FIN417- lecture 8 (exam 2)

FIN417- lecture 8 (exam 2) - INTERNATIONAL FINANCIAL...

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Unformatted text preview: INTERNATIONAL FINANCIAL MANAGEMENT Fifth Edition Instructor’s Notes: Students must augment these materials for their own study purposes! EUN / RESNICK Adapted by M.D. Griffiths Lecture 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 7­2 Lecture 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 7­3 Forwards and Futures: definition Contracts that specify today the price for a future delivery of an asset or a commodity at a specific time. In general they follow equilibrium, no-arbitrage pricing conditions. Forward and futures differ for technical reasons but essentially satisfy the same need and follow the same rules. But: Are Forward and futures price are different? Futures Contracts: Preliminaries A futures contract is like a forward contract: A futures contract is different from a forward contract: 7­5 It specifies that a certain currency will be exchanged for another at a specified time in the future at prices specified today. Futures are standardized contracts trading on organized exchanges with daily resettlement through a clearinghouse. 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). 7­6 Currency Futures Markets The Chicago Mercantile Exchange (CME) is by far the largest followed by Singapore Exchange (SIMEX) Others include: The Philadelphia Board of Trade (PBOT) The Tokyo International Financial Futures Exchange The London International Financial Futures Exchange Contracts on the CME market generally trade according to the following rules Expiry cycle: March, June, September, December. Delivery date third Wednesday of delivery month. Last trading day is the 2nd business day preceding the delivery day. 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. 7­8 Futures Contracts: Preliminaries Some definitions apply: Futures price is the market price of the contract Forward price is the market price for a range of delivery dates Delivery price is the contracted upon price for the contract Settlement price is the price at the end of each trading day before delivery BE CAREFUL: you contract on something but then the market fluctuates and the delivery price remains constant Convergence of Futures to Spot The closer the delivery date, the more the futures contract will converge to the market spot price. Mkt Forward Price Spot Price Mkt Forward Price Spot Price Time (a) (b) Time Forward Contracts vs Futures Contracts FORWARDS FUTURES Private contract between 2 parties Exchange traded Non-standard contract Standard contract Usually 1 specified delivery date Settled at maturity Delivery or final cash settlement usually occurs Range of delivery dates Settled daily Contract usually closed out prior to maturity Futures Contracts Available on a wide range of underlying assets Exchange traded Specifications need to be defined: What can be delivered, Where it can be delivered, When it can be delivered Settled daily Futures Contracts: Preliminaries Standardizing Features: Contract Size Delivery Month Daily resettlement Margins requirements Most contracts are closed out before maturity Delivery If a contract is not closed out before maturity, it usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses. A few contracts (for example, those on stock indices and Eurodollars) are settled in cash Margins A margin is cash or marketable securities deposited by an investor with his or her broker Initial performance bond (about 2 percent of contract value, cash or T-bills held in a street name at your brokerage). Used for margins requirement. Depends on leverage. 2%=50:1 The balance in the margin account is adjusted to reflect daily settlement Margins minimize the possibility of a loss through a default on a contract Some additional definitions Open interest: the total number of contracts outstanding equal to number of long positions or number of short positions 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. Volume of trading: the number of trades in 1 day Reading Currency Futures Quotes OPEN HIGH LOW SETTLE CHG LIFETIME HIGH LOW OPEN INT Euro/US Dollar (CME)—€125,000; $ per € Mar 1.3136 1.3167 1.3098 1.3112 -.0025 1.3687 1.1363 159,822 Jun 1.3170 1.3193 1.3126 1.3140 -.0025 1.3699 1.1750 10,096 Closing price Highest and lowest prices over the life Daily Change of the contract. Opening price Lowest price that day Number of open contracts Highest price that day Expiry month Reading Currency Futures Quotes OPEN HIGH LOW