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 1
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. 2
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) 3
Terminology • Option: Gives the owner the right to buy or sell an asset on or before a specified date for a predetermined price Expiry date, T The underlying asset (Current price: Spot price, S) Exercise price, K or X – Call: the right to buy (C) – Put: the right to sell (P) 4
Terminology cont’d • Owner of an option – Buy – Long position • Seller of an option – Short – Write Has a right Has an obligation (A contingent obligation) • The act of buying or selling the underlying asset via the option contract is exercising the option • European options can only be exercised at the expiry date, while American options 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 5
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) 6
23. 5 Stock Option Quotations RBC’s Nov. 08 options • • • 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) 7
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 8
23. 2 -4 Value of Options Value of a Call Option at Expiration Date (T) 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 Payoff on the expiration date (The third Friday in June) ST > 8, let ST = $9 ST = $8 ST < 8 let ST = $7 Call option value • 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) 9
Value of a Put Option at Expiration Date (T) • A put. Exercise price X = $11 Payoff on the expiration date (The third Friday in Sept. ) ST > 11, let ST = $12 ST = $11 ST < 11 let ST = 10 Put option value • 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) 10
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: 11
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. OPTION VALUE (PREMIUM) = Intrinsic Value + Time value E. g. : (1) The Nov. RY $30 Call. Price 8. 20. Spot 37. 10. (2) The Nov. RY $30 Put. Price 1. 05. 12
Option payoffs cont’d Payoff Profit/loss from Buying a stock, e. g. , at 15 $0 15 Share Price $-15 13
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 14
Payoff/profit to buy a Put option, given a $15 exercise price. $5 10 15 Share Price 15
The buyer and seller always have “mirror image” payoffs Long call 15 Share Price Profit- • When you go short in an option, you sell the option (and receive premium) 16
Exercise • Plot the payoff/profit diagrams for (1) short stock (suppose the sell price is 15) and (2) selling put (suppose X = 15). 17
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. 18
• 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: 19
Writing a Covered Call: Long stock + short call • X = 55, and cost of stock (S 0) = $55. Option premium = $5. • Payoff and Profit: ST 45 50 Buy 1 share 0 5 -5 Profit 5 0 Total payoff 60 -5 Short 1 call 55 • 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. 20
Covered call cont’d • Payoff and profit (When strike = purchase price) • Break-even point • Maximum profit • What if strike ≠ purchase price? 21
Protective Put: Long stock + long put • Use puts to limit downside – Often used with index options to provide portfolio insurance • Let X = 40, S 0=40 ST 35 40 45 50 Buy 1 share 10 Buy 1 put 0 Total payoff 10 Profit 5 22
Protective put cont’d Profit: You start to make $ when Otherwise, you lose at most P. 23
Straddle: Buy 1 call and 1 put (same X, T) • e. g. S 0 = 40, C = 10, P = 10, X = 50 ST 35 40 45 Buy 1 call 0 Buy 1 put 50 55 60 65 0 10 15 10 5 0 0 Total payoff 10 5 10 15 Profit -10 -15 -10 -5 24
Profit Diagram of Long Straddle You do not make $ if 25
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. 26
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 27
Bounds for Call option ST Call Market Value Time value Intrinsic value 0 PV(X) ST The value of a call option Ct must fall within Max(St – PV(X), 0) < Ct < St 28
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). 29
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. 30
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? 31
Arbitrage Bounds on Put Option Values • For put options, we will show that X ≥ Pt ≥ max(0, X – St) 32
Arbitrage Bounds on Put Option Values (cont’d) 33
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) 34
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. 35
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 ST < X 1. Buy stock 2. Buy put 3. Write call 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): 36
