利率衍生性商品的定價及數值方法

Abstract

在世界財務與投資的領域裡，證券的交易工具顯得愈來愈廣大，許多新的金融工具及產品應運而生，而選擇權市場也變得愈來愈重要。在此，我們將致力於利率衍生性商品的定價模型及其數值方法。雖然Black-Scholes公式可以被應用在利率選擇權上，但是對於不同的產品做出不同的假設，它導致了一個特別的定價方法。為了精確且前後一致性地評價出利率衍生性商品的值，我們需要對整個利率期限結構及其相關的利率波動作模型。為了自動地與最初所觀察的市場資料相一致，期限結構一致模型著手於對整個期限結構的動態做模型。 對於大部分的利率模型而言，這些模型或是模型有些易處理性但卻應用於提早執行或是產品有著複雜的最終報酬定價問題，我們不易求得其解析解，因而須藉數值方法來解出。首先，我們建構二項式樹狀圖以代表短期利率的一系列過程，以樹狀圖的結果來評價廣泛的利率衍生性商品。進而延伸到三項式樹狀圖，其中額外的自由度較可捕捉其平均數復歸(mean reversion)的現象。用這些方式所建構出的樹狀圖近似於短期利率的隨機微分方程，並自動地轉成所觀察的折現債券價格和這些債券可能的波動度，因而我們可用來評價出許多的利率衍生性商品。

In the world , the securities have become very popular , with a wide variety of istrument trading in the finance and investment market . And option market becomes more and more important . Here we concentrate on models for pricing interest rate derivatives and its numerical techniques . Although Black-Scholes formula can be used to price interest rate derivatives , different instruments make different assumptions , it leads special pricing methods . In order to value interest rate derivatives accurately and consistently we need to model the whole term structure of interest rates and the associated volatilities of these rates . To be automatically consistent with the initial (observed) market data , term structure consistent models set out to model the dynamics of the entire term structure . For most interest rate models , and for models which have some tractability but applied to pricing products which involve early exercise opportunities or complicated terminal pay-offs , we must use numerical techniques to solve them . First we construct binomial trees to represent a number of processes for short rate , and how the resulting tree can then be used to price a wide range of interest rate derivatives . Furthermore we extend it to building trinomial trees for short rate , the extra degree of freedom which this extension allows, enables us to implement short-rate models that exhibit mean reversion . A tree is constructed in such a way that approximates the stochastic differential equation for short rate and automatically returns the observed prices of pure discount bonds and possibly the volatilities of these bonds . Thus we can use these to price many interest rate derivatives .