Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-0 INTERNATIONAL FINANCIAL MANAGEMENT EUN / RESNICK Second.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-0 INTERNATIONAL FINANCIAL MANAGEMENT EUN / RESNICK Second Edition 9 Chapter Nine Futures and Options on Foreign Exchange Chapter Objective: This chapter discusses exchange-traded currency futures contracts, options contracts, and options on currency futures.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-1 Chapter Outline Futures Contracts: Preliminaries Currency Futures Markets Basic Currency Futures Relationships Eurodollar Interest Rate Futures Contracts Options Contracts: Preliminaries Currency Options Markets Currency Futures Options

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-2 Chapter Outline (continued) Basic Option Pricing Relationships at Expiry American Option Pricing Relationships European Option Pricing Relationships Binomial Option Pricing Model European Option Pricing Model Empirical Tests of Currency Option Models

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-3 Futures Contracts: Preliminaries A futures contract is like a forward contract: It specifies that a certain currency will be exchanged for another at a specified time in the future at prices specified today. A futures contract is different from a forward contract: Futures are standardized contracts trading on organized exchanges with daily resettlement through a clearinghouse.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-4 Futures Contracts: Preliminaries Standardizing Features: Contract Size Delivery Month Daily resettlement Initial Margin (about 4% of contract value, cash or T-bills held in a street name at your brokers).

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-5 Daily Resettlement: An Example Suppose you want to speculate on a rise in the $/¥ exchange rate (specifically you think that the dollar will appreciate). Currently $1 = ¥140. The 3-month forward price is $1=¥150.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-6 Daily Resettlement: An Example Currently $1 = ¥140 and it appears that the dollar is strengthening. If you enter into a 3-month futures contract to sell ¥ at the rate of $1 = ¥150 you will make money if the yen depreciates. The contract size is ¥12,500,000 Your initial margin is 4% of the contract value:

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-7 Daily Resettlement: An Example If tomorrow, the futures rate closes at $1 = ¥149, then your position’s value drops. Your original agreement was to sell ¥12,500,000 and receive $83,333.33 But now ¥12,500,000 is worth $83,892.62 You have lost $559.28 overnight.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-8 Daily Resettlement: An Example The $559.28 comes out of your $3,333.33 margin account, leaving $2,774.05 This is short of the $3,355.70 required for a new position. Your broker will let you slide until you run through your maintenance margin. Then you must post additional funds or your position will be closed out. This is usually done with a reversing trade.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-9 Daily Resettlement: Short Position * Suppose you want to speculate on a rise in the $/¥ exchange rate (specifically you think that the dollar will appreciate). Currently $1 = ¥140. The 3-month forward price is $1=¥150.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-10 What do we have ? Spot rate for yen: $1= ¥ 140 (or, ¥1=$1/140) 3-month futures contract (¥12,500,000 each contract) quotes at: $1= ¥150 (or, ¥1=$1/150) If you expect that dollar will appreciate (that is, yen will depreciate), and wish to make profit from futures speculation, what will you do? Daily Resettlement: An Example * Daily Resettlement: Short Position *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-11 Daily Resettlement: An Example * You will sell ¥ at the rate of $1= ¥150 today. This means that you agree to sell ¥12,500,000 and receive $83,333.33=¥12,500,000×($1/ ¥150) three months later. You deposit your initial margin (4% of the contract value=$83,333.33×0.04=$3,333.33) to meet you margin requirement. On the next day, the futures rate closes at $1=¥149 (¥ appreciates), what happens to your investment? This means that if you want to close your position, you shall buy ¥ at $1=¥ 149. That is, you should pay $83,892.62=¥ 12,500,000×($1/¥ 149). Daily Resettlement: Short Position *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-12 Your profit will be $83,333.33-$83,892.62= -$559.28 if you close your position today. This means that your “opportunity cost” is $559.28 if you do not really close your position. (Or, think it in another way, those who sell ¥ futures today can receive $83,892.62 while you can only receive $83,333.33) This loss in value is than reflected in your margin account, with its new balance $2,774.05=$3,333.33-$559.28. On the next day, if the settlement price is $1=¥ 146, and assume that the maintenance margin is $2,500, what happens then? Daily Resettlement: An Example *Daily Resettlement: Short Position *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-13 Your profit will be ¥12,500,000×[($1/¥149)-($1/¥146)]=-1,723.82 (note that by daily resettlement, your contract rate is renewed to $1=¥149 yesterday, not $1=¥150!) Again, this loss in value will be reflected in your margin account, with its new balance $1,050.23=$2,774.05-$1,723.82. (This also equals to 3,333.33+12,500,000[(1/150)-(1/146)].) Now, your margin account has its balance lower than the maintenance margin ($2,500). You should deposit money into your account until it meets the margin requirement which is now ¥12,500,000×($1/¥146) ×0.04=$3424.66. (You have to deposit $3,424.66-$1,723.82=$1,700.84 into your account.) Daily Resettlement: An Example *Daily Resettlement: Short Position *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-14 What happens if you expect that ¥ will appreciate? * Other things being equal, how will you speculate using futures contract if you expect that ¥ will be appreciating, instead of depreciating? You will buy ¥ at the rate of $1= ¥150 today. This means that you agree to pay $83,333.33=¥12,500,000×($1/ ¥150) to buy ¥12,500,000 three months later. You deposit your initial margin (4% of the contract value=$83,333.33×0.04=$3,333.33) to meet you margin requirement. On the next day, the futures rate closes at $1=¥149 (¥ appreciates, as you expected), what happens to your investment? This means that if you want to close your position, you shall sell ¥ at $1=¥ 149. That is, you will receive $83,892.62=¥ 12,500,000×($1/¥ 149). Daily Resettlement: Long Position *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-15 Your profit will be $83,892.62-$83,333.33= $559.28 if you close your position today. This means that your “opportunity profit” is $559.28 if you do not really close your position. (Or, think it in another way, those who buy ¥ futures today should pay $83,892.62 while you only have to pay $83,333.33) This gain in value is than reflected in your margin account, with its new balance $3,892.61=$3,333.33+$559.28. On the next day, if the settlement price is $1=¥146, and assume that the maintenance margin is $2,500, what happens then? What happens if you expect that ¥ will appreciate? * Daily Resettlement: Long Position *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-16 Your profit will be ¥12,500,000×[($1/¥146)-($1/¥149)]= $1,723.82 (Again, by daily resettlement, your contract rate is renewed to $1=¥149 yesterday, not $1=¥150!) This gain in value will be reflected into your margin account, with its new balance $5,616.43= $3,892.61 +$1,723.82. (This also equals to 3,333.33+12,500,000[(1/146)-(1/150)].) Now, your margin account balance is still greater than the maintenance margin. You do not have to do anything to your margin account. Or, you can withdraw your profit as long as your margin account balance is still greater than its maintenance margin. What happens if you expect that ¥ will appreciate? * Daily Resettlement: Long Position *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-17 What do we learn from this example? * Comparing the gain or loss made by both cases, you will learn that as the textbook stated, futures trading between the long and the short is a zero-sum game. That is, the gain of the long is exactly the loss of the short, and vice versa. Short positionLong position profitMargin accountprofitMargin account Day 0 ($1=¥150) $3,333.33 Day 1 ($1=¥149) -$559.28$2774.05$559.28$3,892.61 Day 2 ($1=¥146) -$1,723.82 $1,050.23 (  3424.66) $1,723.82$5,616.43

