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单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Tang Yincai, Shanghai Normal University,5.,*,Swaps,(,互换,),Chapter 5,1,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Nature of Swaps,A swap is an agreement to exchange,cash flows,(,现金流,),at specified,future times,according to certain specified,rules,2,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Terminology,LIBOR,the,L,ondon,I,nter,B,ank,O,ffer,R,ateIt is the,rate of interest,offered,by banks on,deposits,from other banks in,Eurocurrency markets,3,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,An Example of a “Plain Vanilla” Interest Rate Swap(,大众型利率互换,),An,agreement,by “Company B” to,RECEIVE,6-month,LIBOR,and,PAY,a fixed rate of 5% paevery 6 months for 3 years on a,notional principal,of $100 million,Next slide illustrates cash flows, where,POSITIVE,flows are revenues (,inflows,) and,NEGATIVE,flows are expenses (,outflows,),4,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,-Millions of Dollars-,LIBOR,FLOATING,FIXED,Net,Date,Rate,Cash Flow,Cash Flow,Cash Flow,Mar.1, 1999,4.2%,Sept. 1, 1999,4.8%,+2.10,2.50,0.40,Mar.1, 2000,5.3%,+2.40,2.50,0.10,Sept. 1, 2000,5.5%,+2.65,2.50,+0.15,Mar.1, 2001,5.6%,+2.75,2.50,+0.25,Sept. 1, 2001,5.9%,+2.80,2.50,+0.30,Mar.1, 2002,6.4%,+2.95,2.50,+0.45,Cash Flows to Company B,(See Table 5.1, page 123),5,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,More 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 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 short (long) a fixed-rate bond.,6,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Typical Uses of anInterest Rate Swap,Converting a,liability,from a,FIXED,rate liability to a,FLOATING,rate liability,FLOATING,rate liabilityto a,FIXED,rate liability,Converting an,investment,from a,FIXED,rate investment to a,FLOATING,rate investment,FLOATING,rate investment to a,FIXED,rate investment,7,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Transforming,a,Floating-rate,Loan to a,Fixed-rate,Consider a 3-year swap initialized on March 1, 2000 where,Company 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 + 80bp,CompanyB,CompanyA,5%,LIBOR,LIBOR+0.8%,5.2%,Note:,1 basis point (bp) = one-hundredth of 1%,8,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,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 Swap,In essence, B has transformed its variable rate borrowing at LIBOR + 80bp to a fixed rate of 5.8%,9,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,A and B Transform a Liability,(Figure 5.2, page 125),A,B,LIBOR,5%,LIBOR+0.8%,5.2%,10,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,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 financial institutions earns 3 to 4 basis points on a pair of offsetting transactions,11,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,A and B Transform an Asset,(Figure 5.3, page 125),A,B,LIBOR,5%,LIBOR-0.25%,4.7%,12,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,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%,13,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,The Comparative Advantage Argument,(Table 5.4, page 129),Company A wants to borrow floating,Company B wants to borrow fixed,Fixed,Floating,Company A,10.00%,6-month LIBOR + 0.30%,Company B,11.20%,6-month LIBOR + 1.00%,14,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,The,Comparative Advantage,(continued),One possible swap is,Company 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,gain,Company 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,CompanyB,CompanyA,9.95%,LIBOR,10%,LIBOR + 1%,15,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,The Swap,(Figure 5.6, page 130),A,B,LIBOR,LIBOR+1%,9.95%,10%,16,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,The Swap when a Financial Institution is Involved,(Figure 5.7, page 130),A,F.I.,B,10%,LIBOR,LIBOR,LIBOR+1%,9.93%,9.97%,17,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Total Gain,from an,Interest Rate Swap,The,total gain,from an interest rate swap is always |a-b| where,a,is the,difference,between the,interest rates,in the,fixed-rate,market for the two parties, and,b,is the,difference,between the,interest rates,in the,floating-rate,market for the two parties,In this example a=1.20% and b=0.70%,18,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,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 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 future,19,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,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 bond,Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs),20,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Valuation,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 par immediately after the next payment date (page 132),21,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Valuation,of an,Interest Rate Swap,as a,Package,of,Bonds,(continued),Define,V,swap,:value of the swap to the financial institution,B,fix,:value of the fixed-rate bond underlying the swap,B,fl,:value of the floating-rate bond underlying the swap,L,:notional principal in a swap agreement,t,i,: time when the ith payments are exchanged,r,i,: LIBOR zero rate for a maturity t,i,Then,V,swap,=,B,fix,-,B,fl,and if,k,is the fixed-rate coupon and,k,*,is the floating,22,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Example,of an,Interest Rate Swap,Valued,as a,Package,of,Bonds,Suppose, 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 