Corso di Laurea in International Management FINANCIAL MANAGEMENT McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

Corso di Laurea in International Management Chapter Objective: This chapter discusses exchange-traded currency futures contracts, options contracts, and options on currency futures. 7 Chapter Seven Futures and Options on Foreign Exchange

Corso di Laurea in International Management Chapter Outline Corso di Laurea in International Management Futures Contracts: Preliminaries Currency Futures Markets Basic Currency Futures Relationships Eurodollar Interest Rate Futures Contracts Options Contracts: Preliminaries Currency Options Markets Currency Futures Options

Chapter Outline (continued) Corso di Laurea in International Management 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

Futures Contracts: Preliminaries Corso di Laurea in International Management 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.

Futures Contracts: Preliminaries Corso di Laurea in International Management Futures Contracts: Preliminaries Standardizing Features: Contract Size Delivery Month Daily resettlement Initial performance bond (about 2 percent of contract value, cash or T-bills held in a street name at your brokerage).

Daily Resettlement: An Example Corso di Laurea in International Management Daily Resettlement: An Example Consider a long position in the CME Euro/U.S. Dollar contract. It is written on €125,000 and quoted in $ per €. The strike price is $1.30 the maturity is 3 months. At initiation of the contract, the long posts an initial performance bond of $6,500. The maintenance performance bond is $4,000.

Daily Resettlement: An Example Corso di Laurea in International Management Daily Resettlement: An Example Recall that an investor with a long position gains from increases in the price of the underlying asset. Our investor has agreed to BUY €125,000 at $1.30 per euro in three months time. With a forward contract, at the end of three months, if the euro was worth $1.24, he would lose $7,500 = ($1.24 – $1.30) × 125,000. If instead at maturity the euro was worth $1.35, the counterparty to his forward contract would pay him $6,250 = ($1.35 – $1.30) × 125,000.

Daily Resettlement: An Example Corso di Laurea in International Management Daily Resettlement: An Example With futures, we have daily resettlement of gains an losses rather than one big settlement at maturity. Every trading day: if the price goes down, the long pays the short if the price goes up, the short pays the long After the daily resettlement, each party has a new contract at the new price with one-day-shorter maturity.

Performance Bond Money Corso di Laurea in International Management Performance Bond Money Each day’s losses are subtracted from the investor’s account. Each day’s gains are added to the account. In this example, at initiation the long posts an initial performance bond of $6,500. The maintenance level is $4,000. If this investor loses more than $2,500 he has a decision to make: he can maintain his long position only by adding more funds—if he fails to do so, his position will be closed out with an offsetting short position.

Daily Resettlement: An Example Corso di Laurea in International Management Daily Resettlement: An Example Over the first 3 days, the euro strengthens then depreciates in dollar terms: Settle Gain/Loss Account Balance $1.31 $1,250 = ($1.31 – $1.30)×125,000 $7,750 = $6,500 + $1,250 $1.30 –$1,250 $6,500 $1.27 –$3,750 $2,750 + $3,750 = $6,500 On third day suppose our investor keeps his long position open by posting an additional $3,750.

Daily Resettlement: An Example Corso di Laurea in International Management Daily Resettlement: An Example Over the next 2 days, the long keeps losing money and closes out his position at the end of day five. Settle Gain/Loss Account Balance $1.31 $1,250 $7,750 $1.30 –$1,250 $6,500 $1.27 –$3,750 $2,750 + $3,750 = $6,500 $1.26 –$1,250 $5,250 = $6,500 – $1,250 $1.24 –$2,500 $2,750

Corso di Laurea in International Management Toting Up At the end of his adventures, our investor has three ways of computing his gains and losses: Sum of daily gains and losses – $7,500 = $1,250 – $1,250 – $3,750 – $1,250 – $2,500 Contract size times the difference between initial contract price and last settlement price. – $7,500 = ($1.24/€ – $1.30/€) × €125,000 Ending balance on account minus beginning balance on account, adjusted for deposits or withdrawals. – $7,500 = $2,750 – ($6,500 + $3,750)

Daily Resettlement: An Example Corso di Laurea in International Management Daily Resettlement: An Example Settle Gain/Loss Account Balance $1.30 –$– $6,500 $1.31 $1,250 $7,750 $1.30 –$1,250 $6,500 $1.27 –$3,750 $2,750 + $3,750 $1.26 –$1,250 $5,250 $1.24 –$2,500 $2,750 Total loss = – $7,500 = ($1.24 – $1.30) × 125,000 = $2,750 – ($6,500 + $3,750)

Currency Futures Markets Corso di Laurea in International Management 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

The Chicago Mercantile Exchange Corso di Laurea in International Management The Chicago Mercantile Exchange Expiry cycle: March, June, September, December. Delivery date third 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.

