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1、Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.1,Swaps(互换)Chapter 5,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.2,Nature of Swaps,A swap is an agreement to exchange cash f
2、lows(现金流)at specified future times according to certain specified rules,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.3,Terminology,LIBOR the London InterBank Offer RateIt is the rate of interest offered by banks on deposits from other
3、banks in Eurocurrency markets,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.4,An Example of a“Plain Vanilla”Interest Rate Swap(大众型利率互换),An agreement by“Company B”toRECEIVE 6-month LIBOR andPAY a fixed rate of 5%paevery 6 months for 3 ye
4、ars on a notional principal of$100 millionNext 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,Cash Flows to Company B(See Ta
5、ble 5.1,page 123),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.6,More on Table 5.1,The floating-rate payments are calculated using the six-month LIBOR rate prevailing six month before the payment dateThe principle is only used for the
6、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.,Options,Futures,and Other
7、Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.7,Typical Uses of anInterest Rate Swap,Converting a liabilityfrom a FIXED rate liability to aFLOATING rate liabilityFLOATING rate liabilityto a FIXED rate liability,Converting an investmentfrom a FIXED rate investmen
8、t 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 University,5.8,Transforming a Floating-rate Loan to a Fixed-rate,Consider a 3-year swap initialized on March 1,2000 wher
9、eCompany 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+80bp,LIBOR+0.8%,5.2%,Note:1 basis point(bp)=one-hundredth of 1%,Options,Futures,and Other Derivatives,4th edition 2000 by John
10、C.HullTang Yincai,Shanghai Normal University,5.9,Transforming 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 the swap 3.Pays 5%to Company A in the Swa
11、pIn 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 University,5.10,A and B Transform a Liability(Figure 5.2,page 125),A,B,LIBOR,5%,LIBOR+0.8%,5.2%,Options,Fut
12、ures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.11,Financial Institution is Involved(Figure 5.4,page 126),A,F.I.,B,LIBOR,LIBOR,LIBOR+0.8%,4.985%,5.015%,5.2%,“Plain vanilla”fixed-for-float swaps on US interest rates are usually structured so that the
13、 financial institutions earns 3 to 4 basis points on a pair of offsetting transactions,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.12,A and B Transform an Asset(Figure 5.3,page 125),A,B,LIBOR,5%,LIBOR-0.25%,4.7%,Options,Futures,and Ot
14、her Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.13,Financial Institution is Involved(See Figure 5.5,page 126),A,F.I.,B,LIBOR,LIBOR,4.7%,5.015%,4.985%,LIBOR-0.25%,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal U
15、niversity,5.14,The Comparative Advantage Argument(Table 5.4,page 129),Company A wants to borrow floatingCompany B wants to borrow fixed,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.15,The Comparative Advantage(continued),One possible s
16、wap 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 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 gain,Options,Future
17、s,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.16,The Swap(Figure 5.6,page 130),A,B,LIBOR,LIBOR+1%,9.95%,10%,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.17,The Swap when a Financial Ins
18、titution is Involved(Figure 5.7,page 130),A,F.I.,B,10%,LIBOR,LIBOR,LIBOR+1%,9.93%,9.97%,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.18,Total Gain from anInterest Rate Swap,The total gain from an interest rate swap is always|a-b|wherea
19、 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%,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yinca
20、i,Shanghai Normal University,5.19,Criticism of the Comparative Advantage Argument,The 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 ratesBs fixed rate depends on the spread above LIBO
21、R it borrows at in the future,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.20,Valuation 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 b
22、ondAlternatively,they can be valued as a portfolio of forward rate agreements(FRAs),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.21,Valuation of an Interest Rate Swap as a Package of Bonds,The fixed rate bond is valued in the usual way
23、(page 132)The floating rate bond is valued by noting that it is worth par immediately after the next payment date(page 132),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.22,Valuation of an Interest Rate Swap as a Package of Bonds(contin
24、ued),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 agreement ti:time when the ith payments are exchanged ri:LIBOR zero rate for a maturity tiThen
25、,Vswap=Bfix-Bfl and if k is the fixed-rate coupon and k*is the floating,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.23,Example of an Interest Rate Swap Valued as a Package of Bonds,Suppose,that you agreed to pay 6-month LIBOR and rece