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-18 Currency Futures Markets The Chicago Mercantile Exchange (CME) is by far the largest. Others include: The Philadelphia Board of Trade (PBOT) The MidAmerica commodities Exchange The Tokyo International Financial Futures Exchange The London International Financial Futures Exchange

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-19 The Chicago Mercantile Exchange Expiry cycle: March, June, September, December. Delivery date 3rd Wednesday of delivery month. Last trading day is the second business day preceding the delivery day. CME hours 7:20 a.m. to 2:00 p.m. CST.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-20 CME After Hours Extended-hours trading on GLOBEX runs from 2:30 p.m. to 4:00 p.m dinner break and then back at it from 6:00 p.m. to 6:00 a.m. CST. Singapore International Monetary Exchange (SIMEX) offer interchangeable contracts. There’s other markets, but none are close to CME and SIMEX trading volume.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-21 Basic Currency Futures Relationships Open Interest refers to the number of contracts outstanding for a particular delivery month. Open interest is a good proxy for demand for a contract. Some refer to open interest as the depth of the market. The breadth of the market would be how many different contracts (expiry month, currency) are outstanding.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-22 Reading a Futures Quote Expiry month Opening price Highest price that day Lowest price that day Closing price Daily Change Highest and lowest prices over the lifetime of the contract. Number of open contracts

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-23 Eurodollar Interest Rate Futures Contracts Widely used futures contract for hedging short- term U.S. dollar interest rate risk. The underlying asset is a hypothetical $1,000,000 90-day Eurodollar deposit—the contract is cash settled. Traded on the CME and the Singapore International Monetary Exchange. The contract trades in the March, June, September and December cycle.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-24 Reading Eurodollar Futures Quotes EURODOLLAR (CME)—$1 million; pts of 100% OpenHighLowSettleChgYield Settle Change Open Interest July 94.69 94.68 -.01 5.32 +.0147,417 Eurodollar futures prices are stated as an index number of three-month LIBOR calculated as F = 100-LIBOR. The closing price for the July contract is 94.68 thus the implied yield is 5.32 percent = 100 – 94.68 The change was.01 percent of $1 million representing $100 on an annual basis. Since it is a 3-month contract one basis point corresponds to a $25 price change.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-25 Daily Resettlements: Long Position* Suppose that, the July contract closes at 94.50 today. Thus, the implied yield is 5.50% (=100-94.5). This means that if you buy (long) one 90-day Eurodollar interest rate futures (contract size=$1 million), you are guaranteed to receive $13,750= $1,000,000×5.5%×(90/360) on your July investment. On the next day, the July contract closes at 94.51 (one basis point more than yesterday), what happens to your margin account? The implied yield now is 5.49% (100-94.51). This means that the buyer of the contract is guaranteed to receive $13,725= $1,000,000×5.49%×(90/36) on the same July investment. You earn $25=$13,750-$13,725 more than the new investor. This “opportunity profit” (because you are guaranteed with higher interest rate) will be added into your margin account.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-26 Your profit is calculated by [5.5%-5.49%] ×$1,000,000×(90/360)=$25. Note that this is can also be calculated by [(100-94.50)-(100-94.51)] ×1%×$1,000,000×(90/360). That is, your profit can also be calculated by [94.51-94.5] ×1%×$1,000,000×(90/360)=$25, as the usual formula calculating futures contracts’ payoff. Then you can see why the interest rate futures contracts are quoted this way. Note also that from this example, you shall see why one basis point change in quotation corresponds to a $25 price change. Daily Resettlements: Long Position*

