Currency and Interest Rate Options

Tuesdays 6:10-9:00 p.m. Commerce 260306 Wednesdays 9:10 a.m.-12 noon Commerce 260508 Handout #15 Derivative Security Markets Currency and Interest Rate Options Course web pages: http://finance2010.pageout.net ID: California2010 Password: bluesky ID: Oregon2010 Password: greenland

Reading Assignments for this Week Scan Read Levich Chap 12 Pages Currency and Interest Rate Options Luenberger Chap Pages Solnik Chap 10 Pages 433-483 Fabozzi M. Brandt Chap 27 Pages Interest Rate Options Interest Rate Models Pages 12-2

Midterm Exam: See University Calendar (November 16-20, 2009) Coverage: Chapters 3, 4, 5, 6, 7, 8, 9, 10 + Ben Bernanke s semi-annual testimony It s a closed-book exam. However, a two-sided formula sheet (11 x 8.5) is required; calculator/dictionary is okay; notebook is NOT okay. 75 minutes, 7 questions, 100 points total; five questions require calculation and two questions require (short) essay writing. 12-3

Final Exam See University Calendar (January 8-14, 2010) A Three-hour Exam Open-Book, Open Notes 12-4

Derivative Security Markets Currency and Interest Rate Options MS&E 247S International Investments Yee-Tien Fu

A First Example of No Arbitrage 1 Consider a forward contract that obliges us to hand over an amount $F at time T to receive the underlying asset. Today s date is t and the price of the asset is currently $S(t) (this is the spot price). When we get to maturity, we will hand over the amount $F and receive the asset, then worth $S(T). Is there any relationship among F, S(t), t and T? Derivatives: The Theory and Practice of Financial Engineering. Paul Wilmott. John Wiley & Sons 1998 12-6

A First Example of No Arbitrage 1 Enter into the forward contract, and sell the asset simultaneously. Selling something one does not own is called going short. Put the cash (S(t)) in the bank to get interest. At maturity, we hand over F to receive the asset, hence canceling our short asset position. Our net position is therefore S(t)e r(t-t) F or S(t) (1+r) T-t F F = S(t)e r(t-t) or F = S(t) (1+r) T-t 12-7

A First Example of No Arbitrage 1 Cashflows in a hedged portfolio of asset and forward. Holding Forward -Stock Cash Total Worth today (t) 0 - S(t) S(t) 0 Worth at maturity (T) S(T ) - F - S(T ) S(t)e r(t-t) S(t)e r(t-t) - F 12-8

The Building Blocks of Contingent Decisions (a) Long a bond (invest in a zero-coupon bond) Payoff at maturity + 0 (b) Short a bond (issue a zero-coupon bond) Payoff at maturity 0 12-9

The Building Blocks of Contingent Decisions (c) Purchase of Right to Buy at a Fixed Price Option Payoff (d) Purchase of Right to Sell at a Fixed Price Value of Underlying Asset at Decision Date (e) Sell Right to Buy at a Fixed Price (f) Sell Right to Sell at a Fixed Price $0 $0 Real Options: Amram & Kulatilaka Figure 4.1 12-10

The Building Blocks of Contingent Decisions The top row shows the payoffs from holding an asset that confers the right (but not the obligation) to buy or sell at a fixed price. The payoffs in the bottom row are the mirror images and show the position of the party on the other side of the transaction. Real Options: Amram & Kulatilaka Figure 4.1 12-11

The Building Blocks of Noncontingent Decisions (g) Forward Purchase (long position) (h) Forward Sale (short position) Payoff $0 $0 Value of Underlying Asset at Decision Date A forward contract is the right to buy or sell an asset at a specified date in the future at a specified price. The payoffs to a forward are not contingent on a future decision (hence there is no kink) but do depend on the realized value of an uncertain asset (the line is sloped). Real Options: Amram & Kulatilaka Figure 4.2 12-12

Identifying the Building Blocks in Red & White s Contract Suppose Red & White, a soft drink manufacturer, was offered a new sugar contract with a price floor of 90 cents per pound and a price cap of $1.10 per pound. Using the building-block approach, the contract can be seen as the sum of three parts: 1. Buy sugar on the world spot market. 2. Sell a put option with a strike price of $1.10. 3. Buy a call option with a strike price of 90 cents. Real Options: Amram & Kulatilaka Figure 4.3 12-13

Identifying the Building Blocks in Red & White s Contract Price of Sugar in World Spot Market (per pound) (b) Buy on Spot Market (a) The Payoff to a Contract with a Price Cap and a Price Floor Monthly Contract Payoffs $0.90 $1.10 = + (c) Sell Put Option + $1.10 (d) Buy Call Option Real Options: Amram & Kulatilaka Figure 4.3 $0.90 12-14

Identifying the Building Blocks in Red & White s Contract a This contract requires Red & White to pay a minimum of 90 cents per pound and a maximum of $1.10 per pound. The building-block approach shows that the contract is equivalent to b purchases in the world spot market, plus c a put option with an exercise price at $1.10 per pound, plus d a call option with an exercise price at 90 cents per pound. Real Options: Amram & Kulatilaka Figure 4.3 12-15

Introduction to Options Currency options began trading on the Philadelphia Stock Exchange in 1982, while interest rate options began trading on the Chicago Mercantile Exchange in 1985. Since then, there has been expansion in many directions : more option exchanges around the world, more currencies and debt instruments on which options are traded, option contracts with longer maturities, more styles of option contracts, and greater volume of trading activity. 12-16

Types of Contracts An option is the right, but not the obligation, to buy (or to sell) a fixed quantity of an underlying financial asset or commodity at a given price on (or on or before) a specified date. A call option bestows on the owner the right, but not the obligation, to buy the underlying financial asset or commodity. A put option, conveys to the owner the right, but not the obligation, to sell the underlying financial asset or commodity. 12-17

Types of Contracts With currency options, there are two choices of the underlying financial asset : An option on spot takes spot foreign exchange as the underlying asset. Spot foreign exchange is transferred if such an option is exercised. If an option on futures is exercised, positions in the underlying currency futures contract are created: a long position (at the strike price) for the holder of a call or seller of a put, and a short position (at the strike price) for the holder of a put or seller of a call. 12-18

