1. Black-Scholes Model for European vanilla options

2. Black-Scholes formulas for European vanilla options

3. Pricing American vanilla options

4. Pricing exotic options under Black-Scholes framework
Multi-asset options
Barrier options
Asian options
Lookback options
Forward start option, shout option, compound options

5. Beyond the Black-Scholes World

6. Implied volatility
The value for volatility that makes the theoretical option value and the market price the same

7. Volatility smile
Finance.yahoo.com

8. continued

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

10. Local volatility model

11. No closed form solution
How to identify ?

12. continued

13. 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

14. Stochastic Volatility Model

15. Option Pricing

16. Option pricing with non-traded underlying
So far, the underlying is assumed to be a traded asset.
The underlying is a consumption asset
Oil
Short selling is prohibited
Pricing of forward contract on oil
The underlying is a non-traded asset
Volatility, interest rate
Both long and short positions are prohibited
No arbitrage pricing

17. Continued (stochastic volatility model)

18. Continued

19. The Market Price of Risk

20. Risk Neutral Processes

21. Two Named Models
Hull White
Heston

22. Example 1: Hull-White model

23. Example 2: Heston Model

24. Jump Diffusion Model
Poisson process

25. Jump-diffusion Process

26. Hedging

27. Ito Lemma

28. Two special models Merton (1976) Wilmott et al. (1998)
to hedge the diffusion only
Wilmott et al. (1998)
to hedge both jump and diffusion as much as we can

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

30. Wilmott et al.’s Model
Hedging strategy

31. Continued

32. Continued
Under this best strategy, we let

33. Summary

34. Purpose
Understand the market better
Price options at the OCT market

35. Beyond the Black-Scholes World
Local volatility model
Stochastic volatility model
Jump diffusion model

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

37. Option Pricing at the OTC Market
Model calibration
Extend the model to exotic options
Solution