gamma hedging explanation

Hi guys,
Can someone explain me this
Why gamma hedging results in (x^2 + x)/2
Thank you

 

I've seen the formula for covering theta,where x equals the move in the underlying where one gamma hedges..

What is X in your formula?


x= √[($theta * 2)/100

 

X is price change

 

I assume x is the change in the underlying price? You can derive it from the overall PnL approximation

Code:

pnl ~= dS * delta + 0.5 * gamma * dS^2 + theta * dt + vega * dVol

and the fact that your delta at the end of the move will be

Code:

delta(dS) = delta(0) + gamma * dS

 

I will add it exactly
x² + x) / 2 is when the assumption comes out
(x² - x) / 2 is when the assumption fails
x is the price change and the result of those equations is how many different derivatives when the price changes x
The question is why it is so

 

@taowave, @Same Lazy Element
Thank you very much for your help.
I really appreciate it

 

in your so called approx may i ask why are u including second order dS but truncating all the other derivatives at order 1. surely just delta+theta+vega.

 

Because other second derivatives for most options aren’t going to be significant enough, while gamma will be. I am also completely omitting some other risks like rho and sensitivity to dividends because they are less important in most cases - however, if you are short a 10-year option on SPX you’re gonna learn these two intimately.