2020cfa二级讲义cfa二级-衍生品

1、CONTENTS目录Pricing and Valuation of ForwardCommitmentsValuation of Contingent ClaimsCFA Level IIDerivativesWarm-upt=0Pricingt=tValuationt=TSettlementWarm-up Pricing: 确定远期价格 （t=0）. Valuation: 签订合约期间的某一时刻是否赚钱（t =t）. 合约签

2、订期初时，双方的价值都为0 （forward commitment）。Non-Arbitrage Principle Arbitrage: 在不同市场同时买卖相同资产并获利 （低买高卖）. Arbitrage opportunities: 相同的东西卖不同的价格. The no-arbitrage principle(Law of one price): 不存在任何套利机会. The no-arbitrage principle 可以用来对衍生品进行定价。 FP=S0(1+Rf)TPricing and Val

3、uation of Forward Commitments No-Arbitrage Rule Equity Forward and Futures Interest Rate Forward and Futures (FRA) Fixed-income Forward and Futures Currency Forward and Futures Interest Rate Swap Currency Swap Equity SwapNon-Arbitrage PrincipleAt initiationA

4、t settlement date1. 借钱 S0 2. 买资产3. Short一份远期合约1. 把资产交割给long方 2. 获得FP的现金 3. 偿还本金和利息 Profit=FP-S0(1+Rf)TNon-Arbitrage Principle Cash-and-carry Arbitrage: 正向套利。 If FPS0(1+Rf )TNon-Arbitrage PrincipleAt initiationAt settlement date1. 卖空标的资产，获得S0的现金 2. 存银行 3. Lon

5、g一份远期合约 1. 支付给short方FP的现金 2. 把标的资产还给借出资产的一方 3. 获得本金和利息 Profit=S0(1+Rf )T-FPNon-Arbitrage Principle Reverse-cash-and-carry Arbitrage: 反向套利。 If FPS0(1+Rf )TPricing and Valuation of Forward Commitments No-Arbitrage Rule Equity Forward and Futures Interest Rate

6、Forward and Futures (FRA) Fixed-income Forward and Futures Currency Forward and Futures Interest Rate Swap Currency Swap Equity SwapNon-Arbitrage Principle General pricing formula: FP=(S0-PVbenefit+PVcost)(1+Rf)TExample1Calculate the no-arbitrage forward pri

7、ce for a 120- day forward on a stock that is currently priced at$20.00 and is expected to pay a dividend of $0.5 in 15 days, $0.5 in 75 days, and $0.50 in 180 days. The annual risk-free rate is 3%, and the yield curve is flat. 0.5 0.5PVD = 1.0315/365+1.0375/365 =$0.9964FP = ($20-$0.9964)1.03120/365=

8、$19.19Equity Forward and Futures Forward contracts on a dividend-paying stock Price: FP=(S0-PVD0)(1+Rf)T Value at time t: Vlong=St-PVDt-FP/(1+Rf)(T-t)Equity Forward and Futures EquityIndexForwardContractsWith Continuous Dividends Rfc= ln(1+Rf)c-c）T FP (on an

9、 equity index)=S0e(Rf Where: Rfc=continuously compounded risk-free rate c=continuously compounded dividend yieldcc(T-t) Vlong=St / e (T-t)-FP/ eRfExample2After 70 days, the value of the stock in the previous example is $40.00. Calculate the value of the equity forward contract on the stock to the lo

10、ng position, assuming the risk-free rate is still 3% and the yield curve is flat. 0.5 PVD=1.035/365=$0.4998; 19.19 V(long position)=($40-$0.4998)-=$20.391.0350/365ExampleAfter 90 days, the value of the index in the previous example is 1,030. Calculate the va

11、lue to the long position of the forward contract on the index, assuming the continuously compounded risk-free rate is 4.5% and the continuous dividend yield is 2.0%.Answer:V90(of the long position) =( 1030 )-. /( 1245 )=-209.2. /ExampleThe value of the S&P 500 index is 1,230. The continuously compou

12、nded risk-free rate is 5% and the continuous dividend yield is 2%. Calculate the no-arbitrage price of a 150-day forward contract on the index.Answer:FP =1,230x e(0.05-0.02)X(150/365) =1,245Interest Rate Forward and Futures (FRA) FRA定义: 标的资产是利率的远期合约。 The lon

13、g position: 借入钱的人。 The short position: 借出钱的人。 FRA报价: 14FRA?; 36FRA?Pricing and Valuation of Forward Commitments No-Arbitrage Rule Equity Forward and Futures Interest Rate Forward and Futures (FRA) Fixed-income Forward and Futures Currency Forward and Futures Interest Rate Swap Currency Swap Equity S