SETTLE CHG LIFETIME HIGH LOW OPEN INT Euro/US Dollar (CME)—€125,000; $ per € Mar 1.3136 1.3167 1.3098 1.3112 -.0025 1.3687 1.1363 159,822 Jun 1.3170 1.3193 1.3126 1.3140 -.0025 1.3699 1.1750 10,096 Sept 1.3202 1.3225 1.3175 1.3182 -.0025 1.3711 1.1750 600 Notice that open interest is larger the closer the delivery date, in this case March, 2005. In general, open interest typically decreases with term to maturity of most futures contracts. Basic Currency Futures Relationships OPEN HIGH LOW SETTLE CHG LIFETIME HIGH LOW OPEN INT Euro/US Dollar (CME)—€125,000; $ per € Mar 1.3136 1.3167 1.3098 1.3112 -.0025 1.3687 1.1363 159,822 The holder of a long position is committing himself to pay $1.3112 per euro for €125,000—a $163,900 position. As there are 159,822 such contracts outstanding, this represents a notational principal of over $26 billion (NxFXxSize)! Basic Currency Futures Relationships OPEN HIGH LOW SETTLE CHG LIFETIME HIGH LOW OPEN INT Euro/US Dollar (CME)—€125,000; $ per € Mar 1.3136 1.3167 1.3098 1.3112 -.0025 1.3687 1.1363 159,822 Notice that if you had been smart or lucky enough to open a long position at the lifetime low of $1.1363 by now your gains would have been $21,862.50 = ($1.3112/€ – $1.1363/€) × €125,000 REMEMBER: it’s a zero sum game! Someone took the short position at $1.1363 and lost the same amount! Basic Currency Futures Relationships OPEN HIGH LOW SETTLE CHG LIFETIME HIGH LOW OPEN INT Euro/US Dollar (CME)—€125,000; $ per € Mar 1.3136 1.3167 1.3098 1.3112 -.0025 1.3687 1.1363 159,822 If you had been smart or lucky enough to open a short position at the lifetime high of $1.3687 by now your gains would have been: $7,187.50 = ($1.3687/€ – $1.3112/€) × €125,000 Reading Currency Futures Quotes OPEN HIGH LOW SETTLE CHG LIFETIME HIGH LOW OPEN INT Euro/US Dollar (CME)—€125,000; $ per € Mar 1.3136 1.3167 1.3098 1.3112 -.0025 1.3687 1.1363 159,822 Jun 1.3170 1.3193 1.3126 1.3140 -.0025 1.3699 1.1750 10,096 Sept 1.3202 1.3225 1.3175 1.3182 -.0025 1.3711 1.1750 600 Recall, our interest rate parity condition: 1 + i$ F($/€) = 1 + i€ S($/€) Reading Currency Futures Quotes OPEN HIGH LOW SETTLE CHG LIFETIME HIGH LOW OPEN INT Euro/US Dollar (CME)—€125,000; $ per € Mar 1.3136 1.3167 1.3098 1.3112 -.0025 1.3687 1.1363 159,822 Jun 1.3170 1.3193 1.3126 1.3140 -.0025 1.3699 1.1750 10,096 Sept 1.3202 1.3225 1.3175 1.3182 -.0025 1.3711 1.1750 600 From June to September the euro is at a premium (weaker dollar) therefore we should expect higher interest rates in dollar denominated accounts: if we find a higher rate in a euro denominated account, we may have found an arbitrage. 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. 7­24 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. 7­25 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 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)$6,500 + $1,250 $7,750 = ×125,000 $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. 7­26 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 $1.31 $1,250 $1.30 –$1,250 $1.27 –$3,750 $1.26 –$1,250 $1.24 –$2,500 7­27 Account Balance $7,750 $6,500 $2,750 + $3,750 = $6,500 $5,250 = $6,500 – $1,250 $2,750 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) 7­28 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) 7­29 Sample Problem The March 2008 Mexican peso futures contract has a price of $0.90975 per 10MXN. You believe the spot price in March will be $0.97500 per 10MXN. What speculative position would you enter into to attempt to profit from your beliefs? Calculate your anticipated profits, assuming you take a position in three contracts. What is the size of your profit (loss) if the futures price is indeed an unbiased predictor of the future spot price and this price materializes? Sample Problem If you expect the Mexican peso to rise from $0.90975 to $0.97500 per 10 MXN, you would take a long position in futures since the futures price of $0.90975 is less than your expected spot price. You expect the peso to appreciate and want to receive the benefit in the margin account. Your anticipated profit from a long position in three contracts is: 3 x ($0.097500 - $0.090975) x MP500,000 = $9,787.50, where MXN500,000 is the contractual size of one MXN contract. Sample Problem If the futures price is an unbiased predictor of the expected spot price, the expected spot price is the futures price of $0.90975 per 10 MXN. If this spot price materializes, you will not have any profits or losses from your long position in three futures contracts: 3 x ($0.090975 - $0.090975) x MP500,000 = 0 Other futures: Eurodollar Interest Rate Futures Contracts Widely used futures contract for hedging shortterm 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 OPEN HIGH LOW SETTLE CHG YLD CHG OPEN INT Eurodollar (CME)—1,000,000; pts of 100% Jun 96.56 96.58 96.55 96.56 - 3.44 - 1,398,959 Eurodollar futures prices are stated as an index number of three-month LIBOR calculated as F = 100 – LIBOR. Lenders buy the rate! 