Ch05-Swaps(互換)(金融工程-華東師范大學湯銀才)

1、2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.1Swaps(互換)Chapter 52021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.2Nature of SwapsA swap is an agreem

2、ent to exchange cash flows (現(xiàn)金流) at specified future times according to certain specified rules2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.3TerminologyLIBOR the London InterBank Offer RateIt is the rate of interest offere

3、d by banks on deposits from other banks in Eurocurrency markets2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.4An Example of a “Plain Vanilla” Interest Rate Swap(大眾型利率互換)An agreement by “Company B” toRECEIVE 6-month LIBOR an

4、dPAY a fixed rate of 5% paevery 6 months for 3 years on a notional principal of $100 millionNext slide illustrates cash flows, wherePOSITIVE flows are revenues (inflows) andNEGATIVE flows are expenses (outflows)2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yi

5、ncai, Shanghai Normal University5.5Millions of DollarsLIBORFLOATING FIXED NetDateRateCash FlowCash FlowCash FlowMar.1, 19994.2%Sept. 1, 19994.8%+2.102.500.40Mar.1, 20005.3%+2.402.500.10Sept. 1, 20005.5%+2.652.50+0.15Mar.1, 20015.6%+2.752.50+0.25Sept. 1, 20015.9%+2.802.50+0.30Mar.1, 20026.4%+2.952.50

6、+0.45Cash Flows to Company B(See Table 5.1, page 123)2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.6More on Table 5.1The floating-rate payments are calculated using the six-month LIBOR rate prevailing six month before the p

7、ayment dateThe principle is only used for the calculation of interest payments. However, the principle itself is not exchangedMeaning for “Notional principle” The swap can be regarded as the exchange of a fixed-rate bond for a float-rate bond. Company B (A) is long (short) a floating-rate bond and s

8、hort (long) a fixed-rate bond.2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.7Typical Uses of anInterest Rate SwapConverting a liabilityfrom a FIXED rate liability to aFLOATING rate liabilityFLOATING rate liabilityto a FIXED

9、 rate liabilityConverting an investmentfrom a FIXED rate investment to aFLOATING rate investment FLOATING rate investment to a FIXED rate investment 2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.8Transforming a Floating-rat

10、e Loan to a Fixed-rateConsider a 3-year swap initialized on March 1, 2000 whereCompany B agrees to pay Company A 5%pa on $100 millionCompany A agrees to pay Company B 6-mth LIBOR on $100 millionSuppose Company B has arranged to borrow $100 million LIBOR + 80bpCompanyBCompanyA5%LIBORLIBOR+0.8%5.2%Not

11、e: 1 basis point (bp) = one-hundredth of 1%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.9Transforming a Floating-rate Loan to a Fixed-rate (continued)After Company B has entered into the swap, they have 3 sets of cash flow

12、s 1. Pays LIBOR plus 0.8% to outside lenders 2. Receives LIBOR from Company A in the swap 3. Pays 5% to Company A in the SwapIn essence, B has transformed its variable rate borrowing at LIBOR + 80bp to a fixed rate of 5.8%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C.

13、HullTang Yincai, Shanghai Normal University5.10A and B Transform a Liability(Figure 5.2, page 125)ABLIBOR5%LIBOR+0.8%5.2%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.11Financial Institution is Involved(Figure 5.4, page 126

14、) AF.I.BLIBORLIBORLIBOR+0.8%4.985%5.015%5.2%“Plain vanilla” fixed-for-float swaps on US interest rates are usually structured so that the financial institutions earns 3 to 4 basis points on a pair of offsetting transactions2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C.