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 37
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 38
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? 39
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? 40
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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. 42
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 43
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? 44
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 45
Example cont’d: Matching cash flows Investment strategy 1. buy a call State 1 (future price = State 2 (future price= 2. (i) Buy 0. 5(delta) shares of stock (ii) Borrow 10. 119 @ 5% a year Total from replicating strategy—strategy 2 Value of the option: 46
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 V 0 = S 0 ∆ +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). 47
Solve for ∆ and B: Graph representation t = 0 t =1 To match the cashflows: Note: rf is interest rate for the length of one period. 48
Cont’d Solve for ∆, B: (1) – (2), and • In words: ∆ = (Spread of possible option prices)/ (Spread of possible stock prices) In the previous example 49
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. 50
Optional: Example continued: Call Options are riskier than the stock. • Find the option’s beta. C = S 0 ∆ +B Portfolio beta. (Answer: 5. 25) 51
Valuing the put option • Method 1: Same as before. – Find the option’s delta • Option Delta = (Spread of possible option prices)/(Spread of possible stock prices) – Then find the bank loan that equates the payoff of option and the payoff of (stock position + loan) • Method 2: – Since we already have the call option value, just use put-call parity to infer the put option value. 52
Exercise: Suppose Example I is a put option instead of a call. Use the binomial model to find its delta, replicate the payoffs, and find the value of the put. Put graph here. P = S 0 ∆ +B = Interpret ∆ and B for put: 53
Implication of irrelevance of the underlying’s expected return in the real world • The option value does not directly depend on: – Probabilities and expected returns of the underlying – Individual preferences about risk aversion • If there are just options traded, expected return from the option payoffs should be the same, which is the riskfree rate. – Indirectly, these factors do affect the option value, but only by determining the current stock price S 0 • A very useful trick that can be used to simplify calculations is to exploit the fact that investors’ risk aversion doesn’t matter and just assume that they are risk-neutral • In this case, all assets should earn the risk free rate of return 1) Expected returns of stock = riskfree rate 2) Expected option payoffs = option value (1 + riskfree rate) 54
Binomial option pricing: risk neutral pricing Let S 1, u = u S 0 and S 1, d = d S 0 • The key is to find risk-neutral probabilities for stock price movements. Let q be the “risk-neutral” probability of an “up” move. • To find q, by meaning (1) of previous slide, we have 55
Risk neutral pricing • A minor bit of algebra yields: • By meaning (2) of previous slide, value of option can now be derived as: • A big advantage of risk-neutral pricing is we can circumvent the dynamic hedging nuisance in multi-period pricing. 56
Example of the Risk-Neutral Valuation of a Call Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. The risk-free rate is 5% for one-period. What is the value of an at-the-money call option with a maturity of one-period? The binomial tree 57
Example cont’d • Step 1: Compute risk neutral probabilities • Step 2: Find the value of the call in the up state and down state • Step 3: Find the value of the call at time 0. 58
Exercise Suppose the option is a put in Example I. Use risk-neutrality to value the put. (You should be able to do this without the help of graphs. ) 59
Binomial to Black Scholes Expanding the binomial model to allow more possible price changes 1 step 2 steps 4 steps (2 outcomes) (3 outcomes) (5 outcomes) etc. 60
The Black-Scholes Model • As the number of intervals N , underlying stock price follows a random walk with a positive growth rate. • In this case, solving the option value using the portfolio replicating method yields the famous Black-Scholes equations. • Solving the Black-Scholes equations for the basic version of European call options with no dividends yields the Black-Scholes formula: Where 61
