Chapter 7: Beyond Black-Scholes

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