Question

American Options Models discuss and compare binomial tree, Trinomial Tree method and least squares simulation method (LSM) towards the goal of identifying when they work best and when they are limited

Answer 1

American Options Models

An American option is just like a European option, except the American option carries the right of early exercise. Exercising a call before expiration discards the time value inherent in the option. The only offsetting benefit from early exercise arises from an attempt to capture a dividend. If there is no dividend, there is no incentive to early exercise, so the early exercise feature of an American call on a non dividend stock has no value.

The key factor is an approaching dividend, and exercise of an American call should occur only at the moment before an ex-dividend date. The dividend must be “large” relative to the share price, and the call will typically also be deep-in-the-money.

binomial tree

A binomial tree is a general tree with a very special shape: Definition (Binomial Tree) The binomial tree of order with root R is the tree defined as follows. If k=0, . I.e., the binomial tree of order zero consists of a single node, R.

A binomial tree is a graphical representation of possible intrinsic values that an option may take at different nodes or time periods. The value of the option depends on the underlying stock or bond, and the value of the option at any node depends on the probability that the price of the underlying asset will either decrease or increase at any given node.

Trinomial Tree method

The trinomial tree is a lattice based computational model used in financial mathematics to price options. ... It is an extension of the binomial options pricing model, and is conceptually similar. It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing.

This is where the operator formulation becomes especially useful. By expressing the valuation operator via its singular value decomposition (SVD), we show that under certain conditions, the (left) singular functions present an optimal choice for the basis functions. More precisely, we demonstrate that these singular functions approximate the valuation operator – and, thus, the distribution of relevant capital levels – in an optimal manner. The intuition is that similarly to an SVD for a matrix, the singular functions provide the most important dimensions in spanning the image space of the operator.

Least-Squares Monte-Carlo (LSM) Algorithm As indicated in the previous section, the task at hand is to determine the distribution of Cτ given by Equation (3). Here, the conditional expectation causes the primary difficulty for developing a suitable Monte Carlo technique. This is akin to the pricing of Bermudan or American options, where “the conditional expectations involved in the iterations of dynamic programming cause the main difficulty for the development of Monte-Carlo techniques” (Cl´ement et al., 2002). A solution to this problem was proposed by Carriere (1996), Tsitsiklis and Van Roy (2001), and Longstaff and Schwartz (2001), who use least-squares regression on a suitable finite set of functions in order to approximate the conditional expectation. In what follows, we exploit this analogy by transferring their ideas to our problem.