隱含波動率之研究

A Study on Black-Scholes Model and Implied Volatility

廖宏盛(Liao Hung-sheng)

指導教授 ： 施能仁

亞洲大學/管理學院/經營管理學系碩士班/碩士(2005年)

摘要

衍生性金融商品的評價，大多採用Black-Scholes 評價模型。有關該模型的討論與實證，國內外已經有非常多的研究，但是對於模型中偏微分方程式之推導，以及如何利用邊界條件求出選擇權買權及賣權之公式則較少學者探究，多數應用此模型之學者或投資人只能套用公式做實證，對模型之意義未必深入瞭解，因而無法深入應用。
本研究重新推導B-S模型之買權與賣權評價公式，並以物理及統計學來解釋模型之意義，有助於初學者及一般投資人了解與應用。有關隱含波動率之計算，本研究以牛頓法求解，並提出簡易估計的公式，提供不會使用電腦者快速估計的方法。另推導有關的避險參數，配合隱含波動率，便可將B-S模型做更深入的應用。
在實證方面，本研究以高隱含波動差值配合低風險係數的套利策略，利用台指選擇權實證結果，發現可以有很高的獲利率。

關鍵字

B-S評價模型 ； 隱含波動率 ； 台指選擇權 ； 避險參數 ； 套利

並列摘要

Black-Scholes Model, a famous options pricing theory, has been widely used to evaluate the price of financial derivatives. Despite of the fact that many scholars argued against the model and attempted to modify it, Black-Scholes Model, now an academic research focus, remained the basis of all improved models.
After reviewing the previous literature, the researcher found that only a small number of studies had elaborated the meaning of the pricing theory. Ever since 1973, the researcher hasn’t found any paper in which the formula for call price and put price of European style options was systematically derived.
The purpose of this thesis is to derive the formula through reviewing Fisher Black and Myron Scholes’ theory and then to describe the model in detail. Second, the researcher attempted to elaborate the meaning of the pricing theory from the aspects of physics and statistics. Third, the researcher adopted a simple formula to quickly calculate the implied volatility. In addition, the researcher introduced the Newton’s method to accurately calculate the implied volatility. Based on implied volatility, we can calculate the value of pricing sensitivities for hedging and try to figure out the optimal investment policy. Finally, the researcher has found out an investment policy for arbitrage by using high volatility spread with low risk.
The empirical results on TAIEX Options Market indicated that there have been many opportunities for arbitrages by using this policy and investors could thus have better chances to enjoy a high profit rate. The study would provide some guidelines to investors about TXO options and to the beginners studying B-S Model.

並列關鍵字

Black-Scholes Model ； Implied Volatility ； Taiwain Index Options ； Hedging ； Arbitrage

參考文獻

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