Modelling and Pricing of Variance Swaps for Stochastic Volatility with Delay Anatoliy Swishchuk Mathematical and Computational Finance Laboratory Department of Mathematics and Statistics University of Calgary, Calgary, AB, Canada MITACS Project Meeting: Modelling Trading and Risk in the Market BIRS, Banff, AB, Canada November 11-13, 2004 This research is partially supported by MITACS and Start-Up Grant (Faculty of Science, U of C, Calgary, AB)

Swaps Stock Bonds ( bank accounts ) Option Forward contract Swaps-agreements between two counterparts to exchange cash flows in the future to a prearrange formula Basic SecuritiesDerivative Securities Security -a piece of paper representing a promise

Variance Swaps Variance swaps are forward contract on future realized stock variance Forward contract-an agreement to buy or sell something at a future date for a set price (forward price) Variance is a measure of the uncertainty of a stock price. Variance=(standard deviation)^2=(volatility)^2

Payoff of Variance Swaps A Variance Swap is a forward contract on realized variance. Its payoff at expiration is equal to N is a notional amount ($/variance); K var is a strike price

Realized Continuous Variance Realized (or Observed) Continuous Variance: where is a stock volatility, T is expiration date or maturity.

Types of Stochastic Volatilities Regime-switching stochastic volatility (Elliott & Swishchuk (2004) “Pricing options and variance swaps in Brownian and fractional Brownian markets”, working paper) Stochastic volatility itself (CIR process in Heston model) Stochastic volatility with delay (Kazmerchuk, Swishchuk & Wu (2002) “Continuous-time GARCH model for stochastic volatility with delay”, working paper)

Figure 2: S&P60 Canada Index Volatility Swap

Realized Continuous Variance for Stochastic Volatility with Delay Initial Data deterministic function Stock Price

Equation for Stochastic Variance with Delay (Continuous-Time GARCH Model) Our (Kazmerchuk, Swishchuk, Wu (2002) “The Option Pricing Formula for Security Markets with Delayed Response”) first attempt was: This is a continuous-time analogue of its discrete-time GARCH(1,1) model J.-C. Duan remarked that it is important to incorporate the expectation of log-return into the model

The Continuous-Time GARCH Stochastic Volatility Model This model incorporates the expectation of log-return Discrete-time GARCH(1,1) Model

Stochastic Volatility with Delay Main Features of this Model Continuous-time analogue of discrete-time GARCH model Mean-reversion Does not contain another Wiener process Complete market Incorporates the expectation of log-return

Valuing of Variance Swap for Stochastic Volatility with Delay Value of Variance Swap (present value): To calculate variance swap we need only E P *{V}, whereand where E P * is an expectation (or mean value), r is interest rate.

Continuous-Time GARCH Model or where

Deterministic Equation for Expectation of Variance with Delay There is no explicit solution for this equation besides stationary solution.

Stationary Solution of the Equation with Delay

Valuing of Variance Swap with Delay in Stationary Regime

Approximate Solution of the Equation with Delay In this way

Valuing of Variance Swap with Delay in General Case We need to find E P* [Var(S)]:

Numerical Example 1: S&P60 Canada Index ( )

Dependence of Variance Swap with Delay on Maturity (S&P60 Canada Index )

Variance Swap with Delay (S&P60 Canada Index)

Numerical Example 2: S&P500 ( )

Dependence of Variance Swap with Delay on Maturity (S&P500)

Variance Swap with Delay (S&P500 Index)

Conclusions Variance swap for regime-switching stochastic volatility model; Variance, volatility, covariance and correlation swaps for Heston model; Variance swap for stochastic volatility with delay; Numerical examples: S&P60 Canada Index and S&P500 index

Thank you for your attention!