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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?

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  • $\begingroup$ A callable bond can be defined in many ways. You may need to be more specific about the callable bond that you are referring to. $\endgroup$ – Gordon Apr 21 '17 at 12:51
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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.

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  • $\begingroup$ I believe that the callable bond referred by OP is a coupon bond with a callable feature. $\endgroup$ – Gordon Apr 23 '17 at 0:16
  • $\begingroup$ I believe the price to call the bond at the call date (corresponding to a coupon payment) SHOULD also be 100% (?) If not then the convertion to a standard receiver swaption may not work(?) $\endgroup$ – Yvon. T Sep 20 '17 at 16:58
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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.

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    $\begingroup$ Please merge your accounts. Also this doesn’t really answer the question, do you have a proof? $\endgroup$ – Bob Jansen Sep 20 '17 at 19:31
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    $\begingroup$ If there is a call premium, this is equivalent to a swaption with an exercise fee, which effectively changes the strike rate of the swaption. $\endgroup$ – dm63 Sep 20 '17 at 19:44
  • $\begingroup$ My last answer "I have reconsidered.....every pricing date." is incorrect. Forget it. Sorry.. $\endgroup$ – Y. Tounkap Sep 21 '17 at 6:04
  • $\begingroup$ To be precise I was wondering if it is possible to represent/value a european call option on a fix coupon bond (call date on a coupon payment date) with a strike different from 100% say 102% using a european receiver swaption. I doubt this is without some kind of adjustment possible... $\endgroup$ – Y. Tounkap Sep 21 '17 at 6:27
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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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