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Taking a Look at Exotic Options

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Introduction
Ideally we would want to find closed form solutions for all exotic options. As the payoff structures of the exotic options become more complicated, so does the difficulty in finding closed form solutions. In most cases closed-form solutions do not exist, eg. American barrier options and these must be valued by using numerical methods.
The lattice methods, i.e. binomial and trinomial trees, assume that the underlying stochastic process is discrete, i.e. the underlying asset can change to a finite number of values (each associated with a certain probability) with a small advancement in time.
Pricing barrier options using lattice techniques can be quite delicate. While the use of a large number of time steps may produce accurate solutions for standard options, the use of the same number of time steps in valuing barrier options will often produce erroneous option values. The source of the problem arises from the location of barrier with respect to adjacent layers of nodes in the lattice. If the barrier falls between layers of the lattice, the errors may be quite significant.
Ritchken [1995] provides a highly efficient algorithm which produces a lattice where the nodes hit the barrier.

Options
A plain normal option is called a Vanilla Option. An option is a financial contract which gives its owner a right, but not an obligation to sell or purchase an asset at a later stage during the lifetime of the asset. The asset mentioned herein can be anything from stocks,

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