14、wapExampleCalculate the price of a 14FRA.The current 30- day LIBOR is 2% and 120-day LIBOR is 4%.Theactual30-dayrate(Period): R(30)=0.02 30/360=0.0017Theactual120-dayrate(Period): R(120)=0.04 120/360=0.013The actual 90-day forward rate in 30 days from now (p

15、eriod): 1+R(120) / 1+R(30) -1=1.013/1.0017-1=0.0113.Interest Rate Forward and Futures (FRA) The no-arbitrage forward rate(FR).L(m)/mFR/nL(m+n)/m+n(1+Lm m/360)(1+FR n/360)=(1+Lm+n (m+n)/360)Pricing and Valuation of Forward Commitments No-Arbitrage Rule Equity

16、 Forward and Futures Interest Rate Forward and Futures (FRA) Fixed-income Forward and Futures Currency Forward and Futures Interest Rate Swap Currency Swap Equity SwapExampleThe annualized forward rate, which is the price of the FRA, is: RFRA=0.0113360/90=0.0452=4.52%.http:/www.

17、ExampleCalculate the price of a 240-day forward contract on a 6% U.S. Treasury bond with a spot price of$ 1,020 (including accrued interest) that has just paid a coupon and will make another coupon payment in 180 days. The annual risk-free rate is 5%. Remember that U.S. Treasury bonds mak

18、e semiannual coupon payments:Fixed-income Forward and Futures Coupon bonds: 与分红的股票远期类似，只是现金流是coupon。 Price: FP=(S0-PVC0)(1+Rf)T Value: Vlong=St-PVCt-FP/(1+Rf)(T-t)Pricing and Valuation of Forward Commitments No-Arbitrage Rule Equity Forward and Futures Inter

19、est Rate Forward and Futures (FRA) Fixed-income Forward and Futures Currency Forward and Futures Interest Rate Swap Currency Swap Equity SwapAnswerC= 10000.062=30 30PVC=1.05180/365=29.29 30FP=(1020-1.05180/365) (1+5%)(240/365) =1023Currency Forward and Futur

20、esSt FP Value: V=long(1+R )Tt(1+R )TtFD If continuous interest rates：CC FP=S e(RDR )T0F Vlon S(tc)( FP) eRg=(Tt)eRc(Tt)FDCurrency Forward and Futures Price: covered Interest Rate Parity (IRP): FP=S0(1+RD)T/(1+RF)T RD: risk-free rate of domestic currency RF: risk-free rate of foreign currency DC/FCht

21、tp:/AnswerCHF: rfc=ln(1.06)=0.0583 USD: rc=ln(1.05)=0.0488S0=$0.57T=100/365 FP=($0.57 e0.0583(100/365)(e0.0488(100/365)=$0.5686ExampleThe U.S. risk-free rate is 5 percent, the Swiss risk- free rate is 6 percent, and the spot exchange rate between the United States

22、 and Switzerland is$0.57. Calculate the continuously compoundedU.S. and Swiss risk-free rates. Calculate the price at which you could enter into a forward contract that expires in 100 days. Calculate the value of the forward position 30 days into the contract. Assume that the spot rate is $0.60.http

23、:/Pricing and Valuation of Forward Commitments No-Arbitrage Rule Equity Forward and Futures Interest Rate Forward and Futures (FRA) Fixed-income Forward and Futures Currency Forward and Futures Interest Rate Swap Currency Swap Equity SwapAnswerVt=(0.60e0.0583(70/3

24、65)($0.5686e0.0488(70/365)=$0.03The value of the contract is $0.03 per Swiss franc.Interest Rate Swap 1=CB1+CB2+CB3+Bn(C+1) C=B1B +B1+B2+ C 是一个periodic rate, 必须年化，获得annual swap rate.Interest Rate Swap A plain vanilla swap: 一方支付固定利率，一方支付浮动利率。 Pricing a plain

25、vanilla swap: 计算固定利率 (swap rate)。AnswerStep1: Calculate the discount factorsB1=1/(1+1.5%90/360)=0.9963; B2=1/(1+2%180/360)=0.9901; B3=1/(1+3.5%270/360)=0.9744; B4=1/(1+5%360/360)=0.9524.ExampleCalculate the swap rate of a 1-year quarterly-pay plain-vanilla s

26、wap. The LIBOR spot rates are:R(90-day)=1.5%; R(180-day)=2%; R(270-day)=3.5%; R(360-day)=5%.Interest Rate Swap The market value of a swap to the pay-fixed side is Vswap(pay-fixed)=MVflt-MVfix Vswap(pay-floating)=MVfix-MVfltAnswerStep2: C=(1-B4)/(B1+B2+B3+B4)

