Numerical Valuation Algorithms For Exotic Derivatives. I: The Trinomial Lookback Case

Abstract

The underlying asset of an option can include, e.g., bonds, stocks, stocks indices,
and commodities. The exercise price of the contract and the date of expiration (maturity) are
well defined. The Exotic options are variations of the traditional ones, e.g. Asians, Binary,
Barrier, Compound, Lookback. The Exotic ones are stated over-the-counter. One of the parts
of these contracts has the long position; i.e. who purchases the option. The other has the
short position, i.e. who sells or writes the option. Binomial trees are the basic instruments to
deal with the valuation of European and American stock and bond options. A quite realistic
model is one that supposes that movements in the prices of the assets are made up of a great
amount of small binomial movements. The basic expected return of an asset must be greater
than those provided by the risk free rate of interest, r. The standard deviation s (or volatility)
of the change in the price of an asset in a small time interval (yearly based) is fundamental in
the numerical valuation procedure. As the delta and other indicators change during the life of
the option we need to make periodic adjustments in our portfolio. This is called hedging. The
Lookback options are a class of Exotic options whose price of exercise is the maximum value
(minimum for the call) reached by the price of the stock to the instant of evaluation or
exercise. The valuation process traditionally uses binomial trees and simulation methods that
work backwards in the binomial tree with the purpose of estimating the Lookback value. At
the moment of the expiration it is observed retrospectively. Montecarlo methods are also
used. In this paper we obtain results of the application of a more accurate trinomial method
to this evaluation and study the performance of the special algorithms devised by us to solve
this problem.