Discrete Delta-Hedged PnL and Realized vs. Implied Volatility

Scenario

A quantitative options trader sells a European call option priced with implied volatility $\sigma_I$, and continuously delta-hedges the short position using the underlying asset. Over the holding period, the actual realized volatility of the underlying asset turns out to be $\sigma_R \ne \sigma_I$.

Your Task

1. Using Ito's Lemma under the physical measure $\mathbb{P}$, derive the instantaneous PnL increment $d\Pi$ of the delta-hedged portfolio.
2. Prove that the expected PnL accumulated over $[0, T]$ is given by: $\mathbb{E}[\Pi_T] = \frac{1}{2} \int_0^T S_t^2 \Gamma_t (\sigma_R^2 - \sigma_I^2) dt$
3. Simulate the daily discrete rehedging of a 1-year ATM option with $\sigma_I = 20\%$ when realized volatility is $\sigma_R = 25\%$, and verify that the expected PnL is positive.

Constraints

• Continuous interest rate $r = 0$.
• Daily rebalancing ($N = 252$ steps).

Examples Because an option buyer is long Gamma ($\Gamma > 0$), when realized volatility exceeds implied volatility ($\sigma_R > \sigma_I$), the delta-hedger captures net positive rebalancing gains exceeding the initial option premium.