Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.2004.Jun.29 1 Financial Derivatives The Mathematics Fang-Bo Yeh Mathematics Department System and Control.

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Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.2004.Jun.29 1 Financial Derivatives The Mathematics Fang-Bo Yeh Mathematics Department System and Control Group

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.2004.Jun.29 2 Classic and Derivatives Market Underlying Assets Underlying Assets Cash Market Cash Market Stock Market Stock Market Currency Market Currency Market Contracts Contracts Forward and Swap Market : Forward and Swap Market : FRAs, Caps, Floors, FRAs, Caps, Floors, Interest Rate Swaps Interest Rate Swaps Futures and Options Market: Futures and Options Market: Options, Swaptions, Options, Swaptions, Convertibles Bond Option Convertibles Bond Option

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.2004.Jun.29 3 Main Problem: What is the fair price for the contract? Ans: Ans: (1). The expected value of the discounted (1). 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 4 Problem Formulation Contract F : Underlying asset S, return Underlying asset S, return Future time T, future pay-off f(S T ) Future time T, future pay-off f(S T ) Riskless bond B, return Riskless bond B, return Find contract value F(t, S t )

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.2004.Jun.29 5 Assume 1). The future pay-off is attainable: (controllable) 1). The future pay-off is attainable: (controllable) exists a portfolio exists a portfolio such that such that 2). Efficient market: (observable) 2). Efficient market: (observable) If then If then

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.2004.Jun.29 6 By assumptions (1)(2) Ito’s lemma The Black-Scholes-Merton Equation:

Fang-Bo Yeh, Dept. of Mathematics, Tunghai Univ.2004.Jun.29 7 Main Result The fair price is the expected value of the the expected value of the discounted future stochastic payoff under the new martingale measure.

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