CH-7 Futures and Options on Foreign Exchange – International Financial Management : Book Guide

CHAPTER 7 FUTURES AND OPTIONS ON FOREIGN EXCHANGE ANSWERS & SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS QUESTIONS 1. Explain the basic differences between the operation of a currency forward market and a futures market. Answer: The forward market is an OTC market where the forward contract for purchase or sale of foreign currency is tailor-made between the client and its international bank. No money changes hands until the maturity date of the contract when delivery and receipt are typically made. A futures contract is an exchange-traded instrument with standardized features specifying contract size and delivery date. Futures contracts are marked-to-market daily to reflect changes in the settlement price. Delivery is seldom made in a futures market. Rather a reversing trade is made to close out a long or short position. 2. In order for a derivatives market to function most efficiently, two types of economic agents are needed: hedgers and speculators. Explain. Answer: Two types of market participants are necessary for the efficient operation of a derivatives market: speculators and hedgers. A speculator attempts to profit from a change in the futures price. To do this, the speculator will take a long or short position in a futures contract depending upon his expectations of future price movement. A hedger, on-the-other-hand, desires to avoid price variation by locking in a purchase price of the underlying asset through a long position in a futures contract or a sales price through a short position. In effect, the hedger passes off the risk of price variation to the speculator who is better able, or at least more willing, to bear this risk. 3. Why are most futures positions closed out through a reversing trade rather than held to delivery? Answer: In forward markets, approximately 90 percent of all contracts that are initially established result in the short making delivery to the long of the asset underlying the contract. This is natural because the terms of forward contracts are tailor-made between the long and short. By contrast, only about one percent of currency futures contracts result in delivery. While futures contracts are useful for speculation and hedging, their standardized delivery dates make them unlikely to correspond to the actual future dates when foreign exchange transactions will occur. Thus, they are generally closed out in a reversing trade. In fact, the commission that buyers and sellers pay to transact in the futures market is a single amount that covers the round-trip transactions of initiating and closing out the position. 4. How can the FX futures market be used for price discovery? Answer: To the extent that FX forward prices are an unbiased predictor of future spot exchange rates, the market anticipates whether one currency will appreciate or depreciate versus another. Because FX futures contracts trade in an expiration cycle, different contracts expire at different periodic dates into the future. The pattern of the prices of these contracts provides information as to the market’s current belief about the relative future value of one currency versus another at the scheduled expiration dates of the contracts. One will generally see a steadily appreciating or depreciating pattern; however, it may be mixed at times. Thus, the futures market is useful for price discovery, i.e., obtaining the market’s forecast of the spot exchange rate at different future dates. 5. What is the major difference in the obligation of one with a long position in a futures (or forward) contract in comparison to an options contract? Answer: A futures (or forward) contract is a vehicle for buying or selling a stated amount of foreign exchange at a stated price per unit at a specified time in the future. If the long holds the contract to the delivery date, he pays the effective contractual futures (or forward) price, regardless of whether it is an advantageous price in comparison to the spot price at the delivery date. By contrast, an option is a contract giving the long the right to buy or sell a given quantity of an asset at a specified price at some time in the future, but not enforcing any obligation on him if the spot price is more favorable than the exercise price. Because the option owner does not have to exercise the option if it is to his disadvantage, the option has a price, or premium, whereas no price is paid at inception to enter into a futures (or forward) contract. 