Basis as a Numerical Method

A contingent claim in the telecommunications access market is driven by a complex set stochastic processes. Based on this consideration, this study uses the Monte Carlo Method as part of a two-prong analytical approach. The Monte Carlo method was particularly appealing in the earlier days of the study

when the feasibility of using closed-form analytical solutions seemed unlikely because of the complexity of the dynamics studied. From the standpoint of option pricing theory, the value of a contingent claim is equivalent to the dis- counted risk-neutral expectation of the cash ‡ows from the derivative security. Computing the value of a contingent claim is therefore equivalent to computing an integral over the space de…ned by the underlying value generating process. The computational complexity of evaluating such integral is exponential to the dimensions of the space de…ning value (Barraquand, 1995). The Monte Carlo method is the only tractable approach for computing high dimensional inte- grals (Barraquand and Martineau, 1995). The Monte Carlo method presents a tool for valuing complex derivatives de…ned by, for example, multi-dimensional stochastic processes, path dependence and early exercise, where closed-form analytical solutions may not be feasible

Boyle (1977) was among the …rst proponents of the Monte Carlo method as a numerical method for the valuation of derivative securities. Boyle shows how Monte Carlo simulation, when reinforced with appropriate variance reduction techniques, provides solutions which approximate those arrived at by closed- form analytical solutions. The basis of the Monte Carlo method as a tool for numerical integration is as follows - take a functionh(u)where the realizations

u(i)are independent, identically distributed and de…ned by somepdf f(u):The Monte Carlo method provides a basis for evaluating the following integral

E[h(u)] =

Z

A

h(u)f(u)du (4.1)

The Monte Carlo method in e¤ect computes an integral over the space de…ning value and thereby provides a basis for deriving the value of a contingent claim described as an expectation (see Boyle, 1977; Trigeorgis, 1996; Boyle et al., 1997; Galanti and Jung, 1997). The Monte Carlo method can be extended to compute multi-dimensional integrals.

valuation of contingent claims by a large number of researchers. For example, Hull and White (1987) use the Monte Carlo method to value contingent claims where the volatility of the underlying asset is stochastic. Kemna and Vorst (1990) use the Monte Carlo method to value path-dependant contingent claims where the value of each claim depends on the average value of the underlying asset in a de…ned period preceding the derivative’s maturity. Here the holder of the security is entitled to the higher of such average value and a straight bond value with the exercise price being the nominal value of the bond. Barraquand (1995) apply the Monte Carlo method to value European contingent claims de…ned by multiple sources of uncertainty with the algorithm proposed being capable of valuing claims where the value of the underlying asset is driven by up to 100 sources of risk. Barraquand uses quadratic re-sampling, also referred to in the literature as Moment Matching (see Boyle et al, 1997), to improve the e¢ ciency of their method.

Barraquand and Martineau (1995) develop an approach for valuing an Amer- ican contingent claim whose value depends on multiple sources of uncertainty. They apply the Monte Carlo method to a state space partioning technique that circumvents the curse of dimensionality. Using one underlying asset, Bar- raquand and Martineau, show that their results correspond to the Black and Scholes closed-form analytical solution for both call and put options. Bar- raquand and Martineau, extend their work to 3 and 10 underlying assets and show that their method generates results which correspond to classical integra- tion methods. Barraquand and Martineau successfully apply their method to the valuation of contingent claims with over 400 dimensions of uncertainty.

Broadie and Glasserman (1997) propose an algorithm based on the Monte Carlo method for valuing contingent claims with early exercise features. Their algorithm, which combines the use of two estimators (with high and low bi- ases), is capable of valuing contingent claims with multiple state variables, path dependencies and early exercise. Grant et al. (1997) use the Monte Carlo method to price contingent claims based on the average price of the underly-

ing assets and where early exercise is feasible. They combine forward-looking simulation with backward-moving recursive dynamic programming. Longsta¤ and Schwartz (2001) develop a method for valuing options, by simulation, using a Simple Least-Squares Approach. This method is capable of valuing options de…ned by multiple factors. Ibanez (2004) applies the Monte Carlo method to value multiple exercise contingent claims. These claims in essence represent a portfolio of buying and selling rights where a speci…ed number of rights can be exercised in a de…ned window, and where one right can be exercised per period for a …nite number of exercise dates. Under these contracts the value of contingent claims are a function of the stochastic processes that drive value but also necessarily a function of the number of exercise rights conferred to the purchaser.