Corso di Laurea in International Management 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. The Singapore Exchange offers interchangeable contracts. There are other markets, but none are close to CME and SIMEX trading volume.

Reading Currency Futures Quotes Corso di Laurea in International Management Reading Currency Futures Quotes OPEN HIGH LOW SETTLE CHG OPEN INT Euro/US Dollar (CME)—€125,000; $ per € Mar 1.4748 1.4830 1.4700 1.4777 .0028 172,396 Jun 1.4737 1.4818 1.4693 1.4763 .0025 2,266 Closing price Expiry month Thursday March 3, 2005 Daily Change Opening price Lowest price that day Number of open contracts Highest price that day

Basic Currency Futures Relationships Corso di Laurea in International Management 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.

Reading Currency Futures Quotes Corso di Laurea in International Management OPEN HIGH LOW SETTLE CHG OPEN INT Euro/US Dollar (CME)—€125,000; $ per € 1.4748 1.4830 1.4700 1.4777 .0028 Mar 172,396 1.4737 1.4818 1.4693 1.4763 .0025 Jun 2,266 Notice that open interest is greatest in the nearby contract, in this case March, 2008. Thursday March 3, 2005 In general, open interest typically decreases with term to maturity of most futures contracts.

Basic Currency Futures Relationships Corso di Laurea in International Management OPEN HIGH LOW SETTLE CHG OPEN INT Euro/US Dollar (CME)—€125,000; $ per € 1.4748 1.4830 1.4700 1.4777 .0028 Mar 172,396 1.4737 1.4818 1.4693 1.4763 .0025 Jun 2,266 The holder of a long position is committing himself to pay $1.4777 per euro for €125,000—a $184,712.50 position. As there are 172,396 such contracts outstanding, this represents a notational principal of over $31.8 billion!

Reading Currency Futures Quotes Corso di Laurea in International Management OPEN HIGH LOW SETTLE CHG OPEN INT Euro/US Dollar (CME)—€125,000; $ per € 1.4748 1.4830 1.4700 1.4777 .0028 Mar 172,396 1.4737 1.4818 1.4693 1.4763 .0025 Jun 2,266 Recall from chapter 6, our interest rate parity condition: Thursday March 3, 2005 1 + i€ 1 + i$ F($/€) S($/€) =

Reading Currency Futures Quotes Corso di Laurea in International Management OPEN HIGH LOW SETTLE CHG OPEN INT Euro/US Dollar (CME)—€125,000; $ per € 1.4748 1.4830 1.4700 1.4777 .0028 Mar 172,396 1.4737 1.4818 1.4693 1.4763 .0025 Jun 2,266 From March to June 2008 we should expect lower interest rates in dollar denominated accounts: if we find a higher rate in a euro denominated account, we may have found an arbitrage. Thursday March 3, 2005

Eurodollar Interest Rate Futures Contracts Corso di Laurea in International Management 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.

Reading Eurodollar Futures Quotes Corso di Laurea in International Management OPEN HIGH LOW SETTLE CHG OPEN INT YLD Eurodollar (CME)—1,000,000; pts of 100% 96.56 96.58 96.55 - 3.44 Jun 1,398,959 Eurodollar futures prices are stated as an index number of three-month LIBOR calculated as F = 100 – LIBOR. The closing price for the June contract is 96.56 thus the implied yield is 3.44 percent = 100 – 96.56   Since it is a 3-month contract one basis point corresponds to a $25 price change: .01 percent of $1 million represents $100 on an annual basis.

Trading irregularities Corso di Laurea in International Management Futures Markets are also a great place to launder money The zero sum nature of futures is the key to laundering the money.

