In this dissertation, we create a portfolio of simple vanilla put and call options as an optimal approximation of nonlinear payoffs by using static replication (1995, 1998) [1, 2] under certain measure which is called E(a,b,N,f). More specifically, we focus on the static replication of variance swaps payoffs because of their popularity in current financial market [3]. The analysis is motivated by the following reasons. Due to the limited availability of strike prices with traded vanilla options, static replication is only an approximation [1]. Bradie and Jain (2008) [4] used Black-Scholes and Heston stochastic volatility model to find the optimal approximation. Liu (2010) [5] created three approximation methods. In order to improve the approximation, we use a new measure for the static replication to construct the replicating portfolio with lower cost compared with the current methods.