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Chapter 7: Beyond Black-Scholes

Chapter 7: Beyond Black-Scholes. Black-Scholes Model for vanilla options. Implied volatility and volatility smile. Implied volatility Volatility smile. Continued. Improved models. Local volatility model Stochastic volatility model Jump diffusion model

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Chapter 7: Beyond Black-Scholes

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  1. Chapter 7: Beyond Black-Scholes

  2. Black-Scholes Model for vanilla options

  3. Implied volatility and volatility smile • Implied volatility • Volatility smile

  4. Continued

  5. Improved models • Local volatility model • Stochastic volatility model • Jump diffusion model • Others: discrete hedging, transaction cost

  6. Local volatility model

  7. A special case: Identification of

  8. How to use the local volatility model • Calibration of the model: Identify the volatility function from the market prices of vanilla options • Price non-traded contracts by using the model

  9. Stochastic volatility model

  10. Pricing model

  11. Continued

  12. The Market Price of Risk

  13. Risk neutral processes

  14. Derivatives on a single underlying variable

  15. Pricing equation

  16. Two Named Models • Hull White • Heston

  17. Example 1: Hull-White model

  18. Example 2: Heston Model

  19. Jump-diffusion model • Poisson process

  20. Jump-diffusion Process

  21. Hedging

  22. Ito Lemma

  23. Merton’s Model (1976) • Jump risks are diversified

  24. Summary: purpose • Understand the market better • Price options at the OCT market

  25. Black-Scholes world • Beyond the Black-Scholes World • Local volatility model • Stochastic volatility model • Jump diffusion model

  26. Parameters • Local volatility model: • =(S,t) • Stochastic volatility model: • Hull-White model (3 parameters) • Heston model (2 parameters) • Jump diffusion model • , J

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