Joint modeling of VIX and SPX options at a single and common maturity with risk management applications

Abstract

A double gamma model is proposed for the VIX. The VIX is modeled as gamma distributed with a mean and variance that respond to a gamma-distributed realized variance over the preceeding month. Conditional on VIX and the realized variance, the logarithm of the stock is variance gamma distributed with affine conditional drift and quadratic variation. The joint density for the triple realized variance, VIX, and the SPX is in closed form. Maximum likelihood estimation on time series data addresses model adequacy. A joint calibration of the model to SPX and VIX options is employed to illustrate a risk management application hedging realized volatility options.

Keywords:

• Ask price minimizing hedge
• distorted expectation
• variance gamma model

Additional information

Notes on contributors

Peter Carr

Peter Carr is a Managing Director at Morgan Stanley with 15 years of experience in the derivatives industry. He was also a finance professor for 8 years at Cornell University, after obtaining his Ph.D. from UCLA in 1989. He is presently the Executive Director of the Math Finance program at NYU’s Courant Institute, the Treasurer of the Bachelier Finance Society, and a trustee for the Museum of Mathematics in New York. He has over 70 publications in academic and industry-oriented journals and serves as an Associate Editor for eight journals related to mathematical finance. He was selected as Quant of the Year by Risk Magazine in 2003 and shared in the ISA Medal for Science in 2008. Last December, the International Association of Financial Engineers (IAFE) and Sungard jointly announced that they selected Dr. Carr as its 2010 Financial Engineer of the Year.

Dilip B. Madan

Dilip Madan is Professor of Finance at the Robert H. Smith School of Business. He specializes in mathematical finance. Currently he serves as a consultant to Morgan Stanley, Meru Capital, Norges Bank Investment Management, and MarketToppers. He has also consulted with Citigroup, Bloomberg, the FDIC, Wachovia Securities, and Caspian Capital. He is a founding member and Past President of the Bachelier Finance Society. He received the 2006 von Humboldt Award in Applied Mathematics, was the 2007 Risk Magazine Quant of the year, received the 2008 Medal for Science from the University of Bologna, and held the 2010 Eurandom Chair. He has served as the Managing Editor of Mathematical Finance and currently is co-editor of the Review of Derivatives Research and an Associate Editor for numerous other journals.