Asymptotic and Exact Pricing of Options on Variance
22 Pages Posted: 20 Nov 2010 Last revised: 15 Dec 2010
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: Suggested Citation
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