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  • 學位論文

含權股票選擇權的定價-價值二元樹

Dividend-Stock Option Pricing–Binomial Value Tree

指導教授 : 陳文魁

摘要


為了容許股票選擇權含有股利,Black (1976)僅依BS選擇權公式就調整股權以及常態位數,CRR二元樹卻以窮舉模型對應,而Hull (2009)讓股票先除權再處理股價模式,但這些方法皆有武斷之嫌。本研究為了區分股票價格和股票價值,特安排價值二元樹模型將連續股利納入,故價值模型做了幾個安排:設定債券酬率與股利酬率的相對酬率、分割存續期間為n期並選定只為奇數、指定價外跌價次數為(n-1)/2及利用BS常態位數估算二項分配之修正機率和跌價機率;由於CRR模型為右尾累積機率與BS模型顯有不同,故本研究按照跌次安排股價運動;此外,針對CRR的收斂問題,本研究亦特別安排:(1)讓股價中心緊盯選擇權履約價、(2)讓BS模型的常態位數決定股票的跌價機率、(3)讓跌價機率與修正機率決定股價的跌率與漲率。本研究之價值二元樹模型不僅改善CRR模型的缺點,還提供美式選擇權二元樹,並於Excel開發VBA函數,以利運用。 本研究研究期間為西元2013年1月至2013年12月;經過篩選後,取研究期間中成交量最多之前10檔契約,契約多以3檔標的為主,分別是友達、聯電和華新;研究結果發現,無論評價含有股利之友達、聯電和華新選擇權時,價值二元樹模型求出之價格均比CRR模型求出之價格較為快速地收斂於BS定價公式的價格,證明價值二元樹模型確實具有單調遞減之良好收斂性質。本研究之價值二元樹模型不僅納入股利,使得清楚地區分價格和價值的差異,還改善CRR模型之收斂問題以及右尾累積問題,使得二元樹模型評價含權股票或是歐式美式選擇權時,都能為投資人提供相當精準的計算工具。

並列摘要


The research focuses on pricing stock option with dividend. In order to incorporate dividend, Black (1976) directly adjusted stock yield and normal quantiles in BS option pricing model. The enumeration approach by CRR binomial model is somewhat complicated, while Hull (2009) dealt cases of ex-dividend. Unfortunately, these methods seem too arbitrary. This research introduces binomial value tree model that incorporate dividend to separate between stock price and stock value. Therefore, the value model arrange to setup relative return rate including bonds rate and dividend rate, divide duration into n times, appoint down-pricing number of in-the-money equal to (n-1)/2 and only for odds, and use normal quantiles in BS model to estimate down-pricing probability and adjusted probability in binomial distribution. Moreover, the research (1) makes the exercise of stock follow strike price, (2) the down-pricing probability of stock-exercise decided by the normal distribution probability in BS model, and (3) the up-pricing and down-pricing probability decided by the down-pricing probability and adjusted probability to avoid the low-efficiency of CRR convergence. Because the right-tail form cumulative probability in CRR model is apparently as different as in BS model, the paper sets up exercise of price based on number of down-pricing. Therefore, Binomial Value Tree model not only improves the drawbacks in CRR model but develops American options trees of VBA function in Excel to use. We chose the most top 10 volumes of stock options (AU Optronics Corp., United Microelectronics Corp., and WALSIN LIHWA Corp.) in Taiwan during the whole year of 2013. The research discovers that the convergence of Binomial Value Tree model is more efficient than CRR model. It proves that there is an efficient convergence that has monotone increasing in Binomial Value Tree model. Finally, Binomial Value Tree in this research not only considers dividends but also improves the convergence and right-tail form problems in CRR. Binomial Value Tree model can provide investors with an accurate tool when evaluating stocks or options with dividends.

參考文獻


3. 黃冠閔 (2011),在二項式模型中的高階誤差分析,國立台灣大學數學所未出版碩士論文。
2. 馮小蕙 (2004),台灣股票選擇權理性定價之探討,國立成功大學高階管理碩士在職專班未出版碩士論文。
1. 程言信 (2007),不同波動度模型下臺指選擇權價格和風險值的評估,第八屆管理學域學術研討會論文集。
4. Black, Fischer (1976), “The pricing of commodity contracts”, Journal of Financial Economics 3: 167-179.
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