Types of Contracts The owner of a futures call has the right to buy a futures, so the seller of a futures call must deliver a long position if the call is exercised (i.e., the seller takes a short position). The owner of a put has the right to sell a futures, so the seller of a futures put must deliver a short position (i.e., the seller takes a long position). Because futures contracts have zero-net supply, the underlying positions must be created before they can be delivered. 12-19

Types of Contracts With interest rate options, the underlying contract is typically an interest rate futures. The interest rate can be taken from a 3-month Eurodollar deposit, a long-term Treasury bond, or some other interest-bearing instrument. The price specified in the contract is called the strike price or the exercise price. The maturity date specified in the option is called the expiration date. The price paid for the option is called the option premium. 12-20

Examples Types of Contracts An American call option on spot DM : The right to buy DM 1 million for $0.63 per DM from today until expiration on Dec 15, 1999. A European put option on Swiss franc futures : The right to sell SFr 10 million March 1999 futures for $0.76 per SFr on (and only on) Mar 15, 1999. Note that US$ is used as the second currency or numeraire currency in both the above examples. 12-21

Types of Contracts With the rise of significant trading volume in cross-exchange rates - notably the /, DM/, DM/, and DM against other European currency rates - options written in currency pairs that do not involve the US$ have been introduced. These cross-rate options gained further popularity after 1991 as exchange rate volatility among European currencies increased. 12-22

Location of Trading Currency options are traded either among banks on an over-the-counter (OTC) basis or on organized futures and options exchanges. Over-the-counter (OTC) market -- An informal network of brokers and dealers who negotiate sales of securities. 12-23

Contract Specifications OPTIONS PHILADELPHIA EXCHANGE Calls Puts Vol. Last Vol. Last German Mark 63.19 62,500 German Marks EOM-cents per unit. 54 Aug 6350 0.47 61 Jul 26 2.52 61 Aug 6350 0.38 63½ Jul 2 0.88 4 1.03 62,500 German Marks-European Style. 61½ Jul 1350 0.19 61½ Sep 340 0.70 63 Jul 57 0.97 63 Aug 10 1.25 10 0.95 62,500 German Marks-cents per unit. 57 Sep 10 0.06 58 Sep 20 5.56 61½ Aug 80 2.10 25 0.41 63 Aug 65 1.17 This newspaper extract reports prices of Philadelphia Stock Exchange options on spot currency. For the German mark, there are three sections of figures. Prices on end-of-month (EOM) options that mature on the last Friday of the given month. Prices of European-style German mark options. Prices of American-style German mark options. 12-24

Contract Specifications OPTIONS PHILADELPHIA EXCHANGE Calls Puts Vol. Last Vol. Last German Mark 63.19 62,500 German Marks EOM-cents per unit. 54 Aug 6350 0.47 61 Jul 26 2.52 61 Aug 6350 0.38 63½ Jul 2 0.88 4 1.03 62,500 German Marks-European Style. 61½ Jul 1350 0.19 61½ Sep 340 0.70 63 Jul 57 0.97 63 Aug 10 1.25 10 0.95 62,500 German Marks-cents per unit. 57 Sep 10 0.06 58 Sep 20 5.56 61½ Aug 80 2.10 25 0.41 63 Aug 65 1.17 The closing spot price of the DM ($0.6319/DM) is listed across from the word German Mark. The strike prices (in US cents per DM). The German mark contract size (62,500 DM) is written at the top of each section. Consider this call option. The closing or settlement price was reported as 1.17 cents per DM. Since the contract size is 62,500 DM, the buyer of this call option would expect to pay 62,500x$0.0117=$731.25 plus commission charges. 12-25

Spot Currency Options: Prices at Maturity Consider the following call and put options on the : Each option has a strike price (K) of $1.50/ and an option premium of about $0.10. If we designate C and P as the values of the call and put prices, then at maturity: C = Max [0, S-K ] and P = Max [0, K-S ] (12.1) where S is the spot price of the at the maturity of the option. It is clear from the above equations that the value of an option can never be negative. 12-26

Buyer and Seller of a Spot Currency Call on with Strike at $1.50/ Profit or Loss in $ 0.15 0.10 0.05 0.00-0.05-0.10 Profit and Loss Positions Seller of a Call Buyer of a Call -0.15 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 Figure 12.4A $/ 12-27

Buyer and Seller of a Spot Currency Call on with Strike at $1.50/ For spot prices $1.50, the expiration value of the call is 0. The buyer suffers a loss - the $0.10 premium he originally paid for the call. At spot prices $1.50, the expiration value of the option is positive (= S - $1.50), which reduces the call buyer s losses. The breakeven exchange rate is $1.60. The seller keeps the premium when the spot rate $1.50, because the call is not exercised. At spot rates $1.50, the opportunity cost for the seller is. 12-28

Buyer and Seller of a Spot Currency Put on with Strike at $1.50/ Profit or Loss in $ 0.15 0.10 0.05 0.00-0.05-0.10 Profit and Loss Positions Seller of a Put Buyer of a Put -0.15 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 Figure 12.4B $/ 12-29

Buyer and Seller of a Spot Currency Put on with Strike at $1.50/ For spot prices $1.50, the expiration value of the put is 0. The buyer suffers a loss of $0.10. At spot prices $1.50, the expiration value of the put is positive (= $1.50 - S) which reduces the put buyer s losses. The break-even exchange rate is $1.40. The seller earns the option premium when the spot rate $1.50, because the put is not exercised. At spot rates $1.50, the seller s opportunity cost becomes. 12-30

Spot Currency Options: Prices at Maturity Note that the buyer of either a put or call faces limited liability in that her loss is capped at the initial option premium paid. The seller of a call faces unlimited liability as the underlying asset could appreciate without limit. The seller of a put faces a large liability, which is limited by the fact that the price of the underlying cannot fall below zero. 12-31

Interest Rate Futures Options: Prices at Maturity Consider the following call and put options on a Eurodollar interest rate futures contract : Each option has a strike price (K) of 96.00 (corresponding to a 4.00 % Euro$ interest rate) and an option premium of about 0.25. If we designate C f and P f as the values of the call and put prices, then at maturity: C f = Max [0, F-K ] and P f = Max [0, K-F ] (12.2) where F is the price of the underlying Eurodollar futures contract on the maturity of the option. 12-32

