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I am reading this article by R. Ahmad and P. Wilmott:

Which Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal Portfolios

Let $V^{i}$ the market value of an option such that $V^i = V_{BS}(\sigma_i)$, where $\sigma_i$ is the implied volatity. In formula (1) the authors use the Itô formula to compute the mark-to-market value change of the option: $$ dV^i= \Theta^i dt + \Delta^i dS + \frac 1 2 \sigma^2 S^2 \Gamma^i dt $$ where $\sigma$ is the actual realized volatility and the Greeks are computed from the BS formula with $\sigma=\sigma_i$. This formula assumes that the implied volatility $\sigma_i$ for the market value of the option does not change at all during the time period $dt$ (it is neither a function of $t$, nor of $S$). How is this assumption justified?

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    $\begingroup$ The formula simply does the PnL explain between $t$ and $t+dt$ when $\sigma^i$ is the booking volatility used at time $t$. $\endgroup$ – Antoine Conze Jan 28 at 9:51
  • $\begingroup$ @AntoineConze But in reality $\sigma^i$ will change, so there will be a Vega term in the expression for $dV^i$ $\endgroup$ – Appliqué Jan 28 at 19:12
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    $\begingroup$ If you remark your booking volatility $\sigma^i$ yes there will be a vega term in your PnL explain, as well as a vanna term and a volga term. $\endgroup$ – Antoine Conze Jan 29 at 7:56

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