For a delta-hedged option, it is known that the PnL is proportional to the gamma * stock move ^ 2. As such, it would imply that your PnL would be highest when realized volatility is occurring at the points with highest gamma.
Now, consider we have a 100 strike ATM straddle for stock ABC expiring in 4 days. Consider the following two paths:
1) The stock moves 1% up each day
2) The stock moves 1% up one day, following 1% down the next, and so on
Clearly, as path 2 sees the stock going back and forth near the strike, it should have higher gamma, and hence higher PnL. However, if you were to write it out, you would see that you would make more money from path 1. Why is that, since path 1 has less gamma throughout?
Path Dependency with Delta Hedging
byu/Terrible_Ad5173 inoptions
Posted by Terrible_Ad5173
3 Comments
Only if you’re right, sir , you would be in the negative on theta anyway. Anything with Delta Targeting, you need to be right first before adding any theory.
Just take the greeks and everything out of it for a second and consider the terminal pricing. In scenario 1, the straddle ends worth $4.06 or so, in scenario 2 it ends up worth basically $0. That means that scenario 2, you need to make $4.06 more in scalping the gamma to beat scenario 1. How likely does that seem?
> it is known that the PnL is proportional to the gamma * stock move ^ 2.
No, where in the world did you get this idea?
A delta-hedged portfolio would get the returns of the risk-free rate.