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1、2021/9/11Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University5.1Swaps(互换互换)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 agreement to e
2、xchange cash flows(现金流)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 offered by banks on d
3、eposits 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
4、 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 Yincai,Shanghai Normal Univer
5、sity5.5-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(See T
6、able 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 payment dateThe principle is only used fo
7、r 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.2021/9/11Options,Futu
8、res,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 rate liabilityConverting an investmentfrom a FIXED rate
9、 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-rate Loan to a Fixed-rateConsider a 3-year swap initialized on M
10、arch 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%Note:1 basis point(bp)=one-hundredth of 1%2021/9/11Options,Futures,and
11、 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 LIBOR from Company A in
12、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.HullTang Yincai,Shanghai Normal University5.10A and B Transform a Liability(Figure 5.2,pag
13、e 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)AF.I.BLIBORLIBORLIBOR+0.8%4.985%5.015%5.2%“Plain vanilla”fixed-for-float swaps on US interest rat
14、es 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.HullTang Yincai,Shanghai Normal University5.12A and B Transform an Asset(Figure 5.3,page 125)ABLIBOR5%L
15、IBOR-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 126)AF.I.BLIBORLIBOR4.7%5.015%4.985%LIBOR-0.25%2021/9/11Options,Futures,and Other Derivatives,4th edition 2000 b
16、y 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 Floating Company A10.00%6-month LIBOR+0.30%Company B11.20%6-month LIBOR+1.00%2021/9/11Options,Futures,and Other Derivati
17、ves,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 gainCompany B has 3 sets of cash
18、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%2021/9/11Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University5.16The Swap(Figure 5.6,page 130)AB
19、LIBORLIBOR+1%9.95%10%2021/9/11Options,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%2021/9/11Options,Futures,and Other Derivatives,4th e
20、dition 2000 by John C.HullTang Yincai,Shanghai Normal University5.18Total Gain from anInterest Rate SwapThe total gain from an interest 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
21、rates in thefloating-rate market for the two partiesIn this example a=1.20%and b=0.70%2021/9/11Options,Futures,and Other Derivatives,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
22、 fixed rate markets are 5-year ratesThe LIBOR+0.3%and LIBOR+1%rates available in the floating rate market are six-month ratesBs 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 N
23、ormal University5.20Valuation of an Interest Rate SwapInterest rate swaps can be valued as the difference between -the 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 Deriv
24、atives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University5.21Valuation of an Interest Rate 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
25、 132)2021/9/11Options,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 Bfix:value of the fixed-rate bond underlying the
26、 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*is the floating2021/9/11Options,Futures,and Other Derivat
27、ives,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(with semiannual compounding)on a notional amount of$100 million.The swap has a remaining life of 15
28、 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 last payment was 10.2%(with semi-annual compounding).In this case k=$4 million and k*=$5.1 million,so that Bfix=
29、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 million2021/9/11Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University5.24Valuation in Terms of FRAsEach exc
30、hange of payments in an interest rate swap 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
31、 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.Calculate swap cash flows on the assumption that the LIBOR rates will equal the forward rates3.Set the swap rates equal to the present valu
32、e of these cash flows2021/9/11Options,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
33、%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,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%
34、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 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
35、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 John C.HullTang Yinca
36、i,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$15,000,000 -every year for 5 years2021/9/11Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal Univ
37、ersity5.29Exchange of PrincipalIn an interest rate swap the principal is not exchangedIn a currency swap the principal is exchanged at -the beginning&-the end of the swap2021/9/11Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University5.30The Cash F
38、lows(Table 5.5,page 137)YearsDollarsPounds$-millions-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 2021/9/11Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University5.31Typical Uses of a Currency SwapConversionfrom a l
39、iability 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 University5.32Comparative Advantage Arguments for Currency
40、 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 Normal University5.33Valuation of Currency SwapsLike interest rate s
41、waps,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 University5.34Example for a Currency SwapSuppose that the term structure of int
42、erest 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 on$10 million.The current exchange rate is 110 yen=$1.Evaluate the swap unde
43、r 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 have a domestic and a foreign bond BD=0.8e-0.09x1+0.8e-0.09x2+10.8e-0.09x3=$9.644 million
44、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,Shanghai Normal University5.36Example Valued as FRAsThe current spot rate is 110 yen per dollar or 0.009091 dolla
45、rs 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=0.0106The value of the forward contracts corresponding to the exchange of interest are therefore(60*0.0096)-0.8)e-0.09x1=-0.21
46、(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,Shanghai Normal University5.37Example Valued as FRAs(continued)The final exchange of principal involves receiving 1,200 million yen for$10 million.The
47、 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&Forward
48、sA swap can be regarded as a convenient way of packaging forward contractsThe“plain vanilla”interest rate swap in our example consisted of 6 FRAs(page 133)The“fixed for fixed”currency swap in our example consisted of a cash transaction&5 forward contracts(ex.5.4)2021/9/11Options,Futures,and Other De
49、rivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University5.39Valued as Forward ContractsThe value of the swap is the sum of the valuesof the forward contracts underlying the swapBoth swaps and forwards are normally“at-the-money”initiallyThis means that it costs NOTHING to enter
50、 intoa forward or swapIt does NOT mean that each forward contractunderlying a swap is“at-the-money”initially2021/9/11Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University5.40Credit Risk(page 143)A swap is worth zero to a company initiallyAt a fut