### 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.

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.

#### Full Text:

PDF

Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**