Definition of interest rates in binomial tree model

Question

I'm studying financial mathematics from Shreve's text. I have two problems.

1) "for a binomial tree with three steps, where $S_0=20$, $u=1.05$, $d=.95$ and continuously compounded risk-free interest rate is 5% (note, we need to transfer continuously compounded interest rate first to be equivalent to interest rate compounded annually)..."

2) For a three step binomial tree in which $S_0=10$, $u=1.15$, $d=.9$ and the annual-effective risk free rate is seven percent...

I'm not exactly what these interest rates mean mathematically. I'm familiar with the very basic setup for a binomial tree model, where we assume there is no arbitrage and $d<1+r<u$, and this $r$ is the interest rate one accumulates on investment in the money market for each step in time. In quest 1, what does it even mean for interest to be compounded continuously if the binomial model is discrete time? For the second question, I don't know what the annual-effective interest rate means or how it relates to this $r$ (I don't believe it's defined in the text). Any help on clarifying these definitions would be greatly appreciated.

Answer 1

Even though the steps are discrete, we can still express the risk free rate $ r $ as a continuously compounded rate - all this implies is that, if each time step is $ \tau $ units of time long, the discount rate you need to apply to the payoffs at the next node is $ e^{-r \tau} $

We can also describe the risk free rate as its annual effective rate, typically denoted $ R $, which is just the simple annual rate that implies the same force of interest as a compounded rate. If $ r $ is the continuously compounded rate, then we have $ 1 + R = e^{r} $, whereas for a quarterly compounded rate $ r_{q} $, $ R = (1 + \frac{r_{q}}{4})^{4} $. If you have rates quoted in different bases (e.g. monthly, quarterly, or semiannually), moving them to their respective annual effective rates lets you more easily compare the actual force of interest for each rate.