1. Paper Review: “On the Pricing and Hedging of Volatility Derivatives” by S. Howison, A. Rafailidis and H. Rasmussen (Applied Mathematical Finance J., 2004) Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C “Lunch at the Lab” Talk February 10, 2006

2. Variance Swap

3. Realized Variance in Continuous Time

4. Payoff Function for Variance Swap

5. Realized Volatility (Discrete Time)

6. Realized Volatility (Continuous Time)

7. Payoff Function for Volatility Swap

8. Payoff Function for Volatility- Average Swap

9. Payoff Function for Implied Volatility Swap

10. Payoff for Variance Swaptions

11. Payoff Functions for Volatility Swaptions

12. Payoff Function for Volatility and Asset Swaption

13. Risk-Neutral Pricing Technique

14. Three Approaches to the Risk- Neutral Pricing Pricing Independently of the Volatility Model Pricing by Expectations in a SV Framework Pricing via Partial Differential Equations

15. 1 st Approach: Pricing Independently of the Volatility Model

16. 1 st Approach: Pricing Independently of the Volatility Model (cntd)

18. 2 nd Approach: Pricing by Expectations in a SV Framework

19. 2 nd Approach: Pricing by Expectations in a SV Framework (cntd)

20. 2 nd Pricing Approach: Pricing by Expectations in a SV Framework (cntd)

22. 3d Approach: Pricing via PDE

23. 3d Approach: Pricing via PDE (Model)

24. 3d Approach: Pricing via PDE (Payoffs)

26. 3d Approach: Pricing via PDE (PDE Itself for the Value V of Derivative)

27. 3d Approach: Pricing via PDE (Mean-Reverting Model)

28. General Stochastic Volatility Models

29. Derivation of Certain Expectations

30. Derivation of Certain Expectations.I.

31. Derivation of Certain Expectations.II.

32. Derivation of Certain Expectations.III.

33. Derivation of Certain Expectations.IV.

34. Derivation of Certain Expectations.V.

35. Derivatives Pricing

36. Mean-Reverting-Like Process

37. Mean-Reverting-Like Process.I.

38. Mean-Reverting-Like Process.II.

39. Popular SV Models. I.

40. Popular SV Models. II.

41. Popular SV Models. III.

42. Popular SV Models. IV.

43. Popular SV Models. V.

44. Asymptotical Analysis for Fast Mean-Reversion. I.

45. Asymptotical Analysis for Fast Mean-Reversion. II.

46. Asymptotical Analysis for Fast Mean-Reversion. III.

47. Asymptotical Analysis for Fast Mean-Reversion (Summary).

48. Examples: 1. The Variance Swap

49. Examples: 2. The Standard- Deviation Swap

50. Examples: 3. The Volatility- Average Swap

51. Examples: 4. The Implied Volatility- Swap

52. Examples: 5. The Volatility-Average Swaption

53. References. I.

54. References. II.

55. The End Thank you for Your Attention!