27、=(1-0.9524)/(0.9963+0.9901+0.9744+0.9524)= 1.22%Step3:Swap rate=1.22%360/90=4.88%.AnswerStep1: B1=1/(1+2.5%60/360)=0.9959;B2=1/(1+3%150/360)=0.9877; B3=1/(1+3.5%240/360)=0.9772; B4=1/(1+4%330/360)=0.9646;Step2: PV(fixed)=1.22%(0.9959+0.9877+0.9772+0.964 6)+1

28、0.9646=1.01249ExampleCalculate the value of the plain vanilla swap to the pay-fixed side in the previous example after 30 days. The swap rate is 4.88% and the notional principal is $1million. Assume after 30 days the LIBOR spot rates are:R(60day)=2.5%; R(150-day)=3%; R(240-day)=3.5%; R(330-day)=4%.h

29、ttp:/AnswerStep 4: Calculate the swap value to the pay-fixed side: V= PV(floating)-PV(fixed) notional principal=(0.999635-1.01249)$1 million=$-12,855.375AnswerStep3:Thefirstpaymentatinitiation: 1.5%90/360=0.00375.PV(floating)=(1+0.00375)0.9959=0.999635http:/www.ze

30、Currency Swap 假设dollar 和Euro 之间进行Currency Swap: Swap1: pay dollar fixed and receive Euro fixed. Swap2: pay dollar fixed and receive Euro floating. Swap3: pay dollar floating and receive Euro fixed. Swap4: pay dollar floating and receive Euro floating. In the swap4 (floa

31、ting for floating)，没有pricing问题。Pricing and Valuation of Forward Commitments No-Arbitrage Rule Equity Forward and Futures Interest Rate Forward and Futures (FRA) Fixed-income Forward and Futures Currency Forward and Futures Interest Rate Swap Currency Swap Equity Swaphttp:/www.ze

32、ExampleNow move forward 120 days. The new exchange rate is $1.32 per pound, and the new U.S. term structure is:L$(60)=0.0623120The comparable set of rates areL(60)=0.0527 120L$(240)=0.0639120L(240)=0.0552 120L$(420)=0.0643120L(420)=0.0567 120L$(600)=0.0687120L(600)=0.0585 120ExampleConsider

33、 a two-year currency swap with semiannual payments. The domestic currency is the U.S. dollar, and the foreign currency is the U.K. pound. The current exchange rate is $1.51 per pound.A. Calculate the annualized fixed rates for dollars and pounds:L0$(180)=0.0545The comparable set of rates areL0(180)=

34、0.0483L0$(360)=0.0625L0(360)=0.0526L0$(540)=0.0635L0(540)=0.0539L0$(720)=0.0675L0(720)=0.0561AnswerA: First calculate the fixed payment in dollars and pounds. The dollar present value factors for 180, 360, 540, and 720 days are as follows:ExampleB. Assume th

35、at the notional principal is $1 or the corresponding amount in British pounds. Calculate the market values of the following swaps:Pay fixed and receive $ fixed; Pay floating and receive $ fixed; Pay floating and receive $ floatingAnswerB:The new dollar and p

36、ound factors for 60, 240, 420, and 600 days are as follows:AnswerThe semiannual and annualized fixed payment of per $1 of notional principal is 10.8811=0.0321 or0.9735+0.9412+0.9130+0.88110.0641 for annualized.Similarly the semiannual and annualized fixed payment of per 1 of notional principal is: 0

37、.0269 or 0.0538 for annualized.AnswerThe present value of the remaining fixed payments plus the 1 notional principal is 0.0269(0.9913+0.9645+0.9379+0.9112)+1(0.9112)=1.0136. Convert this amount to the equivalent of$1 notional principal and convert to dollars

38、 at the current exchange rate:1/1.511.321.0136 = $0.8861.AnswerThe present value of the remaining fixed payments plus the $1 notional principal is 0.0321(0.9897 + 0.9591 + 0.9302 + 0.8973) + 1(0.8973) = 1.0185.The present value of the floating payments plus hypothetical $1 notional principal discoun

39、ted back 120 days is 1.02725(0.9897)= 1.0167AnswerConvert this amount to the equivalent of $1 notional principal;and convert to dollars at the current exchange rate 1/1.511.321.015=$0.8873. The market values based on notional principal of $1 are as follows:P