Euro-$ Interest Rate Call with Strike at 96.00 Profit or Loss in % Figure 12.5A 2.0 1.5 1.0 0.5 0.0-0.5-1.0 Profit and Loss Positions Seller of a Call Buyer of a Call -1.5-2.0 94.00 94.50 95.00 95.50 96.00 96.50 97.00 97.50 98.00 Interest Rate Futures Price at Maturity 12-33

Euro-$ Interest Rate Call with Strike at 96.00 Note the inverse relationship between interest rates and futures prices. For futures prices 96.00 (interest rates 4.00%), the expiration value of the call is 0. The buyer suffers a loss equal to the original premium paid for the option (25 basis points x $25/b.p. = $625). If the price at maturity 96.00, the expiration value is positive (= F - 96.00 basis points), which reduces the call buyer s losses. The break-even Euro$ futures rate is 96.25 (3.75% interest rate). 12-34

Euro-$ Interest Rate Put with Strike at 96.00 Profit or Loss in % Figure 12.5B 2.0 1.5 1.0 0.5 0.0-0.5-1.0 Profit and Loss Positions Seller of a Put Buyer of a Put -1.5-2.0 94.00 94.50 95.00 95.50 96.00 96.50 97.00 97.50 98.00 Interest Rate Futures Price at Maturity 12-35

Euro-$ Interest Rate Put with Strike at 96.00 For futures prices 96.00 (interest rates 4.00%), the expiration value of the put is zero. The buyer incurs a loss equal to the original premium paid for the option. If the Euro$ futures price at maturity 96.00, the expiration value of the put is positive (equal to 96.00 - F basis points), which reduces the put buyer s losses. The break-even Euro$ futures rate is 95.75 (4.25% interest rate ). If interest rates rise above 4.25%, the put buyer earns a profit. 12-36

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Circumstance Strategy Very bullish Buy call Slightly bullish Write put Slightly bearish Write call Very bearish Buy put 12-40

Google Earnings Gave Options Traders a 17,530% Gain (Update1) By Michael Patterson and Jeff Kearns April 18, 2008 (Bloomberg) -- Options traders who predicted Google Inc. would beat estimates earned as much as 17,530 percent on their investments today, the most-profitable bet among all U.S. equity derivatives. Contracts giving the right to buy Google shares for $530 before the close of trading today jumped as high as $17.63 from their 10-cent closing price yesterday. That gain almost matched the 18,760 percent advance in the Dow Jones Industrial Average since the beginning of 1900, according to Bloomberg data. 12-41

``Today in Google you see the power of leverage in options, especially going into earnings,'' said Peter Bottini, executive vice president of trading at OptionsXpress Holdings Inc., a Chicago-based online brokerage. ``We were swamped with customers who were calling in at the open of the market.'' Google shares climbed 20 percent, the most since its initial public offering in 2004, to $539.41 after the owner of the most popular Internet search engine beat the average analyst profit estimate by 7.1 percent. Google had dropped 35 percent in 2008 on concern the U.S. economic slump would hurt spending on online advertising. Google call-option volume jumped to 311,139 contracts, the most since January 2006. Those contracts outnumbered trading in bearish bets, or puts, by 1.6-to-1. Call options give the right to buy a security for a certain amount, called the strike price, by a given date. Puts convey the right to sell. 12-42

A Short Straddle in the $/ : betting on convergence 0.10 0.05 Sell Call Profit and Loss Positions Sell Put Profit or Loss in $ 0.00-0.05-0.10-0.15 Straddle Sell Put Sell Call Straddle -0.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 Box 12.1 Figure A $/ 12-43

A Long Straddle in the $/ : betting on divergence 0.20 0.15 Buy Put Profit and Loss Positions Buy Call Profit or Loss in $ 0.10 0.05 0.00-0.05 Straddle Buy Call -0.10 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 Straddle Buy Put Box 12.1 Figure B $/ 12-44

Using a Call to Hedge a Short Pound: hedge an A/P Profit or Loss in $ 0.3 0.2 0.1 0-0.1-0.2 Combined Position Buy Call Profit and Loss Positions Short Pound Buy Call Short Pound -0.3 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 Box 12.1 Figure C $/ 12-45

Using a Put to Hedge a Long Pound: hedge an A/R 0.3 Profit and Loss Positions 0.2 Buy Put Long Pound Profit or Loss in $ 0.1 0-0.1-0.2 Long Pound Buy Put Combined Position -0.3 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 Box 12.1 Figure D $/ 12-46

Combining a Floor and Cap : A Collar Beginning cash outlay is minimized, customized hedge of an A/R 0.3 0.2 Profit and Loss Positions at Maturity Long Pound Profit or Loss in $ 0.1 0-0.1-0.2 Buy a Put Sell a Call A Collar : Sell Call + Buy Put -0.3 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 Box 12.1 Figure E $/ 12-47

Protective Put Strategy (Payoffs) Value at expiry Protective Put payoffs Portfolio value today = p 0 + S 0 $50 Buy the stock Buy a put with an exercise price of $50 $0 $50 Value of stock at expiry 12-48

Protective Put Strategy (Profits) Value at expiry $40 Buy the stock at $40 Protective Put strategy has downside protection and upside potential $0 -$10 -$40 $40 $50 Buy a put with exercise price of $50 for $10 Value of stock at expiry 12-49

Covered Call Strategy Value at expiry Buy the stock at $40 $10 Covered Call strategy $0 Value of stock at expiry -$30 $40 $50 Sell a call with exercise price of $50 for $10 -$40 12-50

Use of a Butterfly Spread From the Trader s Desk A stock is currently selling for $61. The prices of call options expiring in 6 months are quoted as follows: Strike price = $55, call price = $10 Strike price = $60, call price = $7 Strike price = $65, call price = $5 An investor feels it is unlikely that the stock price will move significantly in the next 6 months. The Strategy The investor sets up a butterfly spread: 1. Buy one call with a $55 strike. 2. Buy one call with a $65 strike. 3. Sell two calls with a $60 strike. The cost is $10 + $5 - (2 x $7) = $1. The strategy leads to a net loss (maximum $1) if the stock price moves outside the $56-to-$64 range but leads to a profit if it stays within this range. The maximum profit of $4 is realized if the stock price is $60 on the expiration date. 12-51

