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A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES

Authors


  • Presented at the 19th Annual Derivatives Securities and Risk Management Conference 2009, the Nordic Finance Network Workshop 2009, the Courant Mathematical Finance Seminar (October 2009), Research in Options 2009, Quantitative Methods in Finance 2009, Global Derivatives 2010, the Fields Workshop on Derivatives and Risk Management 2010, and the Bachelier Congress 2010. Thomas Kokholm wishes to thank Bjørn Jørgensen and the Columbia Graduate School of Business for their hospitality. We also thank Bjarne Astrup, Peter Carr, Peter Løchte Jørgensen, and Elisa Nicolato for comments.

Rama Cont, IEOR Deptartment, Columbia University, New York; e-mail: Rama.Cont@columbia.edu.

Abstract

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.

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