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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

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Variance Swap

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Realized Variance in Continuous Time

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Payoff Function for Variance Swap

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Realized Volatility (Discrete Time)

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Realized Volatility (Continuous Time)

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Payoff Function for Volatility Swap

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Payoff Function for Volatility- Average Swap

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Payoff Function for Implied Volatility Swap

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Payoff for Variance Swaptions

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Payoff Functions for Volatility Swaptions

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Payoff Function for Volatility and Asset Swaption

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Risk-Neutral Pricing Technique

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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

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1 st Approach: Pricing Independently of the Volatility Model

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

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2 nd Approach: Pricing by Expectations in a SV Framework

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

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

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3d Approach: Pricing via PDE

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3d Approach: Pricing via PDE (Model)

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3d Approach: Pricing via PDE (Payoffs)

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

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

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General Stochastic Volatility Models

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Derivation of Certain Expectations

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Derivation of Certain Expectations.I.

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Derivation of Certain Expectations.II.

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Derivation of Certain Expectations.III.

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Derivation of Certain Expectations.IV.

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Derivation of Certain Expectations.V.

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Derivatives Pricing

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Mean-Reverting-Like Process

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Mean-Reverting-Like Process.I.

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Mean-Reverting-Like Process.II.

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Popular SV Models. I.

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Popular SV Models. II.

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Popular SV Models. III.

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Popular SV Models. IV.

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Popular SV Models. V.

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Asymptotical Analysis for Fast Mean-Reversion. I.

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Asymptotical Analysis for Fast Mean-Reversion. II.

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Asymptotical Analysis for Fast Mean-Reversion. III.

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Asymptotical Analysis for Fast Mean-Reversion (Summary).

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Examples: 1. The Variance Swap

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Examples: 2. The Standard- Deviation Swap

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Examples: 3. The Volatility- Average Swap

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Examples: 4. The Implied Volatility- Swap

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Examples: 5. The Volatility-Average Swaption

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References. I.

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References. II.

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The End Thank you for Your Attention!

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