金融工程課件04newpde(f)

1、Primbs, MS&E 3451Applications of the Return Form of Arbitrage Pricing:Equity DerivativesPrimbs, MS&E 3452Deriving Equations for Derivative Assets:Three step algorithm:Derive factor models for returns of tradable assets.(often involves Itos lemma.)(2) Apply absence of arbitrage condition.(m=1

2、 sl)(3) Apply appropriate boundary conditions and solve.Primbs, MS&E 3453Black-ScholesPoisson(Cox and Ross)Jump diffusion model(Merton)DividendsOptions on futures(Black)Multiple factorsStochastic Volatility(Hull and White)Exchange one assetfor another (Margrabe)Option on Max, MinOption on an ave

3、rage(Asian options)Path DependentPrimbs, MS&E 3454Black-Scholes (Again)Step 1: Derive factor models for returns of tradable assets.Risk Free Asset:rdtBdB=Underlying Stock:dzdtSdSsm=Itos Lemma:dzScdtcSSccdcSSSStssm=)(2221),(tSctWhat is a factor model for the return on ),(tSctLet be a security der

4、ivative to the stock. ?dzcScdtccSScccdcSSSStssm=)(2221Return:Primbs, MS&E 3455=cScccSSccrSSSStssllsmm0111)(102221Black-Scholes (Again)Derivative:dzcScdtccSScccdcSSSStssm=)(2221Step 1: Derive factor models for returns of tradable assets.Risk Free Asset:rdtBdB=Underlying Stock:dzdtSdSsm=lmK1=Step

5、2: Apply Primbs, MS&E 3456r=0lFirst EquationSecond Equationsml)(1r=Third Equationsmssm)()(2221rcScrccSSccSSSSt=cScccSSccrSSSStssllsmm0111)(102221lmK1=Step 2: Apply Black-Scholes (Again)Primbs, MS&E 3457Third EquationrccSrSccSSSSt=2221sThe Black-Scholes EquationThis is for an option on a non-

6、dividend paying assetwhich follows a geometric Brownian motion.smssm)()(2221rcScrccSSccSSSSt=Step 3: Apply appropriate boundary condition and solve!Black-Scholes (Again)Primbs, MS&E 3458Step 3:European Calls and PutsrccSrSccSSSt=2221s=)(),(KSTSc0), 0(=tcrppSrSppSSSt=2221s=)(),(SKTSp)(), 0(tTrKet

7、p=Solution:)()(),(2)(1dNKedSNtSctTr=tTtTrKSd=ss)()/ln(2211tTdd=s12where:)(Ndistribution function for a standard Normal (i.e. N(0,1)()(),(12)(dSNdNKetSptTr=These formulas are basic.know them!Primbs, MS&E 3459European Calls and Puts)()(),(2)(1dNKedSNtSctTr=)()(),(12)(dSNdNKetSptTr=We derived the e

8、quation for geometric Brownian motion.But, the equation doesnt depend on the mean return.The Black-Scholes formula holds even if the underlying assetlooks like:SdzdttSdSsm=),(0), 0(=tmandYou should check this!Step 3:Primbs, MS&E 3451075808590951001051101151201250510152025758085909510010511011512

9、0125051015202530SScpprice of callprice of put)25. 0%,20%,5,100(=TrKsEuropean Calls and Puts)()(),(2)(1dNKedSNtSctTr=)()(),(12)(dSNdNKetSptTr=Step 3:Primbs, MS&E 34511American Calls and Puts:We have the option to exercise early.Hence, we only keep the option alive if:)0 ,max(),(KStScfor a call)0

10、,max(),(SKtSpfor a putIt is not difficult to show that early exercise of anAmerican call on a non-dividend paying stock is never optimal, hence it has the same value as a European call. Early exercise for a put, however, can be optimal. In general, we need to use numerical techniques to solve for Am

11、erican options.The boundary condition given above is nasty!Step 3:Primbs, MS&E 34512Terminology:European and American call and put options are often referred to as plain vanilla options.Other derivatives are then called exotics.There are (too) many:Binary or digital optionsBarrier optionsCompoun

12、d optionsChooser options.Just because they are called “exotic” doesnt mean they are difficult.Often they are just a different boundary condition for the Black-Scholes equation.Primbs, MS&E 34513Black-ScholesPoisson(Cox and Ross)Jump diffusion model(Merton)DividendsOptions on futures(Black)Multip