Payoff from a Butterfly Spread Stock price Payoff from Payoff from Payoff from Total range first long call second long call short calls payoff * S T < X 1 0 0 0 0 X 1 < S T < X 2 S T X 1 0 0 S T X 1 X 2 < S T < X 3 S T X 1 0 2(S T X 2 ) X 3 S T S T > X 3 S T X 1 S T X 3 2(S T X 2 ) 0 * These payoffs are calculated using the relationship X 2 = 0.5(X 1 + X 3 ). 12-52

Butterfly Spread Using Call Options Profit X 1 X 2 X 3 S T 12-53

Butterfly Spread Using Put Options Profit X 1 X 2 X 3 S T 12-54

Short Straddle Option payoffs ($) 20 This Short Straddle only loses money if the stock price moves $20 away from $50 (diverges). 30 40 60 70 $50 Sell a put with exercise price of $50 for $10 Stock price ($) 30 40 Sell a call with an exercise price of $50 for $10 12-55

What are the similarities and differences between the payoffs of the butterfly spread and the short straddle? 12-56

Option Pricing : Formal Models Binomial Currency Option Example Period 1 Period 2 Spot Rates Option Values S 1 = 1.40 S 2,u = 1.5015 Call 2,u = 0.1015 Put 2,u = 0.0 S 2,d = 1.2293 Call 2,d = 0.0 Put 2,d = 0.1707 Figure 12.8 12-57

Option Pricing : Formal Models Binomial Currency Option Example : Multiple Periods to Expiration Period 1 Period 2 Period 3 Period 4 Period 5 S 5,5 = 1.6594 S 5,4 = 1.5015 S 5,3 = 1.3586 S 5,2 = 1.2293 S 5,1 = 1.1123 Figure 12.9 12-58

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Option Pricing : Formal Models Marginal Effect of a Parameter Change on Option Prices Variable Price Effect on Call Option Spot Price (S ) Call Price Exercise price (K ) Call Price Domestic interest rate (r d ) Call Price Foreign interest rate (r f ) Call Price Spot rate volatility ( ) Call Price Time to maturity (t ) Ambiguous effect, depends on r d, r f, and Price Effect on Put Option Put Price Put Price Put Price Put Price Put Price Ambiguous effect, depends on r d, r f, and Table 12.6 12-60

Chapter 11 Hedging and Insuring 11.9 Options as Insurance Options are another ubiquitous form of insurance contract. Reference: Finance (First Edition) by Zvi Bodie and Robert C. Merton 1999, ISBN 0-13-310897-X, Prentice Hall 12-61

Put-Call Parity: p 0 + S 0 = c 0 + E/(1+ r) T Portfolio value today = c 0 + (1+ r) T E Portfolio payoff a+c Option payoffs ($) 25 Call bond c a 25 Stock price ($) Consider the payoffs from holding a portfolio consisting of a call with a strike price of $25 and a bond with a future value of $25. 12-62

Put-Call Parity: p 0 + S 0 = c 0 + E/(1+ r) T Portfolio value today = p 0 + S 0 Portfolio payoff d+ g Option payoffs ($) 25 25 Stock price ($) Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike. 12-63

Put-Call Parity: p 0 + S 0 = c 0 + E/(1+ r) T Option payoffs ($) Portfolio value today E = c 0 + (1+ r) T Option payoffs ($) Portfolio value today = p 0 + S 0 25 25 25 Stock price ($) Since these portfolios have identical payoffs, they must have the same value today: hence 25 Stock price ($) Put-Call Parity: c 0 + E/(1+r) T = p 0 + S 0 12-64

Stocks and Bonds as Options It all comes down to put-call parity. c 0 = S 0 + p 0 E (1+ r) T Value of a call on the firm Value of a put on the firm Value of = the firm + Value of a risk-free bond Stockholder s position in terms of call options Stockholder s position in terms of put options 12-65

Observe: Buy Call and sell Put, you obtain a Forward. 12-66

What did IRP say about Forward??? Yes! (in terms of cost concept) F 1,T = borrow in local & pay r d invest in foreign currency & receive r f = S 1 exp[(r d -r f )T] 12-67

In terms of net cash inflow, buying a call and selling a put nets C + P A synthetic foreign currency forward is borrowing local currency, Exchange for foreign currency, and lend foreign currency till maturity. Or, in terms of net cash inflow, K(1+r d ) T S(1+r f ) T, or in continuous term K exp(-r d T) S 1 exp(-r f T). 12-68

A synthetic foreign currency forward is borrowing local currency, Exchange for foreign currency, and lend foreign currency till maturity. Or, in terms of net cash inflow, K(1+r d ) T S(1+r f ) T, or in continuous term K exp(-r d T) S 1 exp(-r f T) from leaving no room for arbitrage. Now equate C + P = K exp(-r d T) S 1 exp(-r f T). What did IRP say about Forward??? Yes! (in terms of cost concept) F 1,T = borrow in local & pay r d invest in foreign currency & receive r f = S 1 exp[(r d -r f )T] 12-69

Now maneuver to equate C + P = K exp(-r d T) S 1 exp(-r f T) C P = S 1 exp(-r f T) - K exp(-r d T) ={exp(r d T) [S 1 exp(-r f T) - K exp(-r d T)]}/exp(r d T) = [S 1 exp(r d -r f )T K]/exp(r d T) = (F 1,T K)/exp(r d T) 12-70

Leaving No Room for Arbitrage argument! Organic: Exchange rate S 1 evolves to S 2!!! 12-71

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Assignments from Chapter 12 Exercises 5, 6, 7, 8. 12-73

PUT-CALL-FORWARD PARITY 12.5. Using the Put-Call-Forward Parity, demonstrate that the price of a call with the strike price equal to the futures price is equal to the price of a put with the same strike price and the same maturity. HINTS: Put-Call-Forward Parity tells us that: C - P = (F 0,T - X) / exp(r d T) If the strike price equals the futures price, or X = F 0,T, then C - P = 0, or C = P. Therefore, for a strike price equals to the futures price, the price of the call and the price of the put are equal. 12-74