26、ive 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 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
27、 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 million,Options,Futures,and Other Derivatives,4th edi
28、tion 2000 by John C.HullTang Yincai,Shanghai Normal University,5.24,Valuation in Terms of FRAs,Each exchange 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),Options,Futures,and Other Derivatives,4t
29、h edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.25,Valuation of an Interest Rate Swap as a Package of FRAs,A 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 assumpti
30、on that the LIBOR rates will equal the forward rates3.Set the swap rates equal to the present value of these cash flows,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.26,Example of an Interest Rate Swap Valued as a Package of FRAs,Same p
31、roblem 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,we first need to calculate the forward rates corre
32、sponding 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,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal Universit
33、y,5.27,Example 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 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=-4.27,
34、Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.28,An 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$15,000,000-every year for 5 years,Options,Futures,and Other Deri
35、vatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.29,Exchange of Principal,In an interest rate swap the principal is not exchangedIn a currency swap the principal is exchanged at-the beginning&-the end of the swap,Options,Futures,and Other Derivatives,4th edition 2000 b
36、y John C.HullTang Yincai,Shanghai Normal University,5.30,The Cash Flows(Table 5.5,page 137),Years,Dollars,Pounds,$,-millions-,0,15.00,+10.00,1,+1.20,1.10,2,+1.20,1.10,3,+1.20,1.10,4,+1.20,1.10,5,+16.20,-11.10,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai N
37、ormal University,5.31,Typical Uses of a Currency Swap,Conversionfrom a liability inone currencyto a liability inanother currency,Conversionfrom an investment in one currencyto an investment in another currency,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai
38、Normal University,5.32,Comparative Advantage Arguments for Currency Swaps(Table 5.6,pages 137-139),Company A wants to borrow AUDCompany B wants to borrow USD,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.33,Valuation of Currency Swaps,L
39、ike interest rate swaps,currency swaps can be valued either as the-difference between 2 bonds or as a-portfolio of forward contracts,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.34,Example for a Currency Swap,Suppose that the term stru
40、cture 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 on$10 million.The current exchange rate is 110 yen=$1.Evaluate t
41、he swap under the assumption that payments are made just once per year.,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.35,Example Valued as Bonds,Here we have a domestic and a foreign bond BD=0.8e-0.09x1+0.8e-0.09x2+10.8e-0.09x3=$9.644 m
42、illion BF=60e-0.04x1+60e-0.04x2+1260e-0.04x3=1,230.55 millionThus,the value of the swap is Vswap=S0BF-BD,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.36,Example Valued as FRAs,The current spot rate is 110 yen per dollar or 0.009091 dol
43、lars 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.
44、21(60*0.0100)-0.8)e-0.09x2=-0.16(60*0.0106)-0.8)e-0.09x3=-0.13,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.37,Example Valued as FRAs(continued),The final exchange of principal involves receiving 1,200 million yen for$10 million.The va
45、lue 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 million,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.38,Swaps&Forwards,A swap
46、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),Options,Futures,and Other Derivatives,4th ed
47、ition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.39,Valued as Forward Contracts,The 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 intoa forwar
48、d or swapIt does NOT mean that each forward contractunderlying a swap is“at-the-money”initially,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.40,Credit Risk(page 143),A swap is worth zero to a company initiallyAt a future time its value
49、 is liable to beeither POSITIVE or NEGATIVEThe company has credit risk exposure only when its value is POSITIVE,Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.41,Examples of Other Types of Swaps,Amortizing&step-up swaps(本金分期减少方式互换与本金逐步增加
50、的互换)Extendible&puttable swaps(可延长与可赎回互换)Index amortizing rate swaps(指数递减比率互换)Swaption:Options on swaps(互换权)Equity swaps(股权的互换)Commodity swaps(商品的互换)Differential swaps(差异互换),Options,Futures,and Other Derivatives,4th edition 2000 by John C.HullTang Yincai,Shanghai Normal University,5.42,Assignments,5.
