Multi-Factor Stochastic Volatilities with Delay: Modelling and Pricing of Variance Swaps Anatoliy Swishchuk Mathematical and Computational Finance Laboratory Department of Mathematics and Statistics University of Calgary, Calgary, AB, Canada ‘Lunch at the Lab’ Talk October 17th, 2006

Outline Reminder: One-Factor SV with Delay Multi-Factor SV with Delay: Two-Factor and Three-Factor Variance Swaps for MFSVD Numerical Examples

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

Reminder: One-Factor Stochastic Volatility with Delay II 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

Reminder: One-Factor SVD (Continuous-Time GARCH Model) III where

Reminder: One-Factor SVD (Deterministic ODE for Expectation of Variance with Delay) IV There is no explicit solution for this equation besides stationary solution.

Reminder: One-Factor SVD (Stationary Solution of the ODE with Delay) V

Reminder: One-Factor SVD (Approximate Solution of the ODE with Delay) VI In this way

Multi-Factor Stochastic Volatilities with Delay (SVD) Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) Three-Factor SVD (Pilipovich Mean-Reversion)

Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Incomplete Market)

Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (Incomplete Market)

Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (Incomplete Market)

Three-Factor SVD (Pilipovich Mean-Reversion) (Incomplete Market)

Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Under Risk-Neutral Measure) I

Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Under Risk-Neutral Measure) II

Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (ODE with Delay for Variance) II

Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Solution of ODE with Delay for Variance) III

Two-Factor SVD (Geometric Brownian Motion Mean-Reversion) (Price of Variance Swap) IV

Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (Under Risk-Neutral Measure) I

Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (ODE with Delay for Variance) II

Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (Solution of ODE with Delay for Variance) III

Two-Factor SVD (Ornstein-Uhlenbeck Mean-Reversion) (Price of Variance Swap) III

Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (Under Risk-Neutral Measure) I

Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (ODE with Delay for Variance) II

Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (Solution of ODE with Delay for Variance) III

Two-Factor SVD (Pilipovich One-Factor Mean-Reversion) (Price of Variance Swap) III

Three-Factor SVD (Pilipovich Mean-Reversion) (Under Risk-Neutral Measure) I

Three-Factor SVD (Pilipovich Mean-Reversion) (Under Risk-Neutral Measure) II

Three-Factor SVD (Pilipovich Mean-Reversion) (ODE with Delay for Variance) III

Three-Factor SVD (Pilipovich Mean-Reversion) (Solution of ODE with Delay for Variance) IV

Three-Factor SVD (Pilipovich Mean-Reversion) (Price of Variance Swap) V

Variance for Two-Factor SVD and GBM Mean-Reversion

The Price of Variance Swap for Two-Factor SVD and GBM Mean-Reversion

Variance for Two-Factor SVD and OU Mean-Reversion

Price of Variance Swap for Two-Factor SVD and OU Mean-Reversion

Variance for Two-Factor SVD and Pilipovich One-Factor Mean-Reversion

Price of Variance Swap for Two-Factor SVD and Pilipovich One-Factor Mean-Reversion

Variance for Two-Factor SVD and Pilipovich Two-Factor Mean-Reversion

Price of Variance Swap for Two-Factor SVD and Pilipovich Two-Factor Mean-Reversion

Challenging Problems Volatility Swap for One-Factor SVD (also, Covariance, Correlation Swaps) Volatility Swap for Multi-Factor SVD (also, Covariance, Correlation Swaps) Variance and Volatility Swaps for One-Factor SVD and with Jumps (also, Covariance and Correlation Swaps)

The End Thank you (multi-factor!) for your attention!