Pricing and Hedging American Barrier Options by a Modified Binomial Method

Abstract

The aim of this work is to present a modification of the standard binomial method which allows to price American barrier options improving the efficiency of the trinomial methods. Our approach is based on a suitable interpolation of binomial values and allows to price and hedge such options also in the critical case of near barriers. All the different types of single barrier options are considered, in the case of knock-in barriers a new implementation of the binomial method is provide

The Binomial Interpolated Lattice Method for Step Double Barrier Options

“….., 0 with N − i even (see Remark 5.1 in [8]). In the backward procedure, as usual, we need to compare the early exercise with the continuation value at each node of the tree.…”

“…The numerical results show that the BIL method converges faster than the bino-trinomial tree also in the case in which s 0 is chosen not so much close to the barriers. We also observe that when s 0 = 92 we need M ≥ 104 to satisfy (7) and when s 0 = 138 we have to choose M ≥ 289 to verify condition (8). Moreover, also if M is such that (7) or (8) is satisfied, the Binomial Interpolated Lattice converges faster than the bino-trinomial tree.…”

“…Later Cheuck-Vorst [3] present a modification of the trinomial method (based on a change of the geometry of the tree) which allows to set a layer of nodes exactly on the barrier for every choice of the number of time steps. Gaudenzi-Lepellere [8] introduce suitable interpolations of binomial values and Gaudenzi-Zanette [9] construct a tree where all the mesh points are generated by the barrier itself. However, all the previous methods are not able to price efficiently double barrier options.…”

“…In fact in the case of Table 1 with s 0 = 90.05, we have to choose M ≥ 201840. Instead, in the case in which s 0 = 139.95 we need to choose M ≥ 489104 in order to satisfy condition (8). Similarly, in Table 2, we need to choose M ≥ 55987 if s 0 = 95 and M ≥ 122107 if s 0 = 139.9 to satisfy conditions (7) and (8) respectively.…”

Pricing American barrier options with discrete dividends by binomial trees

“…The main idea is to construct a tree where all singular points are generated by the barrier itself. Such method will be applied first in the case of no discrete dividends providing a new technique which simplifies and improves the known evaluation methods for barrier options (see Boyle and Lau 1994;Ritchken 1995;Cheuk and Vorst 1996;Gaudenzi and Lepellere 2006). These previous techniques are based on the idea that a good precision is obtained when the barrier lies (or it is close) on a line of nodes of the tree.…”