短期 vol 上升等价于延长多久

day European call

专题: Finance / 金融
难度: L4
来源: QuantQuestion

题目详情

You hold a 100- day European call option on a stock with implied volatility 20 . Suppose that you know right now that tomorrow the implied volatility will increase to 25 , but that after that it will return to 20 for the remainder of its life. What extension to the life of the call would produce the same change in the present value of the call as the above- mentioned single- day increase in volatility (assuming a constant implied volatility at 20)? That is, other things being equal, what change in the term to maturity is equivalent to the quoted one- day change in the implied volatility? Explain carefully.

解析

把总方差视为 $\sigma^2T$ （按天计）。

原本 $\sigma=0.2,T=100$ 。

若第 1 天 $\sigma=0.25$ 、其余 99 天 $\sigma=0.2$ ，则总方差为

0.25^2\cdot 1+0.2^2\cdot 99.

令其等价于全程 0.2、但期限变为 $100+\Delta$ ：

0.2^2(100+\Delta)=0.25^2+0.2^2\cdot 99\Rightarrow \Delta=\frac{0.25^2-0.2^2}{0.2^2}=0.5625.

所以约等价于

\boxed{延长约 0.6 天}.