For a american call option on an underlying $S$ with payoff process $\left(\varphi(S_t)\right)_t$ the price and expiry $T$ exercisable on $[0,T]$ the time $t$ price $\pi_t$ conditional to no having exercised the option priori to $t$ is $$\pi_t = N_t \sup_{\tau\in\mathscr{T}_t} \mathbf{E}^{\mathbf{Q}^N}\left[\left. \frac{\varphi(S_{\tau})}{N_{\tau}} \right| \mathscr{F}_t \right]$$ where $N$ is some numéraire and $\mathbf{Q}^N$ some martingale measure associated to $N$ (supposed unique) and $\mathscr{T}_t$ is set of stopping times with values in $[t,T]$. (Note that in incomplete markets the convention is to take the same formula for $\pi_t$ for some equivalent (local) martingale probability measure.)
(Recall that in this concrete case a stopping time for the filtration $\left(\mathscr{F}_s\right)_s$ is a random variable $\theta$ with values in $[0,T]$ such that for each $s\in [0,T]$ the measurable set $\left\{\theta \leq s\right\}$ is in $\mathscr{F}_s$.)
Hence the price of the american option is $$\pi_0 = N_0 \sup_{\tau\in\mathscr{T}_0} \mathbf{E}^{\mathbf{Q}^N}\left[ \frac{\varphi(S_{\tau})}{N_{\tau}}\right].$$
Now if the option be bermudan and exercisable on dates $T_1,\ldots,T_d$ with $T_d = T$ we have $$\pi_0 = N_0 \sup_{\tau\in\mathscr{T}} \mathbf{E}^{\mathbf{Q}^N}\left[ \frac{\varphi(S_{\tau})}{N_{\tau}}\right]$$ where $\mathscr{T}$ is the set of discrete stopping times with values in $\left\{T_1,\ldots,T_d\right\}$.
Now let's move to your case.
First you have to realize that the usual exposé above concerns options where the holder of the option has the exercise rights. These american or bermudan options are called putable options. (No connexion to put options though, it is purely semantic and the semantic will be clear soon). In your case it is the issuer of the bond who owns the exercise rights. Such early exercise claims are deemed callable because the issuer can recall the option. This term comes from the bonds world. And the opposite case (where the option's owner owns exercise rights) has in turn been called putable simply because of the call/put "semantic" symmetry in the mind of finance people.)
Does it change something to the theory above ? Yes, the price is not a $\sup$ anymore but an $\inf$. (The $\sup$ in $\pi_0$'s expression comes from the fact that if you own exercise rights you try with $\tau$ to maximise the value of your claim. You can guess why in the callable case it is an $\inf$.)
Does it really change something ? No, as we have the general formula $$\inf_{\alpha} f(\alpha) = - \sup_{\alpha} (-f(\alpha)),$$ you can reverse the payoff sign, price as in the putable case and reverse the sign of the result.
This being said it is now just a matter of finding what the right underlying and payoff process are for the bermudan option the issuer is short of. (He is short as he makes the payments would he exercise.) The underlying is the issuer's instantaneous rate of borrowing money $R_t$ and the payoff process is simply the PV at $T_i$ of all remaining cashflows ($T_i$ inclued) up to $T_d$ (notional inclued) calculated, that is discounted, with $R_t$. (Note that you could also say that you don't have one underlying but many, namely the whole issuer's instantaneous forward curve but that would lead to another class of models than the one of short rate models where you model only the instaneous rate.)
After having stressed all of that, you need to model $(R_t)_t$ as well as you needed to model $(S_t)_t$ in the equity introductory case. Then you have to chose a numerical method, and same methods as in the equity case should apply -- probably an LSM. Note that in this case as in the equity case the first "trick" of the LSM works, meaning that you can do a brownian bridge to generate the MC paths "backwards".
References :
- for the early exercise in the putable case you can have a look at the first chapter section 1.10 of Volume 1 of Piterbarg's and Andersen's Interest Rate Modeling or to chapter 6 of Henry-Labordère's far more complete and general (regarding early exercise theory) Nonlinear Option Pricing.
- for the callable setup, you can have a look at the chapter 18 of volume 3 of Piterbarg's and Andersen's Interest Rate Modeling. It deals with callable LIBOR exotics (CLE's) while you are only concerned with fixed rate but the rationale is the exact same ; the 18.2.2 presents the exact recursion formula (in the CLE case) you'll have to use to price your callable bond.