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. Other futures: Eurodollar Interest Rate Futures Contracts Eurodollar futures are a way for companies and banks to lock in an interest rate today, for money it intends to borrow or lend in the future. CME Eurodollar futures prices are determined by the market’s forecast of the 3-month USD Libor interest rate expected to prevail on the settlement date. The settlement price of a contract is defined to be 100.00 minus the official British Bankers Association fixing of 3-month Libor on the contract settlement date. For example, if 3month Libor sets at 5.00% on the contract settlement date, the contract settles at a price of 95.00 What do we want? Amy wants to lock in an interest rate so that the correct amount of money will be ready when we need it. Therefore, we need the money at the instrument’s maturity. But, the amount available at maturity is a function of the interest rate which is determined at the beginning of the investment period . So, for certainty in July, we want the future to mature in April! And we want to trade it back there! M A M J J A S O Cash Flows For a 3-mth instrument, the interest rate is known with certainty here! We need the money here so, this is when the instrument has to mature! Making Futures Work Buying a future is an obligation to buy, selling a future is an obligation to sell. To hedge with futures, borrowers sell to lock in their interest rate. When hedging with futures, investors buy to lock in their rates. If interest rates rise, then what will happen to the futures price? Amy is worried about interest rates falling, so does she want to buy or sell a futures contract? Why is she worried about interest rates falling? What month futures contract does she want? The Variation Margin Let’s say Amy buys 100 June futures contract at 93.51. If we look at the first week of trading, it might look like this: Day Mon trade Mon close Tue close Wed close Thur close Fri close Closing Price 93.51 93.54 93.53 93.48 93.50 93.53 Tick Change 0 +3 -1 -5 ___ +2 ___ +3 ___ Margin Change 0 7,500 -2,500 -12,500 ______ ______ 5,000 7,500 ______ Number of ticks * tick value * Number of contracts = Margin change Did the Hedge Work? Fixed rate of interest _____% 6.49 Futures profit/loss _____MM Return on Investment _____MM Total return _____MM Rate of return _____% Rate that futures locks in to. If rate this = lock rate then hedge worked. Amy was concerned in Mar that when she received the money in June interest rates would have declined, so she bought a futures contract to fix the interest rate from June to September. Let’s say, Amy bought the future at $93.51, so she locked in what rate? Did the Hedge Work? 6.49 Fixed rate of interest _____% 0.235 Futures profit/loss _____MM Return on Investment _____MM Total return _____MM Rate of return _____% Rate that futures locks in to. If this rate = lock rate then hedge worked. Assume that in June, the futures contract settles at 94.45. What is the profit or loss on the contract? Number of ticks * Tick value * No. of contract = Profit/loss 94 $25 100 _______ _____ _______ _$235,000 ________ Did the Hedge Work? Fixed rate of interest _____% 6.49 0.235 Futures profit/loss _____MM Return on Investment _____MM Total return _____MM Rate of return _____% Rate that futures locks in to. If this rate = lock rate then hedge worked. Assume the June futures contract settles at 94.45 Now let’s look at the cash flow on the investment: When Amy gets the money in June, she immediately invests it at the then current 3-month money market rate which is _5.55% ______ Did the Hedge Work? Fixed rate of interest _____% 6.49 0.235 Futures profit/loss _____MM 1.3875 Return on Investment _____MM Total return _____MM Rate of return _____% Rate that futures locks in to. If this rate = lock rate then hedge worked. Assume the June futures contract settles at 94.45 Based on this, Amy’s cash flow from the money market would be: $100,000,000 * 5.55% * 3 = $1,387,500 12 Did the Hedge Work? 6.49 Fixed rate of interest _____% 0.235 Futures profit/loss _____MM 1.3875 Return on Investment _____MM 1.6225 Total return _____MM