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-27 Daily Resettlements: Short Position* Suppose that, the July contract closes at 94.50 today. Thus, the implied yield is 5.50% (=100-94.5). This means that if you sell (short) one 90-day Eurodollar interest rate futures (contract size=$1 million), you are guaranteed that you will have to pay $13,750= $1,000,000×5.5%×(90/360) on your July borrowing. On the next day, the July contract closes at 94.51 (one basis point more than yesterday), what happens to your margin account? The implied yield now is 5.49% (100-94.51). This means that the seller of the contract is guaranteed with interest cost of $13,725= $1,000,000×5.49%×(90/36) on the same July borrowing. You loss $25=$13,750-$13,725 because you have to pay more interest payment on the same borrowing. This “opportunity cost” will be deducted from your margin account.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-28 Your profit is calculated by [5.49%-5.5%] ×$1,000,000×(90/360)= -$25. Note that this is can also be calculated by [(100-94.51)-(100-94.50)] ×1%×$1,000,000×(90/360). That is, your profit can also be calculated by [94.50-94.51] ×1%×$1,000,000×(90/360)= -$25, as the usual formula calculating futures contracts’ payoff. Now, it’s your term to think which position you will take if you expect that the 90-day LIBOR will be rising, and wish to speculate from interest rate futures. Also, think what if you expect that the 90-day LIBOR is declining? Daily Resettlements: Short Position*

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-29 Eurodollar Futures Hedge if You are an Investor * You are a treasurer of a MNC, and expect to receive $2,000,000 on Sept. 17, 2003, because of your merchandise sale. You wish to invest these $2,000,000 in Eurodollar deposit because you know that the money will not be needed for a period of 90 days. Today, the settlement price of the Sept03 interest rate futures is 94.00. If you are uncertain about what interest rate you are going to receive, and wish to lock the return of your investment. How will you hedge against this interest rate risk? (You are worrying that the prevailing interest rate on that day will be very low.)

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-30 You will buy (long) 2 Euro dollar interest rate futures contracts at 94.00. This means that you can lock your return from investment at 6% p.a. (that is, you lock your return from 90-day euro dollar deposit at $2,000,000×6%×(90/360)=$30,000.) We will demonstrate that no matter what the realized interest rate at Sept. 17 2003 is, your return is not altered since you’ve hedged your interest rate risk. Eurodollar Futures Hedge if You are an Investor *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-31 1. What happens if the 3-month LIBOR is 5% p.a. on Sept. 2003? a. You deposit your $2,000,000 at that realized interest rate 5% p.a., which gives you $2,000,000×5%×(1/4)=$25,000 interest payment. b. You profit from your futures investment. The settlement price of your futures contract is 95 (=100- 5). Your profit earned on the futures position is calculated as [95-94] ×(1/100) ×(1/4) ×$1,000,000×2=$5,000. Your total return is then $25,000+$5,000=$30,000. Eurodollar Futures Hedge if You are an Investor *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-32 2. What happens if the 3-month LIBOR is 7% p.a. on Sept. 2003? a. You deposit your $2,000,000 at that realized interest rate 7% p.a., which gives you $2,000,000×7%×(1/4)=$35,000 interest payment. b. You lose from your futures investment. The settlement price of your futures contract is 93 (=100- 7). Your profit earned on the futures position is calculated as [93-94] ×(1/100) ×(1/4) ×$1,000,000×2=-$5,000. Your total return is then $35,000-$5,000=$30,000. Eurodollar Futures Hedge if You are an Investor *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-33 Eurodollar Futures Hedge if You are a Borrower * You are a treasurer of a MNC, and expect to borrow $2,000,000 for 90 days on Sept. 17, 2003, because of your imports on that day.. Today, the settlement price of the Sept03 interest rate futures is 94.00. If you are uncertain about what your interest cost will be, and wish to lock the cost of your loan. How will you hedge against this interest rate risk? (You are worrying that the prevailing interest rate on that day will be very high.)

