The main goal of this work is to develop a method of static replication of multi-variate products, such as realized variance swaps, using more standard products, such as bivariate binary or barrier European calls and puts. In Section 1, the concept of a static strategy, based on a continuum of traded call and/or put options, is formally introduced in terms of a measure-valued portfolio, and relevant quantities are defined and analyzed. We put emphasis on the fact that the existence of a wealth process of a static strategy is not always obvious or easy to prove, due to insuficient information about options prices.
In Section 2, static replication is applied to univariate claims. We first examine, following Carr and Madan (2002), replication of path-independent claims with twice differentiable payoffs. Next, we extend this approach to payoffs given by differences of two convex functions and subsequently to payoffs given by functions of finite variation. Since in the last case we consider claims with a possibly discontinuous payoff, we need to postulate that all binary call and put options are traded (for a similar observation, see Carr and Chou (2002)).
In Section 3, the static replication of bivariate claims is introduced and examined. Initially, bivariate functions of finite variation are defined and it is consequently shown how to replicate claims with payoffs given by bivariate functions of finite variation. The work concludes by an application of general results to static hedging of a realized variance swap.