13、le factorsStochastic Volatility(Hull and White)Exchange one assetfor another (Margrabe)Option on Max, MinOption on an average(Asian options)Path DependentPrimbs, MS&E 34514A stock paying a continuous dividend (Merton).Step 1: Derive factor models for returns of tradable assets.Risk Free Asset:rd

14、tBdB=Underlying Stock:SdzSdtdSsm=Assume the price follows a geometric Brownian motion:and that it also pays a continuous dividend at a rate q.Lets think about the proper dynamics for this tradable asset.Primbs, MS&E 34515A stock paying a continuous dividend (Merton).Dividend:Over time dt, the di

15、vidend is qSdt.dtStock price:SdzSdtdSsm=You purchase a portfolio of 1 share: value is vtttdtttdttSdtqSSvvdv=dtqSdStt=dzSdtSqttsm=)(DividendStvt=PriceSt+dt+qStdt = vt+dtdzvdtvqttsm=)(vt = StPrimbs, MS&E 34516Dividend:Over time dt, the dividend is qSdt.dtStSt+dt+qStdtStock price:SdzSdtdSsm=You pur

16、chase a portfolio of 1 share: value is vtdzvdtvqdvtttsm=)(PriceDividendvt= vt+dtA stock paying a continuous dividend (Merton).Primbs, MS&E 34517An important principle:You cannot just buy the price of the stock!You must buy a portfolio which consists of a single share!This portfolio follows:dzvdt

17、vqdvtttsm=)(Since this is what we will purchase, it must satisfy ourabsence of arbitrage conditions. The price of the stockdoes not have to satisfy the conditions because it cannotbe purchased! This is a very important point.Primbs, MS&E 34518Note, value of the derivative depends on price of the

18、 stock S, not v!A stock paying a continuous dividend (Merton).),(tSctLet be a security derivative to the stock price. dzcScdtccSScccdcSSSStssm=)(2221By Itos lemma, its return is:Risk Free Asset:rdtBdB=Underlying Asset Price:dzdtSdSsm=Dividend rate:qValue dynamics:dzdtqvdvsm=)(Primbs, MS&E 34519A

19、 stock paying a continuous dividend (Merton).Now we have models of returns for our tradable assets.Go to Step 2: Apply lmK1=Risk Free Asset:rdtBdB=Underlying Asset Price:dzdtSdSsm=Dividend rate:qValue dynamics:dzdtqvdvsm=)(Derivative:dzcScdtccSScccdcSSSStssm=)(2221Primbs, MS&E 34520=cScccSSccqrS

20、SSStssllsmm0111)(102221Return form of AOA:lmK1=r=0lFirst EquationSecond Equationsml)(1rq=Third Equationsmssm)()(2221rqcScrccSSccSSSSt=Primbs, MS&E 34521Third EquationrccSScqrcSSSt=2221)(sThis is for a derivative on a stock paying a continuous dividend.smssm)()(2221rqcScrccSSccSSSSt=This is the o

21、nly difference from Black-Scholes.Step 3: Apply appropriate boundary condition and solve!Primbs, MS&E 34522This is the only change.European Calls and PutsrccSScqrcSSSt=2221)(s=)(),(KSTSc0), 0(=tcrppSSpqrpSSSt=2221)(s=)(),(SKTSp)(), 0(tTrKetp=tTtTqrKSd=ss)()/ln(2211Solution:)()(),(2)(1)(dNKedNSet

22、SctTrtTq=tTdd=s12where:)(Ndistribution function for a standard Normal (i.e. N(0,1)()(),(1)(2)(dNSedNKetSptTqtTr=These formulas are basic.know them!Step 3:Primbs, MS&E 34523Which assets pay continuous dividends?How about:A stock indexForeign currenciesCommodities with a convenience yieldStep 3:Pr

23、imbs, MS&E 34524Do dividends make an option more or less valuable?00.010.020.030.040.050.060.070.080.090.13.23.43.63.844.24.44.64.8500.010.020.030.040.050.060.070.080.090.13.23.43.63.844.24.44.64.8qqcpcallput)25. 0%,20%,5,100(=TrKsStep 3:Dividends bleed away the price.Primbs, MS&E 34525Black

24、-ScholesPoisson(Cox and Ross)Jump diffusion model(Merton)DividendsOptions on futures(Black)Multiple factorsStochastic Volatility(Hull and White)Exchange one assetfor another (Margrabe)Option on Max, MinOption on an average(Asian options)Path DependentPrimbs, MS&E 34526A stock paying a cash divid

