# How does P&L of delta hedged option position accumulate over time?

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 accumulates randomly, depending on the specific path the stock price takes. The profit rate is determined by the current value of $S_t^2 \Gamma$ . – Alex C Feb 10 '17 at 4:06
• Thank you, @AlexC. Can I say that P&L is path-dependent, but expected P&L is path-independent? – Jack JackGuRae Feb 10 '17 at 4:18
• Well an expectation is by definition an average over all paths so the last part does not really make sense to me. See this related question here: quant.stackexchange.com/questions/30397/… – Quantuple Feb 10 '17 at 8:05
• @Jack JackGuRae, also please have a look at Exhibit 2.1.1. here: trade2win.com/boards/attachments/f112/… – Quantuple Feb 10 '17 at 14:50