Heston模型樹評價選擇權

Tree-Based Methods for Option Pricing in the Heston Model

林宣延(Hsuan-Yen Lin)

指導教授 ： 呂育道

國立臺灣大學/電機資訊學院/資訊工程學研究所/碩士(2020年)

https://doi.org/10.6342/NTU202000085

摘要

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。

關鍵字

隨機波動率模型 ； Heston模型 ； 三元樹

並列摘要

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.

並列關鍵字

Stochastic volatility ； Heston model ； Trinomial tree

參考文獻

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[2] Ming-Hsin Chou. “An Efficient Tree for the Heston Stochastic-Volatility Model.” Master’s Thesis, Department of Finance, National Taiwan University, Taipei, Taiwan, (January 2016)

[3] John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross. “A Theory of the Term Structure of Interest Rates.” Econometrica , 53, No. 2 (Mar 1985), 385-407.

[4] John C. Cox, Stephen A.Ross Mark Rubinstein. “Option pricing: A simplified approach.” Journal of Financial Economics, 7, No. 3 (September 1979), 229-263.

[5] Tian-Shyr Dai and Yuh-Dauh Lyuu. “The Bino-Trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing.” Journal of Derivatives, 17, No. 4 (Summer 2010), 7–24.