Estimating Hedged Option P/L Volatility from Realized Volatility Error

Summary

The document asks how to derive an approximation for the volatility of the profit and loss on a hedged short at-the-money option. It describes a simulation setup in which an underlying follows geometric Brownian motion, the option is priced using the process volatility, and realized volatility is measured at discrete intervals. The question cites a relationship between the measured volatility’s sampling error and the resulting dispersion in option P/L, with the latter expressed in terms of vega, volatility, and the number of observations.

No derivation or answer is included in the document, so it does not explain the assumptions or steps that lead to the stated P/L approximation. The material is useful as a pointer to the connection between volatility estimation error and hedged option outcomes, but the formula is presented as a question rather than a demonstrated result. Its scope is limited to the stated simulation premise and should not be treated as a general derivation for every hedging scheme or option setup.

Key ideas

• The question considers a short at-the-money option under a geometric Brownian motion model.
• It distinguishes the process volatility from volatility estimated using discrete observations.
• The cited P/L volatility approximation scales with option vega and underlying volatility.
• The document asks for a derivation but does not provide one or discuss its assumptions.

Tags

# How to derive the volatility of options PL (hedged) as a function of implied volatility and measured realized volatility


# How to derive the volatility of options PL (hedged) as a function of implied volatility and measured realized volatility












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.

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