Optimal Approximations of Nonlinear Payoffs in Static Replication

Abstract

Static replication of nonlinear payoffs by line segments (or equivalently vanilla options) is an important hedging method, which unfortunately is only an approximation. If the strike prices of options are adjustable, two optimal approximations can be defined for replication by piecewise chords. The first is a naive minimum area approach, which seeks a set of strike prices to minimize the area enclosed by the payoff curve and the chords. The second improves on the first by taking the conditional distribution of the underlying into consideration, and minimizes the expected area instead. When the strike prices are fixed, a third or the approach of least expected squares locates the minimum for the expected sum of squared differences between the payoff and the replication portfolio, by varying the weights or quantities of the options used in the replication. For a payoff of variance swap, minimum expected area and least expected squares are found to produce the best numerical results in terms of cost of replication. The average of the costs of replication by chords and by tangents can be suggested as a better estimate for the price of a derivative with a nonlinear payoff.

Keywords: static replication, nonlinear payoff, replication by line segments, optimal approximation, minimum expected area, least expected squares

JEL Classification: G13, G12