrage will appear.bitrage will appear.Thus, the pricing of equity can be compute by the methoThus, the pricing of equity can be compute by the method of call option pricing.d of call option pricing.
E*
X V*
P
X S
DebtDebt
via V = E + D via V = E + D D = V – ED = V – E
By the result of equity pricingBy the result of equity pricing
- =- =
}{}{ )2/()/ln()2/()/ln( 22
T
TrXVrT
T
TrXV XNeVND
V*
X V*
E*
X V*
D*
X V*
Debt can be expressed asDebt can be expressed as
and and rTXVDD ,,,, 2
0VD
0XD
0TD
02 D
0rD
※ ※ The option pricing model The option pricing model and the CAPMand the CAPM
The pricing of the equity is consistent The pricing of the equity is consistent with the continuous time CAPMwith the continuous time CAPM
Instantaneous return to the Instantaneous return to the stockholderstockholder
)( rrrr mj
e
j
EdE
Er
By ito’s lemmaBy ito’s lemma
Substituting into the definition of the systematic Substituting into the definition of the systematic risk of equityrisk of equity
That isThat is
dtVdVdEVE
tE
VE )( 22
21
2
2
Edt
VE
tE
vEV
VE
EdE
E Vrr )( 2221
2
2
vEV
VE
r
rrEV
VE
r
rrE
m
mv
m
mE
)(
)~,~cov(
)(
)~,~cov(22
vE VE ),(
Elasticity is greater than 1, thus the absoElasticity is greater than 1, thus the absolute value of the systematic risk of the stlute value of the systematic risk of the stock is greater than the absolute value of ock is greater than the absolute value of the systematic risk of the firm.the systematic risk of the firm.
1
}{),(
}{}{
}{
)2/()/ln(
)2/2()/ln()2/2()/ln(
)2/2()/ln(
2
T
TrXVrT
T
TrXV
T
TrXV
NXeVN
VN
EV
T
TrXVEV
VE NVE
※ ※ Risk structure of interest Risk structure of interest ratesrates
is the expected return rate of the bondis the expected return rate of the bond The risk premiumThe risk premium
DXe TTr )(ˆ
)(ˆ TrXDe TTr /)(ˆ
rTXDrTr )/)/ln(()(ˆ
0ˆ Vr 0ˆ
Xr 02
ˆ r 0ˆ
rr 0?ˆ
Tr
※ ※ Coupon BondsCoupon Bonds
With required interest payments the With required interest payments the stockholders’ equity is like an option stockholders’ equity is like an option on an option on…an option on the on an option on…an option on the assets of the firm. By paying the last assets of the firm. By paying the last coupon, the stockholder buy the coupon, the stockholder buy the option to purchase the firm by option to purchase the firm by paying the face value of the debt.paying the face value of the debt.
3.2 Convertible bond pricing3.2 Convertible bond pricing
AssumptionsAssumptions
The convertible bond and the stock are The convertible bond and the stock are the only liabilities issued by the the only liabilities issued by the company.company.
Value of convertible bond at maturity dateValue of convertible bond at maturity date
]],max[,min[ *** VXVB
By payoff graphBy payoff graphB*
V*X/aX
),,,,(),,,,(),,,,,( 222 rTXVCrTXVDrTXVB
),,,,(),,,,(),,,,,( 222 rTXVCrTXVDrTXVB
0
0?
0?
0
0
0
222
rC
rD
rB
CDB
TC
TD
TB
XC
XD
XB
VCB
VC
VD
VB
V
3.3 The pricing of subordinated 3.3 The pricing of subordinated debtdebt
The firm issues two debt, one is senior and the The firm issues two debt, one is senior and the other is junior.other is junior.
Assumption: The issues contain restrictions against Assumption: The issues contain restrictions against dividend payments until after both the bond dividend payments until after both the bond issues are paid off.issues are paid off.
]0],,max[min[
],min[
)](,0max[
**
**
**
jsj
ss
js
XXVD
XVD
XXVE
Pay off graphPay off graphE*
Xs+Xj V*
Dj
Xs Xs+Xj V*
Ds
Xs Xs+Xj V*
The pricing of equity and senior debt are The pricing of equity and senior debt are both unchangedboth unchanged
The pricing of junior debtThe pricing of junior debt
can be expressed ascan be expressed as
}]{}{[
])()()([
)2/())/(ln()2/()/ln(
*****
22
T
TrXXV
T
TrXVj
XX
X XX jsrT
j
jss
js
s js
NNVD
cdVVLXdVVLXVeD
}{)(
}]{}{[
)2/())/(ln(
)2/())/(ln()2/()/ln(
2
22
T
TrXXVrTjs
T
TrXXV
T
TrXVrTs
js
jss
NeXX
NNeX
0;0?2
,,;0,
sXjDjD
TjD
rjD
jXjD
VjD
),,,( 2rTXVDD sjj
3.4 Pricing of warrants and 3.4 Pricing of warrants and rightsrights
AssumptionAssumption
The only liabilities issued by the firm are The only liabilities issued by the firm are its common stock and the warrantsits common stock and the warrants
])(,0max[ ** XXVW
T
TRXVXNe
T
TrXVVNW
rT
)2/())1/(ln()1(
)2/())1/(ln(
2
2
Payoff graphPayoff graph
E*
(1-a)X/a V*
W*
(1-a)X/a V*
00,,,, 2
XW
rWWW
TW
VW and
4. The pricing of other 4. The pricing of other contingent claimscontingent claims4.1 The pricing of 4.1 The pricing of
underwriting contractunderwriting contract Underwriters submit a bid, , today which specifies Underwriters submit a bid, , today which specifies
that on the offer date, T time periods from now, the that on the offer date, T time periods from now, the underwriter will pay dollars and receive shares of underwriter will pay dollars and receive shares of stock representing fraction of the total shares of stock representing fraction of the total shares of the firm. He can sell the securities at the offer price the firm. He can sell the securities at the offer price and receive , or if the share price is below the and receive , or if the share price is below the offer price at the market price, . If his bid is offer price at the market price, . If his bid is accepted, he will be notified immediately. accepted, he will be notified immediately.
