Brownian盗版win8.1升级win10

CHAPTER8MarketModels8.
1.
LognormalForward-LIBORModelRemark8.
1.
RecallfromExample3.
10thatF(t;T,S)isamartingaleunderQS.
Definition8.
2(Forward-ratedynamicsintheLFMmodel).
Inthelognormalforward-LIBORmodel,brieyLFMmodel,thesimply-compoundedforwardinterestratefortheperiod[T,S]isassumedtosatisfythestochasticdierentialequationdF(t;T,S)=σ(t;T,S)F(t;T,S)dWS(t),whereσisdeterministicandWSisaBrownianmotionundertheS-forwardmea-sure.
Theorem8.
3(PricingofcapletsintheLFMmodel).
IntheLFMmodel,thepriceofacapletwithnominalvalueN,caprateK,expirytimeT,andmaturitytimeSisgivenbyCpl(t,T,S,N,K)=NP(t,S)τ(T,S)BlK,F(t;T,S),Ttσ2(u;T,S)du,whereBl(K,F,v)=FΦlnFK+v22vKΦlnFKv22v.
Theorem8.
4(PricingofcapsintheLFMmodel).
IntheLFMmodel,thepriceofacapwithnominalvalueN,caprateK,andthesetoftimesT,isgivenbyCap(t,T,N,K)=Nβi=α+1P(t,Ti)τiBlK,Fi(t),Ti1tσ2i(u)du,whereforα+1≤i≤βFi(t)=F(t;Ti1,Ti),τi=τ(Ti1,Ti),σi(t)=σ(t;Ti1,Ti).
51528.
MARKETMODELSTheorem8.
5(BrownianmotionsintheLFMmodelunderdierentforwardmeasures).
LetS>T.
LetWTandWSbeaQT-BrownianmotionandaQS-Brownianmotion,respectively.
IntheLFMmodel,wethenhavedWS(t)dWT(t)=τ(T,S)F(t;T,S)σ(t;T,S)1+τ(T,S)F(t;T,S)dt.
Theorem8.
6(Forward-ratedynamicsintheLFMmodel).
Leti,k∈{α,β}.
IntheLFMmodel,Fksatisesthefollowingstochasticdierentialequations:(i)Ifk=i,thendFk(t)=σk(t)Fk(t)dWTi(t).
(ii)Ifk>i,thendFk(t)=σk(t)Fk(t)dWTi(t)+kj=i+1τjσj(t)σk(t)Fj(t)Fk(t)1+τjFj(t)dt.
(iii)Ifk8.
2.
LognormalForward-SwapModelRemark8.
7.
RecallfromExample3.
11thatSα,β(t)isamartingaleunderQα,β.
Definition8.
8(ForwardswapratedynamicsintheLSMmodel).
Inthelognormalforward-swapmodel,brieyLSMmodel,theforwardswaprateforT={Tα,Tβ}isassumedtosatisfythestochasticdierentialequationdSα,β(t)=σα,β(t)Sα,β(t)dWα,β(t),whereσisdeterministicandWα,βisaBrownianmotionunderQα,β.
Theorem8.
9(PricingofswaptionsintheLSMmodel).
IntheLSMmodel,thepriceofaEuropeanpayerswaptionwithswaptionmaturityT=TαonanIRSdependingonthenominalvalueN,thexedrateK,andthesetoftimesTisgivenbyPS(t,T,T,N,K)=Nβi=α+1τiP(t,Ti)BlK,Sα,β(t),Ttσ2α,β(u)du.