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.