Rate of return _____% Rate that futures locks in to. If this rate = lock rate then hedge worked. Now, we can figure out what Amy’s total return would be: Futures profit/loss + ROI = $235,000 + $1,387,500 = $1,622,500 Did the Hedge Work? Fixed rate of interest _____% 6.49 0.235 Futures profit/loss _____MM 1.3875 Return on Investment _____MM 1.6225 Total return _____MM 6.49 Rate of return _____% Rate that futures locks in to. If this rate = lock rate then hedge worked. Now, we can figure out what Amy’s rate of return would be: $1,622,500 12 * = 6.49% $100,000,000 3 Hedging with Futures Interest rate futures are quoted as 100 minus the rate of interest. This is known as back-to-front pricing. Back-to-front pricing affects the decision of whether to buy or sell. A borrower is concerned about interest rates rising. When hedging with FRAs, borrowers buy to lock in their interest rate. To hedge with futures, borrowers sell to lock in their interest rate. Since the pricing of futures is inverse, so is the trading. When hedging with futures, investors buy to lock in their rates. Trading irregularities Futures Markets are also a great place to launder money 7­47 The zero sum nature of futures is the key to laundering the money. Money Laundering: Hillary Clinton’s Cattle Futures James B. Blair outside counsel to Tyson Foods Inc., Arkansas' largest employer, gets Hillary’s discretionary order. 7­48 winners losers Submits identical long and short trades Robert L. "Red" Bone, (Refco broker), allocates trades ex post facto. Options Contracts: Preliminaries 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 7­49 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 European vs. American options 7­50 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 In-the-money At-the-money The exercise price is less than the spot price of the underlying asset. The exercise price is equal to the spot price of the underlying asset. Out-of-the-money 7­51 The exercise price is more than the spot price of the underlying asset. Options Contracts: Preliminaries 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 7­52 = Intrinsic Value + Speculative Value Currency Options Markets 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. 7­53 PHLX Currency Option Specifications Currency Australian dollar British pound Canadian dollar Euro Japanese yen Swiss franc 7­54 Contract Size AD10,000 £10,000 CAD10,000 €10,000 ¥1,000,000 SF10,000 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 – E. T If the call is out-of-the-money, it is worthless. CaT = CeT = Max[ST - E, 0] 7­55 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. PaT = PeT = Max[E – ST, 0] 7­56 Basic Option Profit Profiles Profit Owner of the call If the call is in-themoney, it is worth ST – E. If the call is out-ofthe-money, it is worthless and the –c 0 E + c0 buyer of the call E loses his entire investment of c0. Out-of-the-money In-the-money loss 7­57 Long 1 call ST Basic Option Profit Profiles Profit Seller of the call If the call is in-themoney, the writer loses ST – E. If the call is out-ofthe-money, the writer keeps the option premium. c0 ST E loss 7­58 Out-of-themoney E + c0 In-themoney short 1 call Basic Option Profit Profiles Profit If the put is inthe-money, it is E – p 0 worth E – ST. The maximum gain is E – p0 If the put is outof-the-money, it – p0 is worthless and the buyer of the put loses his entire investment loss of p0. 7­59 Owner of the put ST E – p0 long 1 put E In-the-money Out-of-themoney Basic Option Profit Profiles If the put is inthe-money, it is worth E –ST. The maximum loss is – E + p0 Profit Seller of the put p0 If the put is outof-the-money, it is worthless and the seller of the put keeps the option premium– E + p 0 of p0. loss 7­60 ST E – p0 E short 1 put Example Profit Consider a call option on €31,250. The option premium is $0.25 per € The exercise price is $1.50 per €. –$0.25 Long 1 call on 1 euro ST $1.75 $1.50 loss 7­61 Example Profit Consider a call option on €31,250. The option premium is $0.25 per € The exercise price is $1.50 per €. –$7,812.50 Long 1 call on €31,250 ST $1.75 $1.50 loss 7­62 Example Profit What is the maximum gain on this put option? $42,187.50 $42,187.50 = € 31,250 × ($1.50 – $0.15)/ € Consider a put option on €31,250. The option premium is $0.15 per € At what exchange rate do you break even? ST –$4,687.50 $1.35 The exercise price is $1.50 per euro. loss 7­63 $1.50 Long 1 put on €31,250 $4,687.50 = € 31,250 × ($0.15)/ € Put-Call Parity Relationship (equities) An investor has two ways to achieve