認購權證的B-S評價對實際價格的比例與現貨價格對履約價的比例之關係

雖然在學術界已普遍認同B-S模型對於處在價內的選擇權高估價格的現象，而對價外者有低估的傾向，但缺少價內（外）程度高（低）估價格的定量研究，即現貨價格(S(下标 t))對履約價格(X(下标 t))的比值和理論價格(Ĉ(下标 t))對實際價格(C(下标 t))的比值兩者關係的研究。為探討此一主題，本研究以B-S模型配合歷史波動率，在2002到2006年發行之權證中，隨機每年選出十檔權證，共40檔權證進行評價分析，以探討上述兩個比值是否存在－固定關係。本研究首先對權證實際價格(C(下标 t))與B-S模型理論價格(Ĉ(下标 t))作線性迴歸分析：Ĉ(下标 t)/X(下标 t)=aC(下标 t)/X(下标 t)+b，結果發現b值均趨近於0，因此乘係數a可視為Ĉ (下标 t)/C(下标 t)之比值。其次提出a值與C(下标 t)/X(下标 t)之「鐘形」的普遍化關係式，結果顯示此式具有顯著的解釋能力。值得注意的是此式表明當權證處於深度「價外」或深度「價內」時，a值都會變小，即實際價格都會高於理論價格；而正好處於價平附近之權證，實際價格會接近理論價格。

關鍵字

B-S模型；認購權證；迴歸分析；價內；價外

並列摘要

Although the phenomenon that the price of the in-the-money option is overestimated, and the price of the out-of-the-money option is underestimated by the Black-Scholes model is generally accepted in academic circles, there are no quantitative investigations between the magnitude of overestimation (underestimation) of the price and the in-the-money (out-of-the-money) magnitude; that is the study of the relation between the ratio of B-S theoretic price to the real price (C(superscript t)/C(superscript t)) and the ratio of current price to the exercise price (C(superscript t)/X(superscript t)). In order to explore this subject, using B-S model and historical standard deviation, and every year randomly selecting 10 warrants released in 2002 to 2005; therefore, there are 40 warrants in our appraisal analysis, we explored whether there is the fixed relation between the two above mentioned ratios or not. This research first built the relations of B-S theoretic price (Ĉ(superscript t)) to the real price (C(superscript t)) for every warrant by linear regression analysis: Ĉ(superscript t)/X(superscript t)=aC(superscript t)/X(superscript t)+b. The results discovered that the 'b' value is close to 0; therefore, the 'a' bavalue may be regarded as the ratio of B-S theoretic price to the real price (a ≈ Ĉ(superscript t)/C(superscript t)). Then, we proposed the ”bell-shape” general regression relation of C(superscript t)/X(superscript t) to 'a' value, and estimated the regression coefficient. The results showed that this relation has remarkable explanation ability. It is noteworthy that this relation indicates that when the warrant is in deep-in-the-money or deep-out-of-the-money, the 'a' value will become small; that is, the real price is greater than the price estimated by the by B-S model, and when the warrant is in near-the-money, the real price will be close to the B-S model price.