金融工程课件10applications4(f)

1、Primbs, MS&E 3451More Applications of Linear PricingPrimbs, MS&E 3452Exchange one asset for anotherA generalization of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaptionsInterest rate derivativesPrimbs, MS

2、&E 3453Exchange one asset for anotherThis is equivalent to the payoff at time T.)()(12TSTS)()()()0(112101TSTSTSEScSQThen our pricing formula iswhere )(11221212dzdzSSSSdfrom previous calculationsTo value this, we will let S1 be the numeraire.111111dzSdtSdS222222dzSdtSdSConsider two assets:and the

3、 option to exchange asset 2 for asset 1 at time T.dtdzdzE211)()(121TSTSESQPrimbs, MS&E 34541)()()0(12101TSTSEScSQwhere )(11221212dzdzSSSSdNote that:)2, 0()(2122211122Ndzdz21212222So letting:zdSSSSd12121TfEHence, letting we just need to evaluate:12SSf zfddfwhereExchange one asset for anotherPrimb

4、s, MS&E 34551TfEzfddfwhere)()(1210dNdNffETEvaluating givesTTfd2/)ln(201Tdd12where:Substituting in terms of S1 and S2 gives the final answer:)()0()()0(21120dNSdNScTTSSd2/)0(/ )0(ln(2121Tdd1221212222whereExchange one asset for anotherPrimbs, MS&E 3456Exchange one asset for anotherA generalizat

5、ion of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaptionsInterest rate derivativesPrimbs, MS&E 3457Futures contracts and the risk neutral measure:time 0time Ttime/positionmark to market(at time t+dt)time T va

6、lueTtTsdfdsr)exp(0 t)exp(0dttsdsrTtdttsdfdsr)exp(0)exp()exp(0TdttsTtdttsdsrdfdsrt (start)t+dtPrimbs, MS&E 3458Futures contracts and the risk neutral measure:time 0time Ttime/positionmark to market(at time t+dt)time T valuet (start) t)exp(0dttsdsrTtTsdfdsr)exp(0Ttdttsdfdsr)exp(0 T-dt)exp(0TsdsrTd

7、tTTsdfdsr)exp(0TdtTTsdfdsr)exp(0Total Cost: = 0)(exp(0TtTTTsffdsrTotal Payoff:TtTsTsdfdsr)exp(0)(exp(0TtTTsfSdsrSince fTT=ST t+dtPrimbs, MS&E 3459Plug into our risk neutral pricing formula:The futures price is the expected price of the stock at time T in a risk neutral world.)(exp()exp(000TtTTsT

8、sQfSdsrdsrE)(0TtTQfSETQTtQSEfETt 0payoff )exp(price0TsQdsrEIn particular:TQTtQTSEfEf0Tt 0Primbs, MS&E 34510Exchange one asset for anotherA generalization of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaptionsI

9、nterest rate derivativesPrimbs, MS&E 34511Tt0payoffoccursForward contracts and the bond forward risk neutral worldForward prices are a tradable (St) divided by B(t;T) TQTtQTSEFEFBB0Tt 0In particular:The forward price is the expected price of the stock at time T in a bond (B(t;T) forward risk neu

10、tral world.Hence, under B(t;T) as the numeraire, forward prices are martingales. );( TtBFSTtt);( TtBSFtTtRecall: Primbs, MS&E 34512Exchange one asset for anotherA generalization of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates,

11、and swap ratesSwaptionsInterest rate derivativesPrimbs, MS&E 34513Spot and Forward Ratestimespot rates); 0 ( tRspot rate curvet1t2); 0(1tR); 0(2tR);| 0(21ttF0Rates are quoted as yearly rates. Hence, the actual rate applied over time t1 is t1R(0;t1). If P is the principal on a zero coupon bond:);

