# Replication of "recovery bond"

I just started learning about credit products.

Let $$B_t^T$$ be the price of a risk free zero coupon bond at time $$t$$. Similarly, let $$C_t^T$$ be the price of a zero coupon risky bond from some fixed issuer (no recovery). Both pay one dollar at maturity.

Let $$T_1 < T_2$$ and define a recovery bond as an asset that pays $$\mathbf{1}_{C_{T_1}^{T_1} = 1, C_{T_2}^{T_2} = 0} = C_{T_1}^{T_1} \cdot \left(1 - C_{T_2}^{T_2}\right)$$ at time $$T_2$$.

Is it possible to replicate this asset using $$B^{T_1}$$, $$B^{T_2}$$, $$C^{T_1}$$ and $$C^{T_2}$$?

I guess I'm generally confused as to what the basic building blocks are when dealing with credit products.

I think I need these "recovery bonds" to price coupon paying bonds.

References would be much appreciated!

Here's what I tried.

Let $$\Pi$$ be a replicating portfolio for this asset: \begin{aligned} \Pi_t &= x_t B_t^{T_1} + y_t C_t^{T_1} + z_t B_t^{T_2} + w_t C_t^{T_2} \\ d \Pi_t &= x_t \mathrm{d} B_t^{T_1} + y_t \mathrm{d} C_t^{T_1} + z_t \mathrm{d} B_t^{T_2} + w_t \mathrm{d} C_t^{T_2} \end{aligned}

Then \begin{aligned} \mathbf{1}_{C_{T_1}^{T_1} = 1, C_{T_2}^{T_2} = 0} = \Pi_{T_2} = \Pi_0 + x_0 \left(1-B_0^{T_1}\right) + y_0 \left(C_{T_1}^{T_1} - C_0^{T_1}\right) + z_0 \left(B_{T_1}^{T_2} - B_0^{T_2} \right) + w_0 \left(C_{T_1}^{T_2} - C_0^{T_2} \right) + z_1 \left(1 - B_{T_1}^{T_2} \right) + w_1 \left(C_{T_2}^{T_2} - C_{T_1}^{T_2} \right) \end{aligned} I broke this into three equations based on the cases $$C_{T_1}^{T_1} = 1, C_{T_2}^{T_2} = 1$$ and $$C_{T_1}^{T_1} = 1, C_{T_2}^{T_2} = 0$$ and $$C_{T_1}^{T_1} = 0, C_{T_2}^{T_2} = 0$$ and tried to solve for $$\Pi_0$$ to no avail.