# 金融中的随机方法II|Stochastic Methods in Finance II代写STAT0018代考

To estimate the volatility of a stock price empirically, the stock price is usually observed at fixed intervals of time (e.g., every day, week, or month).
Define:
$n+1$ : Number of observations
$S_{i}$ : Stock price at end of $i$ th $(i=0,1, \ldots, n)$ interval
$\tau$ : Length of time interval in years
and let
$$u_{i}=\ln \left(\frac{S_{i}}{S_{i-1}}\right)$$

The usual estimate, $s$, of the standard deviation of the $u_{i}$ ‘s is given by
or
$$s=\sqrt{\frac{1}{n-1} \sum_{i=1}^{n}\left(u_{i}-\bar{u}\right)^{2}}$$
$$s=\sqrt{\frac{1}{n-1} \sum_{i=1}^{n} u_{i}^{2}-\frac{1}{n(n-1)}\left(\sum_{i=1}^{n} u_{i}\right)^{2}}$$
where $\bar{u}$ is the mean of the $u_{i}$ ‘s.
From equation (12.2), the standard deviation of the $u_{i}$ ‘s is $\sigma \sqrt{\tau}$. The variable $s$ is therefore an estimate of $\sigma \sqrt{\tau}$. It follows that $\sigma$ itself can be estimated as $\hat{\sigma}$, where
$$\hat{\sigma}=\frac{s}{\sqrt{\tau}}$$

## STAT0018 COURSE NOTES ：

The estimate of the standard deviation of the daily return is
$$\sqrt{\frac{0.00326}{19}-\frac{0.09531^{2}}{380}}=0.01216$$
or $1.216 \%$. Assuming that there are 252 trading days per year, $\tau=1 / 252$ and the data give an estimate for the volatility per annum of $0.01216 \sqrt{252}=0.193$, or $19.3 \%$. The standard error of this estimate is
$$\frac{0.193}{\sqrt{2 \times 20}}=0.031$$
or $3.1 \%$ per annum.
This analysis assumes that the stock pays no dividends, but it can be adapted to accommodate dividend-paying stocks. The return, $u_{i}$, during a time interval that includes an ex-dividend day is given by
$$u_{i}=\ln \frac{S_{i}+D}{S_{i-1}}$$
where $D$ is the amount of the dividend. The return in other time intervals is still
$$u_{i}=\ln \frac{S_{i}}{S_{i-1}}$$