12.6. Suppose call options on the DM with a strike price of $0.63/DM and maturity of one month are traded at $0.01/DM. One-month futures on the DM are traded at $0.624/DM. One-month US Treasury Bills yield 5.5%. One-month German government securities yield 7.5%. The spot $/DM exchange rate is $0.625/DM. a. Using the Put-Call-Forward Parity, determine the value of the put option with a strike price of $0.63 and one month maturity. b. How would you take advantage of arbitrage opportunities if you find that the actual price of the put is below the theoretical price determined using the Parity condition? HINTS: P = C - (F 0,T -X)/ exp(r d T) C =.010 X =.63 F 0,T =.624 r d = 5.5% T = 1/12 P =.010 - (.624 -.63) / exp(.055* 1 / 12) P =.016 b. I believe in my model price (i.e., theoretical price), so I ll buy from the market if the market price is lower than what my model suggest and I will sell C, buy F, and sell bond to capitalize the difference in model price and market price. 12-75

b. How would you take advantage of arbitrage opportunities if you find that the actual price of the put is below the theoretical price determined using the Parity condition? HINTS: P = C - (F 0,T -X) / exp(r d T) C =.010 X =.63 F 0,T =.624 r d = 5.5% T = 1/12 P =.010 - (.624 -.63) / exp(.055* 1 / 12) P =.016 I believe in my model price (i.e., theoretical price), so I ll buy from the market if the market price is lower than what my model suggest and I will sell C, buy F, and sell bond to capitalize the difference in model price and market price. 12-76

HEDGING 12.7. The treasurer of the XYZ company is expecting a dividend payment of DM 10,000,000 from a German subsidiary in two months. His/her expectations of the future DM spot rate are mixed: The DM could strengthen or stay flat over the next two months. The current exchange rate is $0.63/DM. The two-month futures rate is at $0.6279/DM. The two-month German interest rate is 7.5%. The two-month US T-Bill yields 5.5%. Puts on the DM with maturity of two months and strike price of $0.63 are traded on the CME at $0.0128/DM. Compare the following choices offered to the Treasurer: Sell a futures on the DM for delivery in two months for a total amount of DM 10 million. Buy 80 put options on the CME with expiration in two months and strike price equal to the current price. Set up a forward contract with the firm's bank XYZ. a. What is the respective cost of each strategy? b. Which strategy would best fit the treasurer's mixed forecast for the future spot rate of the DM? 12-77

HINTS: a. Strategy One: Sell futures contracts at the current price of.6279. In this case, the US firm is assured to get $ 6,279,000 (DM 10,000,000 * $.6279/DM). However, it has to deposit the margin requirements and risks having to make payments to maintain the maintenance margin, introducing cash-flow issues in the future. b. Strategy Two: Buy 80 put options on the DM at a strike price of 63. Total cost: 80 * 125,000 * $0.0128 = $128,000. The firm is assured to get at least $6,172,000 but it reserves itself the right to sell the DM on the spot market and not to exercise its options. Break-even rate is $.6428/DM. If the DM falls below that rate, the firm will be able to sell on the spot market and get more for its DM. 12-78

12.8. Refer to the previous question. Suppose the DM actually rose in value to $0.67/DM when the dividend payment is made. a. Which of the three strategies enables the treasurer to take advantage of the rise in the DM against the dollar? b. What is the final gain (loss) incurred in each case? 12-79

HINTS: a. Strategy Three: Set up a forward contract with bank XYZ. No cash-flows are involved until maturity. However, the firm needs a credit line with its bank. The only strategy that allows for a potential future gain from a falling DM is the second strategy: b. If the DM rises to 67, the firm can forego its put option at 63 and actually sell its DM on the spot market. The firm loses the initial $128,000 spent on the put option. It gets $6,700,000 for its DM. Total cash proceeds from the dividend payment: $6,700,000-128,000 = $6,572,000. 12-80

A Real-World Way to Manage Real Options By Tom Copeland and Peter Tufano Harvard Business Review March 2004 12-81

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1,000 x 1.2 = 1,200, 1,000 / 1.2 = 833, etc. 12-83

Copano s Decision Tree 12-84

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Two Years Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by M (1,440) - B : Contingent claims of a synthesized / replicated call option: M (1,728) (1 +.08) (B) = 928 M (1,200) (1 +.08) (B) = 400 M=1, B=741 Asset value at current time: 1 * 1,440 741 = 699 12-88

Two Years Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by M (1,000) - B : Contingent claims of a synthesized / replicated call option: M (1,200) (1 +.08) (B) = 400 M (833) (1 +.08) (B) = 33 M=1, B=741 Asset value at current time: 1 * 1,000 741 = 259 12-89

Two Years Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by M (694) - B : Contingent claims of a synthesized / replicated call option: M (833) (1 +.08) (B) = 33 M (579) (1 +.08) (B) = 0 M=0.13, B=70 Asset value at current time: 0.13 * 694 70 = 20 12-90

One Year Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by M (1,200) - B : Contingent claims of a synthesized / replicated call option: M (1,400) (1 +.08) (B) = 699 M (1,000) (1 +.08) (B) = 259 M=1, B=686 Asset value at current time: 1 * 1,200 686 = 514 12-91

One Year Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by M (833) - B : Contingent claims of a synthesized / replicated call option: M (1,000) (1 +.08) (B) = 259 M (694) (1 +.08) (B) = 21 M=.777, B=479 Asset value at current time:.777 * 833 479 = 169 12-92

Today Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by M (1,000) - B : Contingent claims of a synthesized / replicated call option: M (1,200) (1 +.08) (B) = 114 M (833) (1 +.08) (B) = 0 M=.31, B=239 Asset value at current time:.31 * 1,000 239 = 71 12-93

