0
$\begingroup$

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 $\sigma$.

  • 0
    This model is flawed inasmuch as it permits negative interest rates. – 2017-02-22
  • 0
    Rates are already negative for many major currencies. – 2017-02-22

1 Answers 1

1

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

$$\sigma^2=\frac{1}{n-1}\sum_{k=0}^{n-1}{y_k^2}$$