Download - Session 02 Amrut
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Session 2: Options I
C15.0008 Corporate Finance
TopicsSummer 2006
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Outline
Call and put options
The law of one price
Put-call parity Binomial valuation
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Options, Options Everywhere!
Compensationemployee stock options
Investment/hedgingexchange traded and OTCoptions on stocks, indexes, bonds, currencies,
commodities, etc., exotics Embedded optionscallable bonds, convertible
bonds, convertible preferred stock, mortgage-backed securities
Equity and debt as options on the firm Real optionsprojects as options
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Example..
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Options
The right, but not the obligation to buy (call) or sell(put) an asset at a fixed price on or before a givendate.
Terminology:Strike/Exercise Price
Expiration Date
American/EuropeanIn-/At-/Out-of-the-Money
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An Equity Call Option
Notation: C(S,E,t)
Definition: the right to purchase one shareof stock (S), at the exercise price (E), at orbefore expiration (tperiods to expiration).
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Where Do Options Come From?
Publicly-traded equity options are notissued by the corresponding companies
An options transaction is simply a
transaction between 2 individuals (thebuyer, who is long the option, and thewriter, who is short the option)
Exercising the option has no effect on thecompany (on shares outstanding or cashflow), only on the counterparty
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Numerical example
Call option
Put option
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Option Values at Expiration
At expiration date T, the underlying (stock) has marketprice ST
A call option with exercise price Ehas intrinsic value(payoff to holder)
A put option with exercise price Ehas intrinsic value
(payoff to holder)
),0max(if0
ifpayoff ES
ES
ESES
T
T
TT
),0max(if0
ifpayoff
T
T
TTSE
ES
ESSE
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Call Option Payoffs
Payoff
STE
Long Call
Payoff
STE
Short Call
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Put Option Payoffs
Payoff
STE
Long Put
Payoff
STE
Short Put
E
E
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Other Relevant Payoffs
Payoff
ST
Stock
Payoff
ST
Risk-Free Zero Coupon Bond
Maturity T, Face AmountE
E
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The Law of One Price
If 2 securities/portfolios have the same payoffthen they must have the same price
Why? Otherwise it would be possible to make an
arbitrage profit Sell the expensive portfolio, buy the cheapportfolio
The payoffs in the future cancel, but the
strategy generates a positive cash flow today(a money machine)
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Put-Call Parity
Stock + PutPayoff
STE
Payoff
STE
E=
Payoff
STE
Call +Bond
Payoff
STE
E=
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Put-Call Parity
Payoffs:
Stock + Put = Call + Bond
Prices:
Stock + Put = Call + Bond
Stock = Call Put + BondS = C P + PV(E)
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Introduction to binomial trees
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What is an Option Worth?
Binomial Valuation
Consider a world in which the stock can take ononly 2 possible values at the expiration date of the
option. In this world, the option payoff will alsohave 2 possible values. This payoff can bereplicated by a portfolio of stock and risk-freebonds. Consequently, the value of the option must
be the value of the replicating portfolio.
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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 year
Can the call option payoffs be replicated?
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Replicating Strategy
Buy share of stock, borrow $35.78 (at the risk-free rate).
Cost
(1/2)100 - 35.78 = 14.22
Payoff()137 - (1.02) 35.78 = 32
Payoff
()73 - (1.02) 35.78 = 0
The value of the option is $14.22!
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Solving for the Replicating Strategy
The call option is equivalent to a levered position in thestock (i.e., a position in the stock financed by borrowing).
137 H - 1.02 B = 32
73 H - 1.02 B = 0 H (delta) = = (C+ - C-)/(S+ - S-)
B = (S+ H - C+ )/(1+ rF) = 35.78
Note: the value is (apparently) independent of probabilitiesand preferences!
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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
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Solving Backwards
Start at the end of the tree with each 1-step binomialmodel and solve for the call value 1 period before theend
Solution: H = 0.911, B = 90.21 C+
= 23.68 C- = 0 (obviously?!)
125
100
156.25
0
51.25rF = 1%
C+
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The Answer
Use these call values to solve the first 1-step binomialmodel
Solution: H = 0.526, B = 41.68 C = 10.94
The multi-period replicating strategy has no intermediatecash flows
100
80
125
0
23.68
rF = 1%
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Building The Tree
S
S+
S-
S--
S+-
S++ S+ = uS
S- = dS
S++ = uuS
S-- = ddS
S+- = S-+ = duS = S
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The Tree!
u =1.25, d = 0.8
100
80
125
100
156.25
64
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Binomial Replication
The idea of binomial valuation viareplication is incredibly general.
If you can write down a binomial assetvalue tree, then any (derivative) assetwhose payoffs can be written on this treecan be valued by replicating the payoffs
using the original asset and a risk-free,zero-coupon bond.
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An American Put Option
What is the value of a 1-year put option withexercise price 105 on a stock with current price100?
The option can only be exercised now, in 6 monthstime, or at expiration.
= 31.5573% rF = 1% (per 6-month period)
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Multi-Period Replication
Stock
100
80
125
100
156.25
64
Put (E=105)
5
0
41
P+
P-
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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!
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The Answer
100
80
125
25.00
2.64
rF = 1%
H = -0.497, B = -64.11 P = 14.42
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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