Options Contracts: Preliminaries Corso di Laurea in International Management An option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset in the future, at prices agreed upon today. Calls vs. Puts Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset at some time in the future, at prices agreed upon today. Put options gives the holder the right, but not the obligation, to sell a given quantity of some asset at some time in the future, at prices agreed upon today.

Options Contracts: Preliminaries Corso di Laurea in International Management 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.

Options Contracts: Preliminaries Corso di Laurea in International Management In-the-money The exercise price is less than the spot price of the underlying asset. At-the-money The exercise price is equal to the spot price of the underlying asset. Out-of-the-money The exercise price is more than the spot price of the underlying asset.

Options Contracts: Preliminaries Corso di Laurea in International Management Intrinsic Value The difference between the exercise price of the option and the spot price of the underlying asset. Speculative Value The difference between the option premium and the intrinsic value of the option. Option Premium Intrinsic Value Speculative Value + =

Currency Options Markets Corso di Laurea in International Management PHLX HKFE 20-hour trading day. OTC volume is much bigger than exchange volume. Trading is in six major currencies against the U.S. dollar.

PHLX Currency Option Specifications Corso di Laurea in International Management Currency Contract Size Australian dollar AD10,000 British pound £10,000 Canadian dollar CAD10,000 Euro €10,000 Japanese yen ¥1,000,000 Swiss franc SF10,000 http://www.phlx.com/products/xdc_specs.htm

Basic Option Pricing Relationships at Expiry Corso di Laurea in International Management 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 ST – E. If the call is out-of-the-money, it is worthless. CaT = CeT = Max[ST - E, 0]

Basic Option Pricing Relationships at Expiry Corso di Laurea in International Management 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 - ST. If the put is out-of-the-money, it is worthless. PaT = PeT = Max[E – ST, 0]

Basic Option Profit Profiles Corso di Laurea in International Management Profit Owner of the call If the call is in-the- money, it is worth ST – E. If the call is out-of- the-money, it is worthless and the buyer of the call loses his entire investment of c0. Long 1 call ST –c0 E + c0 E Out-of-the-money In-the-money loss

Basic Option Profit Profiles Corso di Laurea in International Management Profit Seller of the call If the call is in-the- money, the writer loses ST – E. If the call is out-of- the-money, the writer keeps the option premium. c0 E E + c0 ST short 1 call Out-of-the-money In-the-money loss

Basic Option Profit Profiles Corso di Laurea in International Management Basic Option Profit Profiles Profit If the put is in- the-money, it is worth E – ST. The maximum gain is E – p0 If the put is out- of-the-money, it is worthless and the buyer of the put loses his entire investment of p0. E – p0 Owner of the put E – p0 E ST – p0 long 1 put In-the-money Out-of-the-money loss

Basic Option Profit Profiles Corso di Laurea in International Management Basic Option Profit Profiles Profit If the put is in- the-money, it is worth E –ST. The maximum loss is – E + p0 If the put is out- of-the-money, it is worthless and the seller of the put keeps the option premium of p0. Seller of the put p0 E E – p0 ST short 1 put – E + p0 loss

Corso di Laurea in International Management Example Corso di Laurea in International Management Profit Consider a call option on €31,250. The option premium is $0.25 per € The exercise price is $1.50 per €. Long 1 call on 1 pound $1.75 ST –$0.25 $1.50 loss

Corso di Laurea in International Management Example Profit Consider a call option on €31,250. The option premium is $0.25 per € The exercise price is $1.50 per €. Long 1 call on €31,250 $1.75 ST –$7,812.50 $1.50 loss

Corso di Laurea in International Management Example Corso di Laurea in International Management Profit What is the maximum gain on this put option? At what exchange rate do you break even? $42,187.50 = €31,250×($1.50 – $0.15)/€ $42,187.50 Consider a put option on €31,250. The option premium is $0.15 per € The exercise price is $1.50 per euro. $1.35 ST –$4,687.50 Long 1 put on €31,250 $1.50 $4,687.50 = €31,250×($0.15)/€ loss

American Option Pricing Relationships Corso di Laurea in International Management American Option Pricing Relationships With an American option, you can do everything that you can do with a European option AND you can exercise prior to expiry—this option to exercise early has value, thus: CaT > CeT = Max[ST - E, 0] PaT > PeT = Max[E - ST, 0]

