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.

到期還本的指數連動型證券為一種連結權益(equity)的債權證券，所連結的權益部分通常以隱含選擇權的方式建立。指數連動證券具有自動資產配置調整的特性，當股票市場表現不錯時，此契約給予投資人較高的股票市場風險暴露（因為股票上漲時，Delta值增加）。若股票市場表現不佳，則契約收益特徵接近債券的型式。所以是保守型投資得以參與部分股票市場表現之設計。 本論文所研究之中短期連動型票券，係以零息債券持有至到期（其面額等於到期還本金額），期間不可贖回或申購，並以期初零息債券貼現的部分來購買不同的請求權，以做為連動股票市場表現的機制。在推導多重標的資產請求權評價模型上，係採Martingale方式，其中並證明在Gisanov轉換機率測度下，多重標的之隨機項轉換的規則。 本文主要研究Rainbow Call與Spread Call的評價模型與避險參數；進一步研究標的資產間相關係數對選擇權價值之影響與避險上的財務經濟意義。另一方面，運用Martingale此一有力的工具，佐以現金流量分析，來推導跨國標的之評價模型，並提出跨國之避險操作方法，與說明標的資產與匯率間相關係數在避險上的財務經濟意義。 本文最後就兩套請求權設計之指數連動票券，模擬比較在不同相關係數下，與其他選擇權設計之指數連動票券的表現。並嘗試提出該設計票券之較佳表現時機。