CHAPTER 23 OPTIONS

1. CHAPTER 23 OPTIONS Topics: • 23.1 Background • 23.5 Stock Option Quotations • 23.2 - 23.4 Value of Call and Put Options at Expiration • 23.6 Combinations of Options • 23.7 Valuing Options • 23.8 An Option Pricing Formula

2. 23.1 Background • A derivative security is simply a contract which has a value that is dependent upon (or derived from) the value of some other asset(s) • Derivatives that we consider in this course: • Futures and forward contracts (Chapter 26) • Options • there are many different kinds of options: stock options, index options, futures options, foreign exchange options, interest rate caps, callable bonds, convertible bonds, retractable/extendable bonds, etc.

3. Why option is important in corporate finance • Important financial markets besides stock and bond markets • Many employees are compensated in stock options • You need to what they’re worth • You need to know how their value might depend upon the actions chosen by those employees • Many corporate finance projects have implicit real options in them. A static valuation (i.e., that ignores this option value) can give misleading results • In fact, almost any security can be thought of in terms of options (including stocks and corporate bonds)

4. Terminology • Option: Gives the owner the right to buy or sell an asset on or beforea specified date for a predetermined price • Call: the right to buy (C) • Put: the right to sell (P) Expiry date, T The underlying asset (Current price: Spot price, S) Exercise price, K or X

5. Terminology cont’d Has a right • Owner of an option • Buy • Long position • Seller of an option • Short • Write • The act of buying or selling the underlying asset via the option contract is exercising the option • Europeanoptions can only be exercised at the expiry date, while Americanoptions can be exercised at any time up to and including the expiry date (unless otherwise stated, the options we consider will be European options) • Value of the option is called premium Has an obligation (A contingent obligation)

6. Terminology cont’d • Option contracts are either exchange-traded or available on a customized basis in the over-the-counter market • Standardized exchange-traded option contracts (on the CBOE): • Usually are for 100 options (1 contract) • Have strike prices $2.50 apart in the range of $5 to $25, $5 apart in the range of $25 to $200, and $10 apart for strike prices above $200 • Expiration months are two near term months plus two additional months from the Jan, Feb, or Mar quarterly cycle • The expiration date is usually the Saturday following the third Friday of the expiration month (which is the last trading day)

7. 23.5 Stock Option Quotations • Strike: strike price • Exp: Expiry month • Bid: the bid price (The price at which somebody bids to buy) • Ask: the ask price (The price at which somebody asks to sell) • Open int.: Open interest, or number of outstanding contracts (one contract usually equals 100 options) RBC’s Nov. 08 options

8. Option quotations cont’d • Options are not usually protected against cash dividends but are protected against stock-splits and stock dividends • Suppose in Oct. you buy 1 call option of RY with expiry of Jan. and a strike of $30. • In Nov. RY pays $50 cash dividend. • In Nov. RY does a 2:1 split. Your 1 call option will become 2 options with strike price halved

9. 23.2-4 Value of Options Value of a Call Option at Expiration Date (T) • If ST = 9 at the expiry date, exercise option, purchase share at 8 and then sell share for 9. Payoff = C = max(ST – X, 0) Example: An option on a common stock: ST= market price of the common stock on the expiration date, T. Suppose the exercise price, X = $8

10. Value of a Put Option at Expiration Date (T) • A put. Exercise price X = $11 • If ST = 10 at the expiry date, exercise option, purchase share at 10 and then sell share for 11. Payoff = P = max(X - ST, 0)

11. Moneyness • an in the money option is one that would lead to a positive cash flow if exercised immediately • an at the money option is one that would lead to a zero cash flow if exercised immediately • an out of the money option is one that would lead to a negative cash flow if exercised immediately • Let X denote the strike price of the option and St denote the current price of the underlying asset:

12. Value components • Intrinsic Value: the intrinsic value of an option is the maximum of zero and the option’s immediate exercise value • Time Value • The difference between the option premium and the intrinsic value of the option. Time value Intrinsic Value OPTION VALUE (PREMIUM) + = E.g.: (1) The Nov. RY $30 Call. Price 8.20. Spot 37.10. (2) The Nov. RY $30 Put. Price 1.05.

