Options on Realized Variance and Convex Orders

28 Pages Posted: 28 Jan 2010

See all articles by Peter Carr

Peter Carr

New York University Finance and Risk Engineering

Hélyette Geman

University of London - Economics, Mathematics and Statistics

Marc Yor

Universite Paris

Dilip B. Madan

University of Maryland - Robert H. Smith School of Business

Date Written: October 8, 2009

Abstract

Realized variance option and options on quadratic variation normalized to unit expectation are analyzed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk neutral densities are said to be increasing in the convex order. For Lévy processes such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability then the resulting risk neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally we consider modelling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Lévy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order.

Keywords: reverse martingale, quadratic variation, stochastic volatility

JEL Classification: G1, G12, G13

Suggested Citation

Carr, Peter P. and Geman, Helyette and Yor, Marc and Madan, Dilip B., Options on Realized Variance and Convex Orders (October 8, 2009). Available at SSRN: https://ssrn.com/abstract=1540805 or http://dx.doi.org/10.2139/ssrn.1540805

Peter P. Carr

New York University Finance and Risk Engineering ( email )

6 MetroTech Center
Brooklyn, NY 11201
United States
9176217733 (Phone)

HOME PAGE: http://engineering.nyu.edu/people/peter-paul-carr

Helyette Geman

University of London - Economics, Mathematics and Statistics ( email )

Malet Street
London, WC1E 7HX
United Kingdom

Marc Yor

Universite Paris ( email )

223 Rue Saint-Honore
Paris, 75775
France

Dilip B. Madan (Contact Author)

University of Maryland - Robert H. Smith School of Business ( email )

College Park, MD 20742-1815
United States
301-405-2127 (Phone)
301-314-9157 (Fax)

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