This is my first time asking a questions. Apologies in advance if I mess something up. If this happens, please let me know if I do and I'll try to fix it.
My question is regarding the equation Euan Sinclair gave in his Option Trading: Pricing and Volatility Strategies Chapter 11 equation (11.22). The premise is, simulate $N$ geometric Brownian motions with volatility $\sigma$ as the price of the underlying. Calculate the P/L of selling 1 ATM option on that underlying priced at the real volatility $\sigma$. Sinclair argued that due to us observing the process at discrete intervals, although $\sigma$ represents the volatility of the true process, our measured realized volatility is given by $\sigma_{measured} \approx \sigma \pm \frac{\sigma}{\sqrt{2N}} $ and the volatility of the P/L is given by $\sigma_{PL} \approx \sqrt{\frac{\pi}{4}}Vega\frac{\sigma}{\sqrt{N}}$. My question is, how is the last equation $\sigma_{PL} \approx \sqrt{\frac{\pi}{4}}Vega\frac{\sigma}{\sqrt{N}}$ derived? Thank you in advance.
