Document Type


Date of Degree


Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Jorgensen, Palle

First Committee Member

Li, Yi

Second Committee Member

Durumeric, Oguz

Third Committee Member

Seaman, Walter

Fourth Committee Member

Meurice, Yannick


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.


curvature arbitrage, multiple asset model, multidimensional Black-Scholes formula, geometric invariance, Ito's formula, rainbow option


vii, 60 pages


Includes bibliographical references (pages 59-60).


Copyright 2007 Yang Ho Choi