How a Callable Bond Relates to a Receiver Swaption
Summary
The document asks whether a receiver swaption is equivalent to a callable bond. The main response distinguishes the instruments: a callable bond is a security exposed to issuer default, while a swaption is an interest-rate option. It then describes an economic approximation: a bond with a call after a year can be compared with a noncallable bond whose coupon is reduced by the value of a receiver swaption starting in one year and running for the remaining term, struck at the bond coupon.
The response cautions that credit spread affects callable-bond valuation but does not affect the receiver swaption in the same way. Another contribution discusses adjusting a swaption coupon to reflect a call price different from par, with recalibration over time. The document mentions a derivation and an implementation elsewhere but does not reproduce their details, so it does not establish a general formal equivalence.
Key ideas
- A receiver swaption and a callable bond are not identical instruments, partly because the bond carries issuer default risk.
- A callable bond can be approximated using a noncallable bond and a receiver swaption matching the call timing and coupon.
- Credit spread affects callable-bond value but is not captured by the swaption in the same way.
- Non-par call prices may require adjusting the swaption’s coupon assumption as valuation conditions change.
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# Receiver Swaption and Callable Bond - Literature Proof? # Receiver Swaption and Callable Bond - Literature Proof? I'm looking for a formal proof that a receiver swaption is equivalent to a callable bond. I have only found some CFA Internet pages so far where this statement is considered as proven, tough I haven't seen any papers or books which proof that explicetly. Any suggestions? ## Answer by dm63 (score 2) https://quant.stackexchange.com/a/33838 A receiver option is definitely not the same thing as a callable bond. The latter is a security, with an issuer which may default, for one thing. However, there is a connection, as follows: a 10yr bond with a coupon of K and a single call option after one year is economically similar to a 10 year non callable bond with a coupon of K minus a 1yr into 9yr receiver swaption struck at K. Thus, the issuer's option to call the bond is economically similar to a receiver swaption. This is only an approximation - the issuer's "credit spread" is a factor in valuing callable bonds but it does not affect the value of the receiver swaption. ## Answer by user29680 (score 0) https://quant.stackexchange.com/a/36106 I have reconsidered my previous answer and I think it is possible for any Call price. If the call price ist different from 100% say 102% than one could use a Swaption to receive the fixed coupon and pay a calibrated coupon2 to the corresponding forward Bond price 102%. The calibrated coupon2 would need to be updated at every pricing date. ## Answer by user9403 (score 0) https://quant.stackexchange.com/a/36107 I have written brief derivation on page 6 of the following document: https://github.com/phillyfan1138/PaperMarketRisk/blob/master/MarketRiskDocumentation.pdf. I've implemented a pricing model using the fact that they are equivalent (and unit tested that they are equal) in the following repository: https://github.com/phillyfan1138/HullWhite. See especially line 392 of HullWhite.h and test "Payer Swaption" in test.cpp.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.