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A Review of the Derivative Pricing Theory

A Review of the Derivative Pricing Theory. Basic Derivatives. Options Non-linear Payoffs Futures and Forward Contracts Linear Payoffs. No-Arbitrage Principle (1). Application: If A(T)<=B(T), then A(0)<=B(0) If A(T)=B(T), then A(0)=B(0) Forward price of gold: F0=S0*exp(r*T)

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A Review of the Derivative Pricing Theory

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  1. A Review of the Derivative Pricing Theory

  2. Basic Derivatives • Options • Non-linear Payoffs • Futures and Forward Contracts • Linear Payoffs

  3. No-Arbitrage Principle (1) • Application: • If A(T)<=B(T), then A(0)<=B(0) • If A(T)=B(T), then A(0)=B(0) • Forward price of gold: F0=S0*exp(r*T) • Forward price of oil: F0<=S0*exp(r*T)

  4. No-Arbitrage Principle (2) • Model independent results • C<=S-X*exp(r*T), P>=X*exp(r*T)-S • An American call option on a non-dividend payment should never be exercised early. • Put-Call Parity • Model dependent results • Put-Call Symmetry

  5. The Black-Scholes Model • The underlying asset is a traded asset • GBM assumptions • Delta hedging and no-arbitrage principle

  6. Binomial Tree Method • A discrete model.

  7. Linkage between PDE and Expectation • V(S,t)=exp(-r*(T-t))*E[max(S(T)-X,0)|S(t)=S] where dS/S=rdt+sigma dW

  8. Three Methods • PDE • BTM • Monte-Carlo Simulation

  9. The Risk Neutral Valuation • Traded asset • Non-traded underlying

  10. Exotic Options Under the Black-Scholes framework. • American style • Multi-asset options • Barrier options • Asian options and lookback options • Shouting options • Forward start options • Compound options

  11. Similarity Reduction • PDE • BTM

  12. Beyond Black-Scholes • Implied volatility and volatility smile phenomenon • Improved model: • Local Vol • Stochastic Vol • Jump-diffusion

  13. Interest Rate Model • Black model • Spot rate model (short-term rate model) • Yield curve fitting • HJM model

  14. Concerning the Final Exam • 4 compulsory questions (55%) • 5 optional questions: choose no more than 3 questions (45%) • This course aims to establish pricing models of financial derivatives. So the exam is mainly focused on the derivation of those models. • Concerning the jump-diffusion model, you are required to know the assumption of the model only, instead of the derivation process. • We introduce a lot of interest rate derivatives. The index amortizing swap is not required.

  15. Other Topics • Credit risks and derivatives • Real options

  16. Some Related Modules • MA4265 Stochastic Analysis in Financial Mathematics - Lou Jiann Hua • MA5247 Computational Methods in Finance- Li Xun • MA4221 Partial Differential Equations - Dai Min • MA4255 Numerical Partial Differential Equations - Liu Jian Guo

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