Paper Review On the Pricing and Hedging of

Paper Review: “On the Pricing and Hedging of Volatility Derivatives” by S. Howison, A. Rafailidis and H. Rasmussen (Applied Mathematical Finance J. , 2004) Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C “Lunch at the Lab” Talk February 10, 2006

Variance Swap

Realized Variance in Continuous Time

Payoff Function for Variance Swap

Realized Volatility (Discrete Time)

Realized Volatility (Continuous Time)

Payoff Function for Volatility Swap

Payoff Function for Volatility. Average Swap

Payoff Function for Implied Volatility Swap

Payoff for Variance Swaptions

Payoff Functions for Volatility Swaptions

Payoff Function for Volatility and Asset Swaption

Risk-Neutral Pricing Technique

Three Approaches to the Risk. Neutral Pricing • Pricing Independently of the Volatility Model • Pricing by Expectations in a SV Framework • Pricing via Partial Differential Equations

1 st Approach: Pricing Independently of the Volatility Model

1 st Approach: Pricing Independently of the Volatility Model (cntd)

1 st Approach: Pricing Independently of the Volatility Model (cntd)

2 nd Approach: Pricing by Expectations in a SV Framework

2 nd Approach: Pricing by Expectations in a SV Framework (cntd)

2 nd Pricing Approach: Pricing by Expectations in a SV Framework (cntd)

2 nd Pricing Approach: Pricing by Expectations in a SV Framework (cntd)

3 d Approach: Pricing via PDE

3 d Approach: Pricing via PDE (Model)

3 d Approach: Pricing via PDE (Payoffs)

3 d Approach: Pricing via PDE (Payoffs)

3 d Approach: Pricing via PDE (PDE Itself for the Value V of Derivative)

3 d Approach: Pricing via PDE (Mean-Reverting Model)

General Stochastic Volatility Models

Derivation of Certain Expectations

Derivation of Certain Expectations. I.

Derivation of Certain Expectations. II.

Derivation of Certain Expectations. III.

Derivation of Certain Expectations. IV.

Derivation of Certain Expectations. V.

Derivatives Pricing

Mean-Reverting-Like Process

Mean-Reverting-Like Process. I.

Mean-Reverting-Like Process. II.

Popular SV Models. I.

Popular SV Models. II.

Popular SV Models. III.

Popular SV Models. IV.

Popular SV Models. V.

Asymptotical Analysis for Fast Mean-Reversion. I.

Asymptotical Analysis for Fast Mean-Reversion. II.

Asymptotical Analysis for Fast Mean-Reversion. III.

Asymptotical Analysis for Fast Mean-Reversion (Summary).

Examples: 1. The Variance Swap

Examples: 2. The Standard. Deviation Swap

Examples: 3. The Volatility-Average Swap

Examples: 4. The Implied Volatility. Swap

Examples: 5. The Volatility-Average Swaption

References. I.

References. II.

The End • Thank you for Your Attention!