^
B
^
B
)(^
* BV
T
TrBVNB
T
TrBVVNeBVeU rTrT
)2/())ˆ/(ln()ˆ(
)2/())ˆ/(ln(ˆ)1(
2
2
)ˆ;,()1( BTVceBVeU rTrT
BTVeBVe rTrT ˆ;,()1(
0);,(1
ˆ
BTVCVeB rT
VeB rT
1
ˆ
BBBVU ,ˆ)(min **
V*
U
4.2 The pricing of collateralized 4.2 The pricing of collateralized loansloans
Assumption:Assumption:1.1. There are homogeneous expectations There are homogeneous expectations
about the dynamic behavior of the value of about the dynamic behavior of the value of the collateral. The distribution at the end of the collateral. The distribution at the end of any finite time interval is log normal. The any finite time interval is log normal. The variance rate of return is constant.variance rate of return is constant.
2.2. The collateral provides a continuous flow of The collateral provides a continuous flow of service to the borrower. The net value of service to the borrower. The net value of the flow of the service, S, is a constant the flow of the service, S, is a constant fraction, s, of the market value of the fraction, s, of the market value of the assets: s=S/Vassets: s=S/V
3.3. The dynamic behavior of the value of the The dynamic behavior of the value of the assets is independent of the value of the assets is independent of the value of the probability of bankruptcy.probability of bankruptcy.
4. There are no costs to voluntary liquidation or 4. There are no costs to voluntary liquidation or bankruptcy. Bankruptcy is defined as the stabankruptcy. Bankruptcy is defined as the state in which the borrower’s assets are less thte in which the borrower’s assets are less than the promised repayment amount of a matan the promised repayment amount of a maturing loan.uring loan.
5. Capital markets and the market for the collate5. Capital markets and the market for the collateral are perfect. There are no transactions cosral are perfect. There are no transactions cost or taxes. All participants have free access to t or taxes. All participants have free access to all available information. Participants are priall available information. Participants are price takers.(efficient market)ce takers.(efficient market)
6. There is a known constant riskless rate, r.6. There is a known constant riskless rate, r.
WhereWhere
T
TsrXVNXe
T
TsrXVNVeD rTsT
)2/()/ln()2/()/ln( 22
),,,,,( rsTXVD
0,,,0,2
r
D
s
DD
T
Dand
X
D
V
D
4.3 The pricing of leases4.3 The pricing of leases
suggested:suggested:
The value of the borrower’s equity in the The value of the borrower’s equity in the collateral is equivalent to a call option to collateral is equivalent to a call option to purchase the collateral with the exercise purchase the collateral with the exercise price equal to the promised repayment on price equal to the promised repayment on the loan, plus a lease. Therefore, the value the loan, plus a lease. Therefore, the value of the lease equal the value of the of the lease equal the value of the collateral minus the value of the debt collateral minus the value of the debt minus the value of the call minus the value of the call
CDVL
**'*
**"*
0
**'**'*
)(
)()(
)()(
dVVLVeV
dVVLXVe
dVVXLedVVLVeV
X
rT
X
rT
XrTeT
This equation has an intuitive This equation has an intuitive interpretation: the value of the lease interpretation: the value of the lease equals the value of the asset minus a equals the value of the asset minus a claim on the value of the asset T claim on the value of the asset T periods from now.periods from now.
]1[ sTeVL
4.4 The pricing of insurance4.4 The pricing of insurance The insurance contract calls for the The insurance contract calls for the
payment of a premium, P, at the current payment of a premium, P, at the current date, t. If at the expiration date of the date, t. If at the expiration date of the contract, t*, the market value of the contract, t*, the market value of the insured asset, V*, is less than its insured insured asset, V*, is less than its insured value, X, then the insurance contract will value, X, then the insurance contract will pay the holder of the policy the difference, pay the holder of the policy the difference, X-V*. If the market value of the insured X-V*. If the market value of the insured asset is greater than its insured value, asset is greater than its insured value, then there is no payment. then there is no payment. 0,max ** VXP
where where
T
TrXVNXe
T
TrXVVNP rT
)2/()/ln()2/()/ln( 22
),,,( 2rTXVPP
0,,;0,2
P
T
P
X
P
r
P
V
P
P*
X
X V*
45o
