Modified Ritchken and Trevor tree於GARCH Option 評價之應用

Abstract

一般而言，用來評價選擇權的方式大部分為Black-Scholes Model與數值分析方法，其中數值分析方法又分為多種不同的模型。例如: 蒙地卡羅法、二項式法等等。雖然Black-Scholes Model在早期被各界廣泛採用，但它的缺點是有太多的假設，隨著今日日新月異的多種選擇權的發明，Black-Scholes Model在實證分析時出現了一些不合理的問題；我們可以知道Black-Scholes Model面對這些新奇選擇權的評價時並不適用。 Duan (1995)發表了GARCH 選擇權定價模型，論文中指出根本資產之價格動態過程，在服從GARCH模型的行程下，引入經濟學上均衡概念的主張，經過適當的風險測度轉換之後，可以導出歐式選擇權的價格。但是在此條件狀態下的選擇權訂價理論，其數值分析方法，仍不夠完備，以致於實務上未能完全地擷取而加以運用。其問題的主要癥結在於GARCH模型，其本質上必然會產生路徑相依(path dependence)的問題，導致運算與處理上的困難程度增加。而所謂的路徑相依，是指在選擇權存續期間，其價格會受到標的資產價格本身波動性的影響。反之，路徑獨立(path independence)是指選擇權價格只受到標的資產其到期日時之價格影響。GARCH模型的路徑相依的性質，會使得欲用樹狀圖來刻劃價格的波動過程中，各時點的可能狀態個數會因時間的往前推移，而呈現指數的遞增情形，而使得樹狀圖陣列非常的龐大，使得GARCH選擇權定價模型在實務上的應用並不理想。而Ritchken和Trevor (1999)針對在非連續時間的GARCH模型，對歐式選擇權和美式選擇權的訂價，建構一個所謂的樹狀演算法。且說明此一樹狀演算法可以進一步擴展到標的資產服從一般化GARCH模型之下，建立出有效的運算方法，此一具體運算方法，不僅適用於GARCH模型之下選擇權的訂價，而且，也可以用來處理很多雙變數的擴散模型。RT 演算法的優點在於可以捕捉各個時點的條件變異數，可以解決GARCH模型路徑相依的問題，使得評價能更有效率。 於1999，S. Figlewski與B. Gao提出了適應性網狀模型(Adaptive Mesh Model, AMM), 同時解決了分配誤差(distribution errors)與非線性誤差(non-linearity errors)，並且提升了評價模型運算的效率。由於AMM在評價上表現出不錯的彈性以及效率，後來，有不少研究將AMM應用於權證的評價上。 本篇論文將AMM中的概念應用在RT模型上, 我們稱之為AMM-RT 模型。由於非線性誤差大部分出現在執行價格附近，因此AMM執行價格增加網格節點的密度來提升估計的精確度和減少非線性誤差。我們將這種想法應用於RT模型的到期日前一天，在到期日的前一日與到期日之間，我們仍然使用RT模型的演算法，將這段期間切割的較細。這樣的方式可以達到跟AMM一樣的效用，同時也可以如同之前的RT模型一樣具有捕捉條件變異數的能力。波動性(volatility)對於任何一種金融商品而言，都有相當顯著的關係存在，因此我們選擇RT模型搭配GARCH模型來預測選擇權價格，然而，我們又希望增加其精確度與減少其誤差，故到期日前一天增加切割期數以期能達到我們想要的效果。本論文將嘗試分別以傳統的BS模型(在不同的volatility下)與RT模型以及AMM-RT模型再搭配GARCH (1,1)模型去模擬並比較股票選擇權價格。

Evaluating stock option price with traditional predictive techniques have proven to be difficult. GARCH option pricing model proposed by Duan has been proven to be more suitable for the task. BS model have so many assumptions that it cannot be suitable in some exotic option. GARCH option pricing model solve the problem which may occur while using the BS model. This thesis focuses on the stock option price estimating based on GARCH (1, 1) model, which have been surveyed by earlier researcher as well as the comparison between each model is discussed. Derived from the first GARCH option price model proposed by Duan (1995), the Ritchken-Trevor Model offers more accurate pricing than CRR model and traditional trinomial tree model. AMM proposed by S. Figlewski and B. Gao adds the mesh point density partially to modify the inefficiency and calculating error of the CRR and trinomial lattice model, which addresses the problems of distribution errors and non-linearity errors as well as upgrade the efficiency of the pricing model. We apply the idea of AMM in the date T (i.e. the day before the maturity day). Rather than the fine mesh structure like AMM, we develop another fine mesh by the same approach of RT model. We just increase the number of time step by changing parameter m (Here m is the segmental level of the last trading day; m=2, 3, 5 will be discussed) in the last date T. We call this justified model “Modified RT Model (AMM-RT)” in this thesis. The same as AMM, the AMM-RT model solve the nonlinearity error around the strike price while evaluating exotic price like, barrier option. By this modified RT model, we also solve the nonlinearity error as well as increase the accuracy. In this thesis, we demonstrate a comparison of accuracy between BS model (with different volatility), RT model and AMM-RT model. With their ability to discover patterns in nonlinear and chaotic financial systems, the GARCH option pricing model with AMM-RT algorithm not only offer the ability to predict market directions more accurately than current techniques bur also reduce the complexity of computing of the original RT model. Numerical analysis via above methods are discussed and compared with performance. Finally, future directions for applying the AMM-RT model to the financial markets are also disserted.