Please see below graph showing the average hedging P&L from periodically delta hedging a short call position on SPY using a long SPY ETF position and cash. I varied the volatility by multiplying it with factors ranging from 0.5 to 2, and varied the hedging frequency from 1 day (i.e. everyday) to 5 days (i.e. once every 5 days). I was hoping to see that under high volatility regimes, the disparity between total hedging P&L would be pronounced between the frequent and less frequent hedging, and that under low volatility regime that difference won't be as prominent. Instead, what I see below doesn't make much sense.

I have done several sanity checks including checking that my self-financing portfolio value is very (very close) to the option price at Charles Schwab brokerage. The maturity of the option is 0.25 years, number of short calls is 100. I am simulating future price paths using Geometric Brownian Motion, and ensuring that the P&L from delta-hedging is being calculated correctly.

What am I missing here, please? Shouldn't there be a pattern below?

Okay -- fixed a small bug. I wasn't actually changing the volatility factor and the hedging frequency in my nested for-loop. After the fix, there is a slight pattern. Any other thoughts / comments anyone, please?

How do you handle the transaction costs (which are the higher the more frequent hedging, right?)? Secondly, isn't the volatility adjustment you use too simplistic?
Disclaimer (before Bearish starts to laugh that I'm the fist to comment): all I know about trading i learnt "via osmosis" thanks to this forum!

Thanks Katastrofa. I spent more time on this and came up with below. It's true I'm not accounting for transaction costs, but compared to the magnitude of negative P&L on below Exhibits I and IV, transaction costs should rachet up to only a relatively small number. So, that shouldn't be a factor.

That said, it's reassuring to observe as expected that the negative impact on P&L from delta hedging a short call position is much higher when volatility is higher. Not only that, it's also evident that the variance (jitter) in P&L is greater during higher vol periods. That's probably not a surprise.

But (ignoring transaction costs) how does the frequency of delta-hedging impact P&L? From my simulation, it appears there's no specific trend (please see the surface along the frequency axis below). However, P&L does become more negative when frequency = 1 day. I think that could be because delta-hedging a short call position by design requires us to "sell low and buy high" the underlying asset, which any strategy that aspires to make a profit wouldn't do! And the more frequently we buy high and sell low, the worse is the negative profit. 

There could be other dynamics I'm missing here, but this was an interesting exercise nevertheless. I simulated SPY prices for the next 90 days using Geometric Brownian Motion, and did that 250 times (thus that many N(d1), stock price, and profit paths) for each volatility multiple vs. hedging frequency combination. Exhibit I below is the aggregate of all 180 vol and frequency combinations, while Exhibits II, III and IV are from a single random combination to demonstrate the progression of the paths.

Looks really cool! I'm surprised the transaction costs didn't matter much. I'm wondering about dividends.

Hopefully some of the local finance volatility vandals will comment on your exercise

With no transaction costs and a GBM process, then you should see perfect Black-Scholes' hedging emerging as the hedging frequency increases. That means the hedged position tends to a money market position and the limiting profit should be the risk-free earnings from that, with zero volatility in the limit. If you don't see that, you're probably making a mistake. 

The convergence may be slow and it make take some extrapolation to get the correct limit. But the tendency of the distribution of profits to narrow -- ultimately to a Dirac delta -- as the hedging interval [$]\Delta t \rightarrow 0[$], should be fairly evident.

Do you validate these models on out of sample data? Can help to see what they are missing.

Thanks Alan and Katastrofa. There's no model involved here.

From what I understood, possibly incorrectly, you used historical data on the ETF, call options details and volatility to evaluate a delta hedging strategy. A model in my understanding is everything from GBM to the details of the hedging strategy, to PnL calculations.
I was wondering if the performance could be tested on an exogenous data (also a way to validate your result) or compared with other strategies. But maybe that was not your goal.

Once you are able to confirm the limiting result of zero P&L in the fantasy BS world, try seeing what happens when you change the underlying drift to something other than the risk less rate.

Thanks Katastrofa. No, I didn't use any historical data. The delta-hedging is assumed to be carried out 90 days into the future. But yes, on day one of the delta hedging setup, I used historical (realized) volatility and not implied volatility in coming up with a value for the call premium (from Black-Scholes). Ideally, I should have observed that premium in the market and used that premium to calculate cash account as premium received minus cash spent to hold delta shares of SPY ETF. But apart from that, I didn't use historical data or volatility anywhere else in the simulation. 

To David's comment above, technically I needn't have used B-S for day one. I could have simply observed the market price of the call and used that as the premium. 

Incidentally, the call value I calculated using B-S for day one was very close (almost matching) with the premium showing on my brokerage platform that day. It was spooky, but I was pleasantly relieved.