Should I include or not the days a strategy has no open positions (thus no returns) in the Sharpe ratio calculation?
No, don't include them. Otherwise you'll just wind-up with zero-value returns (or worse, forward-filled returns), which will make your Sharpe ratio reflect a performance that didn't actually occur.
If you really think about the actual meaning of Sharpe ratios then you should come to the right conclusion yourself:
- It is a measure of excess risk-adjusted return (whether realized or unrealized)
In that you obviously only want to calculate actual returns. You do not have any actual returns on days with no open positions. Hence, why would you want to include such days?
=> The core thought here is to only measure the nature of the returns not whether returns occurred or not.
Remember that Sharpe ratio includes a risk-free rate of return ("RFR").
Unless the RFR is zero, then excluding days when you have no position is not correct and will technically overstate your Sharpe ratio.
And if you're using a RFR of zero then what you're actually providing is the signal-to-noise ratio. (Although yes, I acknowledge that in recent years the RFR is practically zero. But that doesn't justify these other incorrect answers.)