15、 HullTang Yincai, Shanghai Normal University5.12A and B Transform an Asset(Figure 5.3, page 125) ABLIBOR5%LIBOR-0.25%4.7%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.13Financial Institution is Involved(See Figure 5.5, page

16、 126) AF.I.BLIBORLIBOR4.7%5.015%4.985%LIBOR-0.25%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.14The Comparative Advantage Argument (Table 5.4, page 129)Company A wants to borrow floatingCompany B wants to borrow fixedFixed

17、 Floating Company A10.00%6-month LIBOR + 0.30%Company B11.20%6-month LIBOR + 1.00%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.15The Comparative Advantage (continued)One possible swap isCompany A has 3 sets of cash flows 1

18、. Pays 10%pa to outside lenders 2. Receives 9.95%pa from B Pays LIBOR + 0.05% 3. Pays LIBOR to B a 25bp gainCompany B has 3 sets of cash flows 1. Pays LIBOR + 1.00%pa to outside lenders 2. Receives LIBOR from A Pays 10.95%pa 3. Pays 9.95% to A a 25bp gainCompanyBCompanyA9.95%LIBOR10%LIBOR + 1%2021/9

19、/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.16The Swap (Figure 5.6, page 130) ABLIBORLIBOR+1%9.95%10%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.17T

20、he Swap when a Financial Institution is Involved (Figure 5.7, page 130) AF.I.B10%LIBORLIBORLIBOR+1%9.93%9.97%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.18Total Gain from anInterest Rate SwapThe total gain from an interes

21、t rate swap is always |a-b| wherea is the difference between the interest rates in thefixed-rate market for the two parties, andb is the difference between the interest rates in thefloating-rate market for the two partiesIn this example a=1.20% and b=0.70%2021/9/11Options, Futures, and Other Derivat

22、ives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.19Criticism of the Comparative Advantage ArgumentThe 10.0% and 11.2% rates available to A and B in fixed rate markets are 5-year ratesThe LIBOR+0.3% and LIBOR+1% rates available in the floating rate market are six-month r

23、atesBs fixed rate depends on the spread above LIBOR it borrows at in the future2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.20Valuation of an Interest Rate SwapInterest rate swaps can be valued as the difference between -t

24、he value of a fixed-rate bond & -the value of a floating-rate bondAlternatively, they can be valued as a portfolio of forward rate agreements (FRAs)2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.21Valuation of an Interest Ra

25、te Swap as a Package of BondsThe fixed rate bond is valued in the usual way (page 132)The floating rate bond is valued by noting that it is worth par immediately after the next payment date (page 132)2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shang

26、hai Normal University5.22Valuation of an Interest Rate Swap as a Package of Bonds (continued)Define Vswap:value of the swap to the financial institution Bfix:value of the fixed-rate bond underlying the swap Bfl:value of the floating-rate bond underlying the swap L:notional principal in a swap agreem

27、ent ti: time when the ith payments are exchanged ri: LIBOR zero rate for a maturity tiThen, Vswap = Bfix - Bfl and if k is the fixed-rate coupon and k* is the floating2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.23Example

28、of an Interest Rate Swap Valued as a Package of BondsSuppose, that you agreed to pay 6-month LIBOR and receive 8% pa (with semiannual compounding) on a notional amount of $100 million. The swap has a remaining life of 15 months and the next payment is due in 3 months. The relevant rates for continuo

29、us compounding over 3, 9, and 15 months are 10.0%, 10.5%, and 11.0%, respectively. The six-month LIBOR rate at the last payment was 10.2% (with semi-annual compounding).In this case k = $4 million and k* = $5.1 million, so that Bfix= 4e-0.25x0.10 + 4e-0.75x0.105 + 104e-1.25x0.11= $ 98.24 million Bfl

30、 = 5.1e-0.25x0.10 + 100e-0.25x0.10= $102.51 million Hence, Vswap = 98.24 - 102.51 = -$4.27 million2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.24Valuation in Terms of FRAsEach exchange of payments in an interest rate swap

31、is an FRAThe FRAs can be valued on the assumption that todays forward rates are realized (See section 4.6, page 97)2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.25Valuation of an Interest Rate Swap as a Package of FRAsA sim

32、ple three step process1. Calculate each of the forward rates for each of the LIBOR rates that will determine swap cash flows2. Calculate swap cash flows on the assumption that the LIBOR rates will equal the forward rates3. Set the swap rates equal to the present value of these cash flows2021/9/11Opt

33、ions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.26Example of an Interest Rate Swap Valued as a Package of FRAsSame problem as before.The cash flows for the payment in 3 months have already been set. A rate of 8% will be exchanged for a r