Real Option Calculation By YTF M (1,728) (1 +.08) (B) = 928 M (1,200) (1 +.08) (B) = 400 M=1, B=741 1 * 1,440 741 = 699 M (1,200) (1 +.08) (B) = 400 M (833) (1 +.08) (B) = 33 M = 1, B=741 1 * 1,000 741 = 259 M (833) (1 +.08) (B) = 33 M (579) (1 +.08) (B) = 0 M = 0.13, B = 70 0.13 * 694 70 = 20 M (1,440) (1 +.08) (B) = 699 M (1,000) (1 +.08) (B) = 259 M = 1, B = 686 1 * 1,200 686 = 514 M (1,000) (1 +.08) (B) = 259 M (694) (1 +.08) (B) = 21 M =.777, B = 479.777 * 833 479 = 169 M (1,200) (1 +.08) (B) = 114 M (833) (1 +.08) (B) = 0 M =.31, B = 239.31 * 1,000 239 = 71 From Event Tree Strategic Decision with flexibility 12-94

Cases for MS&E 247S International Investments Summer 2006 Topic 8: International Financial Innovations Case Deutsche Bank: Discussing the Equity Risk Premium. Case Swedish Lottery Bonds. Case Bank Leu s Prima Cat Bond Fond. Case Catastrophe Bonds at Swiss Re. Case Mortgage Backs at Ticonderoga. Case KAMCO and the Cross-Border Securitization of Korean Non-Performing Loans. Case Nexgen: Structuring Collateralized Debt Obligations (CDOs). Case The Enron Odyssey (A):The Special Purpose of SPEs. 12-95

Description of Pine Street Capital: A technology hedge fund is trying to decide whether and/or how to hedge equity market risk. Its hedging choices are short-selling and options. The fund has just gone through one of the most volatile periods in NASDAQ's history, it is trying to decide whether it should continue its risk management program of short-selling the NASDAQ index or switch to a hedging program utilizing put options on the index. Learning Objective: To introduce students to the use of equity derivatives in the context of money management, particularly the use of deltahedging using options. Also, to learn about how and why risk management decisions are made in a simple levered hedge fund. 12-96

Subjects Covered: Derivatives, Financial strategy, Hedging, Investment management, Options, Risk management. Setting: San Francisco, CA; Financial services; $1.3 million revenues; 4 employees; 2000 12-97

Google Earnings Gave Options Traders a 17,530% Gain By Michael Patterson and Jeff Kearns April 18 (Bloomberg) -- Options traders who predicted Google Inc. would beat estimates earned as much as 17,530 percent on their investments today, the most-profitable bet among all U.S. equity derivatives. Contracts giving the right to buy Google shares for $530 before the close of trading today jumped as high as $17.63 from their 10-cent closing price yesterday. That gain almost matched the 18,760 percent advance in the Dow Jones Industrial Average since the beginning of 1900, according to Bloomberg data. Google call-option volume jumped to 311,139 contracts, the most since January 2006. Those contracts outnumbered trading in bearish bets, or puts, by 1.6-to-1. Call options give the right to buy a security for a certain amount, called the strike price, by a given date. Puts convey the right to sell. 12-98

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Review Ross Corporate Finance 7E Chapter 17 Options 12-101

Key Concepts and Skills Understand option terminology Be able to determine option payoffs and profits Understand the major determinants of option prices Understand and apply put-call parity Be able to determine option prices using the binomial and Black-Scholes models 12-102

Chapter Outline 17.1 Options 17.2 Call Options 17.3 Put Options 17.4 Selling Options 17.5 Option Quotes 17.6 Combinations of Options 17.7 Valuing Options 17.8 An Option Pricing Formula 17.9 Stocks and Bonds as Options 17.10 Options and Corporate Decisions: Some Applications 17.11 Investment in Real Projects and Options 12-103

17.1 Options An option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset on (or before) a given date, at prices agreed upon today. Exercising the Option The act of buying or selling the underlying asset Strike Price or Exercise Price Refers to the fixed price in the option contract at which the holder can buy or sell the underlying asset. Expiry (Expiration Date) The maturity date of the option 12-104

Options European versus American options European options can be exercised only at expiry. American options can be exercised at any time up to expiry. In-the-Money Exercising the option would result in a positive payoff. At-the-Money Exercising the option would result in a zero payoff (i.e., exercise price equal to spot price). Out-of-the-Money Exercising the option would result in a negative payoff. 12-105

17.2 Call Options Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today. When exercising a call option, you call in the asset. 12-106

Call Option Pricing at Expiry At expiry, an American call option is worth the same as a European option with the same characteristics. If the call is in-the-money, it is worth S T E. If the call is out-of-the-money, it is worthless: C = Max[S T E, 0] Where S T is the value of the stock at expiry (time T) E is the exercise price. C is the value of the call option at expiry 12-107

Call Option Payoffs Option payoffs ($) 60 40 20 Buy a call 20 40 60 80 100 50 120 Stock price ($) 20 40 Exercise price = $50 12-108

Call Option Profits Option payoffs ($) 60 40 20 10 Buy a call 10 20 20 40 50 60 80 100 120 Stock price ($) 40 Exercise price = $50; option premium = $10 12-109

17.3 Put Options Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today. When exercising a put, you put the asset to someone. 12-110

Put Option Pricing at Expiry At expiry, an American put option is worth the same as a European option with the same characteristics. If the put is in-the-money, it is worth E S T. If the put is out-of-the-money, it is worthless. P = Max[E S T, 0] 12-111

Put Option Payoffs 60 Option payoffs ($) 50 40 20 0 0 20 40 60 80 100 50 Buy a put Stock price ($) 20 40 Exercise price = $50 12-112

Put Option Profits 60 Option payoffs ($) 40 20 10 10 20 Stock price ($) 20 40 50 60 80 100 Buy a put Exercise price = $50; option premium = $10 40 12-113

Option Value Intrinsic Value Call: Max[S T E, 0] Put: Max[E S T, 0] Speculative Value The difference between the option premium and the intrinsic value of the option. Option Premium = Intrinsic Value + Speculative Value 12-114

17.4 Selling Options The seller (or writer) of an option has an obligation. The seller receives the option premium in exchange. 12-115

Call Option Payoffs 60 Option payoffs ($) 40 20 20 40 60 80 100 50 120 Stock price ($) 20 Exercise price = $50 Sell a call 40 12-116

Put Option Payoffs Option payoffs ($) 40 20 0 0 20 40 50 60 80 100 20 Sell a put Stock price ($) 40 Exercise price = $50 50 12-117

