Pricing swaps and options on quadratic variation under stochastic time change models - discrete observations

We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As such a process we consider a class of Lévy models with stochastic time change. Our analysis reveals a natural small parameter of the problem which allows a general asymptotic method to be developed in order to obtain a closed-form expression for the fair price of the above products. As examples, we consider the CIR clock change, general affine models of activity rates and the 3/2 power clock change, and give an analytical expression of the swap price. Comparison of the results obtained with a familiar log-contract approach is provided. ∗We thank Arthur Sepp and an anonymous referee for useful comments. We assume full responsibility for any remaining errors. †Department of Mathematics, Hill Center for Mathematical Sciences, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road Piscataway, NJ 08854-8019, [email protected] ‡Bloomberg LP & NYU, 731 Lexington Avenue, New York, NY 10022, [email protected]