Help with simple derivation of probability of credit default

Question

I'm going over a chapter in Hull's Options, Futures, and Other Derivatives and am stuck on how the probability of default is derived. Here's the image of the derivation.

I can follow all of it except for one step: how do you derive $V(t) = e^{-\int_0^t \lambda(\tau) \,d\tau}$ from $\frac{dV(t)}{dt} = -\lambda (t)V(t) $ ?

I'm not a quant so I don't really know how to proceed. I can just plug in the formula in my project, but I'd rather understand how/why the derivation works.

Answer 1

First and foremost, assume that the company can not default at time $t=0$, implying that $V(0)=1$.

Now, divide with $V(t)$ on both sides and integrate from 0 to $t$:

$$ \int_0^t \frac{\frac{dV(t)}{dt}}{V(t)} dt = - \int_0^t \lambda(t) dt $$ Calculate the LHS: \begin{align} \ln(V(t)) - \ln(V(0)) &= - \int_0^t \lambda(t) dt\\ &\Updownarrow\\ \ln(V(t)) &= - \int_0^t \lambda(t) dt\\ &\Updownarrow\\ V(t) &= e^{ - \int_0^t \lambda(t) dt}, \end{align} where you can substitute $t$ with $\tau$ in the integrand on the RHS (the hazard rate function) in order to alleviate notational confusion. This will give you the desired result.