Bond Price expression

Question

I've researching some mathematical finance and I've stumbled upon something I can't seems to find sources on. I'm probably overlooking something, but I hope someone can enlighten me and give me some sources which I can take a look at. In Björk 2020 on page 95 it is stated that the a risk free asset with price process $B$ has the following expression: $$B_{t} = B_{0} \exp\left(\int_{0}^{t} r_{s} ds\right).$$ Can someone explain me why it is so or link to some source where they explain why it is given like that. Thanks in advanced.

Answer 1

If $ \delta > 0 $ is very small, then the interest incurred during the small subperiod $[t, t + \delta]$ can be approximated using the simple interest formula. More specifically, the interest incurred in $[t, t + \delta]$ is:

$$ B(t + \delta) - B(t) = B(t) r(t) \delta + o(\delta^2) $$

where $ B(t) r(t) \delta $ is the interest that would be incurred if we were to use simple interest.

Dividing by $\delta$, we have

$$ \frac{B(t + \delta) - B(t)}{\delta} = B(t)r(t) + o(\delta) $$

Taking limit $ \delta \to 0 $, we have

$$ B'(t) = \lim_{\delta \to 0} \frac{B(t + \delta) - B(t)}{\delta} = B(t)r(t) $$

The solution of the differential equation $ B'(t) = B(t)r(t) $ with the initial condition $ B(0) = B_0 $ is

$$ B(t) = B_0 \exp\left( \int_0^t r(s) ds \right) $$