Week 12. Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14.

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1 Week 12 Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch

2 Options on Stock Indices and Currencies Objective: To explain the basic asset pricing techniques used in the valuation of index and currency options. 2

3 Options on Stock Indices and Currencies 1. Inclusion of a Dividend Yield. 2. Pricing Formulas Revisited. 3. Option Hedging in the Black and Scholes Setting. 4. Hedging Portfolios with Index Options. 5. Black-Scholes vs. Binomial Models. 6. Credit Risk Modeling. 3

4 1. Inclusion of a Dividend Yield 4

5 Dividend Yield Dividends cause stock prices to decrease by the amount of the dividend. If in the presence of a dividend yield q, the stock grows from S to S T, in absence of dividends the it would grow from S to S T *exp(qt). Alternatively, in absense of dividends, the stock would grow from S *exp(-qt) to S T at time T. Thus, we can obtain the probability distribution of S T by 1. Starting at S and assuming a dividend yield q. 2. Starting at S *exp(-qt) and assuming no dividends. 5 Jorge Cruz Lopez - Bus 316: Derivative Securities

6 Lower Bounds Revisited c S e -qt Ke -rt p Ke rt S e -qt 6

7 Put-Call Parity Revisited c + Ke -rt = p + S e -qt S e -qt K C P S - Ke -rt 7

8 Binomial Trees Revisited t q r e S d S p u ps ) ( ) (1 d u d e p t q r ) ( d u rt f p pf e f ) 1 ( 8 Jorge Cruz Lopez - Bus 316: Derivative Securities

9 2. Pricing Formulas Revisited 9

10 Pricing Formulas Revisited Black-Scholes formulas for an option on a stock or an index paying a dividend yield q: c S qt rt e d N( d1) Ke N( 2) p Ke rt qt N( d2) Se N( d1) where d d 1 2 ln( S ln( S / / K) ( r K) ( r q T q T 2 2 / / 2) T 2) T 1 Jorge Cruz Lopez - Bus 316: Derivative Securities

11 Pricing Formulas Revisited Black-Scholes formulas for a currency option: ) ( ) ( 2 1 d N Ke d N e S c rt T r f ) ( ) ( 1 2 d N e S d N Ke p T r rt f T T f r r K S d T T f r r K S d 2) / 2 ( ) / ln( 2) / 2 ( ) / ln( where Jorge Cruz Lopez - Bus 316: Derivative Securities

12 3. Option Hedging in the Black and Scholes Setting 12

13 Option Hedging in the Black-Scholes Setting Consider the position of a financial institution that just sold an option. The payoff can be extremely negative! What hedging strategy is open to the financial institution? 13

14 4. Hedging Portfolios with Index Options 14

15 Hedging Portfolios with Index Options To hedge the risk on a portfolio, take a long position on N* puts on the index. V N* A where V is the value of the portfolio. is the beta of the portfolio (obtained through regression). A is the value of the assets underlying one futures contract. 15

16 Hedging Portfolios with Index Options Given a lower limit for the value of the portfolio, it can be shown that the appropriate strike price (K) to be selected is: S VT K 1 V 1 S V V r y q T Risk Dividend yield Current Lower Current Dividend NOTE: r, free Index limit value rate on yield Level of the paid paid r y S r q the value on on portfolio the the of theportfolio(insured portfolio index value) y and q are discrete and effective from t to t T 16 Jorge Cruz Lopez - Bus 316: Derivative Securities

17 Hedging Portfolios with Index Options Proof: (3) 1 * (2) (1) 1 q S K S S S q K R r R r R y R V V M M T 17 Jorge Cruz Lopez - Bus 316: Derivative Securities

18 Hedging Portfolios with Index Options Proof continued: r q K S y r V V r q S K y r V V y r q S K r V V T T T Jorge Cruz Lopez - Bus 316: Derivative Securities

19 Hedging Portfolios with Index Options Proof continued: K q r S y r V V S K S q r y r V V T T Jorge Cruz Lopez - Bus 316: Derivative Securities

20 5. Black-Scholes vs. Binomial Models 2

21 Black-Scholes vs. Binomial Models As you increase the number of time-steps in the binomial model, its estimate converges to the Black-Scholes estimate. 21

22 6. Credit Risk Modeling 22

23 Credit Risk Modeling The Black-Scholes model can be used to value corporate debt securities by providing a Structural Model of Default. Consider a company that has assets that are financed with zero-coupon bonds and equity. Bonds have a face value of $D and mature in T years. No dividends. 23

24 Equity Holders Decision If the firm s assets are worth more than the face value of the debt at time T (if AT > D), the equity holders pay off the debt. If the firm s assets are worth less than the face value of the debt at time T (if AT < D), the equity holders declare bankruptcy and the bond holders get the firm. 24

25 Equity Holders Payoff ET = Max (AT D, ) Equity holders have a long position on a call option on the value of the firm s assets with a strike price D. Therefore, the current value of equity is the current value of this call. 25

26 Equity Holders Payoff c E d BS 1 S A ln(a N(d N(d 1 1 ) /D) ) σ A De (r Ke T rt rt σ 2 A N(d N(d 2 2 /2)T ) ) d 2 d 1 σ A T 26

