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銳角三角函數

VIEWS: 13 PAGES: 8

  • pg 1
									              Financial Derivatives
                The Mathematics
                      Fang-Bo Yeh
                 Mathematics Department
                System and Control Group



2004.Jun.29      Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.   1
  Classic and Derivatives Market
      Underlying Assets             Contracts
       Cash Market                  Forward and Swap Market :
       Stock Market                  FRAs , Caps, Floors,
       Currency Market               Interest Rate Swaps
                                     Futures and Options Market:
                                      Options, Swaptions,
                                      Convertibles Bond Option




2004.Jun.29     Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.    2
    Main Problem:
     What is the fair price for the contract?
     Ans:
         (1). The expected value of the discounted
              future stochastic payoff .
         (2). It is determined by market forces which
              is impossible have a theoretical price.




2004.Jun.29     Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.   3
      Problem Formulation
     Contract F :
          Underlying asset S, return          dSt
                                                  =μ dt+σ dZt
                                              St
          Future time T, future pay-off            f(ST)

          Riskless bond B, return                 dBt
                                                      =r dt
                                                   Bt
     Find contract value
                                   F(t, St)

2004.Jun.29      Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.   4
      Assume
      1). The future pay-off is attainable: (controllable)
          exists a portfolio (δ t ,α t )
                                      π t =δ tSt +α t Bt
              such that
                                   dπ t =δ t dSt +α t d Bt

      2). Efficient market: (observable)

                          If πT =F(T,ST ) then π t =F(t,St )


2004.Jun.29        Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.   5
  By assumptions (1)(2)

              dF(t,S)  δ d S  α d B
                       [(μ  r) δ S  r F] dt  σ δ S dZ

  Ito’s lemma
                         F    F 1 2 2  2 F           F
              dF(t,S)    μ S    2σ S     2 
                                                 dt  σ S    dZ
                         t    S       S              S

  The Black-Scholes-Merton Equation:
                    F      F 1 2 2  2 F
                       r S    2σ S       rF
                    t      S       S  2



                         F(T,ST )  f(ST )

2004.Jun.29        Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.   6
                    Main Result
              F(t,St )=e-r(T-t) E p*[f(ST )]

                The fair price is
           the expected value of the
   discounted future stochastic payoff under
         the new martingale measure.



2004.Jun.29   Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.   7
              Numerical Solution
   Finite Difference                Monte Carlo Simulation
    Method                            Method

   Idea:                            Idea:
     Approximate                       Monte Carlo Integration
     differentials by simple           Generating and sampling
     differences via Taylor            Random variables
     series



2004.Jun.29     Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.   8

								
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