Economics 1723: Capital Markets Lecture 20

Transcription

1 Economics 1723: Capital Markets Lecture 20 John Y. Campbell Ec1723 November 14, 2013 John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

2 Key questions What is a CDS? What information do we get from its price? Portfolios of options allow for quite general payo s. For example, what is a protective put strategy? Why might somebody want to hold this type of portfolio? What is the put-call parity? Should you exercise an American call option on a non-dividend paying stock early? John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

3 Roadmap 1 Credit default swaps 2 Option terminology 3 Payo patterns 4 Restricting option prices 5 Put-call parity 6 No early exercise John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

4 Credit default swaps (CDS) A CDS is di erent from an interest rate swap. A CDS is an insurance contract on the default of a particular bond. For example, suppose you own a corporate bond from company XYZ with principal $1,000. If company XYZ defaults, you might get back $800 instead of $1,000. You may buy a CDS for XYZ from someone (CDS seller). In this case, you will de nitely get $1,000. If XYZ defaults, the CDS seller pays you $1000 (in exchange for the bond) so that your total of $1000 is guaranteed. You swap the default risk with the CDS seller. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

5 Example In October 2008, the 5-year CDS rate on Morgan Stanley debt with face value $10,000 was $1,000. This means that you could pay $1,000 a year for ve years, and in return you get $10,000 if MS defaults (in exchange for the MS bond). This price can be used to infer the probability that MS will default. For example, if the recovery rate on MS debt is 50% (in a default, MS would only pay fty cents on the dollar), this implies: 20% chance that Morgan Stanley would default in the next year, About 70% chance of default in the next ve years. Caveats: risk aversion, credit quality of CDS issuers. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

6 CDS during the global nancial crisis CDS rates measured the health of nancial institutions (Krishnamurthy, 2010). John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

7 CDS during the Euro crisis Estimated probability of default over the next ve years in September 2011 (CNNMoney article on September 16, 2011). John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

9 What is an option? An option is the right, but not the obligation, to buy (call option) or sell (put option) an asset at a xed price on (European option) or by (American option) a given future date. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

10 What is an option? The act of buying or selling the asset at the price speci ed in the option is called exercising the option, and the speci ed price is called the: exercise price or, strike price. Write the exercise price as X, and the underlying asset price as S. The date (or last date) T on which exercise is possible is called the expiration date. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

11 Option payo s The payo on a call option at expiration is Max(S T X; 0), assuming that it will be exercised only if it is worthwhile to do so. The payo on a put at expiration is Max(X same assumption. Selling an option is called writing it. S T ; 0) under the The payo on a written call at expiration is Min(X S T ; 0). The payo on a written put at expiration is Min(S T X; 0). Option prices must be positive as their payo s are never negative and sometimes positive. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

12 Example: Option payo s Suppose a call option on AAPL with strike price 100 has a price of $14. The writer (seller) of the call option wins if AAPL has price of less than $100 at the expiration date. Best case scenario for writer: he sells something for $14 that expires worthless. The buyer of the call bene ts if the AAPL has expiration price higher than $100. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

13 John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

16 In, out, or at the money Before expiration, at date t, an option is in the money if it would be pro table to exercise it today: A call is in the money if S t > X. A put is in the money if X > S t. An option is out of the money if it would be unpro table to exercise it today: A call is out of the money if X > S t. A put is out of the money if S t > X. An option is at the money if X = S t. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

18 Combining options Options and underlying securities can be combined to create a wide variety of payo patterns: Protective put (buy put, buy stock), Covered call (write call, buy stock), Straddle (buy call, buy put), Money spread (buy one call, write one with a di erent X), Time spread (buy one call, write one with a di erent T). John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

20 Protective put provides portfolio insurance John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

22 Covered call provides sell discipline Covered calls are attractive to nancial institutions that own stocks. By writing calls: They boost current income, They commit to sell at a speci ed price, X. Example (from BKM): Pension fund holds 1000 shares of stock with S 0 = $100. Plans to sell all shares if price hits $110. Call options with X = $110 for T = 60 days trades at price $5. By writing 10 call contracts (for 100 shares each), the fund: Collects $5000 cash income. Gives up pro ts if price exceeds $110. This is the sell discipline. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

23 Straddle allows to bet on volatility John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

25 What does no-arbitrage imply for option prices? Rest of this lecture: Restrict option prices using no-arbitrage arguments. Implicit assumptions: There are some market participants for which: 1 No transaction costs. 2 All net trading pro ts are subject to same tax rate. 3 Borrowing and lending are possible at the (nominal) risk-free rate, R f. 4 Short selling is possible (this can be relaxed). The assumptions are approximately (but not exactly) correct for the options we will consider. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

26 No-arbitrage propositions The price of a call option is never greater than the price of the underlying asset. The price of a put option is never greater than the exercise price. Call options with lower exercise prices are more valuable than otherwise identical call options with higher exercise prices. Put options with higher exercise prices are more valuable than otherwise identical put options with lower exercise prices. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

27 No-arbitrage propositions American option prices are at least as great as the prices of otherwise identical European options. American option prices are at least as great as the value if they are exercised immediately (intrinsic value). American options with greater time to expiration are at least as valuable as otherwise identical American options with less time to expiration. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

