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I think the delta-replicating of $\sigma_2$ call is just a fancy way of saying "hedging the call option bought at $\sigma_1$ volatility, with deltas based on $\sigma_2$ volatility". This is full arbitrage in case the hedging/replicating is optimal, and just a statistical arbitrage in real life. You probably do not need such sophisticated proof of why this is ...

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Glasserman on pg 166 has a very accessible introduction to LMM under spot and forward measures

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The list will differ depending on your brokerage relationship. Not every broker's hard-to-borrow list will be the same. And you may be able to use 3rd party relationships purely for locates (finding hard-to-borrows). Your liquidity requirements look reasonable. I'm just going to pull a number out of the air (although I see it, this isn't something I track) ...

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