1. Modeling of Variance and Volatility Swaps for Financial Markets with Stochastic Volatility Anatoliy Swishchuk Department of Mathematics & Statistics, York University, Toronto, ON, Canada Seminar-April 15, 2004 Department of Statistics, University of Toronto

2. Outline Introduction Stochastic Volatility Model: Heston (1993) Model Solution of the Volatility Equation Property of the Solution Variance and Volatility Swaps Calculation of Expectation and Variance Covariance and Correlation Swaps Numerical Example: S&P60 Canada Index

3. Introduction Cox, Ingersoll &Ross (CIR) (1985)-stochastic variance model; Heston (1993)-asset price has variance that follows a CIR model; Brockhaus & Long (2000)-calculation expectation and variance for volatility swap using analytical approach; He & Wang (RBC Financial Group) (2002)- proposed deterministic volatility for variance and volatility swaps: Query Note for the 6 th IPSW PIMS, Vancouver, UBC, May 2002

4. Stochastic Volatility Model

5. Explicit Solution for Variance

6. Properties of the Process

7. Properties of Variance

8. Variance Swaps

9. Volatility Swaps

10. Calculation E[V]

11. Calculation of Var[V]

12. Calculation of Var[V] (continuation)

13. Calculation of E[V] and Var[V] in Discrete Case (sketch)

14. Calculation of E[V] and Var[V] in Discrete Case (sketch) (continuation )

15. Covariance and Correlation Swaps

16. Pricing Covariance and Correlation Swaps

17. Valuing of Covariance Swap

18. Calculation Covariance for S1 and S2

19. Calculation Covariance for S1 and S2 (continuation I)

20. Calculation Covariance for S1 and S2 (continuation II)

21. Calculation Covariance Swap for S1 and S2

22. Numerical Example: S&P60 Canada Index

23. Statistics on Log-Returns of S&P60 Canada Index for 5 years (1997-2002)

24. Estimation of the GARCH(1,1) Process

25. Generating Different Input Variables for the Volatility Swap Model

26. Continuation (Numerical Example )

27. Figure 1: Convexity Adjustment

28. Figure 2: S&P60 Canada Index Volatility Swap

29. Some References

30. Some References (continuation)