25、end.Step 1: Derive factor models for returns of tradable assets.Risk Free Asset:rdtBdB=Underlying Stock:Pays a cash dividend of Dt at time t.dzvdtvdvtttsm=We model the value of a share as being continuous.On the other hand, the price drops when the stock goes ex-dividend.tDtdzSdttDSdStttstmt=)(Primb

26、s, MS&E 34527Risk Free Asset:rdtBdB=Underlying Asset Price:Value dynamics:dzdtvdvsm=Note, value of the derivative depends on price of stock S, not v!A stock paying a cash dividend.),(tSctLet be a security derivative to the stock price. dzcScdtcttSctDSccSScccdcSttSSStstsmt=)(),(),()(2221By Itos l

27、emma, its return is:dzSdttDSdStttstmt=)(Primbs, MS&E 34528Risk Free Asset:rdtBdB=Value dynamics:dzdtvdvsm=A stock paying a cash dividend.dzcScdtcttSctDSccSScccdcSttSSStstsmt=)(),(),()(2221Derivative:lmK1=Step 2: Apply =cSccttSctDSccSSccrSttSSStsslltsmmt0111)(),(),()(102221Primbs, MS&E 34529A

28、 stock paying a cash dividend.r=0lFirst EquationSecond Equationsml)(1r=Third Equationsmstsmt)()(),(),()(2221rcScrcttSctDSccSSccSttSSSt=lmK1=Step 2: Apply =cSccttSctDSccSSccrSttSSStsslltsmmt0111)(),(),()(102221Primbs, MS&E 34530A stock paying a cash dividend.Third EquationrcttSctDSccSrSccttSSSt=)

29、(),(),(2221tstThis is for an option on a stock paying a lump (cash) dividend.Step 3: Apply appropriate boundary condition and solve!smstsmt)()(),(),()(2221rcScrcttSctDSccSSccSttSSSt=Primbs, MS&E 34531Step 3:For European options, only the value of S at time T matters:TSDSo, you can just act like

30、the stock started at a lower price and there were no dividends!xxtPrimbs, MS&E 34532For European options, only the value of S at time T matters:So, you can just act like the stock started at a lower price and there were no dividends!TSDe-rTxxtrDeSS=tStep 3:Primbs, MS&E 34533Black-ScholesPois

31、son(Cox and Ross)Jump diffusion model(Merton)DividendsOptions on futures(Black)Multiple factorsStochastic Volatility(Hull and White)Exchange one assetfor another (Margrabe)Option on Max, MinOption on an average(Asian options)Path DependentPrimbs, MS&E 34534Remember this!=cfcccffccrfffftssllsmm01

32、01)(102221Derivative:dzcfcdtccffcccdcfffftssm=)(2221Options on Futures (Black,1976)Step 1: Derive factor models for returns of tradable assets.Risk Free Asset:rdtBdB=Step 2: Apply AOA but for futures! Underlying Future:dzdtfdfsm=Primbs, MS&E 34535r=0lFirst EquationSecond Equationsml=1Third Equat

33、ionsmssm=cfcrccffccfffft)(2221lmK1=Step 2: Apply =cfcccffccrfffftssllsmm0101)(102221Primbs, MS&E 34536Third Equationrccfcfft=2221sThe Black-Scholes Equation for an option on a futures contract.smssm=cfcrccffccfffft)(2221Primbs, MS&E 34537Step 3:It looks like the equation for an option on an

34、asset that pays a continuous dividend, except here the continuous dividendrate is the risk free rate!Hence, we have closed form solutions!Solution:)()(),(2)(1)(dNKedNfetfctTrtTr=tTtTKfd=ss)()/ln(2211tTdd=s12where:)(Ndistribution function for a standard Normal (i.e. N(0,1)()(),(1)(2)(dNfedNKetfptTrtT

35、r=Primbs, MS&E 34538In some ways Blacks model is more fundamental then the othermodels we have seen. (recall our analysis of market price of risk.) If you can price futures contracts in terms of spot prices, thenyou can use Blacks formula to derive all our previous formulas.As an exercise, you s

36、hould try this for assets paying a continuousor cash dividends. Blacks model is also useful for a number of interest rate derivatives.We will see this later.Step 3:Primbs, MS&E 34539Black-ScholesPoisson(Cox and Ross)Jump diffusion model(Merton)DividendsOptions on futures(Black)Multiple factorsSt

37、ochastic Volatility(Hull and White)Exchange one assetfor another (Margrabe)Option on Max, MinOption on an average(Asian options)Path DependentPrimbs, MS&E 34540Assets with Poisson Jumps (Cox and Ross, 1976)Derivative:mdctSctkScdtcScccdcttSt),(),()(=Be careful. Should set the mean of d to 0. This