limited downside risk with unlimited upside protection: Purchase a stock today (S) and buy a put option (p) at a given exercise price (X). S+p Buy a risk free zero coupon bond with a maturity value of X and buy a call option (c) at a given exercise price (X). X∗ e-r∗ t + c Relation Between Prices of a Put and a Call with Same Features? Strategy 1 Buy Stock; Buy a PUT Investment: Expiration: S>X S<X S+p Strategy 2 Buy Call; Invest PV(X) c + PV(X) Value = S Value = X Value = S Value = X Since the Outcomes are the Same , the Investments must be Equal, i.e., [S + p = c + PV(X)] or p = c - S + PV(X) This is Known as PutCall Parity Put-Call Parity Relationship The put-call parity equation implies the following: The greater the exercise price the more expensive is the put premium. The higher the risk-free rate the less expensive is the put premium. The higher the purchase price of the stock the less expensive is the put premium Put-Call Parity: Example Suppose We Knew a 1 Year Call Option with an Exercise Price of $25 was Worth $5 Today given 10% Interest Rates. If the Stock Price is Currently $21 What is the Put Option Worth? p= = = = c - S + Xe-rt Xe $5 - $21 + $25*(0.904) $5 $5 - $21 + $22.62 $5 $6.62 $6.62 Put-call parity: example On June 20th, ABC Oct 65 puts are quoted at 12, and the Oct 65 calls are quoted at 8.5. If ABC pays no dividend and rates are 7% per annum (continuously compounded), at what stock price were these options quoted. (June 20th is exactly 4 months from Oct expiry). Assume European style options. c - p = S - Xe-rT S = 8.5 - 12 + 65e-0.07x0.3333 = 60.00 Arbitrages with Put-Call Parity Stock ABC trades at $35, and 3 month 30 strike calls trade at 7. If rates are 6% (continuously compounded), where should the put trade? p = c + X e-rT - S = 7 + 30x0.98 - 35 = 1.40 If the put trades at 3 (and everything else is the same) how do you take advantage of the mispricing? Theoretically there should be arbitrage profit of $1.60 Sell the LHS of the above equation, and buy the RHS Arbitrages with Put-call parity Sell the LHS, means sell a put at 3, and invest the proceeds at the risk-free rate. Buy the RHS means buy a call at 7, a risk-free zero coupon bond at Xe-rt = 29.40, and sell the stock at 35. Notice the $1.40 we are short is more than offset by the put sale. On day 0, we have $1.60 in our pocket. Arbitrages with Put-call parity At expiry, if the put finishes in the money, we will have the stock “put” to us at 30. If the call finishes in the money, we will exercise to buy the stock at 30. In either case we are buying the stock at 30. Given this accounting, we can ignore any residual option value. (This is how brokers account for options expiry) The zero coupon bond we purchased for 29.40 is now worth $30 and just covers the cost of the stock purchase We keep the $1.60 + interest. Put-Call Parity Relationship S0(1+ ρ)-T + Pe = Ce + X(1+r)-T Where ρ is the foreign interest rate and the spot rate is being quoted in units of the foreign currency Note similarity to put-call parity for options on stocks. Also, a call to buy one currency denominated in the other with exercise price of X is equivalent to a put to sell X units of the latter currency denominated in the former currency with exercise price of 1/X. American Option Pricing Relationships With an American option, you can do everything that you can do with a European option AND you can exercise prior to expiry—this option to exercise early has value, thus: CaT > CeT = Max[ST - E, 0] PaT > PeT = Max[E - ST, 0] 7­73 Market Value, Time Value and Intrinsic Value for an American Call Profit The red line shows the payoff at maturity, not profit, of a call option. Ma et V rk Note that even an outof-the-money option has value—time value. 7­74 Long 1 call Intrinsic value Time value Out-of-the-money loss e alu E In-the-money ST European Option Pricing Relationships Consider two investments 1 Buy a European call option on the British pound futures contract. The cash flow today is –Ce Replicate the upside payoff of the call by 2 Borrowing the present value of the dollar exercise price of the call in the U.S. at i$ 1 The cash flow today is 2 Lending the present value of ST at i£ The cash flow today is 7­75 E (1 + i$) – ST (1 + i£) 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 than the borrowing and lending strategy. Thus: ST E Ce > Max (1 + i ) – (1 + i ) , 0 £ $ 7­76 European Option Pricing Relationships Using a similar portfolio to replicate the upside potential of a put, we can show that: E ST Pe > Max (1 + i ) – (1 + i ) , 