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Session 2: Options I
C15.0008 Corporate Finance Topics
Summer 2006
Outline
• Call and put options
• The law of one price
• Put-call parity
• Binomial valuation
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
Example..
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
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).
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
Numerical example
• Call option
• Put option
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
Call Option Payoffs
Payoff
STE
Long CallPayoff
STE
Short Call
Put Option Payoffs
Payoff
STE
Long PutPayoff
STE
Short Put
E
E
Other Relevant Payoffs
Payoff
ST
Stock
Payoff
ST
Risk-Free Zero Coupon BondMaturity T, Face Amount E
E
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)
Put-Call Parity
Stock + PutPayoff
STE
Payoff
STE
E=
Payoff
STE
Call +Bond
Payoff
STE
E=
Put-Call Parity
Payoffs:
Stock + Put = Call + Bond
Prices:
Stock + Put = Call + Bond
Stock = Call – Put + Bond
S = C – P + PV(E)
Introduction to binomial trees
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.
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?
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!
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!
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
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+
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%
Building The Tree
S
S+
S-
S--
S+-
S++ S+ = uS
S- = dS
S++ = uuS
S-- = ddS
S+- = S-+ = duS = S
The Tree!
u =1.25, d = 0.8
100
80
125
100
156.25
64
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.
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)
Multi-Period Replication
Stock
100
80
125
100
156.25
64
Put (E=105)
5
0
41
P+
P-
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!
The Answer
100
80
125
25.00
2.64rF = 1%
H = -0.497, B = -64.11 P = 14.42
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
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