Asymptotic and Exact Pricing of Options on Variance

22 Pages Posted: 20 Nov 2010 Last revised: 15 Dec 2010

See all articles by Martin Keller-Ressel

Martin Keller-Ressel

Dresden University of Technology - Department of Mathematics

Johannes Muhle-Karbe

Imperial College London - Department of Mathematics

Date Written: November 19, 2010

Abstract

We consider the pricing of derivatives written on the discretely sampled realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuous-time limit, the quadratic variation of the underlying log-price. Here, we characterize the small-time limits of options on both objects. We find that the difference between them strongly depends on whether or not the stock price process has jumps. Subsequently, we propose two new methods to evaluate the price of options on the discretely sampled realized variance. One of the methods is approximative; it is based on correcting prices of options on quadratic variation by our asymptotic results. The other method is exact; it uses a novel randomization approach and applies Fourier-Laplace techniques. We compare the methods and illustrate our results by some numerical examples.

Keywords: Realized Variance, Quadratic Variation, Option Pricing, Small-Time Asymptotics, Fourier-Laplace Methods

JEL Classification: C63, G19

Suggested Citation

Keller-Ressel, Martin and Muhle-Karbe, Johannes, Asymptotic and Exact Pricing of Options on Variance (November 19, 2010). Available at SSRN: https://ssrn.com/abstract=1711774 or http://dx.doi.org/10.2139/ssrn.1711774

Martin Keller-Ressel (Contact Author)

Dresden University of Technology - Department of Mathematics ( email )

Zellescher Weg 12-14
Willers-Bau C 112
Dresden, 01062
Germany

Johannes Muhle-Karbe

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
Imperial College
LONDON, SW7 1NE
United Kingdom

HOME PAGE: http://www.ma.imperial.ac.uk/~jmuhleka/