Consider the simplest interest rate model $$r(t) = r(0) + h(t) +\sigma B(t),$$ where $r$ is an overnight interest rate, $r(0)$ is its initial level, $h(t)$ is a time-dependent drift and $\sigma$ is a constant volatility parameter, $B(t)$ is brownian motion. How to Estimate parameter $\sigma$ from historical observations overnight interest rate? Assume that we know $r(0), r(1),...,r(t)$, and $h(t)$ is also a known function.
What I got is $$r(t) - r(0) - h(t) = \sigma B(t)$$, and then take quadratic variation of both side to get $\sigma$.
You know one sample of $$\hat{r}(0),\hat{r}(1),...,\hat{r}(n)$$ or more suitable for this case , the following sample
$$y_0=\hat{r}(0)-h(0),y_1=\hat{r}(1)-h(1),...,y_n=\hat{r}(n)-h(n).$$
Indeed, if one define the random variable $$Y_k=[r(k+1)-h(k+1)]-[r(k)-h(k)]=\sigma(B_{k+1}-B_{k})$$ with $0\leq k < n$
You can notice that the $Y_k$ are iid and entirely defined by $\sigma$. Furthermore, We have $$E(Y_k)=0$$ and $$var(Y_k)=\sigma^2$$
Using your sample of $y_k$ and you get
$$\sigma^2=\frac{1}{n-1}\sum_{k=0}^{n-1}{y_k^2}$$