12、 0(1 (); 0(ttRPtBBond Price:);|0()(1 ();|0(211221ttFttPttBForward Bond Price:); 0(1 ();|0()(1)(; 0(1 (22211211tRtttFtttRtForward Rates:Primbs, MS&E 34514Tt0payoffoccursForward interest rates and the bond forward risk neutral worldT+tThe forward rate is tradables divided by B(t;T+t).P),|(1),|(ttt

13、TTtFPTTtB),|(tTTtBA forward bond price1),|(11),|(tttTTtBTTtF),(),(),(11),(),(1),|(ttttttTtBTtBTtBTtBTtBTTtFHence, forward rates are martingales under the numeraire B(t;T+t).Primbs, MS&E 34515Tt0payoffoccursForward interest rates and the bond forward risk neutral worldT+t),|(tTTtBP);();|();|0(ttt

14、TTRETTtFETTFBBQQTt 0In particular:in a forward risk neutral world with respect to B(t;T+t).Primbs, MS&E 34516Exchange one asset for anotherA generalization of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaption

15、sInterest rate derivativesPrimbs, MS&E 34517Swaps and Swap ratesAn interest rate swap is an agreement to exchange a fixed rate of interest S for a floating rate of interest on the same notional principal P. The swap rate S is the fixed rate that makes a swap have zero value. 03t4t5tt2tFixedSPtSP

16、tSPtSPtSPt03t4t5tt2tFloat); 0(ttRP)2 ;(tttRP)3 ;2(tttRP)4 ;3(tttRP)5 ;4(tttRP)0(SAFixedPVFloatPVEquating present values at time 0: where A(0) is the value of an annuityPrimbs, MS&E 34518Tt0Forward swap rates and the annuity forward risk neutral worldT+tT+2tT+ntTo be a bit more precise, let S(t|T

17、) be a forward swap rate where the swap begins at time T.Hence:)|()|(TtAFloatPVTtStwhere the Floating side is tradable (for instance as coupons on a floating rate bond).Therefore, forward swap rates are martingales under the numeraire A(t|T). Then the forward swap rate is defined by the relationship

18、:)|()|(TtATtSFixedPVFloatPVttwhere A(t|T) is the value of an annuity starting at time T.Primbs, MS&E 34519Tt0Forward swap rates and the annuity forward risk neutral worldT+tT+2tT+ntTt 0In particular:in a forward risk neutral world with respect to A(t).)|()|()|0(TTSETtSETSAAQQPrimbs, MS&E 345

19、20Summary:Forward Rates:Numeraire: B(t;T+t);();|();|0(tttTTRETTtFETTFBBQQForward Prices:TQTtQTSEFEFBB0Numeraire: B(t;T)Futures Prices:TQTtQTSEfEf0Numeraire: Money MarketSwap Rates:Numeraire: A(t|T)|()|()|0(TTSETtSETSAAQQPrimbs, MS&E 34521Exchange one asset for anotherA generalization of Black-Sc

20、holesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaptionsInterest rate derivativesPrimbs, MS&E 34522The set-up:Tt0The option allows me to purchase the asset at time T for the strike price KExercisedecisionPayoffoccursIf the

21、 forward price at time t is greater than K, then this option is worth FtT-K at time T. Otherwise, the option is worthless because the forward price is less than the strike.(FtT-K)+Hence, the payoff is at time T.)(KFTtI have a European option on a forward contract with delivery date T,but the option

22、expires at time t.Primbs, MS&E 34523The set-up:Tt0ExercisedecisionPayoffoccurs(FtT-K)+To derive the price of this option, I will use the bond of maturity T as the numeraire:)(); 0(0KFETBcTtQBHence, I just need to calculate in a bond forwardrisk neutral world.)( KFETtQBPrimbs, MS&E 34524The s

23、et-up:Tt0ExercisedecisionPayoffoccurs(FtT-K)+Hence, I just need to calculate in a bond forwardrisk neutral world.)( KFETtQBAssume FtT is log-normal: Then we need its mean and volatility:Its mean in a forward risk neutral world is:TTtQFFEB0Let denote the volatility of FtT at time t.t(recall that the