Market Value, Time Value and Intrinsic Value for an American Call Corso di Laurea in International Management Market Value, Time Value and Intrinsic Value for an American Call Profit The red line shows the payoff at maturity, not profit, of a call option. Note that even an out- of-the-money option has value—time value. Long 1 call Market Value Intrinsic value ST Time value Out-of-the-money In-the-money loss E

European Option Pricing Relationships Corso di Laurea in International Management Consider two investments Buy a European call option on the British pound futures contract. The cash flow today is –Ce Replicate the upside payoff of the call by Borrowing the present value of the dollar exercise price of the call in the U.S. at i$ E (1 + i$) The cash flow today is Lending the present value of ST at i£ ST (1 + i£) The cash flow today is –

European Option Pricing Relationships Corso di Laurea in International Management 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 than the borrowing and lending strategy. Thus: Ce > Max ST E (1 + i£) (1 + i$) – , 0

European Option Pricing Relationships Corso di Laurea in International Management Using a similar portfolio to replicate the upside potential of a put, we can show that: Pe > Max ST E (1 + i£) (1 + i$) – , 0

Corso di Laurea in International Management A Brief Review of IRP Recall that if the spot exchange rate is S0($/€) = $1.50/€, and that if i$ = 3% and i€ = 2% then there is only one possible 1-year forward exchange rate that can exist without attracting arbitrage: F1($/€) = $1.5147/€ 1 Borrow $1.5m at i$ = 3% Owe $1.545m F1($/€) = $1.5147 €1.00 Exchange $1.5m for €1m at spot Invest €1m at i€ = 2% Receive €1.02 m

Binomial Option Pricing Model Corso di Laurea in International Management Binomial Option Pricing Model Imagine a simple world where the dollar-euro exchange rate is S0($/€) = $1.50/€ today and in the next year, S1($/€) is either $1.875/€ or $1.20/€. S0($/€) S1($/€) $1.875 $1.50 $1.20

Binomial Option Pricing Model Corso di Laurea in International Management Binomial Option Pricing Model A call option on the euro with exercise price S0($/€) = $1.50 will have the following payoffs. By exercising the call option, you can buy €1 for $1.50. If S1($/€) = $1.875/€ the option is in-the-money: $1.20 $1.875 S1($/€) S0($/€) $1.50 C1($/€) $.375 $0 …and if S1($/€) = $1.20/€ the option is out-of-the-money:

Binomial Option Pricing Model Corso di Laurea in International Management Binomial Option Pricing Model We can replicate the payoffs of the call option. By taking a position in the euro along with some judicious borrowing and lending. $1.20 $1.875 S1($/€) S0($/€) $1.50 C1($/€) $.375 $0

Binomial Option Pricing Model Corso di Laurea in International Management Binomial Option Pricing Model Borrow the present value (discounted at i$) of $1.20 today and use that to buy the present value (discounted at i€) of €1. Invest the euro today and receive €1 in one period. Your net payoff in one period is either $0.675 or $0. S0($/€) S1($/€) debt portfolio C1($/€) $1.875 – $1.20 = $.675 $.375 $1.50 $1.20 – $1.20 = $0 $0

Binomial Option Pricing Model Corso di Laurea in International Management Binomial Option Pricing Model The portfolio has 1.8 times the call option’s payoff so the portfolio is worth 1.8 times the option value. $.675 $.375 1.80 = S0($/€) debt portfolio C1($/€) S1($/€) $1.20 $1.875 $1.50 – $1.20 = $.675 = $0 $.375 $0

Binomial Option Pricing Model Corso di Laurea in International Management Binomial Option Pricing Model The replicating portfolio’s dollar value today is the sum of today’s dollar value of the present value of one euro less the present value of a $1.20 debt: $1.50 $1.20 (1 + i$) €1.00 (1 + i€) × – S0($/€) debt portfolio C1($/€) S1($/€) $1.20 $1.875 $1.50 – $1.20 = $.675 = $0 $.375 $0

Binomial Option Pricing Model Corso di Laurea in International Management Binomial Option Pricing Model We can value the call option as 5/9 of the value of the replicating portfolio: C0 = × 5 9 $1.50 $1.20 (1 + i$) €1.00 (1 + i€) – S0($/€) S1($/€) debt portfolio C1($/€) $1.875 – $1.20 = $.675 $.375 If i$ = 3% and i€ = 2% the call is worth $0.1697 = × 5 9 $1.50 $1.20 (1.03) €1.00 (1.02) – $1.50 $1.20 – $1.20 = $0 $0

Binomial Option Pricing Model Corso di Laurea in International Management Binomial Option Pricing Model The most important lesson 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.