0 $ £ 7­77 What is the Relation Between Price Changes on Stock and Call? The Call Option is Exchangeable into the Underlying Stock The Value of the Call is Driven (Largely) by the Value of the Underlying Stock When the Stock Price Goes Up so Does the Call Price; When the Stock Price Goes Down so Does the Call Price Perfect Positive Correlation Potential to be Able to Create a Perfect Hedge Valuation European Call - Under Uncertainty Current Stock Price = $ 20; Exercise Price = $ 22.50; Time to Expiration = 1 YR; Risk Free Rate = 10%; Stock Price in One Year is Either $ 25 or $ 15 $25 Option = $ 2.50 $15 Option = $ 0.00 $20 Eliminate Uncertainty? How? The Delta Hedge! Form a Hedge Portfolio Long the Stock and Short the Call; (Reverse Correlation with Respect to Impact on Wealth) 25 H - 2.50 = 15 H => H = 0.25 Long 0.25 Shares and Short 1 Call Option If Stock Price Moves Up to $ 25; Portfolio = 25 * 0.25 - 2.50 = $ 3.75 If Stock Price Moves Down to $ 15; Portfolio = 15 * 0.25 - 0 = $ 3.75 Hedge Ratio = 0.25; Regardless of Whether Stock Goes Up or Down -- the Value of Hedge Portfolio is $ 3.75 Value of Call Option Today ? Value Risk-less Portfolio, Absent Arbitrage Opportunities, Must Earn the Risk-Free Rate of Interest Value of the Portfolio Today = PV(3.75) = 3.75 * e (- 0.1 )( 1)= $ 3.39 The Value of the Stock Today is $20; The Value of the Portfolio Today is 20 * 0.25 - C = 5 - C = $ 3.39 or C = $ 1.61 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/€ 0 1. Borrow $1.5m at i$ = 3% 2. Exchange $1.5m for €1m at spot 3. Invest €1m at i€ = 2% 7­82 1 4. Owe $1.545m $1.5147 F1($/€) = €1.00 5. Receive €1.02 m 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 $1.50 7­83 $1.20 Binomial Option Pricing Model A call option on the euro with exercise price S0($/€) = $1.50 will have the following payoffs. By exercising the call option, you can buy €1 for $1.50. If S1($/€) = $1.875/€ the option is in-the-money: S0($/€) S1($/€) $1.875 $1.50 …and if S1($/€) = $1.20/€ the option is out-of-the-money: 7­84 $1.20 C1($/€) $.375 $0 Binomial Option Pricing Model We can replicate the payoffs of the call option. By taking a position in the euro along with some judicious borrowing and lending. S0($/€) S1($/€) $1.875 C1($/€) $.375 $1.20 $0 $1.50 7­85 Binomial Option Pricing Model Borrow the present value (discounted at i$) 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 $1.50 7­86 $1.20 – $1.20 = $0 $0 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 1.80 = $.375 S0($/€) S1($/€) debt portfolio C1($/€) $1.875 – $1.20 = $.675 $.375 $1.50 7­87 $1.20 – $1.20 = $0 $0 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: €1.00 × $1.50 – $1.20 (1 + i€) €1.00 (1 + i$) S0($/€) S1($/€) debt portfolio C1($/€) $1.875 – $1.20 = $.675 $.375 $1.50 7­88 $1.20 – $1.20 = $0 $0 Binomial Option Pricing Model We can value the call option as 5/9 of the value of the replicating portfolio: 5 €1.00 × $1.50 – $1.20 C0 = 9 × (1 + i€) €1.00 (1 + i$) S0($/€) $1.50 7­89 S1($/€) debt portfolio C1($/€) $1.875 – $1.20 = $.675 $.375 If i$ = 3% and i€ = 2% the call is worth 5 €1.00 $1.50 – $1.20 $0.1697 = × × 9 (1.02) €1.00 (1.03) $1.20 – $1.20 = $0 $0 Binomial Option Pricing Model The most important lesson from the binomial option pricing model is: the replicating portfolio intuition. the Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities. 7­90 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: C up – C down H= S1up – S1 down 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. 7­91 Hedge Ratio 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: $0.375 $0.375 – $0 Cup – C down = = H= = S1up – S1down $1.875 – $1.20 $0.675 The delta of a put option is negative. Deltas change through time. 7­92 5 9 Creating a Riskless Hedge The standard size of euro options on the PHLX is €10,000. In our simple world where the dollar-euro exchange rate is S0($/€) = $1.50/€ today and in the next year, S1($/€) is either $1.875/€ or $1.20/€ an at-the-money call on €10,000 has these payoffs: $1.875 If the exchange rate at maturity goes up × 10,000 = $18,750 € €1.00 – $15,000 to S1($/€) = $1.875/€ then the option up C1 = $3,750 finishes in-the-money. $1.50 × €10,000 = $15,000 €1.00 If the rate goes down, the option finishes out of the money. No one will pay $15,000 for €10,000 worth $12,000 7­93 $1.20 × 10,000 € €1.00 = $12,000 down C1 = $0 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: $3,750 C up– C down = $3,750 – 0 = = H= $18,750 – $12,000 $6,750 S1up – S1 down He can hedge his position with three trades: 1. If i$ = 3% then he could borrow $6,472.49 today and owe $6,666.66 in one period. 