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A Review of the Derivative Pricing Theory

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The Black-Scholes Model. The underlying asset is a traded asset. GBM assumptions ... Beyond Black-Scholes. Implied volatility and volatility smile phenomenon ... – PowerPoint PPT presentation

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Title: A Review of the Derivative Pricing Theory


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)
  • If A(T)B(T), then A(0)B(0)
  • Forward price of gold F0S0exp(rT)
  • Forward price of oil F0

4
No-Arbitrage Principle (2)
  • Model independent results
  • CXexp(rT)-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))Emax(S(T)-X,0)S(t)S
  • where dS/Srdtsigma 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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