29 Put-call parity A less obvious but even more important proposition is the put-call parity theorem: Consider European call and put options with the same expiration date and exercise price, on an asset that pays no dividends during the life of the option. Then, no-arbitrage implies the put-call parity theorem: C {z} 0 + call price X (1 + R f ) T {z } discounted exercise price = S 0 {z} asset price + P {z} 0. (1) put price John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

30 Put-call parity: Proof To show Eq. (1), consider two portfolios: Portfolio A (call-plus-bills): One call option plus bills with face value X and the same maturity as the call. Portfolio B (protective put): One put option plus one share of stock. Both portfolios are worth: max (S T ; X) at expiration. Thus, they must cost the same now. This proves Eq. (1) by no-arbitrage. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

31 Put-call parity: Pictorial proof Payo of protective put at expiration is the same as the payo of a call-plus-bills portfolio. Thus, their price must also be equal. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

32 Put-call parity: Example If Eq. (1) does not hold, then there are arbitrage opportunities. Example (from BKM): S 0 = $110 and and R f = 5%, X = $105 and T = 1, C 0 = $17 and P 0 = $5. Which way is the parity violated? C 0 + X (1 + R f ) T =? S 0 + P 0. How would you take advantage of this arbitrage opportunity? What are your cash ows today and at the expiration date? John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

33 Put-call parity in the data BKM Figure 20.1: Stock options on IBM closing prices as of December 2, John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

34 Put-call parity with dividends If the underlying asset pays dividends, then the relationship becomes: C 0 + X (1 + R f ) + D = S T (1 + R f ) T 0 + P 0. {z } discounted value of dividends John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

36 Lower bound on call value Consider again a European call option on an asset with no dividends. The put call parity implies the following bound on call price. Why? X C 0 S 0 (2) (1 + R f ) T This inequality can also be directly proven by considering an argument similar to above. Consider two portfolios: Portfolio A: One call. Worth max (S T X ; 0) at expiration. Portfolio B (levered equity): One share of underlying asset plus X borrowing of. Worth S (1+R f ) T T X at expiration. Which one is worth more at expiration? What does it imply for the prices of A and B? How does this prove Eq. (2). John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

37 Lower bound on call value The RHS of Eq. (2) is also greater than S 0 X, the value of the call if exercised now (the intrinsic value). Why? Thus, Eq. (2) also implies: C 0 S 0 X {z }. intrinsic value The value of a European call (on a non-dividend paying asset) is always greater than its value if it is exercised immediately. $105 Earlier example: We have shown C 0 $110 1:05 also implies C 0 $110 $105 = $5. = $10. This John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

38 Lower bound on call value: Intuition Call price = Intrinsic value + Time value. What are the two components of the time value? John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

39 No early exercise We have shown that the price of a European call option is always greater than its value if it is exercised immediately. But the price of an American option is always greater than the price of an equivalent European option. So an American call option is always worth more than the value if it is exercised immediately. It is never worthwhile to exercise an American call early. If you want to close the position, sell the option instead! John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

40 No early exercise: Caveats The no early exercise proposition applies only to call options on non-dividend-paying stocks. If a stock pays a dividend at some date, it may be worthwhile to exercise the call just before that date so as to get the dividend. Early exercise may be worthwhile for puts because when you exercise a put you get the sales proceeds earlier, so the time value of money works in your favor. Example: if you have a put on a bankrupt (worthless) stock, exercise it now! John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

41 Some takeaways What is a CDS? What information do we get from its price? A CDS is an insurance contract on an underlying bond. CDS buyer pays a periodic premium to the CDS seller. Receivers the par value (in exchange for the bond) if the bond defaults. The CDS price can be used to infer the probability that the bond will default (subject to caveat about risk aversion). Portfolios of options allow for quite general payo s. For example, what is a protective put strategy? Why might somebody want to hold this type of portfolio? A protective put holds one put and one share of the stock. It provides portfolio insurance: It limits the downside losses in exchange for giving up the price of put (the insurance premium) in upside states. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

42 Some takeaways De ne put-call parity. Consider European call and put options on a non-dividend paying stock with the same strike price and maturity. Put-call parity recognizes that the payo on a portfolio of the stock and a put (protective put) is equal to the payo on a portfolio of call-plus-bills. Therefore, the prices of these two portfolios must be equal: C 0 + X (1+R f ) T = S 0 + P 0. Should you exercise an American call option on a non-dividend paying security early? The price of a European call is always greater than its value if exercised immediately (intrinsic value), C 0 S 0 X. The price of an American call is always greater than the price of a European call. Thus, an American call in this case is always worth more alive than dead. Do not exercise early. John Y. Campbell (Ec1723) Lecture 20 November 14, / 43

43 For next time We will discuss how di erent variables (e.g., stock price, exercise price, interest rate) in uence option values. We will then discuss a variety of nancial situations in which options naturally exist. We will also discuss the binomial approach to option valuation. To prepare, read BKM sections 20.5 through 20.7 and sections 21.1 through John Y. Campbell (Ec1723) Lecture 20 November 14, / 43