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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.

This paper examines the pricing of American put options by a perfectly-hedged market maker when there are transactions costs to be paid on the underlying stock. Thus, it extends European stock option pricing under perfect hedging, transactions costs, and binomial stock returns, formulated by Merton (1989), and extended by Boyle and Vorst (1992) BV. A similar extension, to American options on dividend-paying stocks under transaction costs, was done in an earlier study ( Perrakis and Lefoll, 2000). While the perfect-hedging assumption may appear extreme for organized option markets, it does offer a useful benchmark case for the derivation of option bounds, within which the option bid/ask prices must lie. Further, this assumption makes our results also useful to situations where there is a need for option replication such as, for instance, in portfolio insurance. Last, it allows the creation of options in cases in which no organized options market exists, as in most emergent financial markets. Thus, while most results are expressed in terms of option pricing by a perfectly-hedged market maker, they also extend to these other important cases. In the option pricing models of the classic studies of Black and Scholes (1973) and Merton (1973), the call option is perfectly and continuously replicated by a stock-plus-riskless-asset portfolio. The introduction of fixed transactions costs every time this portfolio is being rebalanced makes such a policy infeasible in a continuous time model. For this reason a number of papers have tackled the problem of portfolio selection and/or option pricing under transactions costs, both in continuous time1 and on a binomial lattice. Since perfect option replication is infeasible in the continuous time models, those studies that dealt with option pricing specified either approximate replication at predetermined and exogenously given times, or expected utility-based portfolio selection under transactions costs. By contrast, the Merton–BV approach replicates both long and short call options at every node of the binomial lattice. While the replication of the long option is feasible in all cases, the replication of the short option requires some restrictions on parameter values. These restrictions are satisfied when transactions costs are ‘small’ for the chosen number of lattice steps, in a sense that will become more precise in Section 4 of this paper. Merton solved the replication problem when the option has only two periods to expiration; BV extended the Merton model to any number of periods. An important study by Bensaı̈d et al. (1992) BLPS, derived an algorithm to compute optimal perfect-hedging policies for an intermediary that issues long or short options (which they named super-replication), for several types of European options under binomial returns without necessarily replicating the option at every node. 2 The BLPS study found contrasting results for the important cases of physical delivery and cash settlement options: while the intermediary finds it optimal to replicate everywhere physical delivery long options, such a policy is suboptimal for cash settlement options, unless transactions costs are ‘small’, in the same sense as in the Merton–BV studies. In spite of its generality and powerful theoretical insights, the BLPS algorithm is rather difficult to apply as stated for a large number of periods to expiration, since it requires the determination of the intersection points of two piecewise-linear convex functions at every node of the binomial tree. Further, it is based on the assumption that the option holder's actions that are being hedged are predetermined at option expiration; such an assumption is not appropriate for hedging American options, where the option holder must decide whether to exercise or defer at every node. In this paper it is shown that in the presence of transactions costs there are cases where the option holder's optimal action (exercise or not) depends on preferences or other holdings. Nonetheless, perfect hedging requires that the stock-plus-riskless-asset super-replicating portfolio be able to hedge all possible holder actions. This remark allows the extension of the BLPS algorithm, to cover the case of American put options with physical delivery of the underlying asset. European put options under transactions costs can be valued through put-call parity3 from the above-cited studies’ call option prices. For American put options, however, it is well-known since Merton (1973) that early exercise may be profitable even in the absence of dividends, implying that their value exceeds the one given by put-call parity. Although closed-form expressions do not exist for American puts, recursive or analytical methods have been derived, among others, by Parkinson (1977), Brennan and Schwartz (1977), Geske and Johnson (1984), MacMillan (1986), and Barone-Adesi and Whaley (1987). The binomial model of Cox et al. (1979) and Rendleman and Bartter (1979) can be used to value an American put in the absence of transactions costs.4 The key element in such a valuation is the derivation of the early exercise boundary, the stock price separating early from deferred exercise at every time period prior to expiration. Here we show that transactions costs makes this boundary more complex, since the holder's optimal action is ambiguous. The main weakness of the binomial model in handling option pricing under transactions costs is that no guidelines are given to determine the appropriate number of steps in the lattice. This is crucial because, for a fixed transactions cost parameter, the total transactions costs increase as the number of steps rises. At the limit the value of the portfolio replicating the long European call option (without dividends) tends to the stock price, while the portfolio replicating the short option becomes equal to the well-known Merton (1973) European lower bound. 5 For this reason some studies such as Henrotte (1993) and Flesaker and Hughston (1994) have proposed replacing the fixed transactions cost parameter with one that declines in proportion to the square root of the number of steps. This assumption provides reasonable non-trivial option prices under transactions costs, both in continuous time and on binomial lattices. Further, as will be discussed in Section 5, it also satisfies the restriction of ‘small’ transactions costs in relation to the binomial parameters for most reasonable cases. In the next section we present the problem along the lines of the earlier Merton (1989), BV and BLPS studies. The long put is presented in Section 3, while Section 4 examines the short put for ‘small’ transactions costs. Some numerical examples and a discussion of the extension of the results to ‘large’ transactions costs are presented at the end.

We have presented algorithms evaluating, at each node of the binomial tree, portfo- lios super-replicating American put options in the presence of transactions costs. Such portfolios also represent the tightest possible bounds on such option prices when they are computed under the assumption that the market maker is perfectly hedged. The expressions are valid for all options held long by the investors and for short options when transactions costs are ‘small’ (i.e. when ( 11 ) holds). In general, the shape of the optimal exercise policy found by our results is fairly similar to the no-transaction-costs case. It is optimal to exercise the option immediately for low values of the stock price, and to defer for high values. An important diBerence, however, is that under transactions costs there exists an intermediate region of stock prices where the exercise policy is ambiguous. Nonetheless, the algorithms developed in the theorems presented in this paper are easy to implement, and allow the quick computation of the optimal super-replicating portfolios for any number of periods. The approach followed in this paper can be extended to other relevant cases of American options under transactions costs. In Perrakis and Lefoll (PL, forthcoming) super-replication algorithms are presented for a physical delivery option with a single constant dividend prior to expiration. Options on assets with a constant payout rate, such as index options, can also behandled in principlein thesameway. Such options, however, are cash-settled and may be exercised once a day. As BLPS showed, the optimal hedging policy for both long and short European cash-settled options does not correspond to simple replication when ( 11 ) does not hold. The matter is currently under study.