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.

本文研究最近在市面上常見的結構型債券，利用Martingale評價方法以及數值方法求出結構型商品的理論價格以及利用情境分析來推估期末可能的報酬，提供投資人與券商對於結構型商品特性與風險的了解，並且提供發行商避險的參考。然而結構型商品的複雜程度往往是來自於隱含的新奇選擇權，本文亦分析商品內含的新奇選擇權，使得投資人更了解結構型商品的組成，發行商也可藉以由組成的概念進而設計新的結構型商品。