modeling of variance and volatility swaps for financial markets with stochastic volatility anatoliy...
TRANSCRIPT
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Modeling of Variance and Volatility Swaps
for Financial Markets with Stochastic Volatility
Anatoliy SwishchukDepartment 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 6th 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)