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American options are considered in the binary tree model under small proportional transaction costs. Dynamic programming type algorithms, which extend the Snell envelope construction, are developed for computing the ask and bid prices (also known as the upper and lower hedging prices) of such options together with the corresponding optimal hedging strategies for the writer and for the seller of the option. Representations of the ask and bid prices of American options in terms risk-neutral expectations of stopped option payoffs are also established in this setting.

In the presence of small proportional transaction costs in the form of a bid-ask spread of the underlying stock prices we shall consider American options exercised by the physical delivery of a portfolio of cash and stock. For example, when a call option with strike price K is exercised by physical delivery, the holder of the option pays K and receives 1 share, that is, the option is exercised by the delivery of a portfolio (−K,1)(−K,1) of cash and stock. If buying or selling stock incurs transaction costs, then physical delivery is not equivalent to cash settlement. Chalasani and Jha (2001) investigated American options with cash settlement only under (not necessarily small) proportional transaction costs. They obtained general representations involving so-called randomised stopping times for the ask and bid prices of American contingent claims. However, no algorithmic procedure for computing the prices was proposed. Indeed, Chalasani and Jha commented that “The computation of the expressions for the upper and lower hedging prices appears non-trivial. It would be useful to design efficient algorithms for approximating the values of these expressions.” (Chalasani and Jha, 2001, p. 72). Here we put forward two algorithms extending the Snell envelope construction, one for computing the ask price (the upper hedging price) and one for the bid price (the lower hedging price) of an arbitrary American contingent claim under small proportional transaction costs in the binary tree model. We also construct optimal hedging strategies for the option writer as well as for the seller, and establish representations of the ask and bid option prices in terms of risk-neutral expectations of stopped payoffs. In contrast to Chalasani and Jha (2001), our results involve ordinary rather than randomised stopping times, and we consider American options with physical delivery rather than ones with cash settlement only. Numerical examples demonstrating the applicability of the pricing algorithms are provided, and certain novel and interesting features are noted, such as the non-equality of bid prices for American and European calls, or different optimal stopping times for the writer and the holder of an American option under transaction costs. Other papers devoted to American options under proportional transaction costs include Davis and Zariphopoulou, 1995, Levental and Skorohod, 1997, Mercurio and Vorst, 1997, Kociński, 1999, Kociński, 2001, Perrakis and Lefoll, 2000, Perrakis and Lefoll, 2004, Constantinides and Zariphopoulou, 2001, Jakubenas et al., 2003 and Constantinides and Perrakis, 2004 and Bouchard and Temam (2005). For a wider context we refer to the extensive literature concerned with pricing and hedging European options under proportional transaction costs. This problem was studied by Merton, 1990, Dermody and Rockafellar, 1991, Boyle and Vorst, 1992, Bensaid et al., 1992, Edirsinghe et al., 1993, Jouini and Kallal, 1995, Kusuoka, 1995, Naik, 1995, Shirakawa and Konno, 1995, Soner et al., 1995, Cvitanić and Karatzas, 1996, Koehl et al., 1996, Koehl et al., 1999, Koehl et al., 2002, Cvitanić et al., 1999, Levental and Skorohod, 1997, Perrakis and Lefoll, 1997, Stettner, 1997, Stettner, 2000, Rutkowski, 1998, Touzi, 1999, Jouini, 2000, Ortu, 2001, Palmer, 2001a, Palmer, 2001b and Kociński, 2004, and many others. In the majority of these papers the authors assume a stock price process StSt under no-arbitrage conditions, and introduce transaction costs by multiplying StSt by constant factors 1+λ1+λ and 1−μ1−μ for some λ,μ≥0λ,μ≥0. Here we follow the more general approach of Jouini and Kallal (1995) involving bid-ask spreads View the MathML sourceStb≤Sta for the stock price. As pointed out by Jouini (2000), the spreads can be interpreted as proportional transaction costs in the above sense, but can also be explained by the buying and selling of limit orders. Accordingly, View the MathML sourceSta and View the MathML sourceStb can be thought of as the prices ensuring liquidity in the stock market, that is, at which stock can be bought or, respectively, sold on demand. The spreads, therefore, include proportional transaction costs, but are not limited to them. The lack of arbitrage in a model with bid-ask spreads was characterised by Jouini and Kallal (1995) in terms of the existence of suitably defined risk-neutral measures and associated stock price processes, see Theorem 2.1. We use their results here as our starting point. See also Ortu (2001). Small proportional transactions costs similar to those considered in the present paper have been studied by a number authors in various contexts, for example, by Bensaid et al., 1992, Koehl et al., 1999, Kociński, 1999 and Kociński, 2001, and Melnikov and Petrachenko (2005). The definitions of small transaction costs differ slightly between these various approaches, but there is a substantial overlap, also with that adopted in the present paper. Even in the relatively simple case of small proportional transaction costs within the binary tree model the results for American options give rise to interesting and perhaps unexpected effects such as, for example, the non-equality of bid prices of American and European calls. The algorithms developed can inform further research aiming to extend the results to American options under general proportional transaction costs and to more general market models. Small transaction costs can be considered as perturbations of the friction-free model. From a practical point of view the assumption of small proportional transaction costs might become a limitation if the time step is chosen to be very short, since transaction costs would then need