I am searching for references on pricing callable bonds.

I've not find any rigorous mathematical approach on the web. All I found was some soft approaches in a discrete framework.

Edit: First of all I would like to stress that I am true beginner on bond market modeling.

So an answer containing just a high level language/terms (in a comparative the sense with programming languages not high register of English) helps (since gives a direction to search, teaches me more terms and gives me a better global visual of bond market) but still doesn't answer my first question directly and multiply my questions instead.

I will be more precise about what I am asking.

What I want to understand is:

  1. how to right mathematically the payoff of a callable
  2. how to derive from this payoff formula the relation that states that the price of a callable bond is equal to the price of straight bond plus a call option (an implicit question is to precise the underlying of this call)

$$P_{\text{Callable Bond}}=P_{\text{Option-Free Bond}}-P_{\text{Option-Free Bond}}$$

Let $D(t,T)$ be the discount factor at the time $t$ of a $T$-claim, $ZC(t,T)$ the value of a zero-coupon at time $t$ and $B(t,T)$ the price of a bearing-coupon bond of maturity $T$ paying coupon $c$ at dates $(T_1,T_2, \cdots, T_N) $.

Then the payoff of a callable bond $\Pi_{CB}$ of maturity $T$ callable at $T^*$ (with $T_j<T^*\leq T_{j+1}$) at a fixed strike $K$ can be written as follows

$$\Pi_{CB}(0,T)= \sum_{i=1}^j c D(0,T_i) - D(0,T^*)\left(B(T^*,T)-k\right)_+ +\left[\sum_{k=j+1}^N c D(0,T_k) + D(0,T)\right]\mathbf 1_{\{B(T^*,T)> K\}}$$

Am I write ?

Could someone help with that please?


3 Answers 3


you can view a bond as a floating rate note plus a swap from floating to fixed. Floating rate notes are always at par after coupon payments (ignoring credit risk...) so the pricing of a bond is the same as that of a swap.

So the pricing of a callable bond is the same as that of a cancellable swap.

A cancellable swap can be viewed as a swap minus the bermudan right to enter into the opposite swap.

So it all comes down to pricing Bermudan swaptions. There is an infinity of papers and books on this.

I have written too many papers and books on this. The biggest standard reference is Andersen-Piterbarg.

  • $\begingroup$ Hi Mark, thanks for your answer. You are talking about the following paper of yours right ? "Vega Control" papers.ssrn.com/sol3/papers.cfm?abstract_id=1398523 $\endgroup$
    – Paul
    Oct 31, 2014 at 11:45
  • $\begingroup$ Also could you write mathematically what you said. My question was indirectly how to right mathematically the payoff of a callable bond in a continuous framework. Then I will be interested in credit risk indeed. But first things first... All equivalences you mentioned interest me even more. So if could please right it down it would be very kind of you. Many thanks $\endgroup$
    – Paul
    Oct 31, 2014 at 11:55
  • $\begingroup$ Mark is too shy to plug his own books, but they are well worth picking up! $\endgroup$
    – Brian B
    Oct 31, 2014 at 12:50
  • $\begingroup$ vega control is one of the less relevant papers. See eg Beveridge--Joshi--Tang Practical Policy iteration. But better to start with books on exotic interest derivatives such as mine or Piterbargs $\endgroup$
    – Mark Joshi
    Oct 31, 2014 at 21:07

There was a pretty good article covering this in Wilmott Magazine a while back. It covered the somewhat more general case of Callable Constant Maturity Swap Steepeners.

You can ignore all the machinery around the CMS coupons if you are just treating standard callable bonds. That is to say, in Equation 8, you just need to set the multiplier $m$ to zero.


That's queer that you found nothing. Perhaps this project will be helpful. Let me know if you have questions about it.

  • $\begingroup$ thanks a lot for your answer! What about continuous models ? Do you know any reference ? I admit I am more interested in continuous models. $\endgroup$
    – Paul
    Oct 30, 2014 at 16:26
  • $\begingroup$ A closed-form solution? $\endgroup$
    – James
    Oct 30, 2014 at 18:42
  • $\begingroup$ Not necessarily. $\endgroup$
    – Paul
    Oct 30, 2014 at 19:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.