布雷克-休斯與莫頓模型應用與校準之研究

Two Essays on Black-Scholes and Merton Models: Application and Calibration

蔡輝煌(Hui-Huang Tsai)

指導教授 ： 李賢源

共同指導教授 ： 陳松男(Son-Nan Chen)

國立臺灣大學/管理學院/財務金融學研究所/博士(2009年)

https://doi.org/10.6342/NTU.2009.03085

摘要

本篇論文包含兩大主題，分別從Black-Scholes和Merton模型（以下以BSM統稱之）的應用與校準來探討其成功之處。BSM堪稱為模型中的模型是因為它的簡潔與健全。簡潔，是因為它使用真實工程，從股票與債券複製出選擇權，因而使我們得知在它所設定的理想環境下，這個選擇權的成本為何。健全，則是因為它允許我們可輕鬆地針對真實世界與它所設定的環境之間的不吻合處做調整。第一篇文章以可延伸到期日的債權做例子來展現它的簡潔。此外，一個常見的迷團—波動率偏斜—之存在則顯示在實務上有修改BSM的需要。第二篇文章從理性投機的角度來調整BSM，進而讓我們見證BSM的健全。

文章一：展延該債權，當它不是深度價外時
在本文內，我們修改Longstaff(1990)的可展延負債模型，以舒緩原本模型內會產生的道德問題。在Longstaff的模型中，即使當借款者還有能力償還時，可展延已違約之負債的到期日等於是給他一個去違約的誘因。在本文裡，我們認為假使違約的情形嚴重，該負債就不應該展延。另外，我們展示了在PDE方法之外，可展延負債的評價可以用複合選擇權來做。我們也在本文裡推導可展延負債的公平利率。

文章二：從理性投機的角度來解釋隱含波動率偏斜：來自台灣股市指數選擇權市場的證據
本文基於De Long等(1990)的不穩定理性投機（Destabilizing Rational Speculation）論點，探索存在於台灣股市指數選擇權(TXO, 簡稱：台指選擇權)市場裡的隱含波動率偏斜（Implied Volatility Skew）。不同於Black-Scholes模型所計算出的隱含波動率，由對市場走勢有看法下之覺察的波動率（Perceived Volatility with a View），也就是本文裡的事實上的造市者(de facto Market Maker)認可的波動率，可以顯示出不同的選擇權（不管它是哪種產品或者它的履約價格）的價值，對造市者而言都是一樣的。更有甚者，他們在交易選擇權前出價，換句話說是決定其覺察波動率，是參考現貨市場上的波動率不對稱（Volatility Asymmetry）型態。這也是為何我們建議台指選擇權市場裡偏斜的隱含波動率曲線之形成，可能是因為理性投機者在正向反餽交易者存在時的行為所致。

關鍵字

選擇權評價 ； 展延債務 ； 到期日之展延 ； 隱含波動率偏斜 ； 理性投機

並列摘要

This dissertation includes two essays to investigate the success of Black-Scholes model (commonly abreast with Merton model, hereafter BSM) from application and calibration. BSM deserves to be named as a model for models because of its clearness and robustness. It is clear because it uses true engineering to replicate an option out of stocks and bonds and tells you the cost of an option under the ideal circumstances defined by it; it is robust because it allows us to adjust easily for those mismatches between the real world and the ideal circumstances. Essay 1 gives an example on the debt with conditional extendable maturity to show its clearness. Besides, the existence of a well-known puzzle: volatility skew shows the need to change for BSM in practice. Essay 2 adjusts BSM from the rational speculation viewpoint and we can witness its robustness in this essay.

Essay 1: Extend the debt as it is not deeply out-of-the-money
In this essay, we modify the extendible debts model proposed in Longstaff (1990) to help relieve the moral problem induced in the original model. In Longstaff’s model, extending the maturity of the defaulted debts gives the borrower an incentive to default even if the borrower is insolvent. In this essay, we argue that the debt should not be extended if it is defaulted severely. We have shown that the extendible debt valuation can be obtained by the compound option pricing besides the PDE approach. We also have derived the fair interest rate of the extendible debts in this essay.

Essay 2: Explaining the implied volatility skew from the rational speculation viewpoint: evidence from Taiwan stock index option market
This essay investigates the implied volatility skew existing in Taiwan stock index option (TXO) market based upon the argument of Destabilizing Rational Speculation of De Long, et al. (1990). Different to the implied volatility from Black – Scholes model, the perceived volatility with a view from de facto market makers especially in this essay shows that the values of options are indifferent to them regardless of product types and strike prices. Furthermore, they quote an option, i.e. deciding the perceived volatility in other word, by referring the volatility asymmetry pattern in spot market. That is why we suggest that the formation of skewed implied volatility curve in TXO market is due to the behavior of rational speculators in the prevalence of positive feedback traders.

並列關鍵字

Option Pricing ； Extendible Debt ； Extension of Maturity ； Implied Volatility

參考文獻

Barber, B.M., Y. Lee, Y. Liu and T. Odean, 2008, Just How Much Do Individual Investors Lose by Trading?, Review of Financial Studies 22, 609-632.

Bates, D. S., 2003, Empirical option pricing: a retrospection, Journal of Economics 116, 387-404.

Black, F., 1976, The pricing of commodity contracts, Financial Economics 3, 167-179

Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy 81:3, 637-654.

Bollen, N. P. and R. E. Whaley, 2004, Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?, Journal of Finance LIX: 2, 711-753.