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-34 You will sell (short) 2 Euro dollar interest rate futures contracts at 94.00. This means that you can lock your cost of borrowing at 6% p.a. (that is, you lock your interest cost at $2,000,000×6%×(90/360)=$30,000.) We will demonstrate that no matter what the realized interest rate at Sept. 17 2003 is, your interest cost is not altered since you’ve hedged your interest rate risk. Eurodollar Futures Hedge if You are a Borrower *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-35 1. What happens if the 3-month LIBOR is 5% p.a. on Sept. 2003? a. You borrow $2,000,000 at that realized interest rate 5% p.a., which yields $2,000,000×5%×(1/4)=$25,000 interest cost. b. You lose from your futures investment. The settlement price of your futures contract is 95 (=100- 5). Your profit earned on the futures position is calculated as [94-95] ×(1/100) ×(1/4) ×$1,000,000×2=-$5,000. Your total cost is then $25,000+$5,000=$30,000. Eurodollar Futures Hedge if You are a Borrower *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-36 2. What happens if the 3-month LIBOR is 7% p.a. on Sept. 2003? a. You borrow $2,000,000 at that realized interest rate 7% p.a., which yields $2,000,000×7%×(1/4)=$35,000 interest cost. b. You profit from your futures investment. The settlement price of your futures contract is 93 (=100- 7). Your profit earned on the futures position is calculated as [94-93] ×(1/100) ×(1/4) ×$1,000,000×2=$5,000. Your total cost is then $35,000-$5,000=$30,000. Eurodollar Futures Hedge if You are a Borrower *

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-37 Options Contracts: Preliminaries An option gives the holder the right, but not the obligation, to buy (call) or sell (put) a given quantity (contract size) of an asset (underlying asset) in the future (expiration date), at prices (strike price) agreed upon today. The price specified in the contract is called the strike or exercise price. The specified maturity date is called the expiration date. The price paid for the option is called the option premium.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-38 Options Contracts: Preliminaries European vs. American options European options can only be exercised on the expiration date. American options can be exercised at any time up to and including the expiration date. Since this option to exercise early generally has value, American options are usually worth more than European options, other things equal.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-39 Types of Contracts Examples  An American call option on spot € : The right to buy € 1 million for $1.10 per € from today until expiration on Dec 15, 2001. This “call on € ” is also a “put on US$”.  A European put option on Swiss franc futures : The right to sell SFr 10 million March 2002 futures for $0.65 per SFr on (and only on) Mar 15, 2002. This “put on SFr” is also a “call on US$”.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-40 Options Contracts: Preliminaries In-the-money (it’s worthwhile exercising your option) Call: S T >E Put: S T <E At-the-money (you’re indifferent whether you exercise the option) Call: S T =E Put: S T =E Out-of-the-money (you will not exercise your option) Call: S T <E Put: S T >E

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-41 Currency Options Currency options began trading on the Philadelphia Stock Exchange (PHLX) in 1982. Since then, the markets have expanded : 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.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-43 Location and Scale of Trading Currency options are traded by banks on an over-the- counter (OTC) basis and on organized futures and options exchanges. According to surveys conducted by the Bank for International Settlements, the volume of trading in terms of billions per day is: OTC Organized Exchanges 1995 1998 1995 1998 Currency Options$41.0$87.1$3.8$1.8

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-44 Currency Futures Options Are an option on a currency futures contract. Exercise of a currency futures option results in a long futures position for the holder of a call or the writer of a put. Exercise of a currency futures option results in a short futures position for the seller of a call or the buyer of a put. If the futures position is not offset prior to its expiration, foreign currency will change hands.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-45 Basic Option Pricing Relationships 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 aT = C eT = Max[S T - E, 0]

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-46 Basic Option Pricing Relationships 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 aT = P eT = Max[E - S T, 0]

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-51 American Option Pricing Relationships With an American option, you can do everything that you can do with a European option—this option to exercise early has value. C aT > C eT = Max[S T - E, 0] P aT > P eT = Max[E - S T, 0]

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-52 Market Value, Time Value and Intrinsic Value for an American Call C aT > Max[S T - E, 0] Profit loss E STST Market Value Intrinsic value S T - E Time value Out-of-the-moneyIn-the-money

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-53 European Option Pricing Relationships Consider two investments 1 Buy a call option on the British pound futures contract. The cash flow today is -C e 2 Replicate the upside payoff of the call by 1 Borrowing the present value of the exercise price of the call in the U.S. at i $ The cash flow today is E /(1 + i $ ) 2 Lending the present value of S T at i £ The cash flow is - S T /(1 + i £ )

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-54 European Option Pricing Relationships When the option is in-the-money both strategies have the same payoff. When the option is out-of-the-money it has a higher payoff the borrowing and lending strategy. Thus:

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-55 European Option Pricing Relationships Using a similar portfolio to replicate the upside potential of a put, we can show that:

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-56 Option Pricing with Risk Neutrality: Call* Notations: C t : the price of the call at time t C T : the call option’s payoff at expiration date T S T : the spot exchange rate of the underlying asset at T X: the strike price of the underlying asset r f : the risk-free rate The value of the option at time t (assuming risk neutral) is : C t =E(C T )/(1+r f )

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-57 Example: Assume the underlying asset is $/ ￡ exchange rate, and the spot exchange rate at T has a uniform distribution between $1/ ￡ and $2/ ￡. Also assume X=$1.5/ ￡ ; r f between t and T is 5%.Then, what’s the value of this call option at time t? Option Pricing with Risk Neutrality: Call* 1.521 Prob. STST

