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# 銳角三角函數

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```									              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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