SFB 303 Discussion Paper No. B - 003

Author: Föllmer, Hans, and Dieter Sondermann
Title: Hedging of Non-Redundant Contingent Claims
Abstract: In this paper, our purpose is to extend the martingale approach of Harrison and Kreps (1979) to contingent claims which are non-redundant. We are less concerned here with valuation formulas than with how to use the existing assets for an optimal hedge against the claim. To this end we introduce a class of admissible portfolio strategies which generate a given contingent claim at some terminal time T. Due to the underlying martingale assumptions, the expected terminal cost does not depend on the specific choice of the strategy. It is therefore natural to look for admissible strategies which minimize risk in a sequential sense. We show that this problem has a unique solution where the risk is reduced to what we call the intrinsic risk of the claim. This risk- minimizing strategy is mean-self-financing, i.e., the corresponding cost process is a martingale. A claim is attainable if and only if its intrinsic risk is zero. In that particular case, the risk-minimizing strategy becomes self-financing, i.e., the cost process is constant, and we obtain the usual arbitrage value of the claim. We then study the dependence on the hedger's subjective beliefs: It is shown how the strategy changes under an absolutely continuous change of the underlying martingale measure.
Finally we illustrate our results by computing explicitly the intrinsic risk and the risk-minimizing strategy for a call option where the underlying stock processes already studies in the literature, see e.g. Cox and Ross (1976b), this model is not complete, and a typical call is non-redundant.
Creation-Date: May 1985
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