Ch05_Swaps(互换)(金融工程学,华东师大).

1、Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.1Swaps(互换互换)Chapter 5Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.2Nature of Swaps A swap is an agreement to exchange

2、 cash flows (现金流) at specified future times according to certain specified rulesOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.3Terminology LIBOR the London InterBank Offer RateIt is the rate of interest offered by banks on deposits

3、from other banks in Eurocurrency marketsOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.4An Example of a “Plain Vanilla” Interest Rate Swap(大众型利率互换大众型利率互换) An agreement by “Company B” toRECEIVE 6-month LIBOR andPAY a fixed rate of 5%

4、paevery 6 months for 3 years on a notional principal of $100 million Next slide illustrates cash flows, wherePOSITIVE flows are revenues (inflows) andNEGATIVE flows are expenses (outflows)Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University

5、5.5-Millions of Dollars-LIBORFLOATING FIXED NetDateRateCash Flow Cash Flow Cash 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+0.45Cash Flows to Company B

6、(See Table 5.1, page 123)Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.6More on Table 5.1 The floating-rate payments are calculated using the six-month LIBOR rate prevailing six month before the payment date The principle is only us

7、ed 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 short (long) a fixed-rate bond.Optio

8、ns, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.7Typical Uses of anInterest Rate Swap Converting a liabilityfrom a FIXED rate liability to aFLOATING rate liability FLOATING rate liabilityto a FIXED rate liability Converting an investmentfr

9、om a FIXED rate investment to aFLOATING rate investment FLOATING rate investment to a FIXED rate investment Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.8Transforming a Floating-rate Loan to a Fixed-rate Consider a 3-year swap init

10、ialized 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 million Suppose Company B has arranged to borrow $100 million LIBOR + 80bpCompanyBCompanyA5%LIBORLIBOR+0.8%5.2%Note: 1 basis point (bp) = one-hundredth of 1%Optio

11、ns, 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 flows 1. Pays LIBOR plus 0.8% to outside lenders 2. Receives

12、 LIBOR from Company A in the swap 3. Pays 5% to Company A in the Swap In essence, B has transformed its variable rate borrowing at LIBOR + 80bp to a fixed rate of 5.8%Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.10A and B Transform

13、 a Liability(Figure 5.2, page 125)ABLIBOR5%LIBOR+0.8%5.2%Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.11Financial Institution is Involved(Figure 5.4, page 126) AF.I.BLIBORLIBORLIBOR+0.8%4.985%5.015%5.2%“Plain vanilla” fixed-for-flo

14、at swaps on US interest rates are usually structured so that the financial institutions earns 3 to 4 basis points on a pair of offsetting transactionsOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.12A and B Transform an Asset(Figure

15、5.3, page 125) ABLIBOR5%LIBOR-0.25%4.7%Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.13Financial Institution is Involved(See Figure 5.5, page 126) AF.I.BLIBORLIBOR4.7%5.015%4.985%LIBOR-0.25%Options, Futures, and Other Derivatives, 4

16、th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.14The Comparative Advantage Argument (Table 5.4, page 129) Company A wants to borrow floating Company B wants to borrow fixedFixed Floating Company A10.00%6-month LIBOR + 0.30%Company B11.20%6-month LIBOR + 1.00%Options, Futures

17、, 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. Pays 10%pa to outside lenders 2. Receives 9.95%pa from B Pays LIBOR + 0.05% 3. Pays LIBOR to B a 25bp gai

18、nCompany 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%Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5

19、.16The Swap (Figure 5.6, page 130) ABLIBORLIBOR+1%9.95%10%Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.17The Swap when a Financial Institution is Involved (Figure 5.7, page 130) AF.I.B10%LIBORLIBORLIBOR+1%9.93%9.97%Options, Futures

20、, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.18Total Gain from anInterest Rate Swap The total gain from an interest rate swap is always |a-b| where a is the difference between the interest rates in thefixed-rate market for the two parties, and b i

21、s the difference between the interest rates in thefloating-rate market for the two parties In this example a=1.20% and b=0.70%Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.19Criticism of the Comparative Advantage Argument The 10.0%

22、and 11.2% rates available to A and B in fixed rate markets are 5-year rates The LIBOR+0.3% and LIBOR+1% rates available in the floating rate market are six-month rates Bs fixed rate depends on the spread above LIBOR it borrows at in the futureOptions, Futures, and Other Derivatives, 4th edition 2000

