Valuation of volatility derivatives as an inverse problem

Abstract

Ground-breaking recent work by Carr and Lee extends well-known results for variance swaps to arbitrary functions of realized variance, provided a zero-correlation assumption is made. We give a detailed mathematical analysis of some of their computations and work out the cases of volatility swaps and calls on variance. The latter leads to an ill-posed problem that we solve using regularization techniques. The sum is divergent, that means we can do something Heaviside† †Quote suggested by Peter Carr. We take this opportunity to thank him, Rama Cont, Jining Han, Bob Kohn and Roger Lee for related discussions.