Heston模型樹評價選擇權

Abstract

自從「微笑波動率」的現象被察覺之後，研究者們紛紛提出各種不同的模型來解釋它。在隨機波動率的模型裡，由於波動率不是常數，用來逼近連續時間股價的樹可能無法重合，導致指數的運算時間，使得樹的方法評價選擇權是不切實際的。此論文在Heston模型的假設下使用網格間隔不均勻的三維網格評價歐式選擇權。變異數樹允許節點分支多格跳躍為了避免負機率法生，但是分支跳躍過大仍可能導致樹中與價格相關的維度出現負機率。本論文使用一個方法來改善這個情況，藉由讓樹的底部對齊某個網格點，該網格點會使得變異數樹的節點分支跳躍幅度變小。

Since the phenomenon of volatility smile has been discovered, researchers proposed many kinds of models to explain it. In the stochastic-volatility model, the tree used to approximate the continuous-time stochastic process of the underlying asset may not recombine duo to the non-constant volatility. An exponential tree leads to exponential running time which is impractical. This thesis prices European options in the Heston stochastic-volatility model on a 3-dimensional grid with a non-uniform grid spacing. The tree nodes of the variance process are allowed to jump a multiple number of nodes in order to avoid negative probabilities. But such large jumps may cause negative probabilities in the price-related dimension of the tree. This thesis uses a way to ameliorate this situation by aligning the bottom of the tree at a certain level to make the size of upward jump smaller.