  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.
