以二元樹模型評價歐式障礙選擇權之收斂

Abstract

障礙選擇權的類型主要可分為向上生效型、向上終止型、向下生效型和向下終止型四種，又還可依其為買權還是賣權、美式還是歐式來區分，因此如何對這些各式各樣類型的障礙選擇權作評價並且探討其收斂性是本文的重點。雖在文獻上對於障礙選擇權的評價方法有很多，但本文主要是建構在風險中立下，將反射原理的概念運用到二元樹模型後再對歐式障礙選擇權作評價。 首先，本文利用反射原理分別去探討向上和向下生效型的路徑數，接著藉由選擇權的價格會等於未來期望值的折現得到歐式障礙選擇權的價格，再利用 Uspensky 的方法推導出其收斂至Black-Scholes 的價格公式。 分析結果顯示，歐式障礙選擇權的收斂速度會依履約價格位置的不同而有所不同，可能是$frac{1}{sqrt{n}}$或者是$frac{1}{n}$。但對於一些特定$n$的情況下，履約價格的位置並不會影響到其收斂速度，所得到的結果皆為$frac{1}{n}$。而在本文的最後也會帶入一些數值做運算來驗證我們的結論。

Barrier options have four types: up-and-in, up-and-out, down-and-in and down-and-out. Barrier options also distinguish between call and put, American and European. In this paper, how to price and discuss the convergence of European barrier options is our goal. First, we use the reflection principle to discuss the number of paths for the down-and-in and up-and-in options. The price is its discounted expected payoff under the risk-neutral probability measures. Thus, we get the formula for the binomial price of European barrier options. Then we use Uspensky's method to discuss the convergence of the binomial price to the Black-Scholes price. The convergence order depends on the strike price. We get the errors are of order $frac{1}{sqrt{n}}$ or $frac{1}{n}$ . But, for some n, the errors are all of order $frac{1}{n}$. Finally, we show the numerical results to verify our conclusions.