The American put under transactions costs

This paper examines the optimal super-replication of American put options with physical delivery of the underlying asset, such as stock options, by means of a stock-plus-riskless asset portfolio. The framework of the analysis is the binomial model with proportional transactions costs on stock transactions. The paper extends the model for European options, originally presented in Merton (Geneva Papers Risk Insurance 14 (1989) 225) and Boyle and Vorst (J. of Finance 47 (1992) 271), and generalized in Bensaid et al. (Math. Finance 2 (1992) 63). The optimizing framework of this latter study is adapted to put options held by investors and perfectly hedged by a market maker, and to put options written by investors and both held and hedged perfectly by a market maker. It is shown that a unique optimal super-replicating portfolio exists at every node of the binomial tree for the long option, as well as for the short option when transactions costs are low.