Electronic Thesis and Dissertation Repository

Degree

Master of Science

Program

Applied Mathematics

Supervisor

Dr. Matt Davison

Abstract

In this paper, we examine the pricing and hedging of an index option where one constituents stock plays an overly dominant role in the index. Under a Geometric Brownian Motion assumption we compare the distribution of the relative value of the index if the dominant stock is modeled separately from the rest of the index, or not. The former is equivalent to the relative index value being distributed as the sum of two lognormal random variables and the latter is distributed as a single lognormal random variable. Since these are not equal in distribution, we compare the two models. The validity of this theoretical result is verified against empirical stock market data. We look at two main models representing these cases: first, we use numerical methods to solve the two-dimensional problem directly; second, we make simplifying assumptions to reduce the two-dimensional Black-Scholes problem to a one-dimensional Black-Scholes problem that can be solved analytically.

Since the terminal conditions of an option are usually non-smooth the numerical methods are verified by comparison to a Monte Carlo simulated solution.

Attributes of the models that we compare are the relative option price differences and expected hedging profits. We compare the models for various volatilities, dominance levels, correlations and risk free rates.

This work is significant in options trading because when a stock becomes dominant in its index the distribution of the returns changes. Even if the effect is small, given the millions of dollars exposed to index option trades, it has a material impact.