to scale down to nil asymptotically. The difficulties inherent in continuous time models under transaction costs, which are known to lead to unrealistic option prices and trivial hedging strategies (see Soner et al., 1995 and Cvitanić et al., 1999 for European options, and Levental and Skorohod, 1997 for American options), indicate that asymptotically short time steps may in fact be incompatible with proportional transaction costs. Leland (1985) and Hoggard et al. (1994) have sought a compromise solution by combining the continuous time model under proportional transaction costs with hedging portfolios rebalanced only at discrete time instances, with a fixed finite time interval h between them. Leland type approaches are limited by the condition View the MathML source2/π(2k/σh)<1, the expression on the left known as the Leland number, where 2k2k is the round-trip transaction cost rate (expressed as a percentage of the stock price) and σσ is the volatility of the underlying asset, see Avellaneda and Parás (1994). (Observe that the round-trip transaction cost rate k in (Avellaneda and Parás, 1994) is the same as 2k2k in our notation.) None of these Leland type approaches have been extended to American options, inviting further research in this area of great practical importance. It will be demonstrated in Section 4 that our small transaction costs assumption asymptotically yields a very similar restriction, View the MathML source2k/σh<1. The results of the present paper can thus be regarded as a discrete version of Leland type approaches as applied to American options. The paper is organised as follows: in Section 2 we describe the model, introduce some notation, basic notions and facts, and specify the small proportional transaction costs assumption. The main results are presented in Section 3, with proofs in Appendix A. Numerical examples demonstrating an application of the pricing algorithms are discussed in Section 4. This section also touches upon the relationship between American and European call prices under transaction costs. Section 5 concludes.

We have established a procedure for computing the ask price π a ( ξ, ζ ) of an American contingent claim ( ξ, ζ ) under small proportional transaction costs in the binary tree setting, resembling the standard construction of the Snell envelope. A suitable algorithm has also been developed for computing the bid price π b ( ξ, ζ ). Moreover, we have constructed optimal strategies to hedge long and short positions in American options, and established representations of the ask and bid option prices in terms of the initial values of these strategies, and also in terms of risk-neutral expectations of stopped payoffs. In addition, optimal stopping times ˆ τ for the writer and ˇ τ for the buyer of the option have been constructed. A distinctive feature of the algorithms developed under small transaction costs is that two quantities need to be tracked at each tree node, rather than a single one as in the standard iterative construction of the Snell envelope in the friction-free case. In the numerical examples the algorithms have been applied to compute the ask and bid prices of American options in a realistic setting (option expiry T = 1 year, daily rehedging of portfolios with N = 250, and round-trip transaction costs 2 k = 1% typical of large capitalisation stocks) satisfying the small transaction costs condition. The numerical results are compared to the earlier work by Boyle and Vorst (1992) for European options under proportional transaction costs. The small transaction costs assumption (2.2) is shown to lead to a very similar restriction on the time step versus transaction cost rate as in Leland type approaches. We have observed that the bid prices of American and European calls are not necessarily the same, and proved that the ask prices are equal to one another. Moreover, the optimal stopping times may be different for traders hedging long and short positions in American options under transaction costs. A natural question arises as to what will happen if the small costs assumption (2.2) is relaxed, so that only the no-arbitrage condition prevails. Results in Tokarz (2004) and Roux et al. (2006) , which apply to European options only, suggest that the algorithms for American options might need to be modified by keeping track of more than two quantities at each tree node whenever the bid-ask spreads for the stock price at adjacent nodes overlap. The work of Chalasani and Jha (2001) indicates that the constructions would also need to accommodate randomised stopping times in place of ordinary ones. The reason why ordinary stopping times are sufficient under small transaction costs is, essen- tially, that every adapted process S such that S a t ≤ S t ≤ S a t for each t = 0 , 1 ,...,T can be regarded as the stock price process in an ordinary (friction-free and arbitrage-free) binomial tree model, and the price and optimal stopping time for any given American option under small transaction costs coincide with those obtained in such an ordinary binary tree model, namely that with the stock price process ˆ S constructed in the proof of Theorem 3.2 . In general, this is no longer the case under arbitrary proportional transaction costs. Finally, with reference to Bouchard and Temam (2005) , we observe that their Proposition 4.1 might at first sight appear to contradict our results. For example, consider the American option with payoff ξ t = S a 0 1 { t = 0 } and ζ t = 1 { t = 1 } . Our approach gives the writer’s price of the option as π a ( ξ, ζ ) = S a 0 . However, according to Proposition 4.1 in Bouchard and Temam (2005) , the initial amount S a 0 is insufficient to construct a superhedging strategy for this option whenever S b 0 <S a 0 . To reconcile this apparent contradiction we observe that in our approach we follow Chalasani and Jha (2001) in that the rebalancing of the hedging portfolio at any node occurs once the decision of whether or not to exercise the option at that node has been made by the buyer. Meanwhile, Bouchard and Temam (2005) follow a different convention, namely in their case portfolio rebalancing occurs immediately before it becomes known whether or not the option will be exercised at any give node. This means that in fact there is no contradiction.