Chapter XV Eurocurrency Futures and Options Ch2

If i < f, the buyer pays the seller the absolute value of the above amount. Since cash settlement is made at the beginning of the FRA period, the denominator discounts the payment back to that point. In general, the settlement rate is LIBOR.  FRA: Just Another Forward Contract An FRA is a cash-settled interbank forward contract on the interest rate. The cash flows at the beginning ot the FRA period might confuse you. You can consider these cash flows, however, as being exchanged when the forward contract expires.  Example XV.13: A bank buys a "three against six" (or "3X6") FRA for GBP 4,000,000 for a threemonth period, beginning three months from now and ending six months from now, at an interest rate of 5.5%. There are actually ninety-two days in the FRA period. Three months from now, at the beginning of the FRA period, LIBOR is 6%. The bank receives cash at the beginning of the FRA-period from the selling party in the amount of GBP 4,000,000 x (.06 - .055) x (92/360) = GBP 5,033.92. 1 + .06x(92/360) The bank's net borrowing cost on GBP 4,000,000 at the end of the FRA period is therefore: GBP 4,000,000 x .06 x (92/360) = GBP 61,333.33 minus GBP 5,033.92 x [1 + .06 x (92/360)] = GBP -5,111.11 Net borrowing cost = GBP 56,222.22 The net borrowing cost is equivalent to borrowing GBP 4,000,000 at 5.5%: GBP 4,000,000 x .055 x (92/360) = GBP 56,222.22. Note that if instead of an FRA, the bank uses Eurosterling futures. Then, in six months the bank receives GBP 25 x 50 x 4 = GBP 5,000. (Recall that the value of a basis point change is GPB 25.) ¶ FRAs are tailor-made contracts. They are usually traded for dates between one of the standard Eurodeposit maturities, such as three, six, nine and twelve months, but nonstandard dates are also available. FRAs are off-balance sheet items and have smaller capital requirements than interbank loans. Therefore, FRAs are regularly used to close maturity gaps in an Eurobank's balance sheet. An FRA is an interbank-traded equivalent of the implied forward rate. Example XV.14: On July 20, Leblanc Bank wants CHF 50 million of three-month deposit but is only offered CHF 50 million of six-month deposit by a customer at the bank's bid rate. At the current market, the other rates are these: Cash FRA bid asked bid asked 3 months 5.2185 5.2500 3X6 5.22 5.29 6 months 5.2500 5.3125 Should the manager of Leblanc Bank take the deposit? The manager of the bank converts the six-month deposit to a three-month deposit by selling a three XV.11 against six FRA. That is, Leblanc Bank sells off (lends) the last three months of the deposit in the FRA market. Then, the manager calculates the profit on this transaction. There are 184 days from settlement of the deposit on July 22 to maturity on January 22. The threemonth FRA period is 92 days from October 22 to January 22. The interest paid at the end of six months to the depositor is: CHF 50,000,000 x .05250 x (184/360) = CHF 1,341,666,67. Interest earned on lending for three months in the interbank market and then for another three months at the FRA rate is: CHF 50,000,000 x [(1 + .052185x(92/360)) x (1 + .0522x(92/360)) - 1] = CHF 1,342,703.56. There is a net profit of CHF 1,036.89 at the end of six months. This profit is possible because the bank was offered a six-month deposit at the bank's bid rate of 5.25%. You should note that Leblanc Bank is not able to do arbitrage at the above rates. Usually, Leblanc Bank would have to borrow for six months in the interbank market at 5.3125%. Thus, the interest paid on the deposit would be CHF 1,357,638.90, and the bank would have a loss. ¶ III. Hedging with Eurocurrencies and FRAs Contrary to an FRA, Eurocurrency future's cash flow is based on exactly ninety days and is not discounted to the beginning of the period. Thus, the interest rate on Eurocurrency futures cannot be identified with the implied forward rate used to calculate break-even loan costs. The FRA rate, by contrast, is the same as the implied forward rate. For hedging purposes, therefore, we must calculate a hedge ratio for Eurocurrency futures against FRA's. (For simplicity, we will not consider the interest opportunity cost associated with the daily cash flows on the futures contract). The cash flow (V) to the buyer of a currency FRA is given by the equation (from above): V = N x (i - f) x (ym/yb) 1 + i x (ym/yb) (XV.1) If we let f=1 -.01Zt also denote the forward interest rate at which a Eurocurrency future was opened (e.g., 0.09 for a futures price of 91.00), and let i=1 - .01Zt+1 also denote the forward interest rate at which the Eurocurrency futures price was closed