# Hedging with implied volatility

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