23、 by John C. HullTang Yincai, Shanghai Normal University5.20Valuation of an Interest Rate Swap Interest rate swaps can be valued as the difference between -the value of a fixed-rate bond & -the value of a floating-rate bond Alternatively, they can be valued as a portfolio of forward rate agreemen

24、ts (FRAs)Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.21Valuation of an Interest Rate Swap as a Package of Bonds The fixed rate bond is valued in the usual way (page 132) The floating rate bond is valued by noting that it is worth

25、par immediately after the next payment date (page 132)Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.22Valuation of an Interest Rate Swap as a Package of Bonds (continued)Define Vswap: value of the swap to the financial institution B

26、fix: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 agreement 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* i

27、s the floating111111)e(eeee*fl1fixtrtrtrtrnit rkLkLBLkBnniiOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.23Example of an Interest Rate Swap Valued as a Package of BondsSuppose, that you agreed to pay 6-month LIBOR and receive 8% pa

28、(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 continuous compounding over 3, 9, and 15 months are 10.0%, 10.5%, and 11.0%, respectively. The six-month LIBOR rate at the la

29、st 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 = 5.1e-0.25x0.10 + 100e-0.25x0.10= $102.51 million Hence, Vswap = 98.24 - 102.51 = -$4.27 millionOptions, Futures, a

30、nd Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.24Valuation in Terms of FRAs Each exchange of payments in an interest rate swap is an FRA The FRAs can be valued on the assumption that todays forward rates are realized (See section 4.6, page 97)Options,

31、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 simple three step process1. Calculate each of the forward rates for each of the LIBOR rates that will determine swap cash flows2. Calcul

32、ate 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 flowsOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.26Example of an Interest Rate S

33、wap 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 rate 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, w

34、e 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-annual compounding.The value of the 9 month FRA is0.5 * 100 * (0.08 - 0.11044)e-0.105*0.75 = -1.41 1075. 05 . 010. 0*25. 0105. 0*75. 093fOptions,

35、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% with continuous compounding which corresponds to 12.102% with semi-annual compounding.The value of the 15 month FR

36、A 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 = -4.271175.05.075.0*105.025.1*11.0159fOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.28An Example of a Currency Swap An agr

37、eement 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 yearsOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.29Exchange of Principal In an interest rate swap th

38、e principal is not exchanged In a currency swap the principal is exchanged at - the beginning & - the end of the swapOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.30The Cash Flows (Table 5.5, page 137)YearsDollarsPounds$-million

39、s-0 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 Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.31Typical Uses of a Currency Swap Conversionfrom a liability inone currencyto a liability inanother curre

40、ncy Conversionfrom an investment in one currencyto an investment in another currencyOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.32Comparative Advantage Arguments for Currency Swaps (Table 5.6, pages 137-139) Company A wants to bor

41、row AUD Company B wants to borrow USDUSDAUDCompany A 5.0%12.6%Company B 7.0%13.0%Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.33Valuation of Currency Swaps Like interest rate swaps, currency swaps can be valued either as the - diff

42、erence between 2 bonds or as a - portfolio of forward contractsOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.34Example for a Currency Swap Suppose that the term structure of interest rates is flat in both US and Japan. Further suppo

43、se 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 on $10 million. The current exchange rate is 110 yen = $1. Evaluate the swap under the assumption that payments are made just

44、 once per year.Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.35Example Valued as BondsHere we have 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-0.0

45、4x3 = 1,230.55 millionThus, the value of the swap is Vswap= S0BF -BDmillion55.1$644.911055.230, 1Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.36Example Valued as FRAsThe current spot rate is 110 yen per dollar or 0.009091 dollars p

46、er 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 = 0.0106The value of the forward contracts corresponding to the exchange of interest are therefore(60 * 0.0096) - 0.8)e

47、-0.09x1 = -0.21(60 * 0.0100) - 0.8)e-0.09x2 = -0.16(60 * 0.0106) - 0.8)e-0.09x3 = -0.13Options, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.37Example Valued as FRAs(continued)The final exchange of principal involves receiving 1,200 million

48、 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 millionOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. HullTang Yincai, Shanghai Normal University5.38Swaps & Forwards A swap can be regarded as a