You're mostly correct. As to your questions, think about it this way... Disclaimer: this is a perspective of a gamma trader in my world, so take with a pinch of salt.I have a long post so please bear with me.
I want to understand what everyone is talking about in layperson's language by using an example:
If I am long a Call, I have a situation with positive delta < = 1 depending on OTM, ATM or ITM. Also, gamma, the second derivative is positive: When the underlying goes up, my call value goes up and the delta also goes up. That was what everyone meant by having a positive gamma?
I have several comments and questions:
1. I do nothing and hold the long Call till expiration. My profit/loss = Final stock price - strike - Call premium.
2. If I delta hedge, when I have a long Call, I hedge with short stocks = delta x # of shares. When the stock price goes up, my delta goes up I sell higher price stock to keep delta = 0. When the price goes down, I buy lower price stocks to keep my delta = 0. In effect I am buying low and selling high so each movement gives me profit.
3. Also, the higher the volatility the bigger the movements of the underlying and the higher my hedging profit (selling way higher and buying way lower). I think botpro is not entirely crazy when he said to go long in a high volatility environment if you want to delta hedge.
4. What is my final profit at expiration if I delta hedge? One factor should still = Final stock price - strike - Call premium. How about hedging profit? Can it be predetermined using IV, etc.? Or is it path dependent like someone mention in a different thread?
Any comments will be greatly appreciated. I am eager to understand and join your club.
Thanks.
Firstly, when if you delta-hedge in the real world you need to make discretionary decisions about when and how much delta to hedge. If we leave aside the transaction costs for the moment, you already know that if you delta hedge continuously and the option is fairly valued, your relative return will be zero. If you don't delta-hedge at all, as you pointed out, your return will be what you mentioned in point 1. Around these two extreme outcomes lies the uncertain path-dependent return that you produce with your specific delta-hedging methodology. Different people have different rules that they follow and there is a lot of variation.
Regardless of the methods, it's easiest to think of the outcome this way (and we're leaving aside the mark to market aspects here, this is accrual accounting; also this is a gross simplification):
1) you have bought a call w/IV of X.
2) on a given day I you delta hedged, such that you bought some quantity at price P1I and sold at price P2I for a PNL of ZI; ZI is a measure of realised variance/volatility/etc that you've "locked in" for the day.
3) if ZI is equal to X*sqrt(T) (where T is time to expiry), congrats, you have broken even. That's why this X*sqrt(T) quantity is known as your "daily breakeven". If ZI is less than the daily breakeven, you didn't do so well on the day and vice versa.
4) Lather, rinse, repeat, until expiration
5) At the end your PNL is the sum total of all the ZIs over the course of the procedure. If that number is greater than the premium you paid, congrats! Otherwise, not so lucky.
This brings us to why our young turbulent friend is not so right. Observe that the premium you pay for the option is a certain, known expenditure. The delta hedging PNL is uncertain and depends on a lot of factors. In a high-volatility environment you will definitely have to pay a lot of premium, but will you be able to delta-hedge to recoup it? My personal preference is to buy "cheap" options and leave the selling/buying of expensive ones to other people.
My Z$2c... I cannot guarantee everything I've said here is correct. Hopefully, peeps can weed out any obvious silliness.
