Optimal Approximations of Nonlinear Payoffs in Static Replication

摘要

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 (for OTC options), 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 (for exchange-traded options), a third or the approach of least expected squares locates the minimum for the expected sum of squared differences between the payoff and the replicating 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. Finally, piecewise tangents can also be utilized in static replication, which together with replication by chords,rnforms a pair of lower or upper bound to a nonlinear payoff.