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Modeling of Variance and Volatility Swaps for Financial Markets with Stochastic Volatility

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. Outline. Introduction

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Modeling of Variance and Volatility Swaps for Financial Markets with Stochastic Volatility

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  1. Modeling of Variance and Volatility Swapsfor 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 6th 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)

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