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-58 Option Pricing with Risk Neutrality: Call* 1.52 1 Prob. STST AB If S T falls in A, S T >X  C T =S T -1.5 If S T falls in B, S T <X  C T =0 E(C T )= {the expected payoff given S T >X} ×Prob. {S T >X} + {the expected payoff given S T 1.5] ×0.5+0×0.5 =0.25*0.5=0.125

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-59 Option Pricing with Risk Neutrality: Call* Then we know that C t =E(C T )/(1+r f )=0.125/1.05=0.1190 Other things being equal, what the value of the option will be if there is another call with higher strike price (say, X=$1.8/ ￡ )? 1.8 21 Prob. STST AB

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-60 Option Pricing with Risk Neutrality: Call* If S T falls in A, S T >X  C T =S T -1.8 If S T falls in B, S T <X  C T =0 E(C T )= {the expected payoff given S T >X} ×Prob. {S T >X} + {the expected payoff given S T 1.8] ×0.2+0×0.8 =0.1*0.2=0.02 C t =E(C T )/(1+r f )=0.02/1.05=0.0190<0.1190 Then we know that a call with higher exercise price would be less valuable. Why? 1. It has lower possibility to be “in-the-money”. 2. Even if it is in-the-money, its expected payoff is lower.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-61 Option Pricing with Risk Neutrality: Call* What if the expected volatility of the S T becomes larger? (say, if S T is distributed between 0.5 and 2.5?) STST 1.521 Prob. 1 0.5 2.5

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-62 Option Pricing with Risk Neutrality: Call* If ST falls in A, ST>X  CT=ST-1.5 If ST falls in B, ST<X  CT=0 E(CT)= {the expected payoff given ST>X} ×Prob. {ST>X} + {the expected payoff given ST 1.5] ×0.5+0×0.5 =0.5*0.5=0.25 1.5 Prob. 0.5 2.5 AB

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-63 Option Pricing with Risk Neutrality: Call* Then we know that C t =E(C T )/(1+r f )=0.25/1.05=0.2381>0.1190 When the variance of the expected spot exchange rate at T becomes larger, the call is more valuable. Why? * the increase in variance leads to the possibility of both higher and lower S T. * the higher price causes expected payoff to be higher, but the lower price does not drag down the expected payoff. * The increase in the variance of the spot exchange rate increases the upside potential for the investment, while leaving the downside potential unaffected. * Or, loosely speaking, when the volatility is higher, people need the option to hedge their risk more eagerly. Thus led to a higher call value.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-64 Option Pricing with Risk Neutrality: Put* Notations: P t : the price of the call at time t P T : the call option’s payoff at expiration date T P T : the spot exchange rate of the underlying asset at T X: the strike price of the underlying asset r f : the risk-free rate The value of the option at time t (assuming risk neutral) is : P t =E(P T )/(1+r f )

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-65 Example: Assume the underlying asset is $/ ￡ exchange rate, and the spot exchange rate at T has a uniform distribution between $1/ ￡ and $2/ ￡. Also assume X=$1.5/ ￡ ; r f between t and T is 5%.Then, what’s the value of this put option at time t? Option Pricing with Risk Neutrality: Put* 1.521 Prob. STST

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-66 Option Pricing with Risk Neutrality: Put* 1.52 1 Prob. STST AB If S T falls in A, S T >X  P T =0 If S T falls in B, S T <X  P T =1.5-S T E(P T )= {the expected payoff given S T >X} ×Prob. {S T >X} + {the expected payoff given S T <X} ×Prob. {S T <X} =0×0.5+E[1.5-S T |given S T <1.5] ×0.5 =0.25*0.5=0.125

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-67 Option Pricing with Risk Neutrality: Put* Then we know that P t =E(P T )/(1+r f )=0.125/1.05=0.1190 Other things being equal, what the value of the option will be if there is another put with higher strike price (say, X=$1.8/ ￡ )? 1.8 21 Prob. STST AB

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-68 Option Pricing with Risk Neutrality: Put* If S T falls in A, S T >X  C T =0 If S T falls in B, S T <X  C T =1.8-S T E(P T )= {the expected payoff given S T >X} ×Prob. {S T >X} + {the expected payoff given S T <X} ×Prob. {S T <X} =0×0.2+E[1.8-S T |given S T <1.8] ×0.8 =0.4 ×0.8=0.32 P t =E(P T )/(1+r f )=0.32/1.05=0.3048>0.1190 Then we know that a call with higher exercise price would be more valuable. Why? 1. It has higher possibility to be “in-the-money”. 2. When it is in-the-money, its expected payoff is greater.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-69 Option Pricing with Risk Neutrality: Put* What if the expected volatility of the S T becomes larger? (say, if S T is distributed between 0.5 and 2.5?) STST 1.521 Prob. 1 0.5 2.5

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-70 Option Pricing with Risk Neutrality: Put* If S T falls in A, P T >X  C T =0 If S T falls in B, S T <X  C T =1.5-S T E(P T )= {the expected payoff given S T >X} ×Prob. {S T >X} + {the expected payoff given S T <X} ×Prob. {S T <X} =0×0.5+E[1.5-S T |given S T <1.5] ×0.5 =0.5*0.5=0.25 1.5 Prob. 0.5 2.5 AB

