常數彈性變異數過程與其應用

Abstract

本研究的目的在於測試Cox (1975) 與 Cox and Ross (1976)發展之常數彈性變異 數過程模型(Constant–Elasticity-of-Variance (CEV) process)實證上之表現。近期關於CEV 過程的研究包括了 path-dependent 選擇權與信用風險金融商品的定價。雖然CEV 模型 並未如隨機波動(stochastic volatiltiy)模型與數量繁多的選擇定價模型一般化，然而其簡 單的特性與較低的計算成本，仍然值得研究探討，特別於美式選擇權與新奇選擇權的定 價應用上。 CEV 模型簡約的設定解釋了眾所熟知的leverage effect，亦即報酬樣本變異數與 股價間的負相關。在應用到美式選擇權定價時，需要高維度的lattice models，其計算成 本在實際應用上幾乎為不被允許的。而CEV 模型卻僅需要單一維度的lattice (Nelson and Ramaswamy (1990) and Boyle and Tian (1999))。此外，在信用風險上，CEV 過程也有優 於幾何步朗運動的特性，直觀上來說，CEV 過程在低股價時波動性增加，因此可接觸到 零，而幾何步朗運動中，股價將永遠維持在零以上。 本研究將首先回顧文獻上關於CEV 選擇權模型的實證分析結果，並整理近期關 於CEV 過程在 path-dependent 選擇權與信用風險金融商品的定價上的研究發展。接下 來我們進行CEV 模型在美式選擇權訂價上的實證分析，並與Black-Scholes 模型與隨機 波動模型做比較。最後，我們將實證檢驗由Carr and Linetsky (2006)所提出在CEV 併跳 躍至違約過程下的整合架構，其於選擇權訂價上的表現與隱含違約機率上的分析。 表

This study is intended to examine the empirical performance of the recent applications of the Constant–Elasticity-of-Variance (CEV) process first proposed by Cox (1975, 19961) and Cox and Ross (1976). Recently, several applications of the CEV process have been developed in path-dependent option and credit derivative pricing. Although the CEV model is not as general and flexible as the stochastic volatility models and other extensions of the Black-Scholes model, its simplicity may still be worth exploring since those generalized models are expensive to implement, especially while one applies to American option or exotic option pricing. The CEV process parsimoniously accommodates the well-known leverage effect, an inverse relationship between the stock price and its variance of return. When applied to American option pricing, high-dimensional lattice models are prohibitively expensive. One distinguish feature of the CEV model as opposed to other stochastic volatility models is that it requires only a single dimensional lattice (Nelson and Ramaswamy (1990) and Boyle and Tian (1999)). Furthermore, in credit risk modeling, the CEV process also has advantage over geometric Brownian motion in that, intuitively, the standard CEV process can hit zero due to the increased volatility of the former process at low stock prices while geometric Brownian motion cannot. In this essay, we first review the CEV option pricing model and the previous empirical studies. Next, the recent development and application in path-dependent and credit derivatives under the CEV process are also presented. Third, the comparison of the American option pricing performance of the CEV, the Black-Scholes, and the stochastic volatility models, will be investigated. Finally, a unified framework under CEV diffusion and jump to default process proposed by Carr and Linetsky (2006) will be examined empirically in terms of option pricing performance and implied default probabilities.