Callable Bond Replication with Swaptions, Curve Basis, and Credit Risk
Summary
The document examines a European callable bond with a call date two years ahead and a five-year remaining term. Its textbook decomposition combines the callable bond with a receiver swaption to replicate a non-callable bond. The response qualifies that relationship: it holds only when the floating leg of the swap is worth par at the swap start, as under a single-curve setup where projection and discounting use the same curve. With OIS discounting and a separate Libor projection curve, the floating leg may not be par, so the replication does not hold exactly.
Issuer credit risk requires further adjustments. A swaption that remains available after issuer default is unnecessary in that state, suggesting the option should be conditional on survival. At exercise, the swap fixed leg should match the value of the credit-risky bond; under the response's deterministic spread approximation, the strike is adjusted from K to K minus the spread. This is a first-order approximation, and the appropriate strike depends on the discount curve at exercise.
Key ideas
- The callable bond and receiver swaption replication requires the swap floating leg to be worth par at the start date.
- Separate OIS discounting and Libor projection can break the simple replication because of the curve spread.
- Issuer default before option expiry can make the swaption unnecessary, so survival conditioning affects its value.
- Credit risk changes the exercise-date value that the swaption must match.
- Under a deterministic spread approximation, the response suggests reducing the swaption strike by that spread.
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Full text
# Replication of risky callable bonds # Replication of risky callable bonds I have the following problem regarding callable bond replication. Let's define: A: 7Y Callable bond with fix coupon K%, which is callable exactly in 2Y @par (European feature) B: 2Y x 5Y Receiver Swaption witk strike=K% C: 7Y non-callable bond (same as A without call feature) Textbook replication: A+B=C However, I'm puzzled with the strike of the swaption. K% coupon is based on a 7Y maturity whereas in 2Y (option expiry) the issuer needs to assess whether there are more favourable conditions than paying K% for the remaining maturity of 5Y. From this perspective, I wonder whether we are really considering the correct strike of K% here? Additionally, how would you incorporate deterministic credit risk in the pricing of A+B? ## Answer by Antoine Conze (score 1) https://quant.stackexchange.com/a/37531 $A+B=C$ only holds if the floating leg of the swap the swaption exercises into is worth exactly par on the swap start date, which is true only in a single curve settings (up to slight discrepancies related to libor business days adjustments), that is if the swap is valued using the same libor curve for projecting floating rates and discounting cash flows. If the swap is collateralized at OIS, as is now the standard, then cash flows should be discounted on the OIS curve, and because of the OIS-libor spread that has appeared since 2008, the floating leg is not worth par and $A+B \neq C$. Assume now that discounting of the swap cash flows is done at libor so so that its floating leg is worth par. In the presence of credit risk on the bond issuer then the swaption needs to be modified : - because if the bond issuer defaults before 2 years then the swaption is unecessary, so ideally it should be made conditional on the issuer not defaulting (that would make the swaption cheaper) - and mostly because upon exercise in two years the PV of its fixed leg should be equal to the PV of the credit risky bond $C$, so its strike should be adjusted. Of course the exact strike would depend on the then libor discount curve, but as a first order approximation say that the issuer credit risk is a deterministic $s$ spread above libor, then the strike should be adjusted to $K - s$ so that the credit riskless $K-s$ swap fixed leg is worth approximately the same than the credit risky bond with fixed rate $K$.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.