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-71 Option Pricing with Risk Neutrality: Put* Then we know that P t =E(P T )/(1+r f )=0.25/1.05=0.2381>0.1190 When the variance of the expected spot exchange rate at T becomes larger, the put is more valuable. Why? * the increase in variance leads to the possibility of both higher and lower S T. * the lower price causes expected payoff to be higher, but the higher price does not drag down the expected payoff. * The increase in the variance of the spot exchange rate increases the upside potential for the investment, while leaving the downside potential unaffected. * Or, loosely speaking, when the volatility is higher, people need the option to hedge their risk more eagerly. Thus led to a higher option value ( either call or put).

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-72 Binomial Option Pricing Model Imagine a simple world where the dollar-euro exchange rate is S 0 ($/ ) = $1 today and in the next year, S 1 ($/ ) is either $1.1 or $.90. $1 $.90 $1.10 S 0 ($/ )S 1 ($/ )

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-73 Binomial Option Pricing Model $1 $.90 $1.10 S 0 ($/ )S 1 ($/ ) $.10 $0 C 1 ($/ ) A call option on the euro with exercise price S 0 ($/ ) = $1 will have the following payoffs.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-74 $1 $.90 $1.10 S 0 ($/ )S 1 ($/ ) $.10 $0 C 1 ($/ ) Binomial Option Pricing Model We can replicate the payoffs of the call option. With a levered position in the euro.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-75 $1 $.90 $1.10 S 0 ($/ )S 1 ($/ ) $.10 $0 C 1 ($/ ) Binomial Option Pricing Model debt -$.90 portfolio $.20 $.00 Borrow the present value of $.90 today and buy 1. Your net payoff in one period is either $.2 or $0.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-76 Binomial Option Pricing Model $1 $.90 $1.10 S 0 ($/ )S 1 ($/ ) $.10 $0 C 1 ($/ )debt -$.90 portfolio $.20 $.00 The portfolio has twice the option’s payoff so the portfolio is worth twice the call option value.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-77 $1 $.90 $1.10 S 0 ($/ )S 1 ($/ ) $.10 $0 C 1 ($/ )debt -$.90 portfolio $.20 $.00 Binomial Option Pricing Model The portfolio value today is today’s value of one euro less the present value of a $.90 debt:

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-78 Binomial Option Pricing Model $1 $.90 $1.10 S 0 ($/ )S 1 ($/ ) $.10 $0 C 1 ($/ )debt -$.90 portfolio $.20 $.00 We can value the option as half of the value of the portfolio:

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-79 Binomial Option Pricing Model The most important lesson from the binomial option pricing model is: the replicating portfolio intuition. 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.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-80 Binominal Option Pricing Model* We have 3 ways to price an option, and they should yield the same pricing formula. These 3 methods are: 1. riskless hedge 2. replication method 3. risk-neutral pricing

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-81 Binominal Option Pricing Model- Assumptions and Notations* The spot rate begins at S 0 and either rises to u S 0 or falls to d S 0 in the next period. (u>d) S0S0 Period 0 S 1,u = u S 0 S 1,d = d S 0 Period 1

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-82 Binominal Option Pricing Model- Assumptions and Notations* Also assume that the forward rate (expires at period 1) of the underlying asset is F. If there is a call with strike price E, its value can be represented as: C0C0 Period 0 C uT = Max{u S 0 -E,0} C dT = Max{d S 0 -E,0} Period 1

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-83 Binominal Option Pricing Model- Assumptions and Notations* Also assume that the interest rate of the domestic currency is r. Our main goal is to determine what C 0 should be, using the 3 methods we are going to introduce.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-84 Binominal Option Pricing Model- riskless hedge* The main idea is to construct a “riskless portfolio” that yields a constant payoff, regardless of which value S 1 is going to be. We can long a call, and short h futures (or forward) contracts. (why short? Because thus your futures position’s payoff is opposite to that of the call!) h is called the hedge ratio, it is the size of the short (or long) position the investor must have in the underlying asset per option to yield a risk-free portfolio.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-85 Binominal Option Pricing Model- riskless hedge* If we form a portfolio that longs one call and shorts h futures, its payoff at period 1 will be: Period 0 C uT +(F-u S 0 ) h=A C dT +(F-d S 0 ) h=B Period 1 Since it is “riskless”, A must be equal to B. Thus we can determine what h should be to make A=B.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-86 Binominal Option Pricing Model- riskless hedge* Substituting it into the period 1 payoff, we are sure that this portfolio’s payoff will be:

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-87 Binominal Option Pricing Model- riskless hedge* Since it is a certain profit, to rule out arbitrage opportunity, it should yield the same return as the riskless bond. That is:

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-88 Digression: What will happen if (a) does not hold? A riskless arbitrage opportunity exists. We will demonstrate how the arbitrage activities would drive prices back to their equilibrium levels. That is, we’ll show that arbitrage makes (a) hold with equality. Case I: if A/C 0 >(1+r) The return (cost) of your riskless portflolio is greater than that of buying (selling) the riskless bond. How will you profit from this? (i.e. what is your arbitrage strategy?)

