Standard deviation of a straddle p/l

filthy · Post by **filthy** » Mon Oct 01, 2007 10:54 pm

I was looking through W****tt at the weekend and cam across this statement.

If we sell a delta neutral straddle at an implied vol, sigmai, and realized vol is sigma our expected p/l is

sqrt(2T/pi).(sigmai-sigma).S

(this makes sense)

and it's standard deviation is

sqrt(1-2/pi).sigma.S.sqrt(T)

This was just stated with no refs, proof or motivation. Where does this come from?

Baltazar · Post by **Baltazar** » Tue Oct 02, 2007 8:53 am

To me, it does not make sense, the PnL of a straddle hedging should depend on the hedging frequency.

Boucheau got something on that i believe (i don't have the book at hand)

thefoolishfour · Post by **thefoolishfour** » Tue Oct 02, 2007 10:26 am

Which book is that??

Baltazar · Post by **Baltazar** » Tue Oct 02, 2007 10:38 am

I would say

Theory of Financial Risks: From Statistical Physics to Risk Management

but I am not sure.

Btw, maybe the guy meant unhedged

filthy · Post by **filthy** » Tue Oct 02, 2007 2:02 pm

i assume he meant unhedged. the hedging induced p/l variance is just due to sampling error and was well covered by derman and kamal. this is different. could be a useful result but i have no idea where it comes from.

filthy · Post by **filthy** » Tue Oct 02, 2007 2:16 pm

ok well how about this?

the derman kamal error for a straddle that is hedged once (at expiry) is sqrt(pi/4).vega.sigma

the ATM straddle is roughly 0.8. S.sigma.sqrt(T) so its vega is 0.8S. sqrt(T)

combining these we get an sd of 0.8.sqrt(pi/4).sigma.S.sqrt(T) which is close.