Imagine the universe where we have one investable volatile asset but with an available liquid options chain for it.
The question: can a portfolio consisted of this asset, and some options have any diversification benefit(higher Sharpe ratio) comparing to a portfolio with just the underlying?
At face value, it should; take an example of a portfolio consisting of the 1 share of the underlying and 1 shorted out-of-money call option contract. The payoff of the option part(option premium) doesn't correlate much with the payoff of the now-kinked payoff of the underlying.
Bonus question: If the answer is yes, then in the real world with many investable assets, will adding options into a portfolio be able to improve its Sharpe ratio?