# Frequency Arbitrage

We know that the volatility is lower when the sampling period is longer, for example $\sigma_{7days} < \sigma_{1day}$, Then I came across this strategy that I cannot quite understand how to exploit this:

It says: If weekly historical vol < daily historical vol:
buy strip of T options, delta hedge daily;
sell strip of T options, delta hedge weekly.

Daily Vol / weekly Vol Arbitrage: On each leg: always keep $a invested in the index and update every Δt; Resulting spot strategy: follow each week a mean reverting strategy; Prove that we should keep each day the following exposure: $$a.(\frac{1}{S_{t_{i,j}}} - \frac{1}{S_{t_{i,1}}})$$ where$t_{i,j}\$ is the j-th day of the i-th week