# Bond Price expression

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.

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

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

• Thank you so much for your detailed answer! It makes much more sense to me now :) Commented Nov 2, 2021 at 18:26