5 $12,000 × 9 2. = $6,666.66 $6,666.66 $6,472.49 = 1.03 Then buy the present value of €5,555.56 = €10,000 × (buy euro at spot exchange rate, €5,555.56 compute PV at i€ = 2%), €5,446.62 = 1.02 3. 5 9 Invest €5,446.62 at i€ = 2%. 7­94 5 9 Net cost of hedge = $1,697.44 Service Loan FV € investment in $ T=0 FV € investment S1($|€) Replicating Portfolio Call on €10,000 K($/€) = $1.50/€ T=1 Step 1 Borrow $6,472.49 at i$ = 3% $1.875 × €5,555.56= $10,416 $6,666 = $3,750 – €1.00 Step 2 the replicating portfolio payoffs and the call Buy €5,446.62 at option payoffs are the same so the call is worth S0($|€) = $1.50/€ €10,000 = $15,000 $1,697.44 = Step 3 Invest €5,446.62 at i€ = 2% Net cost = $1,697.44 7­95 5 × €10,000 $1.20 $1.50 – × 9 (1.02) €1.00 (1.03) $1.20 × €5,555.56 = $6,666.67 – $ 6,666.67 = 0 €1.00 Risk Neutral Valuation of Options 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 7­96 to valuation. Risk Neutral Valuation of Options We can safely assume that IRP holds: $1.5147 $1.50×(1.03) = F1($/€) = €1.00×(1.02) €1.00 $1.875 × €10,000 $18,750 = €1.00 €10,000 = $15,000 $1.20 × €10,000 $12,000 = €1.00 Set the value of €10,000 bought forward at $1.5147/€ equal to the expected value of the two possibilities shown above: $1.5147 €10,000× €1.00 = $15,147.06 = p × $18,750 + (1 – p) × $12,000 7­97 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 $15,147.06 – $12,000 p= $18,750 – $12,000 p = .4662 7­98 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 riskneutral probabilities. $1.875 × €10,000 ←value of €10,000 $18,750 = €1.00 $3,750 = payoff of right to buy €10,000 for $15,000 €10,000 = $15,000 $1,697.44 C0 = $1,697.44 = 7­99 $1.20 × €10,000 ←value of €10,000 $12,000 = €1.00 $0 = payoff of right to buy €10,000 for $15,000 .4662×$3,750 + (1–.4662)×0 1.03 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 $1,697.44 this should be easy in the sense that we already know the right answer. As before, i$ = 3%, i€ = 2%, S0($/€) = $1.50 €1.00 $1.50×1.03 $1.5147 F1($/€) = €1.00×1.02 = €1.00 7­ Test Your Intuition (continued) $1.50×1.03 $1.5147 F1($/€) = €1.00×1.02 = €1.00 €12,500 = €1.00 × $15,000 ←value of $15,000 $1.20 €10,000 = $15,000 €1.00 × $15,000 ←value of $15,000 €8,000 = $1.875 €1.00 $15,000 × = €9,902.91 $1.5147 €9,902.91 = p × €12,500 + (1 – p) × €8,000 €9,902.91– €8,000 p= €12,500 – €8,000 7­ p = .4229 Test Your Intuition (continued) €1.00 × $15,000 ←value of $15,000 €12,500 = $1.20 0 = payoff of right to sell $15,000 for €10,000 €10,000 = $15,000 €1,131.63 €P0 = €1,131.63 = €1.00 × $15,000 ←value of $15,000 €8,000 = $1.875 €2,000 = payoff of right to sell $15,000 for €10,000 .4229×€0 + (1–.4229)×€2,000 1.02 1.50 $P0 = $1,697.44 = €1,131.63 × $1.00 € 7­ 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: $1.50 €1,131.63 × €1.00 = $1,697.44 7­ Take-Away Lessons Convert future values from one currency to another using forward exchange rates. Convert present values using spot exchange rates. Discount future values to present values using the correct interest rate, e.g. i$ 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. 7­ Finding Risk Neutral Probabilities down F1($/€) = p × S1 ($/€) + (1 – p) × S1 ($/€) up 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 $1.5147 – $1.20 = = .4662 $18,750 – $12,000 $1.875 – $1.20 For a put on $15,000 with a strike price of €10,000 we solved €9,902.91 = p × €12,500 + (1 – p) × €8,000 €9,902.91– €8,000 €0.6602– €.5333 p= = = .4229 €12,500 – €8,000 €.8333 – €.5333 Currency Futures Options Are an option on a currency futures contract. Exercise of a currency futures option results in a long futures position for the holder of a call or the writer of a put. Exercise of a currency futures option results in a short futures position for the seller of a call or the buyer of a put. If the futures position is not offset prior to its expiration, foreign currency will change hands. 