1. C15.0008 Corporate Finance Topics Summer 2006
Session 2: Options I
C Corporate Finance Topics
Summer 2006

2. Outline Call and put options The law of one price Put-call parity
Binomial valuation

3. 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
Why so much stuff on options?
Bottom line—options is something every well-educated finance student should understand!

4. Example..

5. 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

6. 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).
eBay Jan o6 call exercise price of $45 C(39, 45, 1/3 year)
Currently out-of-the-money

7. 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

8. Numerical example
Call option
Put option

9. 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”)

10. Call Option Payoffs Long Call Short Call Payoff ST E Payoff E ST
This is just the payoff at expiration—it does not include the price/premium!

11. Put Option Payoffs
Payoff ST E
Long Put
Short Put
Payoff E E ST E

12. Other Relevant Payoffs
Risk-Free Zero Coupon Bond
Maturity T, Face Amount E
Payoff ST
Stock
Payoff E ST

13. 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)
This should be familiar from FFM!

14. Put-Call Parity Stock + Put = Call +Bond = Payoff ST E Payoff ST E
Stock+put S<E S+(E-S)=E
S>E S+0=S
Call+bond S<E 0+E=E
S>E (S-E)+E=S
Payoff ST E =

15. Put-Call Parity Payoffs: Stock + Put = Call + Bond Prices:
Stock = Call – Put + Bond
S = C – P + PV(E)
Value of zero coupon bond is just PV of face amount

16. Introduction to binomial trees

17. 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.

18. Payoffs Stock Bond (rF=2%) Call (E=105) 137 102 32 100 100 C 73 102
1-year call option, S=100, E=105, rF=2% (annual)
1 step per year
Can the call option payoffs be replicated?

19. Replicating Strategy
Buy ½ share of stock, borrow $35.78 (at the risk-free rate).
Cost
(1/2) = 14.22
Payoff
(½)137 - (1.02) = 32
(½)73 - (1.02) = 0
Beware the rounding!!
The value of the option is $14.22!

20. 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 B = 32
73 H 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!
Spreadsheet available (but make sure you can do the calculations by hand)
All the information about probabilities and preferences (discount rates) is contained in the current stock price
Why not use DCF?
What is the correct discount rate for the call option? Equal to stock, greater than, less than?
Calculate expected/required return on the stock (assuming prob. of 0.5) E[r(S)]=0.5(137/100-1)+0.5(73/100-1)=0.5(37%)+0.5(-27%)=5%
Calculate expected/required return on the call (assuming prob. of 0.5) E[r(C)]=0.5(32/ )+0.5(0/ )=0.5(125%)+0.5(-100%)=12.5%

21. Multi-Period Replication
100 80
125
156.25 64
Stock
51.25
Call (E=105) C+ C-
1-year call option, S=100, E=105, rF=1% (semi-annual)
2 steps per year

22. 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 =  C+ = 23.68
C- = 0 (obviously?!)
156.25
51.25
rF = 1%
125 C+
100

23. The Answer
Use these call values to solve the first 1-step binomial model
Solution: H = 0.526, B =  C = 10.94
The multi-period replicating strategy has no intermediate cash flows
125
23.68
100
rF = 1% 80
Spreadsheet available (but make sure you can do the calculations by hand)

24. Building The Tree S++ S+ = uS S+ S- = dS S S+- S++ = uuS S-- = ddS S-
S+- = S-+ = duS = S
You do not need to know how to do this!

25. The Tree!
u =1.25, d = 0.8
100 80
125
156.25 64

26. 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.

27. 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.
 = % rF = 1% (per 6-month period)

28. Multi-Period Replication
100 80
125
156.25 64
Stock 5
Put (E=105) P+ P- 41

29. Solving Backwards 125 100 156.25 5 P+ rF = 1%P+
rF = 1%
H = , B =  P+ = 2.64 80 64
100 41 5 P-
rF = 1%
H = -1, B =  P- = !!
The put is worth more dead (exercised) than alive!

30. The Answer
125
25.00
2.64
100
rF = 1% 80
H = , B =  P = 14.42

31. Assignments Reading Problem sets
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
Next class: Black-Scholes, intro to real options (decision trees)