First, there appears no standard agreement in this subject:
Below is simply my personal opinion.
I think the problem is not about high or low frequency, long or short strategy, single or multi asset. It is about how we interpret the (traditional) definition of Sharpe Ratio (SR).
For instance, as in the traditional setting, if we have a daily pnl, then we can compute SR by
- compute the average daily pnl
- compute the standard deviation of the daily pnl
- compute the SR by dividing the average by the standard deviation
This definition will work as long as we can measure pnl at regularly spaced time intervals, e.g., daily, hourly, etc. This is regardless of the position change over the time intervals.
Now, instead of insisting that the returns be sampled at regular intervals, we allow the definition to include returns be sampled at irregular intervals, even unpredictable intervals, e.g., stopping times, trade times.
Then, for a high frequency trading strategy, we measure the delta return, [tex]dr[/tex] , at each trade time. That is, the returns are the changes in portfolio P&L between the last and this trade times.
[tex]SR = E(dr)/sigma(dr)[/tex]
What do you think?