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-89 Digression: What will happen if (a) does not hold? In period 0, you can borrow C 0 and use the very money you borrowed to buy a call and short h futures simultaneously. (Note that on doing this, your period-0 cash flow is zero! ) In period 1, you can earn a certain profit of A from your riskless portfolio. And, at the same time, you have to repay your loan by paying the bond holder C 0 (1+r). Since A>C 0 (1+r), your period-1 cash flow is A-C 0 (1+r)>0. You earn a certain, riskless profit with zero period 0 investment!

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-90 Digression: What will happen if (a) does not hold? What happens then if all the astute traders like you try to arbitrage from this? Look at the period-0 strategy: 1. You first borrow from the bond market  r goes up. 2. You then use the money you borrowed to long a call  call price C 0 goes up. 3. You also short h futures simultaneously  F goes down and thus A goes down. Now you see 3 forces that drive prices back to the levels that make (a) holds with equality!

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-91 Digression: What will happen if (a) does not hold? Case II: if A/(C 0 )<(1+r) The return (cost) of the riskless portfolio is smaller than that of buying (selling) the riskless bond. How will you profit from this? (i.e. what is your arbitrage strategy?)

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-92 Digression: What will happen if (a) does not hold? In period 0, you can short a call and long h futures to form your riskless portfolio, and invest C 0 from writing the call. (Note that on doing this, your period-0 cash flow is zero! ) In period 1, you will receive C 0 (1+r) from your bond holding. And, at the same time, your riskless portfolio is making “profit” of -A. Since A 0. You earn a certain, riskless profit with zero period 0 investment!

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-93 Digression: What will happen if (a) does not hold? What happens then if all the astute traders like you try to arbitrage from this? Look at the period-0 strategy: 1. You first short a call  call price C 0 goes down. 2. You also long h futures simultaneously  F goes up and thus A goes up. 3. You then deposit C 0 in the bond market  r goes down. Now you see 3 forces that drive prices back to the levels that make (a) holds with equality!

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-94 Binominal Option Pricing Model- replication method* The main idea is to construct a portfolio that will yield exactly the same period 1 payoff as a call. That is, we construct a portfolio that replicates a call’s payoff. Since they have the same payoff, to rule out arbitrage opportunity, their cost at period 0 should also be equal. Thus, we can use this portfolio with known period 0 cost to get the value of the call, C 0.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-95 Binominal Option Pricing Model- replication method* If our goal is to replicate the payoff of a long position of call. We can invest B in bond holding, and long  forward contracts. The first problem facing us is to decide what B and  should be. Then we can use this replicating portfolio to price the call.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-96 Binominal Option Pricing Model- replication method* This portfolio have the following pattern of payoff : B Period 0 B(1+r) +  (u S 0 -F)=G B(1+r)+  (d S 0 -F)=H Period 1 If this portfolio replicates the call’s payoff, we must have G=C uT, and H=C dT. Using these two relationships, we can solve for B and .

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-98 Binominal Option Pricing Model- replication method* We can summerize the cash flows as what follows: Domestic deposit (B)-BB(1+r)B(1+r) long  forwards 0  (u S 0 -F)  (d S0-F) B +  F -BGH Call option purchase-C 0 C uT C dT Transaction Cash Flows Period 0 Cash Flows in Period 1 if S 1 = uS 0 if S1 = dS 0 Since we’ve selected B and  so that G=C uT, H=C dT, the period one cash flows is exactly the same. The period 0 cash flow should also be the same (or you will have a riskless arbitrage opportunity).  C 0 =B

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-99 Binominal Option Pricing Model- replication method* Now we’ve priced the call using replication method: Note that this is the very pricing formula we’ve derived using riskless hedge method!

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 100 Binominal Option Pricing Model- risk-neutral pricing* The main idea is to calculate the “risk-neutral probability” that makes the expected payoff of the period 1 payoff equal to its riskless price ~F. Then we can use this risk-neutral probability to calculate the expected payoff of the call. Discounting this expected payoff of the call, we than get what C 0 should be. To start, assume that there is a artificial probability q, named risk-neutral probability.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 101 Binominal Option Pricing Model- risk-neutral pricing* Assume that S 1 goes to uS 0 with probability q, dS 0 with prob. 1-q. S0S0 Period 0 S 1,u = u S 0 S 1,d = d S 0 Period 1 q 1-q Our first step is to find q that makes S 1 behave as riskless. That is, E q (S 1 )=F.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 102 Binominal Option Pricing Model- risk-neutral pricing* Since this probability makes S behaves as riskless (its expected payoff is the forward exchange rate that we’ve known today.), we can use this probability to make the call “riskless”.Then we only have to discount this expected payoff to period 0, and get the price of the call, C 0.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 103 Binominal Option Pricing Model- risk-neutral pricing* Substitute the q we’ve derived into it, we than get exactly the same pricing formula as the previous two methods!

