# Pricing fixed rate redeemable bond

A redeemable fixed rate bond has a yearly payment schedule $$T_1,\ldots,T_m,\ldots,T_n$$ : at each $$T_i$$ is paid the coupon $$c$$ (I assume the year fraction is approximativately $$1$$) (and a notional $$X$$ if $$i=N$$) and the issuer can recall the bond at price $$X$$ (the notional) at dates $$T_m,\ldots, T_n$$ and has $$5$$ business days prior to each $$T_i$$ to decide to do so, in which case he pays $$X$$ (at $$T_i$$) to the bond holder. (This is the strict translation of the termsheet I have.)

Am I wrong stating this : being long of such a bond is equivalent to having bought the following product from the issuer : the product P that if exercised at date $$T_i$$ (for $$m\leq i \leq n$$) by the issuer will have paid to me the cashflows $$cN$$ at $$T_1,\ldots,T_i$$ and $$X$$ at $$T_i$$ ?

Noting $$B_i$$ the bond paying the cashflows $$cX$$ at $$T_1,\ldots,T_i$$ and $$N$$ at $$T_i$$ is this product P the bermudan option of entering into the bond $$B_i$$ as seller at $$T_i$$ if exercised ?

I cannot wrap my head around what the exercise policy for the issuer is : does he exercise at $$T_i$$ if from $$T_i$$ minus $$5$$ business days the yield to maturity of bond $$B_i$$ is smaller that the RFR at $$T_i$$ ? Or if the RFR is bigger that the coupon rate minus the issuer's spread ? (Nothing is mentioned about that in the term sheet.)

Is it possible to write the price of that bond as the bond $$B_{n-m}$$ (the underlying vanilla bond) plus or minus some option and try to have an approximate price of that option ?

• Do you mean what's usually called a "callable" bond, or something else? Is the strike (call price) fixed? Jul 27 at 21:16
• Well that I don't know : it's either a bermudan callable or bermudan putable bond for sure but I don't know the terminology plus in the termsheet (whose strict translation is given in the 1st paragraph of my question) doesn't mention any option nor string "call" or "put" ... So yeah the "redemption" price (the price at which the issuer can terminate the bond) is fixed equal to the notional : if the issuer terminates the bond on $T_i$ then I get coupon on $T_i$ (in any case) but also the notional. Jul 28 at 6:38
• If by strike price your the rate $K$ such that the position of the RFR relative to $K$ will be used by the issuer to decide if he terminates the bond or not, I don't have it but I can guess it's the coupon rate - the issuer spread. Jul 28 at 6:42
• coupon rate minus issuer's credit spread Jul 28 at 7:00
• I re-read the translation very carefully and am not fully clear, sorry. But is sounds like a Bermudan callable bond. The only unusual feature I see is only 5 days' notice. Usually callables require at least 30 days notice. Maybe other important details got fuzzy in translation. So, my general thoughts on pricing a callable whose price is not observable: yes, most people would price a non-callable bond B, minus the value of the call on B, perhaps using 2D tree with some volatilities of intetest rate and of credit spread . The advantage is that you easily get intetest rate and credit (cont) Jul 28 at 11:59

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.
• "These american or bermudan options are called putable options. (No connexion to put options though," puttable bonds are common in Russia and India. In the U.S., we have municipal Variable-Rate "Demand" notes/bonds, which means puttable. I'm not aware of such bonds being popular in any other markets. The bond holder has the right to sell the instrument back to the bond issuer at some predetermined times and price, usually just the remaining principal, sometimes some other strike. I'm curious why you don't think that it's an embedded "put option". Jul 30 at 16:23
• Fair point indeed. Looking at it from now I think I mixed myself up wanting to say something really simple : you can have putable options that are not puts and well as you can have callable options that are not calls. For instance american call options on SOFR futures are not put options but are putable options in the semantic used in my answer. Jul 30 at 17:42
• I agree, when the underlying already has observable prices, then pricing just the option is easier. This recent paper scirp.org/journal/paperinformation.aspx?paperid=94494 on pricing vrd debt is more complicated than the quesrion because their coupon is floating/stochastic, rather than fixed. Jul 30 at 18:21