Corso di Laurea in International Management The Hedge Ratio In the example just previous, we replicated the payoffs of the call option with a levered position in the underlying asset. (In this case, borrowing dollars to buy euro at the spot.) The hedge ratio of a option is the ratio of change in the price of the option to the change in the price of the underlying asset: H = C – C S1 – S1 down up This ratio gives the number of units of the underlying asset we should hold for each call option we sell in order to create a riskless hedge.

Corso di Laurea in International Management Hedge Ratio Corso di Laurea in International Management 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: H = C – C S1 – S1 down up $0.375 – $0 $1.875 – $1.20 $0.375 $0.675 5 9 = The delta of a put option is negative. Deltas change through time.

Creating a Riskless Hedge Corso di Laurea in International Management Creating a Riskless Hedge The standard size of euro options on the PHLX is €10,000. In our simple world where the dollar-euro exchange rate is S0($/€) = $1.50/€ today and in the next year, S1($/€) is either $1.875/€ or $1.20/€ an at-the- money call on €10,000 has these payoffs: $1.875 €1.00 $1.20 × If the exchange rate at maturity goes up to S1($/€) = $1.875/€ then the option finishes in-the-money. €10,000 = $18,750 – $15,000 C1 = $3,750 up × €10,000 = $15,000 $1.50 €1.00 If the rate goes down, the option finishes out of the money. No one will pay $15,000 for €10,000 worth $12,000 €10,000 = $12,000 C1 = $0 down

Creating a Riskless Hedge Consider a dealer who has just written 1 at-the-money call on €10,000. He calculates the hedge ratio as 5/9: H = C – C S1 – S1 down up = $3,750 – 0 $18,750 – $12,000 $3,750 $6,750 5 9 He can hedge his position with three trades: If i$ = 3% then he could borrow $6,472.49 today and owe $6,666.66 in one period. Then buy the present value of €5,555.56 (buy euro at spot exchange rate, compute PV at i€ = 2%), Invest €5,446.62 at i€ = 2%. $6,472.49 = $6,666.66 1.03 €5,446.62 = €5,555.56 1.02 $12,000 × = $6,666.66 5 9 Net cost of hedge = $1,697.44 = €10,000 ×

Replicating Portfolio Call on €10,000 K($/€) = $1.50/€ $6,666.67 $10,416 FV € investment in $ €5,555.56 FV € investment × Corso di Laurea in International Management – $ 6,666.67 – $6,666 Service Loan Replicating Portfolio Call on €10,000 K($/€) = $1.50/€ $1.875 S1($|€) €1.00 $1.20 T = 0 T = 1 Borrow $6,472.49 at i$ = 3% Step 1 = $3,750 = 0 Buy €5,446.62 at S0($|€) = $1.50/€ Step 2 the replicating portfolio payoffs and the call option payoffs are the same so the call is worth $1,697.44 = × 5 9 $1.50 $1.20 (1.03) €1.00 €10,000 (1.02) – €10,000 = $15,000 Invest €5,446.62 at i€ = 2% Step 3 Net cost = $1,697.44

Risk Neutral Valuation of Options Corso di Laurea in International Management Calculating the hedge ratio is vitally important if you are going to use options. The seller needs to know it if he wants to protect his profits or eliminate his downside risk. The buyer needs to use the hedge ratio to inform his decision on how many options to buy. Knowing what the hedge ratio is isn’t especially important if you are trying to value options. Risk Neutral Valuation is a very hand shortcut to valuation.

