Session 2: Options I
C15.0008 Corporate Finance
Topics
Summer 2006
Outline
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•
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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, mortgagebacked 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”)
 ST  E if ST  E
payoff  
 max( 0, ST  E )
if ST  E
0
• A put option with exercise price E has intrinsic value
(“payoff to holder”)
 E  ST
payoff  
0
if ST  E
if ST  E
 max( 0, E  ST )
Call Option Payoffs
Long Call
Short Call
Payoff
Payoff
E
ST
E
ST
Put Option Payoffs
Long Put
Short Put
Payoff
Payoff
E
E
E
ST
E
ST
Other Relevant Payoffs
Risk-Free Zero Coupon Bond
Maturity T, Face Amount E
Stock
Payoff
Payoff
E
ST
ST
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 + Put
Payoff
Payoff
=
E
E
ST
E
ST
E
ST
Call +Bond
Payoff
=
E
ST
Payoff
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
Bond (rF=2%)
137
100
102
100
73
Call (E=105)
32
C
102
1-year call option, S=100, E=105, rF=2% (annual)
1 step per year
Can the call option payoffs be replicated?
0
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
The value of the option is $14.22!
Payoff
(½)73 - (1.02) 35.78 = 0
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
156.25
Stock
125
Call (E=105)
C+
100
100
51.25
80
0
C-
64
0
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
156.25
51.25
C+
125
100
rF = 1%
0
• Solution: H = 0.911, B = 90.21  C+ = 23.68
• C- = 0 (obviously?!)
The Answer
• Use these call values to solve the first 1-step binomial
model
125
23.68
rF = 1%
100
80
0
• Solution: H = 0.526, B = 41.68  C = 10.94
• The multi-period replicating strategy has no intermediate
cash flows
Building The Tree
S++
S+ = uS
S+
S- = dS
S+-
S
S++ = uuS
S-- = ddS
SS--
S+- = S-+ = duS = S
The Tree!
u =1.25, d = 0.8
156.25
125
100
100
80
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
156.25
Stock
125
Put (E=105)
P+
100
100
0
80
5
P-
64
41
Solving Backwards
156.25
125
0
P+
rF = 1%
100
5
H = -0.089, B = -13.75  P+ = 2.64
100
80
5
P-
64
rF = 1%
41
------- 25!!
H = -1, B = -103.96  P- = 23.96
The put is worth more dead (exercised) than alive!
The Answer
125
2.64
rF = 1%
100
80
25.00
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