out, then i - f = (1 - .01Zt+1) - (1 - .01Zt) = .01(Zt - Zt+1) = -.01 Z. Therefore, the corresponding cash flow on a Eurodollar futures contract (where N = USD 1,000,000, ym = 90) is V = - 1,000,000(.01Z)(90/360) = - 2500 Z. XV.12 (XV.2) (Recall, there is no discount to the beginning of the period for a Eurodollar future's cash flow.) Therefore, the number of Eurodollar futures contracts b needed to hedge an FRA or the implied forward rate is the ratio of (XV.1) to (XV.2): b = [N x (-.01 x Z )(ym/yb)/ (1 + i x (ym/yb)]/[-2500 Z] = N/[250,000(i + yb/ym)] = N/[250,000(1 + yb/ym - .01Zt+1)]. (XV.3) The hedge ratio depends on the current Eurodollar futures price Zt+1. Note that if the forward interest rate were zero Zt+1=100 and if the days in the FRA period were 90 (ym=90), and if the dollar amount of the FRA were N = USD 1,000,000, then b=1. The number of Eurodollar futures contracts needed to hedge an FRA goes down as the discount rate S increases (as the current futures price Zt+1 decreases). Example XV.15: There are ninety-one days in an FRA period beginning at the expiration of a Eurodollar contract. The current price of the futures is 92.00. The year basis, yb, is 360 days. We want to calculate the amount of the FRA hedges a short position in 100 Eurodollar contracts? The Eurodollar hedge ratio, b, is prespecified as b = -100. We only need to calculate N. N = -100 x 250,000 x [1 + 360/91 -.01(92.00)] = USD -100,901,099. Against the short position of 100 Eurodollar futures contracts, the trader should sell an FRA in the amount of USD 100,901,099. To verify this answer, suppose that interest rates fall by 1 basis point (.0001). Then, since rates go from 8.00 to 7.99, the cash flow to the seller of the FRA is: V = USD 100,901,099 x (.0001) x (91/360) = USD 2,500. 1 + .0799 x (91/360) The cash flow on the short Eurodollar futures position is V = USD 100 x 2,500 x (92.00 - 92.01) = USD -2,500. For these value changes to remain equal and offsetting, however, the amount of the FRA would have to be continually adjusted because b changes with the current futures price. ¶ Example XV.16: Go back to Example XV.15. Suppose b is left unchanged in the original position, when Z changes to 93.00 (Z equals to 100 basis points). Now, the cash flow to the FRA seller is: V = USD 100,901,099 x (.01) x (91/360) = USD 250,621. 1 + .07 x (91/360) The cash flow on the short Eurodollar futures position is V = USD 100 x 2,500 x (92.00 - 93.00) = USD -250,000. XV.13 The total position has a net change of USD 621. ¶ Example XV.17: Go back, again, to Example XV.15. Now suppose interest rates rise by 1% from 8.00 to 9.00, the FRA seller has a cash flow of V = USD 100,901,099 x (-.01) x (91/360) = USD -249,382. 1 + .07 x (91/360) The cash flow on the short Eurodollar futures position is V = USD 100 x 2,500 x (93.00 - 92.00) = USD -250,000. The total position has a net change of USD 618. ¶ Thus, the position of being short Eurodollar futures and selling FRAs, at the fixed ratio b, makes money for interest rate moves in either direction. Such a position can be characterized in terms of options as a delta-neutral, long-gamma position. Example XV.18: In Example XV.11, we compared the implied forward rate to the Euribor futures price, and determined that the Swiss bank could hedge its gap risk in the futures market at a profit. Suppose the Swiss bank wants to hedge EUR 20,000,000. We want to determine the number of futures contracts the bank needs to go short. (Recall Zt+1 = 96.13.) The futures price is Zt+1 = 96.13. Thus, the number of contracts to sell is given by equation (IX.3) as b = 20,000,000 / [250,000 x (1 + 360/91 -.01(96.13)]  20. ¶ As noted before, the hedge ratio b given by (XV.3) does not take into consideration the variation margin flows on the Eurocurrency contract. Since variation margin must be paid in cash, there is in most circumstances an interest cost associated with the need to borrow negative variation margin flows, or an interest profit to be earned on positive variation margin flows. To adjust the futures position for this interest rate effect (a process called tailing), we discount the cash flows on the FRA from now to the beginning of the FRA period. That is, we present value the FRA cash flows to make them comparable to the Eurocurrency futures cash flows. Let i be the relevant discount rate from now to the beginning of the FRA period, which occurs in T days. (In Example XV.11, i = 4% and T =92 days). Then, from (XV.3), the adjusted futures hedge ratio, b*, is b* = b/[1 + i x (T/360)]. As T  0, b*  b. Example XV.19: In Example XV.18, i=4%, T=92. Therefore, at the beginning of the