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The beta of an investment strategy corresponds to its relation with the systematic moves of the prices, i.e. the one driven by very common factors. Typically market indexes are benchmarks used to measure the beta against.
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Accepted
Beta vs. Implied Volatility statistical arbitrage using options
This is definitely a valid (and possibly viable) strategy.
I think that your constraint of zero costs is a red herring and serves no useful purpose beside forcing you to take lopsided bets in the di …