5 inputs: 1. 2. 3. 4. 5. Inputs to Black-Scholes X: Strike S: Spot (current stock price) σ = annual standard deviation of underlying stock’s return r = continuously compounded annual risk free interest rate T-t: Time to maturity expressed as fraction of year N(. ) is the cumulative standard normal distribution function, i. e. , N(d 1) is the probability that a variable distributed normally with mean zero and standard deviation of one will be less than or equal to d 1 Note that we don’t need expected return of the underlying asset to figure out the stock option value. 62
Black-Scholes and Binomial Methods 1. N(d 1) is delta, the number of shares held in the replicating portfolio 2. The amount borrowed is 63
Some properties of Standard Normal Distribution N(d 1 = 3) = 0. 9987: There is 99. 87% probability that a drawing from a standard normal distribution will be below 3. N(0) = 0. 5, N(+∞)=1 N(-d 1) = 1 - N(d 1) N(d 1 = 0) = 0. 5 64
Table 23. 3 Cumulative Probabilities of Standard Normal Dist’n d 0. 00 0. 01 0. 02 0. 03 0. 0000 0. 0040 0. 0080 … 0. 0120 0. 09 0. 0359 0. 10 0. 20 0. 0793 0. 0910 … … 3. 00 0. 4987 0. 4988 0. 4990 N(0. 23) = N(-0. 23) = In excel, the function is normsdist( ). 65
Linear Interpolation N (-0. 0886) = ? From Table 23. 3: N ( - 0. 08) = 0. 5 – 0. 0319 = 0. 4681 N ( - 0. 09) = 0. 5 – 0. 0359 = 0. 4641 • Linear interpolation: • Exercise N(0. 0886) = 66
Black-Scholes Value of Put The corresponding Black-Scholes formula for puts is: • Exercise: Prove that the B-S call and put formulae satisfy the put -call parity. 67
Example Call option. Inputs: T-t = 0. 25 (time to expiry 3 months), St = $53, X = $55, r =. 05, = 0. 30. What’s the value of the call? (2. 5895) N(d 2) = N(d 1) = C = 68
Example cont’d • What is the value of the corresponding put option? (3. 9063) 69
Factors determining option values Volatility is probably the most important consideration in options, since option value is very sensitive to volatility. 70
The Black-Scholes Model with Dividends (European options) • Options are protected against stock splits but not cash dividends. • Two possible adjustment to B-S value for dividends: 1. Discrete dividend payments: – For options on a single stock, one approach is to assume that dividends are known up to the option expiry date. The underlying stock is then viewed as consisting of two components: • A risk free series of dividends known up until T • The risky stock value – Effectively the spot price is lower from the perspective of option holder. Replace in the B-S formula S with the new after-dividend S, i. e. , with S – PV(Dividend). 71
Example • S = 50, X = 45, σ = 20%, rf = 6%, t = 3 mths, div. = $0. 50 (payable in 2 mths) S = 50 0 Div = $2 2 X = 45 3 • Adjust the stock price S’ = 50 – 0. 50 e(-0. 06* 2/12) = 49. 505 • Recalculate the BS call value using S’ instead of S throughout. (C = 5. 494) 72
2. Continuous Dividend Yield • In some cases it is reasonable to assume that dividends are paid continuously as a known percentage of the value of the underlying asset. • Examples include stock indexes, foreign currencies and futures contracts • If an asset pays a continuous dividend yield q, we would do the similar adjustment. • Adjust back dividend to St : Replace St with 73
Example • S = 50, X = 45, σ = 20%, rf = 6%, t = 3 mths, q = 4% • Adjust the stock price S’ = 50 e(-0. 04* 3/12) = 49. 502 Recalculate the BS call value using S’ instead of S throughout. ( C = 5. 491) 74
Review Questions 1. Two stocks have identical firm-specific risks but different betas. All else being equal, a put option on the high-beta stock is worth more than one on the low-beta stock. 75
2. Value of portfolio of options and option on portfolios: Suppose you have two stocks (stocks 1 and 2) in a portfolio. The return correlation of the two stocks is 0. 5. Currently, stocks 1 and 2 are both trading at $20, and both stocks have same volatility. The weight of stock 1 in the portfolio is 0. 5, and the weight of stock 2 is also 0. 5. There are three call options, written on stock 1, stock 2 and the portfolio respectively, all with same X and T. (1) Which option is the most expensive? (2) Which option is the least expensive? 76
• Assigned problems: #23. 3, 5 -7, 10, 13, 14, 17, 19, 21, 24, 25, 26, 29, 32, 33 77
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