session 2: options i c15.0008 corporate finance topics summer 2006
TRANSCRIPT
![Page 1: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/1.jpg)
Session 2: Options I
C15.0008 Corporate Finance Topics
Summer 2006
![Page 2: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/2.jpg)
Outline
• Call and put options
• The law of one price
• Put-call parity
• Binomial valuation
![Page 3: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/3.jpg)
Options, Options Everywhere!
• Compensation—employee stock options• Investment/hedging—exchange traded and OTC
options on stocks, indexes, bonds, currencies, commodities, etc., exotics
• Embedded options—callable bonds, convertible bonds, convertible preferred stock, mortgage-backed securities
• Equity and debt as options on the firm• Real options—projects as options
![Page 4: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/4.jpg)
Example..
![Page 5: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/5.jpg)
Options
The right, but not the obligation to buy (call) or sell (put) an asset at a fixed price on or before a given date.
Terminology:
Strike/Exercise Price
Expiration Date
American/European
In-/At-/Out-of-the-Money
![Page 6: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/6.jpg)
An Equity Call Option
• Notation: C(S,E,t)
• Definition: the right to purchase one share of stock (S), at the exercise price (E), at or before expiration (t periods to expiration).
![Page 7: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/7.jpg)
Where Do Options Come From?
• Publicly-traded equity options are not issued by the corresponding companies
• An options transaction is simply a transaction between 2 individuals (the buyer, who is long the option, and the writer, who is short the option)
• Exercising the option has no effect on the company (on shares outstanding or cash flow), only on the counterparty
![Page 8: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/8.jpg)
Numerical example
• Call option
• Put option
![Page 9: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/9.jpg)
Option Values at Expiration• At expiration date T, the underlying (stock) has market price ST• A call option with exercise price E has intrinsic value (“payoff to holder”)
• A put option with exercise price E has intrinsic value (“payoff to holder”)
),0max(if0
ifpayoff ES
ES
ESEST
T
TT
),0max(if0
ifpayoff T
T
TT SEES
ESSE
![Page 10: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/10.jpg)
Call Option Payoffs
Payoff
STE
Long CallPayoff
STE
Short Call
![Page 11: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/11.jpg)
Put Option Payoffs
Payoff
STE
Long PutPayoff
STE
Short Put
E
E
![Page 12: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/12.jpg)
Other Relevant Payoffs
Payoff
ST
Stock
Payoff
ST
Risk-Free Zero Coupon BondMaturity T, Face Amount E
E
![Page 13: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/13.jpg)
The Law of One Price
• If 2 securities/portfolios have the same payoff then they must have the same price
• Why? Otherwise it would be possible to make an arbitrage profit– Sell the expensive portfolio, buy the cheap
portfolio– The payoffs in the future cancel, but the
strategy generates a positive cash flow today (a money machine)
![Page 14: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/14.jpg)
Put-Call Parity
Stock + PutPayoff
STE
Payoff
STE
E=
Payoff
STE
Call +Bond
Payoff
STE
E=
![Page 15: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/15.jpg)
Put-Call Parity
Payoffs:
Stock + Put = Call + Bond
Prices:
Stock + Put = Call + Bond
Stock = Call – Put + Bond
S = C – P + PV(E)
![Page 16: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/16.jpg)
Introduction to binomial trees
![Page 17: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/17.jpg)
What is an Option Worth?
Binomial Valuation
Consider a world in which the stock can take on only 2 possible values at the expiration date of the option. In this world, the option payoff will also have 2 possible values. This payoff can be replicated by a portfolio of stock and risk-free bonds. Consequently, the value of the option must be the value of the replicating portfolio.
![Page 18: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/18.jpg)
Payoffs
Stock
100
137
73
Bond (rF=2%)
100
102
102
Call (E=105)
C
32
0
1-year call option, S=100, E=105, rF=2% (annual)1 step per yearCan the call option payoffs be replicated?
![Page 19: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/19.jpg)
Replicating Strategy
Buy ½ share of stock, borrow $35.78 (at the risk-free rate).
Cost(1/2)100 - 35.78 = 14.22
Payoff(½)137 - (1.02) 35.78 = 32
Payoff(½)73 - (1.02) 35.78 = 0
The value of the option is $14.22!
![Page 20: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/20.jpg)
Solving for the Replicating Strategy
The call option is equivalent to a levered position in the stock (i.e., a position in the stock financed by borrowing).
137 H - 1.02 B = 32
73 H - 1.02 B = 0 H (delta) = ½ = (C+ - C-)/(S+ - S-)
B = (S+ H - C+ )/(1+ rF) = 35.78
Note: the value is (apparently) independent of probabilities and preferences!
![Page 21: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/21.jpg)
Multi-Period Replication
Stock
100
80
125
100
156.25
64
Call (E=105)
0
51.25
0
C+
C-
1-year call option, S=100, E=105, rF=1% (semi-annual)2 steps per year
![Page 22: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/22.jpg)
Solving Backwards
• Start at the end of the tree with each 1-step binomial model and solve for the call value 1 period before the end
• Solution: H = 0.911, B = 90.21 C+ = 23.68• C- = 0 (obviously?!)
125
100
156.25
0
51.25 rF = 1%C+
![Page 23: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/23.jpg)
The Answer
• Use these call values to solve the first 1-step binomial model
• Solution: H = 0.526, B = 41.68 C = 10.94• The multi-period replicating strategy has no intermediate
cash flows
100
80
125
0
23.68rF = 1%
![Page 24: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/24.jpg)
Building The Tree
S
S+
S-
S--
S+-
S++ S+ = uS
S- = dS
S++ = uuS
S-- = ddS
S+- = S-+ = duS = S
![Page 25: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/25.jpg)
The Tree!
u =1.25, d = 0.8
100
80
125
100
156.25
64
![Page 26: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/26.jpg)
Binomial Replication
• The idea of binomial valuation via replication is incredibly general.
• If you can write down a binomial asset value tree, then any (derivative) asset whose payoffs can be written on this tree can be valued by replicating the payoffs using the original asset and a risk-free, zero-coupon bond.
![Page 27: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/27.jpg)
An American Put Option
What is the value of a 1-year put option with exercise price 105 on a stock with current price 100?
The option can only be exercised now, in 6 months time, or at expiration.
= 31.5573% rF = 1% (per 6-month period)
![Page 28: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/28.jpg)
Multi-Period Replication
Stock
100
80
125
100
156.25
64
Put (E=105)
5
0
41
P+
P-
![Page 29: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/29.jpg)
Solving Backwards
125
100
156.25rF = 1%
5
0
P+
H = -0.089, B = -13.75 P+ = 2.64
80
64
100
41
5
P- rF = 1%
H = -1, B = -103.96 P- = 23.96 25!!-------
The put is worth more dead (exercised) than alive!
![Page 30: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/30.jpg)
The Answer
100
80
125
25.00
2.64rF = 1%
H = -0.497, B = -64.11 P = 14.42
![Page 31: Session 2: Options I C15.0008 Corporate Finance Topics Summer 2006](https://reader036.vdocument.in/reader036/viewer/2022081507/551afa7b5503465e7d8b539d/html5/thumbnails/31.jpg)
Assignments
• Reading– RWJ: Chapters 8.1, 8.4, 22.12, 23.2, 23.4– Problems: 22.11, 22.20, 22.23, 23.3, 23.4,
23.5
• Problem sets– Problem Set 1 due in 1 week