Title
Document Type
Dissertation
Date of Degree
2007
Degree Name
PhD (Doctor of Philosophy)
Degree In
Applied Mathematical and Computational Sciences
First Advisor
Palle Jorgensen
First Committee Member
Yi Li
Second Committee Member
Oguz Durumeric
Third Committee Member
Walter Seaman
Fourth Committee Member
Yannick Meurice
Abstract
The Black-Scholes model is one of the most important concepts in modern financial theory. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used today, and regarded as one of the best ways of determining fair prices of options. In the classical Black-Scholes model for the market, it consists of an essentially riskless bond and a single risky asset. So far there is a number of straightforward extensions of the Black-Scholes analysis. Here we consider more complex products where each component in a portfolio entails several variables with constraints. This leads to elegant models based on multivariable stochastic integration, and describing several securities simultaneously. We derive a general asymptotic solution in a short time interval using the heat kernel expansion on a Riemannian metric. We then use our formula to predict the better price of options on multiple underlying assets. Especially, we apply our method to the case known as the one of two-color rainbow ptions, outperformance option, i.e., the special case of the model with two underlying assets. This asymptotic solution is important, as it explains hidden effects in a class of financial models.
Keywords
curvature arbitrage, multiple asset model, multidimensional Black-Scholes formula, geometric invariance, Ito's formula, rainbow option
Pages
vii, 60 pages
Bibliography
Includes bibliographical references (pages 59-60).
Copyright
Copyright 2007 Yang Ho Choi