在隨機波動率下之雙變數樹評價模型

Abstract

隨機波動模型含標的資產價格模型與波動率模型，因此屬雙變數模型。文獻上有許多隨機波動模型，其中 Hull 和 White (1987)的模型允許標的資產價格與波動率之間存在相關性(簡稱為 HW 模型)，Hilliard 和 Schwartz (1996)提出的隨機波動模型(簡稱為 HS 模型)則推廣了 HW 模型。Hilliard 和 Schwartz 先將標的資產價格及波動率轉換到兩個隨機項為常數的隨機過程，此步驟是為了保證接下來建的樹能夠接合(recombining)而有效率。我們首先證明了此轉換的唯一性，此結果本身有獨立的重要性。接著，Hilliard 和 Schwartz (1996)在新的隨機過程底下，建構出有效率的雙變數二元樹(簡稱為 HS 樹狀模型)以評價選擇權，但本論文證明 HS 樹狀模型一定會有錯誤機率，因此是不正確的。本論文提出一解決之道，先對兩隨機過程作正交化(orthogonalization)，再對新產生的兩隨機過程採用雙變數樹狀模型，避免了負機率，因而能正確地評價選擇權，但此樹狀模型大小呈指數成長。最後本論文證明在 HS 模型下，任何樹狀模型之大小皆至少為次指數(sub-exponential)成長。這個隨機波動模型的次指數複雜度的結果是文獻上第一個，不但限制了 HS 模型在實務上的應用性，也對隱含樹 (implied tree)的負機率問題提供合理的解釋。

Stochastic-volatility models are bivariate because they contain two stochastic processes, one for the underlying asset and the other the variance. Many such models have been proposed. An example is Hull and White’s (1987) stochastic-volatility model, which allows nontrivial correlation between the underlying asset and the variance. Hilliard and Schwartz (1996) extend that model and give an efficient bivariate binomial tree (HS tree for short) after the two underlying processes are transformed into ones with constant diffusion terms. This transformation step is needed to make sure that the tree recombines and is efficient. This thesis proves the uniqueness of such transforms, an important result in its own right. However, this thesis proves that negative transition probabilities are inherent for the HS tree; thus it is erroneous. Then it suggests a new tree to correct it. First, it orthogonalizes the two stochastic processes. Then the mean-tracking method is employed to construct a bivariate binomial-trinomial tree. But the tree size grows exponentially. The option prices calculated by this tree are reasonably accurate. We then prove any tree for the HS model must have a sub-exponential size. This sub-exponential lower bound is the first in the literature for stochastic-volatility models. This complexity result places a severe limit on the practicality of the HS model and provides a tantalizing explanation for the implied tree’s failure to avoid negative transition probabilities.