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I don't know from where this reasoning got stuck in my head about call and put options and it's messing up everything:

When interest falls, likelihood of call increases as market rate is lower than the interest rate borrower/issuer is current paying, increasing the demand for call options, which will in turn increase the value of call option. same reasoning may be applied to put and in interest rate rise case. I know this reasoning is definitely wrong but I am not sure why it's wrong. I think I have read it somewhere and conclusion drawn from the reading was call option and prepayment option behave in a similar way. so I far I have been driving the call and put values using this reasoning and without any surprise getting most of the answers wrong. Did anyone remember reading this somewhere? Can someone identify any flaws in this reasoning so that I can just forget about it and follow Andy's reasoning. I remember reading Andy's reasoning as well, probably in a different context. But I am not sure why I remembered the wrong one. Am I mixing interest rate risk and reinvestment risk?

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