# Black-Scholes期權定價模型的缺陷

## 最佳答案

• $r$，利率
• $\ sigma$，volatility

- -編輯 - - 這是巴菲特的一個重要論點：

The final candidate, other than the arbitrage argument on which the model is based, is the volatility. If Mr. Buffett is criticizing the use of the lognormal diffusion assumption when pricing long-term options, he is not alone. Recall that the lognormal assumption implies that volatility increases linearly with respect to the horizon over which it is measured as shown in equation (1) [lognormal distribution of B-S]. There is empirical evidence which indicates that the linearity assumption fails to hold at long horizons. For example, Siegel (2008) reports that the variance of real returns on the S&P 500 historically have failed to rise linearly with the horizon. If the long-run volatility is lower, the value of long-term put options will be less. For instance, a volatility of 15%, instead of 18%, reduces the estimated value of Mr. Buffett’s hypothetical put position to \$1.5 million. It also reduces the probability that the index will be lower at expiration than at initiation.

（強調我的）