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Why a Recovery Bond Cannot Be Replicated from Defaultable Zero-Coupon Bonds

Article Quant Q&A · Author: user357269
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Summary

The document defines a contingent payment at the later date: it pays only if the risky issuer’s earlier-maturing bond pays in full and its later-maturing bond defaults. It asks whether risk-free and risky zero-coupon bonds at the two maturities can replicate that payoff, and sketches a portfolio of those instruments with trading at the intermediate date.

The author divides outcomes into default and survival cases but reports being unable to solve for a replicating portfolio. The text does not provide a proof, pricing model, or resolution. Replication depends on the assumed default states, available trading and rebalancing, and the instruments’ payoff structure; the document itself does not specify these sufficiently to establish whether replication is possible. It frames the question as a learning issue about credit-market building blocks and coupon-bond pricing.

Key ideas

  • The target payoff requires survival through the first maturity and default by the second.
  • The proposed replicating assets are risk-free and risky zero-coupon bonds at two maturities.
  • The author considers rebalancing at the intermediate date but does not derive a solution.
  • The document leaves replication feasibility and coupon-bond pricing implications unresolved.

Tags

Fixed incomeDerivatives pricing
Full text
# Replication of "recovery bond"


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.