Option Diagrams Revisited Option payoffs ($) 40 10 Buy a put Sell a call Buy a call Sell a put 10 Buy a call 40 50 60 100 Stock price ($) Buy a put 40 Sell a put Exercise price = $50; option premium = $10 Sell a call 12-118

17.5 Option Quotes --Call-- --Put-- Option/Strike Exp. Vol. Last Vol. Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼ 138¼ 135 Jul 2365 4¾ 2431 13/16 138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½ 12-119

Option Quotes This option has a strike price of $135; --Call-- --Put-- Option/Strike Exp. Vol. Last Vol. Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼ 138¼ 135 Jul 2365 4¾ 2431 13/16 138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½ a recent price for the stock is $138.25; July is the expiration month. 12-120

Option Quotes This makes a call option with this exercise price in-themoney by $3.25 = $138¼ $135. --Call-- --Put-- Option/Strike Exp. Vol. Last Vol. Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼ 138¼ 135 Jul 2365 4¾ 2431 13/16 138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½ Puts with this exercise price are out-of-the-money. 12-121

Option Quotes --Call-- --Put-- Option/Strike Exp. Vol. Last Vol. Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼ 138¼ 135 Jul 2365 4¾ 2431 13/16 138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½ On this day, 2,365 call options with this exercise price were traded. 12-122

Option Quotes The CALL option with a strike price of $135 is trading for $4.75. --Call-- --Put-- Option/Strike Exp. Vol. Last Vol. Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼ 138¼ 135 Jul 2365 4¾ 2431 13/16 138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½ Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions. 12-123

Option Quotes --Call-- --Put-- Option/Strike Exp. Vol. Last Vol. Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼ 138¼ 135 Jul 2365 4¾ 2431 13/16 138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½ On this day, 2,431 put options with this exercise price were traded. 12-124

Option Quotes The PUT option with a strike price of $135 is trading for $.8125. --Call-- --Put-- Option/Strike Exp. Vol. Last Vol. Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼ 138¼ 135 Jul 2365 4¾ 2431 13/16 138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½ Since the option is on 100 shares of stock, buying this option would cost $81.25 plus commissions. 12-125

17.6 Combinations of Options Puts and calls can serve as the building blocks for more complex option contracts. If you understand this, you can become a financial engineer, tailoring the riskreturn profile to meet your client s needs. 12-126

Long Straddle Option payoffs ($) 40 30 Buy a call with exercise price of $50 for $10 20 30 40 60 70 Stock price ($) Buy a put with exercise price of $50 for $10 $50 A Long Straddle only makes money if the stock price moves $20 away from $50. 12-127

Short Straddle Option payoffs ($) 20 This Short Straddle only loses money if the stock price moves $20 away from $50. 30 40 60 70 $50 Sell a put with exercise price of $50 for $10 Stock price ($) 30 40 Sell a call with an exercise price of $50 for $10 12-128

Put-Call Parity Option payoffs ($) Portfolio value today E = c 0 + (1+ r) T Option payoffs ($) Portfolio value today = p 0 + S 0 25 25 25 Stock price ($) Since these portfolios have identical payoffs, they must have the same value today: hence 25 Stock price ($) Put-Call Parity: c 0 + E/(1+r) T = p 0 + S 0 12-129

17.7 Valuing Options The last section concerned itself with the value of an option at expiry. This section considers the value of an option prior to the expiration date. A much more interesting question. 12-130

American Call Option payoffs ($) Profit 25 Market Value S T Time value Call Intrinsic value loss E Out-of-the-money In-the-money C 0 must fall within max (S 0 E, 0) < C 0 < S 0. S T 12-131

Option Value Determinants Call Put 1. Stock price + 2. Exercise price + 3. Interest rate + 4. Volatility in the stock price + + 5. Expiration date + + The value of a call option C 0 must fall within max (S 0 E, 0) < C 0 < S 0. The precise position will depend on these factors. 12-132

17.8 An Option Pricing Formula We will start with a binomial option pricing formula to build our intuition. Then we will graduate to the normal approximation to the binomial for some real-world option valuation. 12-133

Binomial Option Pricing Model Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S 0 = $25 today and in one year S 1 is either $28.75 or $21.25. The risk-free rate is 5%. What is the value of an at-the-money call option? S 0 S 1 $28.75 = $25 (1.15) $25 $21.25 = $25 (1.15) 12-134

Binomial Option Pricing Model 1. A call option on this stock with exercise price of $25 will have the following payoffs. 2. We can replicate the payoffs of the call option with a levered position in the stock. S 0 S 1 C 1 $28.75 $3.75 $25 $21.25 $0 12-135

Binomial Option Pricing Model Borrow the present value of $21.25 today and buy 1 share. The net payoff for this levered equity portfolio in one period is either $7.50 or $0. The levered equity portfolio has twice the option s payoff, so the portfolio is worth twice the call option value. S 0 ( debt) = S 1 $28.75 $21.25 = portfolio $7.50 C 1 $3.75 $25 $21.25 $21.25 = $0 $0 12-136

Binomial Option Pricing Model The value today of the levered equity portfolio is today s value of one share less the present value of a $21.25 debt: $25 $21.25 (1 r f ) S 0 ( debt) = S 1 $28.75 $21.25 = portfolio $7.50 C 1 $3.75 $25 $21.25 $21.25 = $0 $0 12-137

Binomial Option Pricing Model We can value the call option today as half of the value of the levered equity portfolio: C 0 1 2 $25 $21.25 (1 ) r f S 0 ( debt) = S 1 $28.75 $21.25 = portfolio $7.50 C 1 $3.75 $25 $21.25 $21.25 = $0 $0 12-138

Binomial Option Pricing Model If the interest rate is 5%, the call is worth: 1 $21.25 1 C0 $25 2 (1.05) 2 $25 20.24 $2. 38 C 0 S 0 ( debt ) = S 1 $28.75 $21.25 = portfolio $7.50 C 1 $3.75 $2.38 $25 $21.25 $21.25 = $0 $0 12-139

Binomial Option Pricing Model The most important lesson (so far) from the binomial option pricing model is: the replicating portfolio intuition. Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities. 12-140

Delta This practice of the construction of a riskless hedge is called delta hedging. The delta of a call option is positive. Recall from the example: Swing of call Swing of stock $3.75 0 $28.75 $21.25 $3.75 $7.5 1 2 The delta of a put option is negative. 12-141

