Assume that I long ATM Call option on a stock at 20% implied volatility. This Call option will expire in 1 month. If, over this coming month, the underlying moves with 30% volatility and I delta-hedge this option. I expect to get profit from Gamma scalping. My expected P&L would be positive.

How would my P&L accumulate, on average,

Does P&L grow linearly over time, or does P&L grow exponentially over time?

Does moneyness of option affect accumulated P&L shape?

It might interest you to think about the following question, that was often used at interviews where I used to work to see if

candidates had any intuition beyond the mathematics of option trading.
Consider a market that conforms exactly to the Black-Scholes assumptions (GBM, constant parameters, continuous frictionless trading etc.).

Implied vols are offered exactly 10 vols over historical, realised volatility at all times. You cannot buy options cheaper than this.

You are only allowed to buy options and delta hedge them

**precisely** according to Black-Scholes delta hedging rules. No cheating!

How can you make money in this environment?