The Gap the Hedger Prefers

You buy a one-month at-the-money call on a stock trading at $\$100$, paying $\$4.00$ for it (take the option to cover one share, so all figures are per share). Its delta is $0.50$, and you immediately delta-hedge by shorting $0.50$ shares at $\$100$. You re-hedge to delta-neutral once a day at the close, and you hold the position to expiry.

The stock ends the month at $\$80$ either way, but consider two paths:

- **Path A:** overnight, the stock gaps straight from $\$100$ to $\$80$ on bad news, then sits at $\$80$ for the rest of the month. - **Path B:** the stock drifts down about $\$1$ a day for twenty trading days, arriving at $\$80$ at expiry.

The option expires worthless in both cases. Which path makes you more money, and by roughly how much? What general principle is this?