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I am looking for a minimal way to price callable bond from a defaultable issuer. The idea is to assume that we are in a deterministic world (i.e no volatility).

I tried a methodology but I am not sure if it is correct.

We know that, $$ \text{NPV}_{\text{callable}} = \text{NPV}_{\text{vanilla}} - \text{NPV}_{call}. $$ Therefore one should observe, $$ \text{NPV}_{\text{vanilla}} \geq \text{NPV}_{\text{callable}} $$

First let's assume a vanilla bond with a maturity $T$, paying a coupon $c$ at $T_{\text{1}}$ $(T > T_1)$. The recovery rate is denoted $R$ and the notional is 1. The risk-free is denoted $r$ and the default intensity $\lambda$.

Assuming that the recovery rate only apply to the notional, the NPV is given by, $$\text{NPV}_{\text{vanilla}}(t) = e^{-r(T-t)} \, \Big(S(t, T) + R\,(1 - S(t, T) \Big) + c\,e^{-r(T_1-t)}\,S(t, T).$$ Where $S(t, T)$ is the survival probability between $t$ and $T$ of the issuer and is equal to, $$e^{-\lambda(T-t)}.$$

Now let's add a call date at $T_{\text{call}}$ before the coupon ($T > T_1 > T_{\text{call}}$). The call rate is $K$.

To price the callable bond I first compute the NPV of the bond if it were to called for sure, i.e: $$\text{NPV}_{\text{called}}(t) = K\,e^{-r(T_{\text{call}}-t)} \, \Big(S(t, T_{\text{call}}) + R\,(1 - S(t, T_{\text{call}}) \Big)$$ As the bond is assumed to be called the remanning coupon is not paid and the bond now mature at $T_{\text{call}}$.

Finally the NPV of callable bond is given by the minimum between the two NPVs, $$\text{NPV}_{\text{callable}} = \min \Big( \text{NPV}_{\text{vanilla}}, \text{NPV}_{\text{called}}\Big)$$ Hence the call option is ensure to have a zero or positive value. If both NPVs are equal then the issuer is indifferent between calling the bond or let it live to the maturity.

By analogy, this methodology is the same as pricing an equity option by using the intrinsic value.

However it still seems a bit weird to me because I assume that the recovery rate is paid when the notional would have been paid given the optimal decisions to take today. This notion of recovery is a bit of a pain for me and I am not sure if I handle it correctly.

Would be great to have your advices on that, thanks!

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