FRA period, XV.14 b* =19.80  20. That is, in this case, the initial position should be maintained at 20 contracts as T 0. ¶ The calculations associated with the FRA/Eurocurrency hedge ratio become more complicated once we allow for the general case that the FRA periods and Eurocurrency periods do not correspond. Because the Eurocurrency futures market is more liquid than the FRA market, an FRA period that overlaps two or more Eurocurrency futures periods would result in the FRA rate being calculated by an interpolation of the rates implied in the futures contracts. In addition, the hedge for the FRA would be split up among the Eurocurrency futures used in the interpolation. IV. Interest Rate, Eurocurrency Futures Options and Other Derivatives Options on Eurocurrency futures are traded at many exchanges including the CME, LIFFE and SIMEX. A Eurocurrency futures call gives the buyer the right to go long a Eurocurrency futures contract, while a Eurocurrency futures put gives the buyer the right to go short. More specifically, a CME Eurodollar put is a contract between a buyer and a writer whereby the put buyer pays a price (the put premium) to the writer in order to acquire the right, but not the obligation, to go short one CME Eurodollar futures contract at the opening price given by the put's strike price. If the put buyer exercises the right to go short, the put writer must go long the Eurodollar contract. Once option trades have cleared, the CME Clearing House becomes the writer to every option buyer, and the buyer to every other option writer. The options are American and expire on the last trade date for the futures contract. Strike prices are in intervals of .25 in terms of the CME index. Thus, one may buy a put on June Eurodollar futures with a strike of 9375. If exercised, it gives the right to go short one CME Eurodollar futures contract at an opening price of 93.75. Recall that Eurodollar futures contracts involve a face amount of USD 1 million and are traded for delivery in March, June, September and December. For monthly delivery, the CME trades a futures option on LIBOR. Example XV.20: On Tuesday, November 1, 1994, the Wall Street Journal published the following quotes for Eurodollar futures options. EURODOLLAR (CME) $ million; pts. of 100% Strike Calls-Settle Price Dec Mar Jun 9350 0.56 0.29 0.18 9375 0.33 0.16 0.10 9400 0.14 0.07 0.05 9425 0.03 0.03 0.02 Puts-Settle Dec Mar 0.01 0.18 0.03 0.30 0.09 0.45 0.23 0.66 June 0.53 0.69 0.89 1.11 XV.15 9450 0.00 0.01 0.10 0.45 0.89 9475 0.00 0.00 0.00 0.70 1.14 Est. vol. 56,820; Fri vol. 80,063 calls; 72,272 puts Op. Int. Fri 939,426 calls; 1,016,455 puts 1.36 1.61 Eurodollar futures option premium quotations are in percentage points, with each basis point representing USD 25. Example XV.21: Consider the June 95 put, with a strike price of 93.75, in Example XV.20. A price of .69 would represent USD 25 x 69 = USD 1,725. ¶ Example XV.22: A trader goes short a June 1995 Eurodollar future at a price Z=93.75, simultaneously buying a June 1995 call with a strike price of 93.50. The premium on the call is C = .18 (see Example XV.20). The spot interest rate is 6%. We want to calculate the payoffs on the last day of trading, thirty days from today, for each of the following possible prices for the expiring future: 93.00, 94.00, 95.00, and 96.00. The call premium paid today is USD 25x18=USD 450. Adding the 6% carrying charge for thirty days gives USD 450x[1 + .06x(30/360)] = USD 452.25 at option expiration. If the future expires at 93.00, the payoff will be 93.75-93.00=.75 or USD 1,875 (.75x2,500). The call will expire worthless, so the net payoff is USD 1,875 - USD 452.25 = USD 1422.75. Futures Price 93.00 94.00 95.00 96.00 Future Payoff 0.75 -0.25 -1.25 -2.25 Call Payoff 0.00 0.50 1.50 2.50 Option Cost .18 .18 .18 .18 Carrying Cost .0009 .0009 .0009 .0009 Total Total (USD) .5691 1422.75 .0691 172.75 .0691 172.75 .0691 172.75 By buying the call, the trader has limited his/her possible loss on the future to 6.91 basis points or USD (172.75). This amount can be approximated (ignoring carrying cost, which, in general, are small in magnitude) as Z - X - C = 93.75 - 93.50 - .18 = .07. ¶ 4.A Valuation of futures options To price Eurocurrency futures options, we will use the Black-Scholes formula, derived in Appendix V. The key lies with the forward interest rate f embodied in the futures price. The value of a European (exercise is only allowed at expiration) call on the forward interest rate f is given by: ct = Bt(T)[f N(d1) - X N(d2)], d1 = ln(f/X) + .5 2 T  T.5 (XV.4) and d2 = ln(f/X) - .5 2 T,  T.5 XV.16 where Bt(T) is the price of futures