6. What is meant by the terminology that an option is in-, at-, or out-of-the-money? Answer: A call (put) option with St > E (E > St) is referred to as trading in-the-money. If St  E the option is trading at-the-money. If St Max (1 + E i$) – (1 + STi £) , 0 Binomial Option Pricing Model - Imagine a world where the spot exchange rate is S0($/€) = $1.50/€ today and in the next year S1($/€) is either $1.80/€ or $1.20/€. - €10,000 will change from $15,000 to either $18,000 or $12,000. - A call option on €10,000 with strike price S0($/€) = $1.50 will payoff either $3,000 or zero. - If S1($/€) = $1.800/€ the option is in-the-money since you can buy €10,000 (worth $18,000 at S1($/€) = $1.80/€ ) for only $15,000. $18,000 = €10,000 × $1.80 €1.00 C1up = $3,000 $12,000 = €10,000 × $1.20 €1.00 C1down = $0 Binomial Option Pricing Model We can replicate the payoffs of the call option by taking a long position in a bond with FV = €5,000 along with the right amount of dollardenominated borrowing (in this case borrow the PV of $6,000). The portfolio payoff in one period matches the option payoffs: $ value of bond FV = €5,000 pay debt portfolio $9,000 – $6,000 = $3,000 = C1up $6,000 – $6,000 = $0 = C1down Binomial Option Pricing Model The replicating portfolio’s dollar cost today is the sum of today’s dollar cost of the present value of €5,000 less the cash inflow from borrowing the present value of $6,000: €5,000 $1.50 $6,000 × – (1 + i€ ) €1.00 (1 + i$) When S0($/€) = $1.50/€, i$ = 7.1%, and i€ = 5%, the most a willing buyer should pay for the call option is $1,540.62. That’s what it would cost him today to build a portfolio that perfectly replicates the call option payoffs— why pay more to buy a ready-made option? $1,540.62 = $7,142.86 − $5,602.24 The Hedge Ratio • We replicated the payoffs of the call option with a levered position in the underlying asset (in this case, borrowing $5,602.24 to buy €4,761.90 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 = up down S1 – S1 This ratio gives the number of units of the underlying asset we should hold and the amount of borrowing in order to create a replicating portfolio. Hedge Ratio • This practice of constructing a riskless hedge is sometimes called delta hedging. • The hedge ratio of a call option is positive. – Recall from the example: H = C1up – C1down = $3,000 – $0 = 1 S1up – S1down $18,000 – $12,000 2 The hedge ratio of a put option is negative. These hedge ratios change through time. Currency Futures Options • Currency futures options are options on a currency futures contract. • Exercise of a currency futures option results in a long futures position for the holder of a call or the writer of a put. • Exercise of a currency futures option results in a short futures position for the seller of a call or the buyer of a put. • If the futures position is not offset prior to its expiration, foreign currency will change hands. Binomial Futures Option Pricing A 1-period at-the-money call option on euro futures has a strike price of F1($|€) = $1.5300/€ $1.80×1.071 F1($|€) = €1.00×1.05 = Call Option Payoff = $0.3060 F1($|€) = $€1.001.50××1.0711.05 = Option Price = ? F1($|€) = = Option Payoff = $0 When a call futures option is exercised the holder acquires: 1. A long position in the futures contract. 2. A cash amount equal to the excess of the futures price over the strike price. Binomial Futures Option Pricing Consider the portfolio: Long H futures contracts Short 1 futures call option F1($|€) = = Option Price = $0.1714 Portfolio is riskless when the portfolio payoffs in the “up” state equal the payoffs in the “down” state: H×$0.2700 – $0.3060 = –H×$0.3300 The “right” amount of futures contracts is H = 0.510000. $1.80×1.071 F1($|€) = €1.00×1.05 = Futures Call Payoff = –$0.3060 Futures Payoff = H × $0.2700 Portfolio Cash Flow = H × $0.2700 – $0.3060 $1.20×1.071 $1.2240 F1($|€) = = €1.00×1.05 €1.00 Futures Payoff = –H × $0.3300 Option Payoff = $0 Portfolio Cash Flow = –H × $0.3300 Binomial Futures Option Pricing The payoffs of the portfolio are – $0.1683 in both the up and down states. $1.5300 F1($|€) == €1.00 With futures there is no cash flow at initiation. Without an arbitrage, it must be the case that the call option income today is equal to the present value of $0.1683 discounted at i$ = 7.10%: $1.80×1.071 $1.8360 F1($|€) = = €1.00×1.05 €1.00 Call Option Payoff = –$0.3060 Futures Payoff = H × $0.2700 Portfolio Cash Flow = 0.510 × $0.2700 – $0.3060 = –$0.1683 $1.20×1.071 $1.2240 F1($|€) = = €1.00×1.05 €1.00 Futures Payoff = –0.510 × $0.3300 C0 = $0.1572 =Portfolio Cash Flow = Option Payoff = $0 –0.510×$0.3300 = –$0.1683 Risk Neutral Valuation of Options • Calculating the hedge ratio is vitally important if you are going to use options. – The seller needs to know the hedge ratio if he wants to protect his profits or eliminate his downside risk. – The buyer needs to know the hedge ratio to decide how many options to buy. • Knowing what the hedge ratio is isn’t especially