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Financial Derivatives The Mathematics

Financial Derivatives The Mathematics. Fang-Bo Yeh Mathematics Department System and Control Group. Classic and Derivatives Market. Underlying Assets Cash Market Stock Market Currency Market. Contracts Forward and Swap Market : FRAs , Caps, Floors,

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Financial Derivatives The Mathematics

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

  2. Classic and Derivatives Market • Underlying Assets • Cash Market • Stock Market • Currency Market • Contracts • Forward and SwapMarket : FRAs , Caps, Floors, Interest Rate Swaps • Futures and OptionsMarket: Options, Swaptions, Convertibles Bond Option

  3. 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.

  4. Problem Formulation Contract F : Underlying asset S, return Future time T, future pay-off f(ST) Riskless bond B, return Find contract value F(t, St)

  5. Assume 1). The future pay-off is attainable: (controllable) exists a portfolio such that 2). Efficient market: (observable) If then

  6. By assumptions (1)(2) Ito’s lemma The Black-Scholes-Merton Equation:

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

  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

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