How to find volatility of a 1 day option based on 2 day annualized volatility

Question

first time -I'm curious as to how the following would work:

I have a 2 day(only includes full day of Thursday and Friday) swaption with a volatility of 100 bps.

We also know the weights we've assigned for the next 2 days, i.e thursday has a weight of 2(we multiply the daily bp vol by 2 - this can be due to a market event causing rates to move - i.e Fed meeting), and Friday is just a standard weight of 1.

How would we go about finding the daily bp vol breakeven for Thursday and Friday.

I tried the following but am uncertain if it is correct, because it may not be a linear interpolation:

100/(sqrt(252) =6.2599 avg daily bp vol

2x + x = 2*(6.299) - multiply by 2 because 2 biz days

x= 4.1966(fridays vol; Annualized: 66.67)

2x = 8.399 (thursdays vol; Annualized: 133.33)

Thanks!

Answer 1

That’s pretty close, but you should add the variances not the vols. Thus if the daily vols for Thursday and Fridayvrespectively are $v_1$ and $v_2$, we have the two equations $$v_1^2 + v_2^2= 2*6.2599^2$$ and $$v_1=2v_2$$. These can be solved for $v_1$ and $v_2$.