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 104 An Example Now, you should be able to calculate C 0 using the example in the textbook. Suppose there is a September SF call with strike price E=$0.67/SF. The spot exchange is $63.86/SF. The interest rate from now to expiry date is 0.00908. Also assume the September futures price now is F=$0.6433/SF.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 105 An Example The spot exchange rate behaves in the following way: 63.86 Period 0 S 1,u = u S 0 = 67.78 S 1,d = d S 0 = 60.17 Period 1 The call’s payoffs are thus: C0C0 Period 0 C uT = 67.78-67=0.78 C dT = 0 Period 1

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 106 An Example-riskless hedge* To form a riskless portfolio that longs one call and short h futures. This portfolio has the following payoff pattern: Making A=B, we must long h=0.1025 forward contracts: h=(.78-0)/(67.78-60.17)=0.1025 Thus A=B=0.78+(64.33-67.78)0.1025=0.4264 C 0 =A/(1+r)=0.4264/1.00908=0.42 cents per SF. Period 0 C uT +(F-u S 0 ) h=A=0.78+(64.33-67.78)h C dT +(F-d S 0 ) h=B=0+(64.33-60.17)h Period 1

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 107 An Example-replication method If our goal is to replicate the payoff of a long position of call. We can invest B in bond holding, and long  forward contracts. This portfolio has the following payoff pattern: B Period 0 B(1+r) +  (u S 0 -F)=G=1.00908B+ 3.45  B(1+r)+  (d S 0 -F)=H=1.00908B-4.16  Period 1

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 108 An Example-replication method To replicate the call’s payoff, we must have G=C uT =0.78, H=C dT =0. Solve for B and , we know that we have to deposit B=0.42 cents, and long 0.1025 futures contracts. Domestic deposit (B)-0.420.430.43 long  forwards 0 0.35 -0.43 B +  F -0.420.780 Call option purchase-C 0 0.780 Transaction Cash Flows Period 0 Cash Flows in Period 1 if S 1 = uS 0 if S1 = dS 0

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 109 An Example-replication method Since we’ve selected B and  so that the period one cash flows is exactly the same. The period 0 cash flow should also be the same (or you will have a riskless arbitrage opportunity).  C 0 =B=0.4222

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 110 An Example- risk-neutral pricing 63.86 Period 0 S 1,u = u S 0 = 67.78 S 1,d = d S 0 = 60.17 Period 1 q 1-q Risk-neutral probability makes q*67.88+(1-q)60.17=F=64.33. Solve for q, we have the risk-neutral probability q=0.5466

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 111 An Example- risk-neutral pricing Using this probability, we can calculate the expected payoff of the call, and discount it to get the call price: C 0 =[q*C uT +(1-q)C dT ]/(1+r) =[.5466(0.78)+(1-.5466)(0)]/1.00908 =0.42 cents per SF

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 112 Pricing Spot Currency Options The Discrete Time Binomial Model A put option can be priced similarly. The two-period model can be extended to a multiperiod setting :

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 113 European Option Pricing Formula We can use the replicating portfolio intuition developed in the binomial option pricing formula to generate a faster-to-use model that addresses a much more realistic world. Using the above-mentioned idea, you generalize it to the standard Black-Scholes options pricing formula.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 114 European Option Pricing Formula The model is Where C 0 = the value of a European option at time t = 0 r $ = the interest rate available in the U.S. r £ = the interest rate available in the foreign country—in this case the U.K.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 115 European Option Pricing Formula Find the value of a six-month call option on the British pound with an exercise price of $1.50 = £1 The current value of a pound is $1.60 The interest rate available in the U.S. is r $ = 5%. The interest rate in the U.K. is r £ = 7%. The option maturity is 6 months (half of a year). The volatility of the $/£ exchange rate is 30% p.a. Before we start, note that the intrinsic value of the option is $.10—our answer must be at least that.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 116 European Option Pricing Formula Let’s try our hand at using the model. If you have a calculator handy, follow along. Then, calculate d 1 and d 2 First calculate

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 118 Option Value Determinants Call Put 1.Exchange rate+ – 2.Exercise price– + 3.Interest rate in U.S.+ – 4.Interest rate in other country+ – 5.Variability in exchange rate+ + 6.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 the above factors.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 119 Combining put and call to beat market volatility Portfolios of put and call options are called “strategies”. Straddle: a particular kind of “strategies” that combines a put and a call at the same strike price. Practically, we can use straddle to beat market volatility.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 121 Combining put and call to beat market volatility If S T >U: gain profit from the call If S T <D: gain profit from the put If D<S T <U: loss money So, if you think the market volatility will be very large, you can invest in this kind of straddle to profit from market volatility.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 123 Combining put and call to beat market volatility If S T >U: loss from the call If S T <D: loss from the put If D<S T <U: gain profit So, if you think the market volatility will be very small, you can invest in this kind of straddle.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9- 124 Empirical Tests The European option pricing model works fairly well in pricing American currency options. It works best for out-of-the-money and at-the-money options. When options are in-the-money, the European option pricing model tends to underprice American options.

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-0 INTERNATIONAL FINANCIAL MANAGEMENT EUN / RESNICK Second.

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Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved. 9-0 INTERNATIONAL FINANCIAL MANAGEMENT EUN / RESNICK Second.

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