Valuation of Anerican Put Options: A Comparison of Existing Methods

Abstract

美式賣權已經存在很長的時間，由於沒有公式解，目前只能利用數值分析方法(numerical analysis approach)和解析近似法(analytic approximations) 來評價它。這類的評價方法在文獻中相當多，但對這些方法的完整的比較卻相當貧乏。本文整理了27種評價方法和186種在文獻中常被引用的美式賣權契約，這些契約包含了各種不同狀態(有股利、沒有股利、價內、價平、價外、短到期日、長到期日)，後續的研究者可以用這些美式賣權契約來驗證他們的方法。本文實作其中14種方法並應用於上述的186種美式賣權契約上。這14種方法包含了樹狀法、有限差分法、蒙地卡羅法與解析近似法。從這些數值的結果中，本文根據精確度與計算效率整理出各種方法的優缺點與適用的時機。\n 由本文之數值分析，我們得到下列幾點結論：1.Binomial Black and Scholes with Richardson extrapolation of Broadie and Detemple (1996)與Extrapolated Flexible Binomial Model of Tian (1999)這二種方法在這14種方法中，在速度與精確度的考量下是最好的方法；2.在精確度要求在root mean squared relative error大約1%的情形下，解析近似法是最快的方法；3.Least-Squares Simulation method of Longstaff and Schwartz (2001)在評價美式賣權方面並不是一個有效的方法。

American put option has existed for a long time. They cannot be valued by closed-form formula and require the use of numerical analysis methods and analytic approximations. There exists a great deal of methods for pricing American put option in related literatures. But a complete comparison of these methods is lacking. From literatures, we survey 27 methods and 186 commonly cited option contracts, including options on stock with dividend, non-dividend, in-the-money, at-money and out-of-money, short maturity and long maturity. In addition, we implement 14 methods, including lattice approaches, finite difference methods, Monte Carlo simulations and analytic approximations, and apply these methods to value the 186 option contracts above. From the numerical results, we summarize the advantages and disadvantages of each method in terms of speed and accuracy: 1.The binomial Black and Scholes with Richardson extrapolation of Broadie and Detemple (1996) and the extrapolated Flexible Binomial Model of Tian (1999) are both efficient improvements over the binomial method. 2.With root mean squared relative error about 1%, the analytic approximations are faster than the numerical analysis methods. 3.The Least-Squares Simulation method of Longstaff and Schwartz (2001) is not an effective method for pricing American put options.