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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

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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

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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

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Stochastic Volatility Model

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Explicit Solution for Variance

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Properties of the Process

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Properties of Variance

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Variance Swaps

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Volatility Swaps

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Calculation E[V]

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Calculation of Var[V]

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Calculation of Var[V] (continuation)

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

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

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Covariance and Correlation Swaps

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Pricing Covariance and Correlation Swaps

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Valuing of Covariance Swap

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Calculation Covariance for S1 and S2

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Calculation Covariance for S1 and S2 (continuation I)

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Calculation Covariance for S1 and S2 (continuation II)

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Calculation Covariance Swap for S1 and S2

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Numerical Example: S&P60 Canada Index

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

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Estimation of the GARCH(1,1) Process

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Generating Different Input Variables for the Volatility Swap Model

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Continuation (Numerical Example )

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Figure 1: Convexity Adjustment

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Figure 2: S&P60 Canada Index Volatility Swap

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Some References

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Some References (continuation)

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