Risk Neutral Valuation of Options Corso di Laurea in International Management Risk Neutral Valuation of Options We can safely assume that IRP holds: F1($/€) = $1.5147 €1.00 $1.50×(1.03) €1.00×(1.02) = €10,000 = $15,000 $1.20 €1.00 €10,000 × $12,000 = $1.875 $18,750 = Set the value of €10,000 bought forward at $1.5147/€ equal to the expected value of the two possibilities shown above: $15,147.06 = p × $18,750 + (1 – p) × $12,000 €10,000× $1.5147 €1.00 =

Risk Neutral Valuation of Options Corso di Laurea in International Management Risk Neutral Valuation of Options Solving for p gives the risk-neutral probability of an “up” move in the exchange rate: $15,147.06 = p × $18,750 + (1 – p) × $12,000 p = $15,147.06 – $12,000 $18,750 – $12,000 p = .4662

Risk Neutral Valuation of Options Corso di Laurea in International Management Risk Neutral Valuation of Options Now we can value the call option as the present value (discounted at the USD risk-free rate) of the expected value of the option payoffs, calculated using the risk-neutral probabilities. €10,000 = $15,000 $1.20 €1.00 €10,000 × $12,000 = $1.875 $18,750 = ←value of €10,000 $3,750 = payoff of right to buy €10,000 for $15,000 $1,697.44 $0 = payoff of right to buy €10,000 for $15,000 C0 = $1,697.44 = .4662×$3,750 + (1–.4662)×0 1.03

Corso di Laurea in International Management Test Your Intuition Use risk neutral valuation to find the value of a put option on $15,000 with a strike price of €10,000. Hint: given that we just found that the value of a call option on €10,000 with a strike price of $15,000 was $1,697.44 this should be easy in the sense that we already know the right answer. $1.50 €1.00 S0($/€) = As before, i$ = 3%, i€ = 2%, $1.50×1.03 €1.00×1.02 F1($/€) = $1.5147 €1.00 =

Test Your Intuition (continued) $1.50×1.03 €1.00×1.02 F1($/€) = $1.5147 €1.00 = €10,000 = $15,000 €1.00 $1.875 $15,000 × €8,000 = $1.20 €12,500 = ←value of $15,000 $15,000 × $1.5147 €1.00 = €9,902.91 €9,902.91 = p × €12,500 + (1 – p) × €8,000 p = €9,902.91– €8,000 €12,500 – €8,000 p = .4229

Test Your Intuition (continued) Corso di Laurea in International Management Test Your Intuition (continued) €1.00 $1.20 $15,000 × €12,500 = ←value of $15,000 0 = payoff of right to sell $15,000 for €10,000 €10,000 = $15,000 €1,131.63 €1.00 $1.875 $15,000 × €8,000 = ←value of $15,000 €2,000 = payoff of right to sell $15,000 for €10,000 €P0 = €1,131.63 = .4229×€0 + (1–.4229)×€2,000 1.02 $P0 = $1,697.44 = €1,131.63 × $1.50 €1.00

Test Your Intuition (continued) Corso di Laurea in International Management Test Your Intuition (continued) The value of a call option on €10,000 with a strike price of $15,000 is $1,697.44 The value of a put option on $15,000 with a strike price of €10,000 is €1,131.63 At the spot exchange rate these values are the same: €1,131.63 × €1.00 $1.50 = $1,697.44

Corso di Laurea in International Management Take-Away Lessons Convert future values from one currency to another using forward exchange rates. Convert present values using spot exchange rates. Discount future values to present values using the correct interest rate, e.g. i$ discounts dollar amounts and i€ discounts amounts in euro. To find the risk-neutral probability, set the forward price derived from IRP equal to the expected value of the payoffs. To find the option value discount the expected value of the option payoffs calculated using the risk neutral probabilities at the correct risk free rate.

Finding Risk Neutral Probabilities up F1($/€) = p × S1 ($/€) + (1 – p) × S1 ($/€) down For a call on €10,000 with a strike price of $15,000 we solved $15,147.06 = p × $18,750 + (1 – p) × $12,000 p = $15,147.06 – $12,000 $18,750 – $12,000 = .4662 = $1.5147 – $1.20 $1.875 – $1.20 For a put on $15,000 with a strike price of €10,000 we solved €9,902.91 = p × €12,500 + (1 – p) × €8,000 p = €9,902.91– €8,000 €12,500 – €8,000 = .4229 = €0.6602– €.5333 €.8333 – €.5333

Currency Futures Options Corso di Laurea in International Management 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.

Currency Futures Options Corso di Laurea in International Management Currency Futures Options Why a derivative on a derivative? Transactions costs and liquidity. For some assets, the futures contract can have lower transactions costs and greater liquidity than the underlying asset. Tax consequences matter as well, and for some users an option contract on a future is more tax efficient. The proof is in the fact that they exist.