13. Option payoffs cont’d Profit/loss from Buying a stock, e.g., at 15 Payoff $0 15 Share Price $-15

14. Payoff/profit of Long Call • Let X = $15. What’s your payoff/profit of buying a call at T? • Payoff is just the final value of the option CT, profit takes the option premium into account. • When you long an option, you buy the option at a cost (the option premium) $20 15 35 Share Price

15. Payoff/profit to buy a Put option, given a $15 exercise price. $5 10 15 Share Price

16. The buyer and seller always have “mirror image” payoffs Long call Share Price 15 • Profit- • When you go short in an option, you sell the option (and receive premium)

17. Exercise • Plot the payoff/profit diagrams for (1) short stock (suppose the sell price is 15) and (2) selling put (suppose X = 15).

18. 23.6 Combinations of Options and stocks • Portfolios of stocks and options. • Firms and individuals’ portfolios (asset holdings) may consist of different options and stocks on the same underlying asset. These positions may offset each other or compound the payoff possibilities. • You want to combine them and consider the risk of changes in the price of the underlying asset to the aggregate portfolio.

19. Long Call + Short put with same strike and expiry (let X = 55) Equivalent to: Short call + Long put with same strike and expiry (called synthetic short sale) Equivalent to:

20. Writing a Covered Call: Long stock + short call • X = 55, and cost of stock (S0) = $55. Option premium = $5. • Payoff and Profit: • Long stock position covers or protects a trader from the payoff on the short call that becomes necessary if there is a sharp increase in stock price.

21. Covered call cont’d • Payoff and profit (When strike = purchase price) • Break-even point • Maximum profit • What if strike ≠ purchase price?

22. Protective Put: Long stock + long put • Use puts to limit downside • Often used with index options to provide portfolio insurance • Let X = 40, S0=40

23. Protective put cont’d Profit: You start to make $ when Otherwise, you lose at most P.

24. Straddle: Buy 1 call and 1 put (same X, T) • e.g. S0 = 40, C = 10, P = 10, X = 50

25. Profit Diagram of Long Straddle You do not make $ if

26. 22.7 (Arbitrage) Bounds on Call Option • Objective: derive restrictions on option values which (i) any reasonable option pricing model must satisfy; and (ii) do not depend on any assumptions about the statistical distribution of the price of the underlying asset • To simplify matters, we will assume that no dividends are paid by the underlying asset during the life of the option • Let’s look at a call only: • Fact 1: A call option with a lower exercise price is worth more. • A call option with an exercise price of zero is effectively the same financial security as the stock itself. • An upper bound for the call option price is the price of the stock.

27. Cont’d • Fact 2: The option value is at least the payoff if exercised immediately • The action of exercising gives you the underlying asset • However, if you want the asset • If you exercise now you pay exercise price now • Or better, you can exercise on the expiry day and pay the exercise price later. • Equivalently, if you exercise immediately, you only need to pay the present value of (Exercise price) • Lower bound of call: St – PV(X) • Fact 3: As the stock price gets large (relative to exercise price) • the probability that the option expires worthless vanishes • the value of the call approaches: Value of stock – PV (Exercise Price) ≈St - X

28. Market Value Bounds for Call option ST Call Time value Intrinsic value 0 ST PV(X) The value of a call option Ct must fall within Max(St – PV(X),0)<Ct<St

29. Implications of Call Bounds • If St = 0 then Ct = 0. • If X = 0 then Ct = St. • If T-t = ∞, then Ct = St. • For an American option, it is not optimal to exercise a call option before its expiry date (if the underlying stock will not pay any dividends over the life of the option). • Note that this implies that the value of an American call option will be equal to that of a corresponding European option (same underlying asset, X, T).

30. It’s never optimal to exercise the American call early • Why? • If you exercise a call at any time t, you receive the intrinsic value: (St – X) • But the option is always worth at least the intrinsic value. • Example: St = 60, T - t = 3 months, X = $55, r =10% • if you exercise you receive $5 but you lose time value of the call. • The option is worth at least • At least a dominant strategy is to put $55 in bank and wait until the expiry to exercise. • You get interest income on the $55. • If you want the underlying, hold the option and you can buy it any time you want.

31. Call bounds example What is the lower bound for the price of a six-month call option on a non-dividend-paying stock when the stock price is $80, the strike price is $75, and the risk-free interest rate is 10 percent per annum?

32. Arbitrage Bounds on Put Option Values • For put options, we will show that X ≥ Pt ≥ max(0, X – St)

33. Arbitrage Bounds on Put Option Values (cont’d)

34. Implications of Put Option Bounds • If St = 0 then Pt = X. • If X = 0 then Pt = 0. • For a European put, and if T-t = ∞, then Pt = 0. • It may be optimal to exercise an American put option before its expiry date, even if the underlying asset does not pay dividends during the life of the option (consider what happens if S → 0)

35. An aside: Continuously compounding • In option pricing, people generally use continuously compounding instead of discrete compounding that is used in Chapter 8. • Continuous compounding assumes that compounding period for effective return is as short as possible (think of milliseconds). A dollar with nominal return of r over a period of T will become the following continuous-time proceeds: where m is the compounding frequency.

