# 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?

$A+B=C$ 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 :

1. 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)
2. 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 $K$.
• Hi Antoine, I'm with you on the collateralized issue and let's assume single curve discounting.. However, my question really is on setting the strike of the swaption (as well as incorpoating credit risk in the component pricing). – user31190 Dec 29 '17 at 17:25
• Hi. I have edited my answer to reflect the credit risk issue. – Antoine Conze Dec 29 '17 at 18:15
• Hi, thanks for your explanations. They are plausible. In your 2nd scenario, I think it would also be valid to shift the yield curve (and hence forwards) by +s and leave the strike at K, right? – user31190 Jan 7 '18 at 19:39