離散型亞式選擇權評價及避險係數的封閉解

The Closed–Form Approach to the Valuation and Greeks of Discrete Asian Options

范育誠(Yu-Chen Fan)

指導教授 ： 呂育道

國立臺灣大學/管理學院/財務金融學研究所/碩士(2009年)

https://doi.org/10.6342/NTU.2009.01474

摘要

離散型亞式選擇權有固定履約價(fixed-strike)、浮動履約價(floating-strike)兩種型態，此二類離散型亞式選擇權有對稱的性質，因此我們只需探討前者，而後者可透過此關係間接求得。接著利用平移對數常態分配(shifted log-normal distribution)動差擬合(moment matching)觀察期間的平均價格，求出選擇權價格及避險係數(Greeks)的封閉解。在我們的認知中，之前的文獻是用數值方法去找出該近似分配的參數，但我們發現有封閉解，所以執行起來更方便。最後以蒙地卡羅模擬(Monte Carlo simulation)所得價格當作基準，和以平移對數常態分配、對數常態分配(log-normal distribution)、反伽馬分配(reciprocal gamma distribution)做動差擬合所求出的價格做比較。數值結果顯示我們的評價方法非常接近模擬價格。

關鍵字

離散型亞式選擇權 ； 平移對數常態分配 ； 動差擬合 ； 避險係數 ； 封閉解

並列摘要

There are two types of discrete Asian options, fixed-strike and floating-strike. We focus on the fixed-strike type because the floating-strike type can be calculated through the symmetric property. To derive the closed-form solution to the option price and Greeks, we use the shifted log-normal distribution to match the moments of the arithmetic average value. Furthermore, we find a closed-form solution to the parameters which is new in the literature so that pricing and hedging are faster to execute than before. In the end, we compare three moment matching methods (based on shifted log-normal, log-normal and reciprocal gamma distribution) and Monte Carlo simulation as benchmark. Numerical results show our approach gives results very close to those by Monte Carlo simulation.

並列關鍵字

discrete Asian options ； shifted log-normal distribution ； Greeks ； moment matching ； closed-form solution

參考文獻

[1] Borovkova, S., Permana, F. J. and van der Weide, H. (2007) “A Closed Form Approach to the Valuation and Hedging of Basket and Spread Options.” The Journal of Derivatives, Summer, 14(4), 8–24.

[2] Curran, M. (1994) “Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean Price.” Management Science, 40(12):1705–1711.

[3] Henderson, V., and R. Wojakowski. (2002) “On the Equivalence of Floating and Fixed-Strike Asian Options.” Journal of Applied Probability, 39(2), 391–394.

[4] Hull, J. and A. White. (1993) “Efficient Procedures for Valuing European and American Path-Dependent Options.” Journal of Derivatives 1, 21–23.

[5] Kemna, A. G. Z. and A. C. F. Vorst. (1990) “A Pricing Method for Options Based on Average Asset Values.” Journal of Banking and Finance 14, 113–129.