1. Multi-Factor Stochastic Volatilities with Delay: Modelling and Pricing of Variance Swaps
Anatoliy Swishchuk
Mathematical and Computational Finance Laboratory
Department of Mathematics and Statistics
University of Calgary, Calgary, AB, Canada
‘Lunch at the Lab’ Talk
October 17th, 2006

2. Outline Reminder: One-Factor SV with Delay
Multi-Factor SV with Delay: Two-Factor and Three-Factor
Variance Swaps for MFSVD
Numerical Examples

3. Reminder: One-Factor SVD (The Continuous-Time GARCH Stochastic Volatility Model)
This model incorporates the expectation of log-return
Discrete-time GARCH(1,1) Model

4. Reminder: One-Factor Stochastic Volatility with Delay II
Main Features of this Model
Continuous-time analogue of discrete-time GARCH model
Mean-reversion
Does not contain another Wiener process
Complete market
Incorporates the expectation of log-return

5. Reminder: One-Factor SVD (Continuous-Time GARCH Model) III
where

6. Reminder: One-Factor SVD (Deterministic ODE for Expectation of Variance with Delay) IV
There is no explicit solution for this equation besides stationary solution.

7. Reminder: One-Factor SVD (Stationary Solution of the ODE with Delay) V

8. Reminder: One-Factor SVD (Approximate Solution of the ODE with Delay) VI
In this way

9. Multi-Factor Stochastic Volatilities with Delay (SVD)
Two-Factor SVD (Geometric Brownian Motion Mean-Reversion)
Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion)
Two-Factor SVD (Pilipovich One-Factor Mean-Reversion)
Three-Factor SVD (Pilipovich Mean-Reversion)

10. Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Incomplete Market)

11. Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (Incomplete Market)

12. Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (Incomplete Market)

13. Three-Factor SVD (Pilipovich Mean-Reversion) (Incomplete Market)

14. Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Under Risk-Neutral Measure) I

15. Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Under Risk-Neutral Measure) II

16. Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (ODE with Delay for Variance) II

17. Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Solution of ODE with Delay for Variance) III

18. Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Price of Variance Swap) IV

19. Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (Under Risk-Neutral Measure) I

20. Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (ODE with Delay for Variance) II

21. Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (Solution of ODE with Delay for Variance) III

22. Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (Price of Variance Swap) III

23. Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (Under Risk-Neutral Measure) I

24. Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (ODE with Delay for Variance) II

25. Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (Solution of ODE with Delay for Variance) III

26. Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (Price of Variance Swap) III

27. Three-Factor SVD (Pilipovich Mean-Reversion) (Under Risk-Neutral Measure) I

28. Three-Factor SVD (Pilipovich Mean-Reversion) (Under Risk-Neutral Measure) II

29. Three-Factor SVD (Pilipovich Mean-Reversion) (ODE with Delay for Variance) III

30. Three-Factor SVD (Pilipovich Mean-Reversion) (Solution of ODE with Delay for Variance) IV

31. Three-Factor SVD (Pilipovich Mean-Reversion) (Price of Variance Swap) V

36. Variance for Two-Factor SVD and GBM Mean-Reversion

37. The Price of Variance Swap for Two-Factor SVD and GBM Mean-Reversion

38. Variance for Two-Factor SVD and OU Mean-Reversion

39. Price of Variance Swap for Two-Factor SVD and OU Mean-Reversion

40. Variance for Two-Factor SVD and Pilipovich One-Factor Mean-Reversion

41. Price of Variance Swap for Two-Factor SVD and Pilipovich One-Factor Mean-Reversion

42. Variance for Two-Factor SVD and Pilipovich Two-Factor Mean-Reversion

43. Price of Variance Swap for Two-Factor SVD and Pilipovich Two-Factor Mean-Reversion

44. Challenging Problems
Volatility Swap for One-Factor SVD (also, Covariance, Correlation Swaps)
Volatility Swap for Multi-Factor SVD (also, Covariance, Correlation Swaps)
Variance and Volatility Swaps for One-Factor SVD and with Jumps (also, Covariance and Correlation Swaps)

45. The End
Thank you (multi-factor!) for your attention!