34、ate of 10.2%. The NPV of this transaction is0.5 * 100 * (0.08 - 0.102)e-0.1*0.25 = -1.07To figure out the NPV of the remaining two payments, we first need to calculate the forward rates corresponding to 9 and 15 months or 10.75% with continuous compounding which corresponds to 11.044% with semi-annu

35、al compounding.The value of the 9 month FRA is0.5 * 100 * (0.08 - 0.11044)e-0.105*0.75 = -1.41 2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.27Example Swap as Valued as FRAs(continued)The 15 month forward rate isor 11.75% w

36、ith continuous compounding which corresponds to 12.102% with semi-annual compounding.The value of the 15 month FRA is0.5 * 100 * (0.08 - 0.12102)e-0.11*1.25 = -1.79Hence, the total value of the swap is -1.07 - 1.41 - 1.79 = -4.272021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by J

37、ohn C. HullTang Yincai, Shanghai Normal University5.28An Example of a Currency Swap An agreement to - pay 11% on a sterling principal of 10,000,000 & - receive 8% on a US$ principal of $15,000,000 - every year for 5 years2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. H

38、ullTang Yincai, Shanghai Normal University5.29Exchange of PrincipalIn an interest rate swap the principal is not exchangedIn a currency swap the principal is exchanged at - the beginning & - the end of the swap2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yin

39、cai, Shanghai Normal University5.30The Cash Flows (Table 5.5, page 137)YearsDollarsPounds$millions0 15.00 +10.001 +1.20 1.102 +1.20 1.10 3 +1.20 1.104 +1.20 1.10 5+16.20 -11.10 2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.

40、31Typical Uses of a Currency SwapConversionfrom a liability inone currencyto a liability inanother currencyConversionfrom an investment in one currencyto an investment in another currency2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal Un

41、iversity5.32Comparative Advantage Arguments for Currency Swaps (Table 5.6, pages 137-139)Company A wants to borrow AUDCompany B wants to borrow USDUSDAUDCompany A 5.0%12.6%Company B 7.0%13.0%2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Norma

42、l University5.33Valuation of Currency SwapsLike interest rate swaps, currency swaps can be valued either as the - difference between 2 bonds or as a - portfolio of forward contracts2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal Universi

43、ty5.34Example for a Currency SwapSuppose that the term structure of interest rates is flat in both US and Japan. Further suppose the interest rate is 9% pa in the US and 4% pa in Japan. Your company has entered into a three-year swap where it receives 5% pa in yen on 1,200 million yen and pays 8% pa

44、 on $10 million. The current exchange rate is 110 yen = $1. Evaluate the swap under the assumption that payments are made just once per year.2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.35Example Valued as BondsHere we hav

45、e a domestic and a foreign bond BD= 0.8e-0.09x1 + 0.8e-0.09x2 + 10.8e-0.09x3 = $ 9.644 million BF = 60e-0.04x1 + 60e-0.04x2 + 1260e-0.04x3 = 1,230.55 millionThus, the value of the swap is Vswap= S0BF -BD2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Sh

46、anghai Normal University5.36Example Valued as FRAsThe current spot rate is 110 yen per dollar or 0.009091 dollars per yen. Because the interest rate differential is 5% the one, two, and three year exchange rates are (from eq (3.13) 0.009091e0.05x1 = 0.00960.009091e0.05x2 = 0.01000.009091e0.05x3 = 0.

47、0106The value of the forward contracts corresponding to the exchange of interest are therefore(60 * 0.0096) - 0.8)e-0.09x1 = -0.21(60 * 0.0100) - 0.8)e-0.09x2 = -0.16(60 * 0.0106) - 0.8)e-0.09x3 = -0.132021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Sha

48、nghai Normal University5.37Example Valued as FRAs(continued)The final exchange of principal involves receiving 1,200 million yen for $10 million. The value of the forward contract corresponding to this transaction is(1,200 * 0.0106) - 10)e-0.09x3 = 2.04Hence, the total value of the swap is 2.04 -0.13 - 0.16 - 0.21 = 1.54 million2021/9/11Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.38Swaps & ForwardsA swap can be regarde