GARCH-二元跳躍選擇權的樹狀評價法

Abstract

本篇論文提出文獻上第一個GARCH-跳躍選擇權的樹狀演算法。根據Duan, Ritchken, and Sun (2005)的GARCH-lognormal跳躍模型，修改其中跳躍的分配，我們提出一個新的模型：GARCH-二元跳躍選擇權評價模型。在這個新模型裡，除了變異數異質性，二元跳躍亦發生在pricing kernel，資產報酬以及資產報酬變異數。考慮GARCH-lognormal跳躍模型對於建造樹狀演算法的高度不可追朔性，我們因而採用簡單的二元跳躍。若刪去跳躍的部份，我們的模型將簡化為 Duan (1995)的NGARCH模型。若刪去GARCH效果，在分割越細下，我們的模型將收斂到Amin (1993) 以及 Trippi, Brill, and Harriff (1992)的模型。我們運用類似Amin (1993)的方法，將二元跳躍建構在一個GARCH選擇權評價樹上。而此一GARCH選擇權評價樹，我們的選擇是Lyuu and Wu (2005)的mean-tracking (MT)樹。從而歐式與美式的選擇權皆可以很有效率的利用我們的演算法評價。在這篇論文裡，我們提供此一樹狀演算法不會發生到期日縮短問題的充分條件。此外，我們亦推導出此一樹狀演算法只會隨到期日呈二次方成長的充分條件。這個發現讓我們的演算法效率與Black-Scholes模型下的二元樹狀演算法一致。高效率則代表著此一演算的實用性。另外，我們的演算法將可以簡單推廣到其他的GARCH模型。利用數值分析，我們確認所有的理論推導以及此一演算法的評價的正確性。除此之外，我們也比較了GARCH-lognormal跳躍模型與GARCH-二元跳躍模型所產生的選擇權價格之異同，並由此討論其中的涵義。

This thesis proposes the first tree-based algorithm in the literature for the pricing of options under the GARCH-jump model. Following Duan, Ritchken, and Sun (2005), we propose a new model, the GARCH-binary jump option pricing model, with modifications on their jump distribution. In this new model, besides conditional heteroskedasticity, there are also binary jumps in the pricing kernel and correlated binary jumps in asset returns and volatilities. The high complexity and intractability in constructing a tree-based algorithm for GARCH-lognormal jump processes motivate the simple binary jump distribution assumption. When jumps are suppressed, our model nests Duan’s (1995) NGARCH option pricing model, where conditional returns are posited to be normal. When the GARCH effect is suppressed, the diffusion limit of our model converges to the binary jump-diffusion models of Amin (1993) and Trippi, Brill, and Harriff (1992). Following the binary jump-diffusion tree of Amin (1993), we superimpose binary jumps on a GARCH option pricing tree. Our choice of the tree is the mean-tracking (MT) tree proposed by Lyuu and Wu (2005). Both European and American options can then be priced by our algorithm. We give sufficient conditions for our algorithm to avoid the short-maturity problem inherit in the original GARCH option pricing tree of Ritchken and Trevor (1999) and Cakici and Topyan (2000). Furthermore, the tree size growth is guaranteed to be quadratic if the number of partitions of one day, n, does not exceed a threshold predetermined by the model parameters. This surprising finding places the tree-based GARCH-jump option pricing algorithm in the same complexity class as binomial trees under the Black-Scholes model. The level of efficiency makes the proposed model and algorithm practical. Furthermore, our algorithm can be naturally amended to alternative GARCH specifications. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of our algorithm. Numerical comparisons are also conducted between the GARCH-binary and lognormal jump models. Several implications are drawn from these results.