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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

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Session 2: Options I

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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”)


Long Call

Payoff

E

ST

Call Option Payoffs

Short Call

Payoff

E

ST


Long Put

Payoff

E

ST

Put Option Payoffs

Short Put

Payoff

E

E

ST

E


Stock

Payoff

ST

Other Relevant Payoffs

Risk-Free Zero Coupon Bond

Maturity T, Face Amount E

Payoff

E

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)


Payoff

Payoff

Call +Bond

Stock + Put

E

E

Payoff

Payoff

E

E

ST

ST

E

E

ST

ST

Put-Call Parity

=

=


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%)

Call (E=105)

137

102

32

100

100

C

73

102

0

1-year call option, S=100, E=105, rF=2% (annual)

1 step per year

Can the call option payoffs be replicated?


Payoff

(½)137 - (1.02) 35.78 = 32

Cost

(1/2)100 - 35.78 = 14.22

Payoff

(½)73 - (1.02) 35.78 = 0

Replicating Strategy

Buy ½ share of stock, borrow $35.78 (at the risk-free rate).

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!


156.25

51.25

125

100

100

0

80

64

Multi-Period Replication

Stock

Call (E=105)

C+

C-

0

1-year call option, S=100, E=105, rF=1% (semi-annual)

2 steps per year


51.25

0

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?!)

156.25

rF = 1%

125

C+

100


23.68

0

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

125

rF = 1%

100

80


Building The Tree

S++

S+ = uS

S+

S- = dS

S+-

S

S++ = uuS

S-- = ddS

S-

S--

S+- = S-+ = duS = S


156.25

125

100

100

80

64

The Tree!

u =1.25, d = 0.8


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)


156.25

0

125

100

100

5

80

64

Multi-Period Replication

Stock

Put (E=105)

P+

P-

41


0

5

156.25

100

5

41

80

125

P+

P-

100

64

Solving Backwards

rF = 1%

H = -0.089, B = -13.75  P+ = 2.64

rF = 1%

H = -1, B = -103.96  P- = 23.96 25!!

-------

The put is worth more dead (exercised) than alive!


2.64

25.00

The Answer

125

rF = 1%

100

80

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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