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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

Session 2: Options I

C15.0008 Corporate Finance Topics

Summer 2006


Outline

Outline

  • Call and put options

  • The law of one price

  • Put-call parity

  • Binomial valuation


Options options everywhere

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

Example..


Options

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

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

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

Numerical example

  • Call option

  • Put option


Option values at expiration

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”)


Call option payoffs

Long Call

Payoff

E

ST

Call Option Payoffs

Short Call

Payoff

E

ST


Put option payoffs

Long Put

Payoff

E

ST

Put Option Payoffs

Short Put

Payoff

E

E

ST

E


Other relevant payoffs

Stock

Payoff

ST

Other Relevant Payoffs

Risk-Free Zero Coupon Bond

Maturity T, Face Amount E

Payoff

E

ST


The law of one price

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

Payoff

Payoff

Call +Bond

Stock + Put

E

E

Payoff

Payoff

E

E

ST

ST

E

E

ST

ST

Put-Call Parity

=

=


Put call parity1

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

Introduction to binomial trees


What is an option worth

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

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?


Replicating strategy

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

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

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


Solving backwards

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


The answer

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

Building The Tree

S++

S+ = uS

S+

S- = dS

S+-

S

S++ = uuS

S-- = ddS

S-

S--

S+- = S-+ = duS = S


The tree

156.25

125

100

100

80

64

The Tree!

u =1.25, d = 0.8


Binomial replication

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

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 replication1

156.25

0

125

100

100

5

80

64

Multi-Period Replication

Stock

Put (E=105)

P+

P-

41


Solving backwards1

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!


The answer1

2.64

25.00

The Answer

125

rF = 1%

100

80

H = -0.497, B = -64.11  P = 14.42


Assignments

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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