24、volatility is the same in every risk neutral world)Primbs, MS&E 34525The set-up:Tt0ExercisedecisionPayoffoccurs(FtT-K)+Hence, I just need to calculate in a bond forwardrisk neutral world.)( KFETtQBSo Blacks Formula is:)()()(, 0()(); 0(2100dKNdNFTBKFETBcTTtQBttKFdT)()/ln(22101tdd12wherePrimbs, MS

25、&E 34526Exchange one asset for anotherA generalization of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaptionsInterest rate derivativesPrimbs, MS&E 34527A simple generalization of the Black-Scholes formulaW

26、e can use Blacks model to generalize Black-Scholes by thinking of Black-Scholes as an option on a forward contract where the delivery of the forward, and expiration of the option are at the same time.Hence, we can use Blacks formula with t=T and ST=FTT.Tt0ExercisedecisionPayoffoccurs(FtT-K)+Blacks M

27、odelT0ExercisedecisionPayoffoccurs(ST-K)+ =(FTT-K)+Black-ScholesPrimbs, MS&E 34528A simple generalization of the Black-Scholes formulaT0ExercisedecisionPayoffoccurs(ST-K)+ =(FTT-K)+Black-ScholesPlugging into Blacks formula gives:)()()(, 0(2100dKNdNFTBcTTTKFdT)()/ln(22101Tdd12whereBut, I want the

28、 solution in terms of S0:We can use that:TFTBS00); 0(Additionally, let R(T) be the time T interest rate:(Here, I am using continuous compounding!)TTReTB)(); 0(Then:TTRTeSF)(00(Substitute in)Primbs, MS&E 34529A simple generalization of the Black-Scholes formulaT0ExercisedecisionPayoffoccurs(ST-K)

29、+ =(FTT-K)+Black-ScholesTTTRKSd)()/ln(22101)()(2)(100dNKedNScTTRTdd12where:)(Ndistribution function for a standard Normal (i.e. N(0,1)It looks like standard Black-Scholes, just use the interestrate corresponding to the expiration date!Primbs, MS&E 34530Exchange one asset for anotherA generalizat

30、ion of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaptionsInterest rate derivativesPrimbs, MS&E 34531A Bond OptionT0ExercisedecisionPayoffoccurs(Bc(T)-K)+Consider an option on a bond Bc(t) with strike K and ex

31、piration T. (This bond could pay coupons and have any maturity).Then Blacks model says:)()()|0(); 0(210dKNdNTBTBcCTTKTBdc)()/)|0(ln(2211Tdd12whereLet Bc(0|T) be the forward price of the bond.(This could be calculated using the term structure).Let denote the volatility of the bond price at time T.Pri

32、mbs, MS&E 34532Exchange one asset for anotherA generalization of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaptionsInterest rate derivativesPrimbs, MS&E 34533Picture of a capRKInterest RateA cap is an upp

33、er limit on the interest payments corresponding to the life of a loan. A caplet is an upper limit on a single interest payment in a loan. Hence a cap corresponds to a caplet for every interest payment. Primbs, MS&E 34534T0An interest rate caplet:T+tnatural time lag tInterest rateset R(T;T+t)Inte

34、rest payment occurs);(KRTTRPtt);(); 0(KQRTTRPETBcapletBtttUse B(t;T+t) as the numeraire:Then:)()();|0(); 0(21dNRdNTTFPTBcapletKtttTTRTTFdKt)()/);|0(ln(2211Tdd12where);();|0(ttTTRETTFBQWe know:Assume: R(T;T+t) is log-normal with volatility Primbs, MS&E 34535T0An interest rate caplet (a second app

35、roach):T+tnatural time lag tInterest rateset R(T;T+t)Interest payment occurs);(KRTTRPtt)(1 ()1 (1ttRRPK);()1 (1 (ttTTBRPK);()1 (1)1 (tttTTBRRPKKA put option on a bond. We can use Blacks model to value it:(In practice, the first approach is used more often.)(1)(ttttRRRPKSince this payoff is known at