contract with expiration date T, N(.) is the cumulative normal distribution, and 2 is the variance of Bt. The European put price is obtained from the put-call conversion equation: p = c + B (X-f). (XV.5) The latter equation shows the European put and call will have equal values when the forward interest (or FRA) rate is equal to the strike price. Example XV.23: Table XV.B gives some sample values for European options on interest rates. Assume =.20, T=90/365, discount rate 7% (B=98.28), the option premium is paid today, and the cash value of the option payoff is paid at option expiration. X: f: 6.0 7.0 Table XV.B Value of European Options on Forward Interest Rates Call Put 6.0 7.0 8.0 6.0 7.0 .234 .999 1.966 .234 .016 .016 .272 1.013 .999 .272 8.0 .001 .030 Calculations for the call and put option with X=6 and f=7 A. Call. Substituting into (XV.4): d1 = [ln(7/6) + .5 x (.202) x .2466]/[.20 x .2466.5] = 1.6018 d2 = [ln(7/6) - .5 x (.202) x .2466]/[.20 x .2466.5] = 1.5025 Now, look for the cumulative normal distribution at z=1.6018 The number you get is .4454. But, since d1 (z) is positive, in order to get the whole area under the curve up to z=1.6018, you need to add .50 to .4454. N(d1=.9635) = .9454 N(d2=.8890) = .9335 c = .9828 x [7.0 x .9454 - 6 x .9335] = .99931 B. Put. Substituting into (XV.5): p = .99931 + .9828 (6 -7.0) = .0172. ¶ The following example shows how these values should be interpreted. Example XV.24: Consider the option in Table XV.B with strike X = 6.0 and forward interest rate f = 7.0. It has value of c = .99923. Let's analyze the meaning of this number. Since X and f are in percent, the price c is also stated in percent. In order to translate this price to a dollar amount, we have to know the option size and the duration in days of the forward interest period. XV.17 Suppose the option is based on three-month LIBOR and involves a nominal amount of USD 10 million. Also suppose there are ninety-two days in the three-month period. Then the dollar cost of the option is: .99923 x (1/100) x (92/360) x USD 10,000,000 = USD 25,535.88. The values in Table XV.A also assume that the option premium is paid today, and that the cash in the option payoff is received at expiration, which is the beginning of the forward interest or FRA period. If either of these assumptions is false, there will have to be a further interest adjustment. For example, suppose the cash in the option payoff will not be received until the end of the forward interest period (92 days). Then the table value (for X=6.0, f=7.0) must be discounted by the forward interest rate f=7.0 for 92 days: .99923/[1 + .07 x(92/360)] = .98167. This corresponds to an option premium of USD 25,087.12. ¶ We now proceed to price options on Eurodollar futures. Since the price of a Eurodollar future Z may be written as Z = 100 - f  f = 100 - Z. Note that f  X  Z  100 - X. Thus a call on f, which pays off when f > X, is equivalent to a put on Z, which pays off when Z < 100 - X. Example XV.25: Let X=6. A call on the interest rate f has a positive exercise value when f > 6. But this is equivalent to the condition the Eurodollar futures price Z < 100 - 6 = 94. So the value of an interest rate call with strike price 6 is equal to the value of a Eurodollar futures put with strike price 94. Note: The value of a call on the forward interest rate f with strike price X is equal to the value of a put on the Eurodollar future Z = 100- f with strike price 100 - X. ¶ At the CME, Eurodollar options are American in style, so to price Eurodollar options we need to use the American counterpart to equations (IX.4) and (IX.5), which allow for early exercise of the option. Example XV.26: The Eurodollar future is at 94. To obtain the value of a future call with strike price of 93, we calculate f=100 - 94 = 6 and X=100 - 93 = 7. Thus the value of the Eurodollar futures call is the value of an American put on the forward rate f = 6 with a strike price of X = 7. Table XV.C is the same as Table XV.B, the Eurodollar futures prices and strikes, however, represent their interest rate equivalent, and the options are American. XV.18 Table XV.C X: Z: 93.00 94.00 92 1.020 1.977 Value of Eurodollar Futures Options Call 93 94 92 .277 .017 .001 1.005 .238 .033 Put 93 .277 .018 94 1.009 .238 Thus, the call value of .999, corresponding to X=6 and f=7 in Table XV.B, should be compared to the put value 1.005, corresponding to X=94 and Z = 93 in Table XV.C. The two numbers differ; for example, .999 is the European value while 1.005 is the corresponding American value. ¶ 4.B Caps, Floors, and Collars Many financial contracts in the international financial markets