Correct me if I am wrong, but the binomial model is used to value any bond with or without embedded options. And:

in the practical world, why do we need to derive an arbitrage-free price for a bond, besides the obvious reason that we can take advantage of arbitrage?

In what circumstances should I use the binomial model to derive a bond’s price and when should I not?

Late here to respond, but I’ve got some experience in FI so maybe I can offer a couple thoughts.

In the real word, I’m not sure in most cases if you will ever need to derive an arb-free price on a bond. In trading debt, or managing debt, there is an assumption (correctly or incorrectly perhaps) that the market removes these arb opportunities fairly quickly if. Maybe for illiquid offers it is easier to do and then introducing currency into the mix would open up your chances of finding an arb opportunity in the fixed income space. My experience is somewhat small but that’s my initial take.

I just made it through the Fixed Income readings and my takeway on using binomial tress would be the following;

The binomial model is not necessary when there is no expected interest rate volatilty i.e., the curve is flat. You can simply discount the future cash flows by the appropriate risk free rate to determine the arb-free present value. This assumes though that there is no embedded credit risk as well but so far in my re-reading of fixed income, that’s the takeaway I have.

When you have interest rate volatility, you use a binomial tree

When you want to value a bond with or without a call and have expected interest rate volatility, you use a binomial tree

When you want to calculate the expected exposure to compare a credit-risky bond with a credit-risk free bond using the CVA, and you are doing it over future periods when you expected interst rate volatilty, you will use a binomial tree.