38、 doesntmatter in the current setting, but it can matter in other settings.Bond:rdtBdB=Asset price:mdkdtSdS) 1( =ldt1-ldt100dLets do it anyway.Primbs, MS&E 34541Assets with Poisson Jumps (Cox and Ross, 1976)Bond:Asset price:)(1()1(dtdkdtkSdSllm=Derivative:rdtBdB=)(),(),(),(),()(dtdctSctkScdtctSct

39、kScScccdcttStllm=ctSctkSckctSctkScScckrSt),(),(10111),(),()() 1(10lllmlmlmK1=Step 2: Apply Primbs, MS&E 34542r=0lFirst EquationlmK1=Step 2: Apply =ctSctkSckctSctkScScckrSt),(),(10111),(),()() 1(10lllmlmSecond Equationlml=11krConfusing Notation!Third EquationctSctkSckrrctSctkScSccSt),(),(1),(),()

40、(=lmlmPrimbs, MS&E 34543Third EquationctSctkSckrrctSctkScSccSt),(),(1),(),()(=lmlm)(,(),() 1)(rtSctkSckrcSccSt=mmThis is a partial differential/difference equation.Step 3: Apply appropriate boundary condition and solve!Note the non-local nature of the equation due to jumpsPrimbs, MS&E 34544C

41、losed form solution for a European Call Option:Looks a lot like the Black-Scholes formula but with Poisson instead of Gaussian.)/,(),(),()(kyxKeyxStSctTr=where=iiie!),(1)(=kktTrymand x is the smallest nonnegative integer greater than: )ln()()/ln(ktTSKmStep 3:Primbs, MS&E 34545Black-ScholesPoisso

42、n(Cox and Ross)Jump diffusion model(Merton)DividendsOptions on futures(Black)Multiple factorsStochastic Volatility(Hull and White)Exchange one assetfor another (Margrabe)Option on Max, MinOption on an average(Asian options)Path DependentPrimbs, MS&E 34546Mertons Jump Diffusion Model (1976):Call

43、option:=dtctSctYScEdctSctYScdzcScdtctSctYScEccSScccdcttSSSStlslsm),(),(),(),( ),(),()(2221Bond:rdtBdB=Asset price:) 1) 1()1(dtYEdYdzdtYESdS=lslmldt1-ldtY-100dtdStep 1:Primbs, MS&E 34547Mertons Jump Diffusion Model (1976):dtctSctYScEdctSctYScl),(),(),(),(Merton makes a big assumption:Put another

44、way, the jump risk has zero beta.We know that this also means: l2 and l3 are zero!) 1) 1(dtYEdYl=1000100111),(),()( 132102221llsslllsmlmcScctSctYScEcSSccYErSSSStStep 2:All risk associated with the jump component is diversifiable.Primbs, MS&E 34548Mertons Jump Diffusion Model (1976):=cScctSctYScE

45、cSSccYErSSSStsslllsmlm0111),(),()( 1102221Step 2:r=0lFirst EquationSecond EquationslmlrYE= 11Third EquationcScrYErctSctYScEccSSccSSSStsslmlsm) 1(),(),()(2221=Primbs, MS&E 34549cScrYErctSctYScEccSSccSSSStsslmlsm) 1(),(),()(2221=Third Equation:SSSStScrYErctSctYScEcSScc) 1(),(),()(2221=lmlsmStep 3:

46、 Apply appropriate boundary condition and solve!),(),()1(2221tSctYScErccSScYErcSSSt=lslPrimbs, MS&E 34550When Y is lognormal, there is a closed form solution for a call option=0)(!)(),(nnntTfntTetScllwhere)1 (k=llandnfis the Black-Scholes formula withvolatility)(22tTnsrisk-free rate)(tTnkrlthe s

47、tandard deviation of )ln(Y)1ln(k= 1=YEkMessy.but at least we can calculate it.Primbs, MS&E 34551An easy special case:Complete bankruptcy:0=YQuestion: Does bankruptcy make an option worth more or less?),(),()1(2221tSctYScErccSScYErcSSSt=lslMertons formula reduces from:crcSScrcSSSt)()(2221lsl=to:T

48、he Black-Scholes equation with risk free rate: rlPrimbs, MS&E 34552Option Prices versus the Risk Free Rate0.020.030.040.050.060.070.080.090.10.110.124.24.44.64.855.25.45.65.8rcCall Option0.020.030.040.050.060.070.080.090.10.110.122.62.833.23.43.63.84rPut Optionp)25. 0%,20,100(=TKsPrimbs, MS&