27 Bond Holders Payoff BT = Min (D, AT) = D Max (, D AT) Bond holders have a risk-free loan with a face value of D and a short position in a put option on the value of the firm s assets with a strike price D. Current value of Bond is the PV(D) less the current value of this put. 27

28 Bond Holders Payoff p B B B BS Ke De De De rt rt rt rt N(-d (1 N(d De 2 2 rt N(-d ) ) S N(-d 2 )) A (1- N(-d 2 ) A 1 N(d ) A N(-d 1 N(-d )) 1 ) 1 ) 28

29 Employee Stock Options Objective: To explain the basic issues surrounding the use and valuation of employee stock options. 29

30 Employee Stock Options 1. Contractual Arrangements 2. Alignment of Interests 3. Accounting Issues 4. Valuation of ESOs 5. Backdating Scandals 3

31 1. Contractual Agreements 31

32 Nature of ESOs Employee stock options are call options issued by a company on its own stock. They are often at-the-money at the time of issue. They often last as long as 1 years. 32

33 Typical Features of ESOs There is a vesting period during which options cannot be exercised. When employees leave during the vesting period options are forfeited. When employees leave after the vesting period in-the-money options are exercised immediately and out of the money options are forfeited. 33

34 Typical Features of ESOs Employees are not permitted to sell options. When options are exercised the company issues new shares. 34

35 Exercise Decision To realize cash from an employee stock option the employee must exercise the options and sell the underlying shares. Even when the underlying stock pays no dividends, an employee stock option (unlike a regular call option) is often exercised early. 35

36 2. Alignment of Interests 36

37 Drawbacks of ESOs Gain to executives from good performance is much greater than the penalty for bad performance. Executives do very well when the stock market as a whole goes up, even if their firm does relatively poorly. 37

38 Drawbacks of ESOs Executives are encouraged to focus on short-term performance at the expense of long-term performance. Executives are tempted to time announcements or take other decisions that maximize the value of the options. 38

39 3. Accounting Issues 39

40 Accounting for ESOs Prior to 1995 the cost of an employee stock option on the income statement was its intrinsic value on the issue date: Accounting Principle Opinion No. 25 (APB 25): Intrinsic Value Method ATM options have no intrinsic value, the compensation cost is zero. 4

41 Accounting for ESOs After 1995 a fair value had to be reported in the notes (but expensing fair value on the income statement was optional): Financial Accounting Standard No. 123 (FAS 123): Fair Value Method Value ESO and compensation cost > 41

42 Accounting for ESOs Since 25 both FASB and IASB have required the fair value of options to be charged against income at the time of issue. 42

43 Nontraditional Plans The attraction of at-the-money call options used to be that they led to no expense on the income statement because they had zero intrinsic value on the exercise date. Other plans were liable to lead an expense. Now that the accounting rules have changed some companies are considering other types of plans. 43

44 Possible Nontraditional Plans Strike price is linked to stock index so that the company s stock price has to outperform the index for options to move in the money. Strike price increases in a predetermined way. Options vest only if specified profit targets are met. 44

45 4. Valuation of ECOs 45

46 Valuation of ESOs Use Black-Scholes with time to maturity equal to an estimate of expected life (See Example 14.1). Use a more sophisticated approach involving binomial trees. 46

47 Difficulties when Valuing ESOs with Black-Scholes ESOs cannot be sold, and typically, they cannot be exercised for several years. The employee may resign, be fired, or die. Any of these situations may affect the value of the stock option, either by forcing early exercise or by requiring that the options be forfeited. 47

48 Difficulties when Valuing ESOs with Black-Scholes The company may reduce the strike price if the stock price drops, a practice known as repricing. Between July and December 21, the price of ORACLE fell from $14 to $7. On December 12, the Board of Directors lowered the strike price on a number of ESOs in an effort to retain employees at a time when a significant percentage of ESOs were far OTM. 48

49 Difficulties when Valuing ESOs with Black-Scholes Unusual contractual features of the ESO. For example, an industry index may be the strike price. In their June 23 proxy statement, Level 2 Communications described its Outperform Stock Options (OSO) granted to employees. Participants in the OSO program do not realize any value unless the stock price outperforms the S&P5 stock index. When OSO are exercised, the payoff is based on a multiplier related to how much the stock outperforms the market index. 49

50 Difficulties when Valuing ESOs with Black-Scholes The term of the options can be 1 years or more, which makes volatility and dividend estimation difficult. The company may not have a publicly traded stock, in which case the stock price may not even be known. 5

51 Dilution Effects Employee stock options are liable to dilute the interests of shareholders because new shares are bought at below market price. However this dilution takes place at the time the market hears that the options have been granted (i.e. dilution is anticipated in efficient markets). It does not take place at the time the options are exercised. 51

52 5. Backdating Scandals 52

53 Backdating Scandals Backdating is the practice of marking a document with a date that precedes its inception date. Backdating appears to have been a widespread practice in the United States. A company might take the decision to issue atthe-money options on April 3 when the stock price is $5 and then backdate the grant date to April 3 when the stock price is $42. Why would they do this? 53

54 Backdating Scandals See Eric Lie s web site: 54

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