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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

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”)

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

Payoffs:

Stock + Put = Call + Bond

Prices:

Stock + Put = Call + Bond

Stock = Call – Put + Bond

S = C – P + PV(E)

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?

(½)137 - (1.02) 35.78 = 32

Cost

(1/2)100 - 35.78 = 14.22

Payoff

(½)73 - (1.02) 35.78 = 0

Replicating StrategyBuy 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!

51.25

125

100

100

0

80

64

Multi-Period ReplicationStock

Call (E=105)

C+

C-

0

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

2 steps per year

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

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

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)

5

156.25

100

5

41

80

125

P+

P-

100

64

Solving BackwardsrF = 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!

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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