Binomial Futures Option Pricing A 1-period at-the-money call option on euro futures has a strike price of F1($|€) = $1.5147/€ $1.50×1.03 €1.00×1.02 F1($|€) = $1.5147 €1.00 = $1.875×1.03 $1.8934 $1.20×1.03 $1.2118 Call Option Payoff = $0.3787 Option Price = ? Option Payoff = $0 When a call futures option is exercised the holder acquires 1. A long position in the futures contract 2. A cash amount equal to the excess of the futures price over the strike price

Binomial Futures Option Pricing $1.875×1.03 €1.00×1.02 F1($|€) = $1.8934 €1.00 = Consider the Portfolio: long Η futures contracts short 1 futures call option Futures Call Payoff = –$0.3787 Futures Payoff = H × $0.3603 $1.50×1.03 €1.00×1.02 F1($|€) = $1.5147 €1.00 = Portfolio Cash Flow = H × $0.3603 – $0.3787 Option Price = $0.1714 Portfolio is riskless when the portfolio payoffs in the “up” state equal the payoffs in the “down” state: H×$0.3603 – $0.3787 = –H×$0.3147 The “right” amount of futures contracts is Η = 0.5610 $1.20×1.03 €1.00×1.02 F1($|€) = $1.2118 €1.00 = Portfolio Cash Flow = –H×$0.3147 Option Payoff = $0 Futures Payoff = –H×$0.3147

Binomial Futures Option Pricing $1.875×1.03 €1.00×1.02 F1($|€) = $1.8934 €1.00 = The payoffs of the portfolio are –$0.1766 in both the up and down states. Call Option Payoff = –$0.3787 Futures Payoff = H × $0.3603 Portfolio Cash Flow = 0.5610 × $0.3603 – $0.3787 = –$0.1766 $1.50×1.03 €1.00×1.02 F1($|€) = $1.5147 €1.00 = There is no cash flow at initiation with futures. Without an arbitrage, it must be the case that the call option income is equal to the present value of $0.1766 discounted at i$ = 3% $1.20×1.03 €1.00×1.02 F1($|€) = $1.2118 €1.00 = Futures Payoff = –0.5610×$0.3147 Option Payoff = $0 $0.1766 1.03 C0 = $0.1714 = Portfolio Cash Flow = –0.5610×$0.3147 = –$0.1766

Option Pricing $0.375 = Call payoff $0 = Put payoff C0 = $.169744 – .80 p = 1.25 – 0.80 = .4662 1.03 1.02 Find the value of an at-the-money call and a put on €1 with Strike Price = $1.50 i$ = 3% i€ = 2% u = 1.25 d = .8 $1.50 $1.875 = 1.25 × $1.50 $1.20 = 0.8 × $1.50 $0.375 = Call payoff $0 = Put payoff C0 = $.169744 P0 = $0.15555 $0 =Call payoff $0.30 = Put payoff .4662× $0.375 C0 = 1.03 = $.169744 .5338 × $0.30 P0 = 1.03 = $0.15555

Hedging a Call Using the Spot Market We want to sell call options. How many units of the underlying asset should we hold to form a riskless portfolio? $0.375 – $0 Η = $1.875 – $1.20 = 5/9 $1.50 $1.875 = 1.25 × $1.50 $1.20 = 0.8 × $1.50 $0.375 = Call payoff Sell 1 call option; buy 5/9 of the underlying asset to form a riskless portfolio. If the underlying is indivisible, buy 5 units of the underlying and sell 9 calls. $0 = Call payoff

Hedging a Call Using the Spot Market Cash Flows T = 1 S1($|€) = $1.875 $0.375 – $0 Η = $1.875 – $1.20 = 5/9 C1= $.375 Call finishes in-the-money, so we must buy an additional €4 at $1.875. Cost = 4 × $1.875 = $7.50 Cash inflow call exercise = 9 × $1.50 = $13.50 Portfolio cash flow = $6.00 S0($|€) = $1.50/€ Go long PV of €5. Cost today = €5 1.02 × = $7.3529 $1.50 €1.00 S1($|€) = $1.20 C1= $0 Call finishes out-of-the-money, so we can sell our now-surplus €5 at $1.20. Cash inflow = 5 × $1.20 = $6.00 Write 9 calls: Cash inflow = 9 × $0.169744 = $1.5277 Portfolio cash flow today = –$5.8252 Handy thing to notice: $5.8252 × 1.03 = $6.00

