Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.

2004.Jun.29 11

Financial Derivatives The Mathematics

Fang-Bo YehFang-Bo Yeh

Mathematics DepartmentMathematics Department

System and Control GroupSystem and Control Group

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.

2004.Jun.29 22

Classic and Derivatives Market

Underlying AssetsUnderlying Assets Cash MarketCash Market Stock MarketStock Market Currency MarketCurrency Market

ContractsContracts Forward and SwapForward and Swap Market :Market :

FRAs , Caps, Floors, FRAs , Caps, Floors,

Interest Rate SwapsInterest Rate Swaps Futures and OptionsFutures and Options MarketMarket: :

Options, Swaptions, Options, Swaptions,

Convertibles Bond OptionConvertibles Bond Option

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.

2004.Jun.29 33

Main Problem:

What is the fair price for the contract?What is the fair price for the contract? Ans:Ans: (1). (1). The expected value of the discounted The expected value of the discounted future stochastic payoff .future stochastic payoff .

(2). It is determined by market forces which (2). It is determined by market forces which is impossible have a theoretical price. is impossible have a theoretical price.

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.

2004.Jun.29 44

Problem Formulation

Contract Contract FF : :

Underlying asset S, returnUnderlying asset S, return

Future time T, future pay-off Future time T, future pay-off f(Sf(STT))

Riskless bond B, returnRiskless bond B, return

Find contract valueFind contract value

F(t, SF(t, Stt))

t

t

dS=μ dt+σ dZ

S t

t

t

dB=r dt

B

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.

2004.Jun.29 55

Assume

1)1). . The future pay-off is attainable: (controllable)The future pay-off is attainable: (controllable)

exists a portfolioexists a portfolio

such thatsuch that

2)2). . Efficient market: (observable)Efficient market: (observable)

If thenIf then

t tδ α( , )

t t t t tδ α= S + Bπ

t t t t tδ αd = dS + d Bπ

T Tπ =F(T,S ) t tπ =F(t,S )

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.

2004.Jun.29 66

By assumptions (1)(2)

Ito’s lemmaIto’s lemma

The Black-Scholes-Merton Equation:The Black-Scholes-Merton Equation:

δ α

δ δ

dF(t,S) d S d B

[(μ r) S r F] dt σ S dZ

22 21

2 2μ σ σ

F F F FdF(t,S) S S dt S d Z

t S SS

22 21

2 2σ

F F Fr S S r F

t S S

T TF(T,S ) f(S )

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.

2004.Jun.29 77

Main Result-r(T-t)

t p* TF(t,S )=e E

[f

(S )]

The fair price isThe fair price is

the expected value of thethe expected value of the

discounted future stochastic payoff discounted future stochastic payoff underunder

the new martingale measure.the new martingale measure.

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.

2004.Jun.29 88

Numerical Solution

Finite Difference Method

Idea:

Approximate differentials by simple differences via Taylor series

Monte Carlo Simulation Method

Idea:

Monte Carlo Integration Generating and sampling Random variables