Derivatives Models on Models

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Closed Form Valuation of American Barrier Options^*

Closed form formulae for European barrier options are well known in the literature. This is not the case for American barrier options, for which no closed form formulae have been published. One has therefore had to resort to numerical methods. Lattice models like a binomial or a trinomial tree, for valuation of barrier options are known to converge extremely slowly, compared to plain vanilla options. Methods for improving the algorithms have been described by several authors. However, these are still numerical methods that are quite computer intensive. In this chapter we show how some American barrier options can be valued analytically in a very simple way. This speeds up the valuation dramatically as well as giving new insight into barrier option valuation.

1 Analytical Valuation of American Barrier Options

Closed form solutions and valuation techniques for standard European barrier options are well known from the literature, see for instance Merton (1973); Reiner and Rubinstein (1991); Rich (1994); Haug (1997). No closed form solution for American barrier options exist in the current literature. The technique used to value American barrier options has therefore been numerical methods. Lattice models have been especially popular. Without doing any adjustments lattice models have been shown to converge extremely slowly. Although several methods that improve on the technique have been published, they are still quite computer intensive: ...