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,B,fix,= 4e,-0.25x0.10,+ 4e,-0.75x0.105,+ 104e,-1.25x0.11,= $ 98.24 million,B,fl,= 5.1e,-0.25x0.10,+ 100e,-0.25x0.10,= $102.51 million,Hence,V,swap,= 98.24 - 102.51 = -$4.27 million,23,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Valuation in Terms of FRAs,Each exchange of payments in an interest rate swap is an FRA,The FRAs can be valued on the assumption that todays forward rates are realized (See section 4.6, page 97),24,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Valuation,of an,Interest Rate Swap,as a,Package,of,FRAs,A simple three step process1. Calculate each of the forward rates for each of the LIBOR rates that will determine swap cash flows,2. Calculate swap cash flows on the assumption that the LIBOR rates will equal the forward rates,3. Set the swap rates equal to the present value of these cash flows,25,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Example,of an,Interest Rate Swap,Valued,as a,Package,of,FRAs,Same problem as before.,The cash flows for the payment in 3 months have already been set. A rate of 8% will be exchanged for a rate of 10.2%. The,NPV,of this transaction is,0.5 * 100 * (0.08 - 0.102)e,-0.1*0.25,=,-1.07,To figure out the NPV of the remaining two payments, we first need to calculate the forward rates corresponding to 9 and 15 months,or 10.75% with continuous compounding which corresponds to 11.044% with semi-annual compounding.,The value of the 9 month FRA is,0.5 * 100 * (0.08 - 0.11044)e,-0.105*0.75,=,-1.41,26,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Example,Swap,as,Valued,as,FRAs,(continued),The 15 month forward rate is,or 11.75% with continuous compounding which corresponds to 12.102% with semi-annual compounding.,The value of the 15 month FRA is,0.5 * 100 * (0.08 - 0.12102)e,-0.11*1.25,=,-1.79,Hence, the total value of the swap is,-1.07 - 1.41 - 1.79 = -4.27,27,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,An Example of a Currency Swap,An agreement to,-,pay,11% on a sterling principal of,10,000,000 &,-,receive,8% on a US$ principal of,$15,000,000,- every year for 5 years,28,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Exchange of Principal,In an,interest rate swap,the principal is not exchanged,In a,currency swap,the principal is exchanged at,- the beginning &,- the end of the swap,29,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,The Cash Flows,(Table 5.5, page 137),Years,Dollars,Pounds,$,-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,-11.10,30,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Typical Uses of a Currency Swap,Conversion,from a,liability,in,one,currencyto a,liability,in,another,currency,Conversion,from an,investment,in,one,currencyto an,investment,in,another,currency,31,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Comparative Advantage Arguments for Currency Swaps,(Table 5.6, pages 137-139),Company A wants to borrow AUD,Company B wants to borrow USD,USD,AUD,Company A,5.0%,12.6%,Company B,7.0%,13.0%,32,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Valuation,of Currency Swaps,Like interest rate swaps, currency swaps can be valued either as the,- difference between 2 bonds or as a,- portfolio of forward contracts,33,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Example,for a,Currency Swap,Suppose 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 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.,34,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Example,Valued,as,Bonds,Here we have a domestic and a foreign bond,B,D,= 0.8e,-0.09x1,+ 0.8e,-0.09x2,+ 10.8e,-0.09x3,= $ 9.644 million,B,F,= 60e,-0.04x1,+ 60e,-0.04x2,+,1260e,-0.04x3,= 1,230.55 million,Thus, the value of the swap is,V,swap,=,S,0,B,F,-,B,D,35,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Example,Valued,as,FRAs,The 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),0.009091e,0.05x1,= 0.0096,0.009091e,0.05x2,= 0.0100,0.009091e,0.05x3,= 0.0106,The 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,=,-0.13,36,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Example,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,= 2.04,Hence, the total value of the swap is,2.04 -0.13 - 0.16 - 0.21 = 1.54 million,37,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Swaps & Forwards,A swap can be regarded as a convenient way of packaging forward contracts,The “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),38,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Valued as Forward Contracts,The,value,of the,swap,is the,sum,of the,values,of the,forward contracts,underlying the,swap,Both,swaps,and,forwards,are normally “at-the-money” initially,This means that it costs,NOTHING,to enter intoa,forward,or,swap,It does,NOT,mean that,each,forward contract,underlying a,swap,is “at-the-money” initially,39,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Credit Risk (page 143),A swap is worth zero to a company initially,At a future time its value is liable to beeither,POSITIVE,or,NEGATIVE,The company has credit risk exposure only when its value is,POSITIVE,40,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,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,),41,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,Assignments,5.1, 5.2, 5.3, 5.4, 5.8, 5.9, 5.10, 5.11, 5.12,5.15,Assignment Questions,42,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull,
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