Implied Volatility Changes in Delta-Hedged Option P&L

Summary

The document questions an option mark-to-market formula used in a discussion of delta hedging and volatility arbitrage. It describes valuing an option with Black–Scholes at its market implied volatility, while applying Ito’s formula to express the value change through theta, delta, and gamma using realized volatility. The question is whether this treatment assumes implied volatility stays constant over the time interval, even as the underlying price or time changes.

This is a focused conceptual question rather than a full explanation of the hedging method. It highlights the distinction between realized volatility driving underlying-price dynamics and implied volatility used to mark the option, and points toward the need to account for changes in the implied-volatility surface when those assumptions do not hold. The document contains no answer, derivation, or empirical evidence, so it does not establish when the frozen-implied-volatility approximation is appropriate or how adding volatility dynamics changes hedge P&L.

Key ideas

• The stated option-value change uses theta, delta, and gamma with realized volatility.
• The Greeks are evaluated using the option’s implied volatility in the Black–Scholes formula.
• The question is whether the formula freezes implied volatility during the interval.
• Changes in implied volatility would require additional terms beyond the stated expression.

Tags

# Hedging with implied volatility


# Hedging with implied volatility












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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