49、E 34553Black-ScholesPoisson(Cox and Ross)Jump diffusion model(Merton)DividendsOptions on futures(Black)Multiple factorsStochastic Volatility(Hull and White)Exchange one assetfor another (Margrabe)Option on Max, MinOption on an average(Asian options)Path DependentPrimbs, MS&E 34554Stochastic Vola

50、tility (Hull and White)Risk Free Asset:rdtBdB=Underlying Asset Price:1dzvdtSdS=mVolatility is random:2bdzadtdv=dtdzdzE=21Derivative:),(tvScby Itos lemma:21221221)(dzcbcdzcScvdtcScvbcbcvSacScccdcvSSvvvSSvSt=mStep 1:Primbs, MS&E 34555=cbccScvvcScvbcbvScacSccrvSSvvvSSvSt000111)(21022121lllmmr=0lFir

51、st EquationSecond Equationvr=ml1Third EquationrcScvbcbvSccbarSccSvvvSSvSt=)(221212lWhere l2 is the market price of risk for the volatility factor.Stochastic Volatility (Hull and White)Step 2:I wont go into solutions of this equation.Primbs, MS&E 34556Black-ScholesPoisson(Cox and Ross)Jump diffus

52、ion model(Merton)DividendsOptions on futures(Black)Multiple factorsStochastic Volatility(Hull and White)Exchange one assetfor another (Margrabe)Option on Max, MinOption on an average(Asian options)Path DependentPrimbs, MS&E 34557Option to exchange one asset for another (Margrabe, 1978)Risk Free

53、Asset:rdtBdB= Asset 1:11111dzdtSdSsm=dtdzdzE=21 Asset 2:22222dzdtSdSsm=Derivative:),(21tSScby Itos lemma:222111212122222121212122112121221121)(dzccSdzccSdtccSScScScScSccdcSSSSSSSSSStssssssmm=Step 1:Primbs, MS&E 34558r=0lFirst EquationSecond Equation111smlr=ccSccSccSScScScScScrSSSSSSSSSSt21212211

54、21222211110212122222121212122112100001111)(sslssllssssmmmmThird Equation222smlr=rccSScScScrScrScSSSSSSSSt=)(21221121212122222121212121ssssFourth EquationOption to exchange one asset for another (Margrabe, 1978)Step 2:Primbs, MS&E 34559Solution:)()(),(211221dNSdNStSSc=tTtTSSd=ss)()/ln(221121tTdd=

55、s12where:)(Ndistribution function for a standard Normal (i.e. N(0,1)2122212sssss=Step 3:Option to exchange one asset for another (Margrabe, 1978)0 ,max(),(1221SSTSSc=If you have the option to exchange asset S2 for asset S1 at time Tthis leads to the boundary condition:22), 0(StSc=0), 0 ,(1=tScLooks

56、a lot like Black Scholes.Primbs, MS&E 34560Step 3:Option to exchange one asset for another (Margrabe, 1978)0 ,max(),(1221SSTSSc=A touch of intuition:)0 , 1max(121=SSSWe are measuring S2 in units of S1.This in known as a change of numeraire.We will see this trick later.Looks like a call option on

57、 S2/S1 with strike 1.By the way, can a standard call option be seen as exchanging oneasset for another?Primbs, MS&E 34561Black-ScholesPoisson(Cox and Ross)Jump diffusion model(Merton)DividendsOptions on futures(Black)Multiple factorsStochastic Volatility(Hull and White)Exchange one assetfor anot

58、her (Margrabe)Option on Max, MinOption on an average(Asian options)Path DependentPrimbs, MS&E 34562Path Dependence:Some derivatives depend on the path of the underlying asset.Payoff:)0 ,max(01TtTTdtSSA specific example is the Average strike option.So, ),(0tdSScttttWe need to write an Ito equatio

59、n for this.The dependence on the entire path is a problem!For example: An Asian option depends on the average priceof the underlying stock over a given time period.Primbs, MS&E 34563The problem is the tdS0tttermThe general approach is to try to capture the path dependence with another variable.N

60、ow we can apply Itos lemma and continue in typical fashion.Path Dependence:Lets see how this would work for an Asian option=ttdSI0ttthendtSdItt=Lets assign:),(),(0tISctdSSctttt=ttSowhere S and I are Ito processesPrimbs, MS&E 34564Asset price:dzdtSdSsm=Bond:rdtBdB=Path Dependence:=ttdSI0ttdtSdItt=Path DependenceDerivative:dzcScdtccSScScccdcS