7­ Currency Futures Options Why a derivative on a derivative? Transactions costs and liquidity. For some assets, the futures contract can have lower transactions costs and greater liquidity than the underlying asset. Tax consequences matter as well, and for some users an option contract on a future is more tax efficient. The proof is in the fact that they exist. Binomial Futures Option Pricing A 1-period at-the-money call option on euro futures has a strike price of F1($|€) = $1.5147/€ $1.875×1.03 $1.8934 = F1($|€) = €1.00×1.02 €1.00 $1.5147 F1($|€) = $1.50×1.03 = €1.00×1.02 €1.00 Option Price = ? Call Option Payoff = $0.3787 $1.20×1.03 $1.2118 = F1($|€) = €1.00×1.02 €1.00 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 Consider the Portfolio: long Η futures contracts short 1 futures call option $1.5147 F1($|€) = $1.50×1.03 = €1.00×1.02 €1.00 $1.875×1.03 $1.8934 = F1($|€) = €1.00×1.02 €1.00 Futures Call Payoff = –$0.3787 Futures Payoff = H × $0.3603 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 $1.2118 = F1($|€) = €1.00×1.02 €1.00 Futures Payoff = –H×$0.3147 Option Payoff = $0 Portfolio Cash Flow = –H×$0.3147 Binomial Futures Option Pricing The payoffs of the portfolio are –$0.1766 in both the up and down states. $1.5147 F1($|€) = $1.50×1.03 = €1.00×1.02 €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% C0 = $0.1714 = $0.1766 1.03 $1.875×1.03 $1.8934 = F1($|€) = €1.00×1.02 €1.00 Call Option Payoff = –$0.3787 Futures Payoff = H × $0.3603 Portfolio Cash Flow = 0.5610 × $0.3603 – $0.3787 = –$0.1766 $1.20×1.03 $1.2118 = F1($|€) = €1.00×1.02 €1.00 Futures Payoff = –0.5610×$0.3147 Option Payoff = $0 Portfolio Cash Flow = –0.5610×$0.3147 = –$0.1766 Option Pricing p= 1.03 – .80 1.02 1.25 – 0.80 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 C0 = C0 = $.169744 P0 = $0.11316 .4662× $0.375 1.03 $1.50 = $.169744 = .4662 $1.875 = 1.25 × $1.50 $0.375 = Call payoff $0 = Put payoff $1.20 = 0.8 × $1.50 $0 =Call payoff $0.30 = Put payoff P0 = .4229 × $0.30 1.03 = $0.11316 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? Η= $0.375 – $0 $1.875 – $1.20 = 5/9 $1.875 = 1.25 × $1.50 $0.375 = Call payoff $1.50 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. $1.20 = 0.8 × $1.50 $0 = Call payoff Hedging a Call Using the Spot Market T=0 Η= $0.375 – $0 $1.875 – $1.20 S0($|€) = $1.50/€ Cash Flows T = 1 = 5/9 S1($|€) = $1.875 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 = $6.00 Go long PV of €5. S1($|€) = $1.20 €5 $1.50 C1= $0 × = $7.3529 Cost today = 1.02 €1.00 Call finishes out-of-the-money, so we Write 9 calls: can sell our now-surplus €5 at $1.20. Cash inflow = 9 × $0.169744 = $1.5277 Cash inflow = 5 × $1.20 = $6.00 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? Η= $0 – $0.30 $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 T=0 Η= $0 – $0.30 $1.875 – $1.20 Cash Flows T = 1 = – 4/9 S0($|€) = $1.50/€ Borrow the PV of €4 at i€ = 2%. €4 $1.50 × = $5.8824 Inflow = 1.02 €1.00 Write 9 puts: Cash inflow = 9 × $0.15555 = $1.3992 Portfolio Inflow today = $7.2816 S1($|€) = $1.875 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 = $7.50 S1($|€) = $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 Handy thing to notice: $7.2816 × 1.03 = $7.50 Portfolio cash flow = $7.50 Hedging a Call Using Futures S1($|€) = $1.875 S0($|€) = $1.50/€ 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 = –$1.5735 S ($|€) = 1 Go long 5 futures contracts. $1.20 Futures contracts matures: buy 5 units at forward price. Cost = 5× $1.5147 = $7.5735 Cost today = 0 $1.50 1.03 × = $1.5147Call finishes out-of-the-money, so we Forward Price = €1.00 1.02 Write 9 calls: sell our 5 units of underlying at $1.20. Cash inflow = 9 × $0.169744 = $1.5277 Cash inflow = 5 × $1.20 = $6.00 Portfolio cash flow today = $1.5277 Portfolio cash flow = –$1.5735 Handy thing to notice: $1.5277 × 1.03 = $1.5735 Hedging a Put Using Futures S0($|€) = $1.50/€ S1($|€) = $1.875 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 = –$1.4412 S1($|€) = Go short 4 futures contracts. $1.20 Put finishes in-the-money, we must Cost today = 0 buy €9 at $1.50/€ = 9×1.50 = $13.50 $1.50 1.03 × = $1.5147 Futures contracts matures: sell €4 at Forward Price = €1.00 1.02 forward price $1.5147/€ Write 9 puts: 4× $1.5147 = $6.0588 Cash inflow = 9 × $0.15555 = $1.3992 sell remaining €5 at $1.20 = $6.00 Portfolio Inflow today = $1.3992 Portfolio cash flow = –$1.4412 Handy thing to notice: $1.3992 × 1.03 = $1.4412 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 1.03 – .80 1.02 = .4662 p= 1.25 – 0.80 Sup-up = $2.3438 2 Cup-up = $0.8468 2 S = $1.875 C = $1.0609 up 1 up 1 S0 = $1.50/€ C0 = $0.4802 Sup-down = $1.50 2 Cup-down = $0 2 Sdown $1.20 1= down .4662× $0.8468 C1 = $0 C= = $1.06 1.03 .4662× $1.0609 C0 = = $0.4802 7­ 1.03 up 1 Sdown-down $0.96 = 2 Cdown-down = $0 2 ...
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This note was uploaded on 12/10/2011 for the course FIN 417 taught by Professor Griffith during the Spring '11 term at Miami University.

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FIN417- lecture 8 (exam 2) - INTERNATIONAL FINANCIAL...

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