Asymptotic and Exact Pricing of Options on Variance

TL;DR

This paper analyzes the pricing of options on realized variance, providing asymptotic and exact methods that account for jumps in the underlying asset, and compares their effectiveness through numerical examples.

Contribution

It introduces two new methods for pricing options on discretely sampled realized variance, including an exact approach using Fourier-Laplace techniques and a correction-based approximation.

Findings

The difference between realized variance and quadratic variation depends on jumps.

The exact pricing method employs a novel randomization and Fourier-Laplace approach.

Numerical examples demonstrate the accuracy of the proposed methods.

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.