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?

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    $\begingroup$ In theory you will always have a diversification benefit as long as the correlation of returns does not equal one special value related to the Sharpes of the two assets. $\endgroup$ – steveo'america Apr 23 at 18:06
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    $\begingroup$ Yes. For example assume you buy both the stock and add a delta neutral long or short position in option straddles. Now you are taking both directional stock market risk and volatility risk, which are far from being perfectly correlated. For example adding a short straddle to the portfolio would likely boost the Sharpe ratio (while introducing some tail risk). $\endgroup$ – fesman Apr 24 at 9:04
  • $\begingroup$ Sounds relevant, don’t you want to post it as an answer? $\endgroup$ – Nikolay Rys Apr 24 at 9:14

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