Here's the scheme we're investigating:

Suppose we've written (and sold) a call option on a stock where the current stock parameters are:

      Stock Price = S
      Strike Price = K
      Stock Volatility = V
      Expected Stock Return = R
      Years to Expiry = T
      Risk-free Rate = Rf

We use, as the price of the option, the Black-Scholes formula (as in an Excel spreadsheet):

[A] Option price C = S*NORMSDIST((LN(S/K)+(Rf+V^2/2)*T)/(V*SQRT(T))) - K*EXP(-R*T)*NORMSDIST((LN(S/K)+(Rf+V^2/2)*T)/(V*SQRT(T))-V*SQRT(T))

We calculate Delta, the rate of change of option price with respect the stock price , via:

[B] Delta = dC / dS = NORMSDIST((LN(S/K)+(Rf+V^2/2)*T)/(V*SQRT(T)))

1. We sell an option for $C when the stock price is $S.
2. We buy D shares at the current price of $S per share, borrowing $D S to do so.
3. The weekly interest rate for borrowing is i so the weekly cost of borrowing is i D S (where i = 0.0008 means 0.08%).
4. N weeks pass and the stock price has increased by $ΔS ... and the option is exercised.
5. We then buy an option at a price which is larger than $C by an amount $ΔC ... on the same stock.
6. We collect the shares from the option we bought to cover the option we sold.
         We've just lost   $ΔC on the buying and selling of the options.
7. But then we sell the D shares we bought at step 2, making D ΔS on the sale.
8. We pay the N-week interest on the loan in step 2, namely $N i D S
         Our net gain is:   D ΔS - ΔC - N i D S
9. So as not to lose money, we choose D to make this net gain = $0; that is:
         D ΔS - ΔC - N i D S = 0
10. Hence our purchase of shares in step 2 should be for D shares where:
          D = ΔC / (ΔS - N i S )

Hence the number of shares we should hold (in step 2, above) is: D = ΔC / ΔS.

>And that's delta, for the stock, right?
Yes, for small changes in stock price, since delta is actually dC / dS.

>ΔC / ΔS or dC / dS. What's the difference?
In a car trip, ΔC / ΔS is like the average speed of your car and dC / dS is the instantaneous speed ... as noted on the speedometer.

Anyway, since we are to hold sufficient stock to cover the loss in option trading, we should buy and sell stock as the weeks go by
... since delta changes.

>And you're ignoring the cost of borrowing?
Yes, to make things simple ... just to get a flavour of this delta hedging stuff.

Okay, suppose our parameters are:

      Stock Price = S = $60
      Strike Price = K = $62
      Stock Volatility = V = 20%
      Years to Expiry = T = 20/52 (meaning 20 weeks)
      Risk-free Rate = Rf = 4%
and we sell 10 contracts (worth1000 shares of stock).

Initially:
      C = $2.59
      Delta = 0.481
and we'd buy (initiallly) 1000*0.481 = 481 shares of stock at $60 ... borrowing the money to do so.
As the weeks go by, delta changes.
The stock price S may change and the time to maturity T certainly changes ... see magic formula [B], above.

Figure 1 shows the number of shares we should own with 15, 10 then 5 weeks to go before the option expires. It's a chart of 1000*delta. >But you don't know the future stock price! True, but Figure 5 is interesting nevertheless. We just buy or sell our stock holdings to match the chart ... depending upon what the stock price happens to be at the time. For example: At 15 weeks to go and a stock price is $65 we should own about 730 shares. At 10 weeks to go and a stock price is $67 we should own about 840 shares. At 5 weeks to go and a stock price is $64 we should own about 735 shares. >Yeah, I see the magenta dots.

Figure 1

Now, if we could generate a distribution of stock prices at some time in the future, we could generate a distribution for delta ...

>Can you do that?
Well ... so far I have something like this:

>You don't use the expected return, do you?
No. I just stuck it in the parameters as decoration.
>Where'd you get that stock price distribution?
I stole it from here.
>That delta distribution looks like a cumulative probability distribution.
Doesn't it? Of course, in magic formula [B] above, that NORMSDIST function is a cumulative distribution.
>What if I make a mistake in estimating V, the volatility?
Don't make a mistake else your delta-chart will change ... like this  
>And if you buy the option, what then?
You can short the stock ... delta-shares worth, to hedge against the stock dropping in price. Indeed, it seems to make more sense if you're the guy who bought the option and are worried about its value dropping as the stock price drops.

>And that's the way delta hedging works?
Uh ... how would I know? I just regurgitate what I find on the Net.  
>If the objective is to lose no money then I have another, simpler method
Very funny. However, 99% of delta hedging is done by market makers (Norbert tells me). Their gains are elsewhere.
They just don't want to lose anything when moving options between buyers and sellers.

>What's that funny chart ... Distribution of Gains after 2 weeks?
It's assumed that nobuddy exercises the option you write until the stock price exceeds K + C. In the example illustrated, that's $64.59
If the stock price is less than this, you just make the money you got by selling the option, namely $2.59.

>And you guarantee the spreadsheet?
Always  

Forgot to mention y'all can download the spreadsheet (such as it is) by clicking on the picture above.
If'n that don't hardly work, try a RIGHT-click then Save the target file.