Hedging a Put Using the Spot Market We want to sell put options. How many units of the underlying asset should we hold to form a riskless portfolio? $0 – $0.30 Η = $1.875 – $1.20 = – 4/9 S1($|€) = $1.875 Put payoff = $0.0 S0($|€) = $1.50/€ S1($|€) = $1.20 Put payoff = $0.30 Sell 1 put option; short sell 4/9 of the underlying asset to form a riskless portfolio. If the underlying is indivisible, short 4 units of the underlying and sell 9 puts.

Hedging a Put Using the Spot Market Cash Flows T = 1 S1($|€) = $1.875 $0 – $0.30 Η = $1.875 – $1.20 = – 4/9 Put finishes out-of-the-money. To repay loan buy €4 at $1.875. Cost = 4 × $1.875 = $7.50 Option cash inflow = 0 Portfolio cash flow = $7.50 S0($|€) = $1.50/€ Borrow the PV of €4 at i€ = 2%. Inflow = €4 1.02 × = $5.8824 $1.50 €1.00 S1($|€) = $1.20 put finishes in-the-money, so we must buy 9 units of underlying at $1.50 each = 9×1.50 = $13.50 use 4 units to cover short sale, sell remaining 5 units at $1.20 = $6.00 Portfolio cash flow = $7.50 Write 9 puts: Cash inflow = 9 × $0.15555 = $1.3992 Portfolio Inflow today = $7.2816 Handy thing to notice: $7.2816 × 1.03 = $7.50

Hedging a Call Using Futures Futures contracts matures: buy 5 units at forward price. Cost = 5× $1.5147 = $7.5735 Call finishes in-the-money, we must buy 4 additional units of underlying at S1($/€) = $1.875. Cost = 4 × $1.875 = $7.50 Option cash inflow = 9 × $1.50 = $13.50 Portfolio cash flow = –$1.5735 Go long 5 futures contracts. Cost today = 0 Forward Price = Futures contracts matures: buy 5 units at forward price. Cost = 5× $1.5147 = $7.5735 × = $1.5147 1.03 1.02 $1.50 €1.00 Call finishes out-of-the-money, so we sell our 5 units of underlying at $1.20. Cash inflow = 5 × $1.20 = $6.00 Portfolio cash flow = –$1.5735 Write 9 calls: Cash inflow = 9 × $0.169744 = $1.5277 Portfolio cash flow today = $1.5277 Handy thing to notice: $1.5277 × 1.03 = $1.5735

Hedging a Put Using Futures Futures contracts matures: sell €5 at forward price. Loss = 4× [$1.875 – $1.5147] = $1.4412 Put finishes out-of-the-money. Option cash flow = 0 Portfolio cash flow = –$1.4412 Go short 4 futures contracts. Cost today = 0 Forward Price = Put finishes in-the-money, we must buy €9 at $1.50/€ = 9×1.50 = $13.50 Futures contracts matures: sell €4 at forward price $1.5147/€ 4× $1.5147 = $6.0588 sell remaining €5 at $1.20 = $6.00 Portfolio cash flow = –$1.4412 × = $1.5147 1.03 1.02 $1.50 €1.00 Write 9 puts: Cash inflow = 9 × $0.15555 = $1.3992 Portfolio Inflow today = $1.3992 Handy thing to notice: $1.3992 × 1.03 = $1.4412

– .80 p = 1.25 – 0.80 = .4662 1.03 1.02 2-Period Options Value a 2-period call option on €1 with a strike price = $1.50/€ i$ = 3%; i€ = 2% u = 1.25; d = .8 S2 = $2.3438 S2 = $1.50 up-down up-up C2 = $0.8468 up-up C1 = $1.0609 up S0 = $1.50/€ S1 = $1.875 S1 = $1.20 down up C0 = $0.4802 S2 = $0.96 down-down C2 = $0 up-down C1 = $0 down .4662× $0.8468 C1 = 1.03 = $1.06 up 1.03 .4662× $1.0609 C0 = = $0.4802 C2 = $0 down-down