are based on periodic, but variable or "floating," interest payments, such as six-month LIBOR. Some parties to these contracts are at risk (as borrowers) if these series of interest payments rises dramatically. A solution to reduce this upside interest rate risk is to place a ceiling (or cap) on interest rate levels. Other parties are at risk (as lenders) if the level of interest payments falls. A solution for this downside interest rate risk is to place a floor under interest rates, so they do not fall too low. Caps and floor evolved from interest rate guarantees that fixed a maximum or minimum level of interest on floating-rate loans. The advent of trading in over-the-counter interest rat caps and floor dates back to 1985, when banks began to strip such guarantees from floating-rate notes to sell to the market. The leveraged buyout boom of the 1980s spurred the evolution of the market for interest rate caps. Firms engaged in leveraged buyouts typically took on large quantities of short-term debt, which made them vulnerable to interest rates increases. As a result, lenders began requiring such borrowers to buy interestrate caps to reduce the risk of financial distress. The mechanics of these contracts are very simple. For example, suppose a LIBOR borrower buys an interest rate cap on six-month LIBOR. The buyer of the cap pays the seller a premium in return for the right to receive the difference in the interest cost on some notional principal amount any time LIBOR rises above the stipulated cap. Payments amounts are determined by the value of LIBOR on a series of interest rate reset dates. The market interest rate on the first six-month interval is already known, and is typically excluded from the cap. Cap buyers typically schedule interest rate reset and payment intervals to coincide with interest payments on outstanding floating-rate debt. Example XV.27: A one-year floor. On October 24, 1999 a Euroswiss Franc (LIBOR) borrower buys a one-year interest rate floor of 6.85%, with three-month Euroswiss payments on May 20, August 20, November 20, and February 20. The nominal amount specified in the floor is CHF 5,000,000. A new three-month interval will begin on February 20 and extend to the following May 20. The interest rate for this period will be fixed on February 20, 1995 but interest will be paid on the following May 20. Three-month Euroswiss is fixed at 6.50% on February 20. On May 20 (89 days later) the floor writer will pay the floor buyer: XV.19 CHF 5,000,000 x (6.68 - 6.5)/100 x (89/360) = CHF 2,225. ¶  A cap is a portfolio of calls The cap is a series of European call options on the interest rate (i), where the strike price is the cap rate (ic). The amount of the payment is determined by: Notional amount x Max(0, i - ic) x days to reset date/360. Similarly, a floor is a series of European put options on the interest rate, where put strike price is the floor rate (if). The amount of the payment is determined by: Notional amount x Max(0, if - i) x days to reset date/360. The first option begins at the beginning of the period and expires on the first interest reset date.  Example XV.28: In Example XV.27, the first option begins on November 20, 1995 and expires on February 20, 1996 (a total of 91 days). The underlying variable in the option is the 3-mo implied forward (or FRA) rate from February 20 to the following May 20. The option expires on February 20 because the rate is set (reset) or determined on that date. But the cash value of the option will not be received for another 89 days (on the following May 20): Rate Fixed  February 20 November 20 91 Days Option Begins May 20 89 Days Option matures Cash Received A collar is a combination of calls and puts. The buyer of an interest rate collar purchases an interest rate cap, while selling a floor indexed to the same interest rate. Borrowers with floating-rate loans buy collars to limit effective borrowing costs between some maximum, determined by the cap, and a minimum, determined by the floor. Buying a collar limits a borrower’s ability to benefit from a significant drop in market interest rates. Collars, however, have the advantage of having a significant lower premium. If the strike prices are equal (ic=if) , one is left with a series of regular forward or FRAs with the strike price as the forward rate. Example XV.29: A two-year collar. On April 13, 2000 two parties negotiate a two-year collar to cap six-month Euroyen (LIBOR) at 4.5%, while placing a floor at 3.75%. Euroyen LIBOR is currently at 3.20%. The reset dates are October 15 and April 15. The collar is based on a notional amount of JPY 200,000,000. On October XV.20