important if you are only trying to value options. • Risk Neutral Valuation is a very handy shortcut to valuation. Risk Neutral Valuation of Options F1($/€) = = We can safely assume that IRP holds: $18,000 = × €10,000 €10,000 = $15,000 $12,000 = × €10,000 Set the value of €10,000 bought forward at $1.5300/€ equal to the expected value of the two possibilities shown above: €10,000 × = $15,300 = q × $18,000 + (1 – q) × $12,000 Risk Neutral Valuation of Options Solving for q gives the risk-neutral probability of an “up” move in the exchange rate: $15,300 = q × $18,000 + (1 – q) × $12,000 $15,300 – $12,000 q = $18,000 – $12,000 q = 11/20 = 0.55000 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. $18,000 = × €10,000 ←value of €10,000 $3,000= payoff of right to buy €10,000 for $15,000 €10,000 = $15,000 $1,540.62 $12,000 = × €10,000 ←value of €10,000 $0 = payoff of right to buy €10,000 for $15,000 (11/20) ×$3,000 + (9/20)×$0 C0 = $1,540.62 = 1.071 Test Your Intuition The value of a call option on €10,000 with a strike price of $15,000 is $1,540.62 Call $15,000 Call Option Option Seller €10,000 Owner The value of a put option on $15,000 with a strike price of €10,000 is also $1,540.62 Put $15,000 Put Option Option Seller €10,000 Owner If the options finish in-the-money they have the same cash flows. So they should have the same value today. 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 the value of a call option on €10,000 with a strike price of $15,000 to be $1,540.62, this should be easy in the sense that we already know the right answer. As before, i$ = 7.1%, i€ = 5%, S0($/€) = F1($/€) = = Test Your Intuition (continued) $1.50×1.071 $1.5300 F1($/€) = €1.00×1.05 = €1.00 6/€12,500 =× $15,000 ←€ value of $15,000 when S1 = $1.20/€ €10,000 = $15,000 11/17 €8,333.33 = €1.00 × $15,000 ←€ value of $15,000 $1.80 when S1 = $1.80/€ $15,000 × = €9,803.92 € 9,803.92 = q × €12,500 + (1 – q) × €8,333.33 q = €9,803.92 –€8,333.33 €8,333.33 q = 6/17 €12,500 – Test Your Intuition (continued) €12,500 = €1.00 × $15,000 ←value of $15,000 $1.20 €0 = payoff of right to sell $15,000 for €10,000 €10,000 = $15,000 €1,027.08€1.00 $15,000 ←value of $15,000 €8,333.33 = × $1.80 €1,666.67= payoff of right to sell $15,000 for €10,000 €P0 = €1,027.08 = 6/17× €0 + (11/17)×€1,666.67 1.05 $P0 = $1,540.62 = €1,027.08 × Risk Neutral Valuation Practice Use risk neutral valuation to value a PUT option on £8,000 with a strike price of €10,000. S0(£/€) = £0.80/€, i£ = 15½% and i€ = 5% €10,666.67 In the next year, there are two possibilities: p = €0 S1(£/€) = £1.00/€ or S1(£/€) = £0.75/€ €10,000 Step 1: Calculate risk neutral probabilities. Step 2: Calculate option value as the present value of the expected value of the option payoffs.€8,000 down €9,090.91 = q × €10,666.67+ (1 – q) × €8,000 p1 = €2,000 €10,666.67€9,090.91–– €8,000 = 9/22 p0 = €1,125.54= 13/22 1.05× €2,000 q = €8,000 Risk Neutral Valuation Practice Check your work by finding the value of an at-the-money 1-period call option on €10,000 with a strike of £8,000. S0(£/€) = £0.80/€, i£ = 15½%, i€ = 5%, so F1(£/€) = £0.8800/€ In the next year, there are two possibilities: S1(£/€) = £1.00/€ or S1(£/€) = £0.75/€£10,000 = × €10,000 cu1 = £2,000 £0.80= max[0, £10,000 − £8,000] €10,000 × = £8,000 c0 = £900.43 q × c1u + (1 – q) × c1d£0.75 c0 = 1 + r₤ £7,500 =d €1.00 × €10,000 £.88/€ − £.75/€c1 = £0 q = 0.5200 = £1.0/€ − £.75/€ = max[0, £7,500 − £8,000] 0.52 × £2,000 + 0.48 × £0 c0 = = £900.43 = €1,125.54 × £0.80/€ 1.155 Things to be Careful About • Convert future values from one currency to another using forward exchange rates. • Convert present values using spot exchange rates. • Discount future values to present values using the correct interest rate, e.g. i$ 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 calculated using q and solve for q. • 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. Black–Scholes Pricing Formulae The Black-Scholes formulae for the price of a European call and a put written on currency are: c=S0e−riT N(d1)−Ee−r$T N(d2) p=[EN(−d2)−FT N(−d1)]e−r$T lnFT + 1 σ2T d1 =  E  2 2 1 N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d. Black–Scholes Pricing Formula Use the European option pricing formula to find the value of a six-month call option on Japanese yen. The strike price is $1 = ¥100. The volatility is 25 percent per annum; r$ = 5.5% and r¥ = 6%. F = Ste(r$−r£ )T = (1/100)e(.055−.06)0.50 =1/100.2503 ln1/100.2503 d1 = ln(F / E) +.52T =  1/100 +.5(0.25)2.5 = −.0025+ 0.156 = 0.074246  T .25 .5 0.1768 d2 =d1 − T = 0.074246−.25 .5 =−0.10253 Ce = [FN(d1) −EN(d2 )]e−r$T Ce = [1/100.2503N(0.074246) − (1/100)N(−0.10253)]e−0.0550.5 Ce = $0.006137 Solution Manual for International Financial Management Cheol S. Eun, Bruce G. Resnick 9780077861605