Delta Determining the Amount of Borrowing: 1 $21.25 1 C0 $25 $25 $20.24 $2. 2 (1.05) 2 Value of a call = Stock price Delta Amount borrowed $2.38 = $25 ½ Amount borrowed Amount borrowed = $10.12 38 12-142

The Risk-Neutral Approach q S(U), V(U) S(0), V(0) 1- q S(D), V(D) We could value the option, V(0), as the value of the replicating portfolio. An equivalent method is risk-neutral valuation: V (0) q V ( U ) (1 q) V ( D) (1 ) r f 12-143

The Risk-Neutral Approach q S(0), V(0) 1- q S(0) is the value of the underlying asset today. S(U), V(U) q is the risk-neutral probability of an up move. S(D), V(D) S(U) and S(D) are the values of the asset in the next period following an up move and a down move, respectively. V(U) and V(D) are the values of the option in the next period following an up move and a down move, respectively. 12-144

The Risk-Neutral Approach S(0), V(0) q S(U), V(U) V (0) q V ( U ) (1 q) V ( D) (1 ) r f 1- q S(D), V(D) The key to finding q is to note that it is already impounded into an observable security price: the value of S(0): S(0) q S( U ) (1 q) S( D) (1 ) (1 A minor bit of algebra yields: q r f r f ) S(0) S( D) S( U ) S( D) 12-145

Example of Risk-Neutral Valuation 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%. What is the value of an at-the-money call option? The binomial tree would look like this: $ 28.75 $25 (1.15) q $28.75,C(U) $25,C(0) $ 21.25 $25 (1.15) 1- q $21.25,C(D) 12-146

Example of Risk-Neutral Valuation The next step would be to compute the risk neutral probabilities q (1 r f ) S(0) S( D) S( U ) S( D) q (1.05) $25 $21.25 $28.75 $21.25 $5 $7.50 2 3 2/3 $28.75,C(U) $25,C(0) 1/3 $21.25,C(D) 12-147

Example of Risk-Neutral Valuation After that, find the value of the call in the up state and down state. C( U ) $28.75 $25 2/3 $28.75, $3.75 $25,C(0) C( D) max[$25 $28.75,0] 1/3 $21.25, $0 12-148

Example of Risk-Neutral Valuation Finally, find the value of the call at time 0: C(0) q C( U ) (1 q) C( D) (1 ) r f C(0) 2 3 $3.75 (1 3) $0 (1.05) $2.50 C( 0) (1.05) $2.38 2/3 $28.75,$3.75 $25,$2.38 $25,C(0) 1/3 $21.25, $0 12-149

Risk-Neutral Valuation and the Replicating Portfolio This risk-neutral result is consistent with valuing the call using a replicating portfolio. C 2 3 $3.75 (1 (1.05) 3) $0 $2.50 1.05 0 $2.38 1 $21.25 1 C0 $25 2 (1.05) 2 $25 20.24 $2. 38 12-150

The Black-Scholes Model Where C 0 S d N( d1) Ee rt N( 2) C 0 = the value of a European option at time t = 0 r = the risk-free interest rate. d d 1 2 ln( S d 1 / E) ( r T T σ 2 2 ) T N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d. The Black-Scholes Model allows us to value options in the real world just as we have done in the 2-state world. 12-151

The Black-Scholes Model Find the value of a six-month call option on Microsoft with an exercise price of $150. The current value of a share of Microsoft is $160. The interest rate available in the U.S. is r = 5%. The option maturity is 6 months (half of a year). The volatility of the underlying asset is 30% per annum. Before we start, note that the intrinsic value of the option is $10 our answer must be at least that amount. 12-152

The Black-Scholes Model Let s try our hand at using the model. If you have a calculator handy, follow along. First calculate d 1 and d 2 2 ln( S / E) ( r.5σ ) T d1 T ln(160 /150) (.05.5(0.30) d 0.30.5 ).5 2 1 0.52815 Then, d 0.52815 0.30.5 2 d1 T 0.31602 12-153

The Black-Scholes Model C d 1 0.52815 d 0.31602 2 0 S d N( d1) Ee rt N( 2) N(d 1 ) = N(0.52815) = 0.7013 N(d 2 ) = N(0.31602) = 0.62401 C C 0 0 $160 0.7013 150e $20.92.05.5 0.62401 12-154

17.9 Stocks and Bonds as Options Levered equity is a call option. The underlying asset comprises the assets of the firm. The strike price is the payoff of the bond. If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an in-the-money call. They will pay the bondholders and call in the assets of the firm. If at the maturity of the debt the shareholders have an out-of-the-money call, they will not pay the bondholders (i.e. the shareholders will declare bankruptcy) and let the call expire. 12-155

Stocks and Bonds as Options Levered equity is a put option. The underlying asset comprises the assets of the firm. The strike price is the payoff of the bond. If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an in-the-money put. They will put the firm to the bondholders. If at the maturity of the debt the shareholders have an out-of-the-money put, they will not exercise the option (i.e. NOT declare bankruptcy) and let the put expire. 12-156

Stocks and Bonds as Options It all comes down to put-call parity. c 0 = S 0 + p 0 E (1+ r) T Value of a call on the firm Value of a put on the firm Value of = the firm + Value of a risk-free bond Stockholder s position in terms of call options Stockholder s position in terms of put options 12-157

Mergers and Diversification Diversification is a frequently mentioned reason for mergers. Diversification reduces risk and, therefore, volatility. Decreasing volatility decreases the value of an option. Assume diversification is the only benefit to a merger: Since equity can be viewed as a call option, should the merger increase or decrease the value of the equity? Since risky debt can be viewed as risk-free debt minus a put option, what happens to the value of the risky debt? Overall, what has happened with the merger and is it a good decision in view of the goal of stockholder wealth maximization? 12-158

Example Consider the following two merger candidates. The merger is for diversification purposes only with no synergies involved. Risk-free rate is 4%. Market value of assets Face value of zero coupon debt Debt maturity Asset return standard deviation Company A $40 million $18 million 4 years 40% Company B $15 million $7 million 4 years 50% 12-159