# 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.

We do not buy nor sell any option;
play intra-week mean reversion until T;

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

It amounts to follow an intra-week mean reversion strategy

Could someone helps to explain a little?

• Bruno Dupire is the author of this strategy. Feb 25 '18 at 22:02
• yes! but I don't quite understand @Alex C
– Qing
Feb 25 '18 at 22:04