40、ay fixed and receive $ fixed=$0.0464=1.0185-0.8861 Pay floating and receive $ fixed=$0.0424=1.0185-0.8873Pay floating and receive $ floating=$0.0461=1.0167-0.8873 Pay fixed and receive $ floating=$0.0501=1.0167-0.8861AnswerThe present value of the floating payments plus hypothetical 1 notional princ

41、ipal: 1.02415(0.9913)=1.015Equity Swap 三种equity swaps: Pay fixed rate and receive equity return; Pay floating rate and receive equity return; Pay one equity return and receive another equity return. C= B1B n1+B2+B nPricing and Valuation of Forward Commitment

42、s No-Arbitrage Rule Equity Forward and Futures Interest Rate Forward and Futures (FRA) Fixed-income Forward and Futures Currency Forward and Futures Interest Rate Swap Currency Swap Equity SwapAnswerStep1: Calculate the new discount factors 30days later: B1=

43、1/(1+2.5%60/360)=0.9958; B2=1/(1+3%150/360)=0.9877;B3=1/(1+3.5%240/360)=0.9772; B4=1/(1+4%330/360)=0.9646Step2: PV(fixed)=0.955%(0.9958+0.9877+0.9772+0.9646)+ 10.9646=1.002087ExampleAn equity swap has the annual swap rate of 3.82% and the notional principal of $1million.The underlying is an index, c

44、urrently trading at 1,000. Assume after 30 days the index becomes1,200 and the LIBOR spot rates are:R(60-day)=2.5%; R(150-day)=3%; R(240day)=3.5%; R(330day)=4%.Calculate the value of the equity swap to the fixed-rate payer.CONTENTS目录Pricing and Valuation of

45、ForwardCommitmentsValuation of Contingent ClaimsAnswerStep3: PV(index)=11200/1000=1.2Step4:V= PV(index)-PV(fixed) notional principal=(1.2-1.002087)$1million=$197,913Stock Binomial Model Stock Binomial Model (one-period) Stock Binomial Model (two-period) Hedg

46、ing RatioValuation of Contingent Claims Stock Binomial Model Interest rate Binomial Model Black-Scholes-Merton Model (BSM) Black Model Option Greeks and Implied VolatilityStock Binomial Model (one-period) Risk-neutral probability of an up move is u; Risk-neu

47、tral probability of an down move is =1- , 1+Rfd duu=ud. C1+=Max (0, S1 +-X); C1 = Max (0, S1 -X) C = C + C 10u 1d 11+RTfStock Binomial Model (one-period)S1+=S0uS0S1-=S0dStock Binomial Model (two-period) Call optionS+=S UU0+c+ =MAX (0, S+-X)S =S0Uc+SS+-(=S-+

48、)=S0UD0c=?c+- = MAX (0, S+-X)S-=S0Dc-S-=S0DDc-=MAX (0,S-X)美式看涨怎么处理？ Stock Binomial Model Stock Binomial Model (one-period) Stock Binomial Model (two-period) Hedging RatioStock Binomial Model (two-period) Rf=2%，U=1.2，D=0.8 1+R fd 1+2%0.8u=ud= 1.20.8 =0.55 p+=

49、 0.550+0.452.4 /(1+2%)=1.06 p- = (0.552.4+0.455.6)/(1+2%)=3.76 p= (0.551.06+0.453.76)/(1+2%)=2.23Stock Binomial Model (two-period) Put option (X=12, Rf=2%，U=1.2，D=0.8)S0U=101.2=12 p+ =0S0UU=101.21.2=14.4 p+=0 S =10 S0UD=120.8=9.60p+- =12-9.6=2.4 S0D=100.8=8.S0DD=100.80.8=6.4p- =12-8=4p- =12-6.4=5.6

50、Hedging Ratio Hedging ratio=(C+-C-)/(S+-S-) 如果overpriced, 卖出，买入Hedging ratio份股票； 如 果underpriced, 买 入，卖 出 Hedging ratio份股票。Stock Binomial Model Stock Binomial Model (one-period) Stock Binomial Model (two-period) Hedging Ratiohttp:/www.zej

51、Interest rate Binomial Model Interest rate caplet: 类似于一个call option on interest rates. Expiration value of caplet=max 0, (one year rate-cap rate) notional principal Interest rate floorlet: 类似于一个put option on interest rates. Expiration value of floorlet=max 0, (floor rate- one year rate) notional principalValuation of Contingent Claims Stock Binomial Model Interest rate Binomial Model Black-Scholes-Merton Model (BSM) Black Model Option Greeks and Implied VolatilityExampleInt. Rate=8.58%Int. Rate=6.40%Int. Rate=5.13%Int. Rate=5.90%Int. Rate=4