Okay, here's what we want to do: Beware!: The notation may have changed from Part I

1. We pick a stock we're interested in and look at historical data, extractin g a Mean Annual Return, Rt and Volatility V.
2. We note the current stock price, Po.
3. Then, for any time T in the future, we use Ito's magic formula to give the Price Distribution:

   f(P) = 1/(V*P*SQRT(2*PI()*T))*EXP(-(1/(2*T*V^2))*(LN(P/Po)-(Rt-0.5*V^2)*T)^2)

4. That'd give something like this:
   
   where the right chart is the cumulative distribution at time T in the future.
5. Now we calculate the Expected Stock Price (at time T), using this distribution.
   To do this, we dividethe range of (future) stock prices into subintervals of length dP and pick equally spaced prices: P₁, P₂, ... P_n. (See the above picture.)
   Then we calculate:

   E[T] = { P1 f(P1) + P2 f(P2) + ... + Pn f(Pn) } dP

   Note that, in the picture, the initial price Po is shown red.
   The prices P₁, P₂ etc. run from left to right and include Po ... somewhere in the middle
6. Now we consider a Call Option with Strike Price K and expiry time Te.
   At time T in the future, the time to expiry is: TT = Te - T.
   The value of the Option will depend upon P (the stock price at time T) and TT (the time left to expiry) and some Risk-free Rate Rf ... and other stuff
   We assume it's given by the magic Black-Scholes formula:

   C(P) = P*NORMSDIST((LN(P/K)+(Rf +V^2/2)*TT)/(V*SQRT(TT))) - K*EXP(-Rf *TT)*NORMSDIST((LN(Po/K)+(Rf +V^2/2)*TT)/(V*SQRT(TT))-V*SQRT(TT))

   For example, putting TT = Te and P = Po will give the (initial) option premium. We'll call that:

   Co = C(Po) = Po*NORMSDIST((LN(Po/K)+(Rf +V^2/2)*Te)/(V*SQRT(Te))) - K*EXP(-Rf *Te)*NORMSDIST((LN(Po/K)+(Rf +V^2/2)*Te)/(V*SQRT(Te))-V*SQRT(Te))

7. Our intention is to keep track of the value of our Call as Time T progresses, and sell it when the Gain is acceptable.
   Indeed, at each future time T, we can calculate an Expected stock price and the corresponding Call value, C(E).
   We look intently at C(E) - Co, the Gain in option premium, and sell when it's a maximum.

For a given stock, say GE, we have no control over the price Po or the annual return Rt or Volatility V or risk-free rate Rf, but we have lots of Strikes to choose from and .... >And lots of Time to expiry, right? Yes. That's Te. So we look over the available choices for K and Te and follow each option for 0 < T < Te, with the help of Ito and Black-Scholes and ... >And pick the best. That's our intention. >And you believe in all this stuff, right? The option premium, from Figure 1, doesn't even match Black-Scholes! Uh ... well ... it's close enough. For example, GE is now at Po = $34.33 and if we use Rt and V generated by historical data over the last 5 years and Rf = 4% and Te = 14 weeks (from TODAY to June 15, 2007) and K = $32.50, we'd get Co = $2.78, so ... >But Figure 1 says 2.70. Besides, you took the option that had the "best" fit to Black-Scholes !! Well ... we should learn something from our analysis. >Speak for yourself !

Figure 1

>Are you sure about that?
You think I goofed?
>Yes.
Well ... play with the spreadsheet anyway. It's great fun!

Click on picture to download spreadsheet

See? There are charts of probability distributions f, and the cumulative distribution F, for both the stock price P and option premium C at some time T in the future.
>zzzZZZ
Just fill in the red boxes and click a button and ...
>zzzZZZ

Oh, one other thing. I mentioned the Expected Stock Price (at time T). There'll also be an Expected Call premium as well ... and that'll change with T. The spreadsheet has this chart, too ... somewhere. See? As T increases to the Expiry time Te, you can get neato charts By "Gain", I mean the Expected % change in Call premium. Got it? >zzzZZZ