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Convexity and Volatility

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Hi everyone,

It's my first time posting but I've been reading the forums since I enrolled for the FRM part I more than a year ago and I wanted, before anything else, to thank you all, particularly David, for all the help I've gotten from such a knowledgeable and supportive community while studying for the exams. Really, you all have been incredible.

I imagine it being weird, a theory question right after the exam date, but this is something I couldn't really get while reading the curriculum but had to let go in order to finish the readings on time. Now, more free, I've been trying to make sense of it without much success, and that's why I finally decided to register and hope that some of you can make things clear for me.

So, onto the subject.
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Tuckman.Ch8 explains, among other things, how the uncertainty of interest rate expectations (volatility in the binomial tree) lowers yields and produces Convexity. The author even goes on to say that "the value of convexity increases with volatility". This one was one of the hardest concepts from the curriculum for me to understand (and a totally new one), but I think I got it after reading some threads here, playing with a couple of trees, and looking deeply at Jensen's inequality.

What seemed weird to me and didn't seem to fit with what I knew about convexity and options was what all of this implied for Gamma. I had the idea that both exposure to bond convexity and option gamma was profitable when the underlying (it being interest rates or anything else) was volatile (at least when its volatilty was greater than what the market expected/what was priced into the bond/option). I understood that one paid for convexity/gamma because of this, that a bond with more convexity would be more attractive (and more expensive) than an otherwise identical one, and that when one was long a bond the positive convexity always made the effect of any rate move better (or less bad). I remember that I read somewhere something along the lines of "having (positive) exposure to gamma (convexity) means being long realized volatily, while having (positive) exposure to vega means being long implied volatility". I really liked that phrase and how it explained in a simple way something that was initially difficult for me to grasp when first studying the greeks for the CFA.

Of course, none of that really contradicted Tuckman. After all, the volatility he was saying convexity arises from was (in my mind, at least) more related to implied (again, uncertainty in expectations) than to realized volatilty. The "causaility arrows" could go like: more implied volatility -> more convexity -> better outcomes if more realized volatility. What made all of this crumble was the fact that Gamma is supposed to be inversely related to (Implied) Vol, according to BSM. That was definitely incompatible with what Tuckman was saying, so I figured I was wrong in assuming that the uncertainty he talks about is the same as ImpliedVol.

But if it's not, what is it? Is it more related to RealizedVol? The "uncertainty about future rate states" component makes me think that it isn't.
Or am I mixing up different models? I understood BSM like a continuous case of the binomial trees and, since I was thinking about the rates as the underlying and not the assumed-constant-BSM-rate (deltas and gammas, not rhos), I thought the same analysis could be applied.

Am I wrong about something obvious I'm not seeing? Or am I just mixing things that shouln't be mixed?

I'm sorry about all the text and I really hope I'm making myself clear (not native speaker), I'm just really frustrated about not being able to reconcile all of this. I'll be eternally grateful for any answer or idea that lead us to one.
 
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