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?