Callable Bond Valuation Through Cancellable Swaps and Bermudan Swaptions

# (Beginer on bond market) References on callable bond's pricing


# (Beginer on bond market) References on callable bond's pricing












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: how to right mathematically the payoff of a callable 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?

## Answer by Mark Joshi (score 4)

https://quant.stackexchange.com/a/15231

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.

## Answer by Brian B (score 1)

https://quant.stackexchange.com/a/15225

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.

## Answer by James (score 0)

https://quant.stackexchange.com/a/15221

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