36、time T, we can discount it back to time T)(1)1 ()(1()(1)(ttttttttRRRPRRRPKKBut:Primbs, MS&E 34536Blacks Model is used to price a number of interest ratederivatives:We showed how to price a caplet, which puts an upper limiton interest rates at a specific time in the future.When we place an upper

37、limit on all the interest payments for a loan, this is called a cap. Hence, a cap is just a portfolio of caplets. We price it by linearity. Price each caplet and add them together.A floor is a lower limit on interest rates over the life of a loan. A floorlet is for a single interest payment.A collar

38、 is a cap and a floor.Primbs, MS&E 34537Exchange one asset for anotherA generalization of Black-ScholesBlacks model with stochastic interest ratesBond optionsCaplets, etc.Futures, forwards, forward rates, and swap ratesSwaptionsInterest rate derivativesPrimbs, MS&E 34538Tt0A swap option (swa

39、ption) T+tT+2tT+nt)|()|(TTASTTSPKtWe have an option to enter into a swap at time T, where we pay a fixed swap rate SK.Assume the swap rate at time T is S(T|T). If this is greater than SK, then we exercise the option. This is worth PtS(T|T)-SK) at each swap date. total payoff)|(KSTTSPt)|(KSTTSPtSince

40、 the swap rate is known at time T, we can discount the payoff back to time T.)|()|0(KQSTTSEPTAAt)|()|()|()|0(TTATTASTTSPETAswaptionKQAtUse an annuity as the numeraire:Primbs, MS&E 34539Tt0)|()|0(TTSETSAQWe know:Assume: S(T|T) is log-normal with volatility Then:)()()|0()|0(21dNSdNTSPTAswaptionKtT

41、TSTSdK)()/)|0(ln(2211Tdd12whereT+tT+2tT+nt)()|(TASTTSPKttotal payoff)|(KSTTSPt)|(KSTTSPtA swap option (swaption)|()|0(KQSTTSEPTAAt)|()|()|()|0(TTATTASTTSPETAswaptionKQAtPrimbs, MS&E 34540Tt0T+tT+2tT+ntA swap option (swaption) (a second approach)Since a swap can be thought of as exchanging a bond

42、 with fixed interest for a bond with floating interest, a swaption can be thought of as an option on a bond. payoff(fixed bond - float bond)+The strike price is the value of the floating rate bond, which is always its principal P at every reset date. =(fixed bond-P)+Blacks model for a bond option ca

43、n be used to price this. However, this approach is not used often in practice.Primbs, MS&E 34541Appendix: Alternate proofs of martingale property for forwards, forward rates, and forward swap rates.Primbs, MS&E 34542ForwardsForward RatesSwap RatesPrimbs, MS&E 34543Tt0payoffoccurstotal pa

44、yoffTtTFS Buy a forwardTtTFS total cost0); 0(0TtTQFSETBBForward contracts and the bond forward risk neutral worldConsider the bond forward risk neutral world with B(t;T) as the numeraire TQTtQSEFEBBTt 0TQTtQTSEFEFB0Tt 0In particular:The forward price is the expected price of the stock at time T in a

45、 bond (B(t;T) forward risk neutral world.Primbs, MS&E 34544ForwardsForward RatesSwap RatesPrimbs, MS&E 34545Spot and Forward Ratestimespot rates); 0 ( tRspot rate curvet1t2); 0(1tR); 0(2tR);| 0(21ttF0Rates are quoted as yearly rates. Hence, the actual rate applied over time t1 is t1R(0;t1). If P is the principal on a zero coupon bond:); 0(1 (); 0(ttRPtBBond Price:);|0()(1 ();|0(211221ttFttPttBForward Bond Price:); 0(1 ();|0()(1)(; 0(1 (22211211tRtttFtttRtForward Rates:Primbs, MS&E 34546Tt0payoffoccursForward interest rates and the bond forward risk neutral worl