Session 02 Amrut

7/28/2019 Session 02 Amrut

1/31

Session 2: Options I

C15.0008 Corporate Finance

TopicsSummer 2006


2/31

Outline

Call and put options

The law of one price

Put-call parity Binomial valuation


3/31

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


4/31

Example..


5/31

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


6/31

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).


7/31

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


8/31

Numerical example

Call option

Put option


9/31

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


10/31

Call Option Payoffs

Payoff

STE

Long Call

Payoff

STE

Short Call


11/31

Put Option Payoffs

Payoff

STE

Long Put

Payoff

STE

Short Put

E

E


12/31

Other Relevant Payoffs

Payoff

ST

Stock

Payoff

ST

Risk-Free Zero Coupon Bond

Maturity T, Face AmountE

E


13/31

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)


14/31

Put-Call Parity

Stock + PutPayoff

STE

Payoff

STE

E=

Payoff

STE

Call +Bond

Payoff

STE

E=


15/31

Put-Call Parity

Payoffs:

Stock + Put = Call + Bond

Prices:

Stock + Put = Call + Bond

Stock = Call Put + BondS = C P + PV(E)


16/31

Introduction to binomial trees


17/31

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.


18/31

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?


19/31

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!


20/31

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!


21/31

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


22/31

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+


23/31

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%


24/31

Building The Tree

S

S+

S-

S--

S+-

S++ S+ = uS

S- = dS

S++ = uuS

S-- = ddS

S+- = S-+ = duS = S


25/31

The Tree!

u =1.25, d = 0.8

100

80

125

100

156.25

64


26/31

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.


27/31

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)


28/31

Multi-Period Replication

Stock

100

80

125

100

156.25

64

Put (E=105)

5

0

41

P+

P-


29/31

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!


30/31

The Answer

100

80

125

25.00

2.64

rF = 1%

H = -0.497, B = -64.11 P = 14.42


31/31

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

Session 02 Amrut

Documents

Transcript of Session 02 Amrut