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

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

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

Stochastic Volatility Model

Explicit Solution for Variance

Properties of the Process

Properties of Variance

Variance Swaps

Volatility Swaps

Calculation E[V]

Calculation of Var[V]

Calculation of Var[V] (continuation)

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

Calculation of E[V] and Var[V] in

Discrete Case (sketch) (continuation)

Covariance and Correlation Swaps

Pricing Covariance and Correlation Swaps

Valuing of Covariance Swap

Calculation Covariance for S1 and S2

Calculation Covariance for S1 and S2 (continuation I)

Calculation Covariance for S1 and S2 (continuation II)

Calculation Covariance Swap for S1 and S2

Numerical Example: S&P60 Canada Index

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

Estimation of the GARCH(1,1) Process

Generating Different Input Variables for the Volatility Swap Model

Continuation (Numerical Example)

Figure 1: Convexity Adjustment

Figure 2: S&P60 Canada Index Volatility Swap

Some References

Some References (continuation)