36. European options: Put-Call Parity (no dividends) Suppose you buy one share of stock, buy one put option on the stock, and sell one call option on the stock (same X, T). Cashflow at expiry • No matter what happens, you end up with a payoff of X • Today’s value of your synthetic investments must equal PV(cf @ expiry):

37. Put-call parity cont’d • Given any three of risk free bond, stock, put, and call, the fourth can always be created synthetically. (In fact, this was done before 1977 on the CBOE to construct puts) • Holding stock and put today is equal to holding a call the PV(X) amount of riskfree bond. • Note that the LHS is the value of protective put. • You can also use the profit graph of protective put to prove put-call parity

38. Put call parity with dividend paying stocks • We can also derive parity relationships when there are dividends paid by the underlying asset, when the options are American, etc. • In the case of European options on a stock paying known dividends before the expiry date, the result is where It is the present value (at t) of the dividends paid before T

39. Put-call parity example 1 The price of a European call which expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and there is no dividend payment. The term structure is flat, with all risk-free interest rates being 10 percent. What is the price of a European put option that expires in six months and has a strike price of $30?

40. Put-call parity example 2 • Several years ago, the Australian firm Bond Corporation sold some land that it owned near Rome for $110 million and as a result boosted its reported earnings for that year by $74 million. The next year it was revealed that the buyer was given a put option to sell the land back to Bond for $110 million and also that Bond had paid $20 million for a call option to repurchase the land for the same price of $110 million. 1. What happens if the land is worth more than $110 million when the options expire? What if it is worth less than $110 million? 2. Assuming that the options expire in one year, what is the interest rate? 3. Was it misleading to record a profit from selling the land?

42. How do we value options? • Why the discounted cash flow approach will not work? • Expected cash flows? • Opportunity cost of capital? • We are going to use a replicating portfolio method • Owning the option is equivalent to owning some number of shares and borrowing money • Therefore the price of the option should be the same as the value of this replicating portfolio.

43. The Binomial Model (Two-State Option Pricing) • In order to use the replicating portfolio method, we need to make some assumptions about how the underlying stock behaves • The simplest possible setting is one in which there are only two possible outcomes of stock price at the expiry date of 1 period (“binomial”) • Also assume • the underlying stock won’t pay a dividend over the life of the option • interest rates are constant over the life of the option

44. Example I • Let’s assume a stock’s current price is $25. It may go up (u) or down (d) by 15% next year, with equal probabilities. • The discount rate for the stock is 10%. • The stock has a beta of 1. • T-bills are yielding 5%. • A call option on the stock is traded on the NYSE. The strike price of this option is 25, and the maturity is one year. • What is the option worth?

45. The method: Replicate the payoffs of the option using only the stock and loan • Step 1: Find the payoffs of the option in both “good” and “bad” states. • Step 2: Choose the right number of shares to make the difference in outcomes equal to the option. • Step 3: Replicate the payoffs by adding the right size loan to your shares

46. Example cont’d: Matching cash flows Value of the option:

47. A generalization of option replicating portfolios: deciding delta and loan • Form a portfolio with ∆ shares of stock and B borrowed or invested in the risk-free bond. The current value of the portfolio V0 = S0∆+B • The basic idea is to pick ∆ and B to replicate the payoffs of the call option at its expiry • The number of shares bought, ∆, is called the hedge ratio or option delta • If B < 0, then it is borrowing; if B > 0, then it is lending (investing in T-bills).

48. Solve for ∆ and B: Graph representation To match the cashflows: t = 0 t =1 Note: rfis interest rate for the length of one period.

49. Cont’d Solve for ∆, B: (1) – (2), and • In words: ∆ = (Spread of possible option prices)/ (Spread of possible stock prices) In the previous example

50. Possibly more intuitive • You might think of option deltas as solving (Option delta) * (Spread in Share price) = Spread in option price • That is, the delta “scales” the amount of variation in the stock price so that the spread of outcomes in your replicating portfolio is the same as in the option. • (The implicit loan in options) Note that we just showed the call option is like a levered version of stock. • It’s like buying the stock “bundled” with personal debt. • Levered equity: higher risk or higher beta.