Here's the collection of charts which we need:

• gains (for any combination of buying and writing) are the sums of individual gains.
• Two charts with opposite slope give a sum with zero slope.
• Two charts with the same slope give a sum with double slope (either positive or negative).
• buying and writing (uncovered) both begin with zero slope whereas writing (covered) is the only graph which begins with positive slope.
• buying and writing (uncovered) both end with non-zero slopes (the former is positive, the latter negative) whereas writing (covered) is the only graph which ends with zero slope.
• Breaks (or should we say bends?) in the gains graph occur at strike prices of the individual buy and sell graphs.

Okay, let's start with a simple gains chart, designed for those who expect the underlying stock to increase, but want protection against drastic decreases in stock price (and are willing to accept limited gains for this protection). There are two calls involved, with strike prices of A and B ('cause that's where the breaks are located). For tiny stock prices (to the left of A) our gains chart has zero slope so each individual call will have zero slope and that means a buy and a write (uncovered) (each of which begins with zero slope). At A the gains graph rises, so A must be the strike price of the buy (cuz it's the only one which rises). At B the gains graph is horizontal, so B must be the strike price of the write (uncovered); its "gains graph" has a negative slope which cancels nicely with the positive slope of the buy leaving a sum with zero slope. Conclusion? BUY a call and WRITE an uncovered call at a higher strike price.

Example 1: Bull Spread

As y'all kin imagine, there's also a Bear Spread where y'all WRITE at a lower strike price.

Now a more complcated example where the gains chart is for those who expect the underlying stock to change somewhat ... maybe up ... maybe down ... but want protection against drastic changes in stock price. There are now three (count 'em) three calls involved, with strike prices of A, B and C. For tiny stock prices (to the left of A) our gains chart has zero slope so each individual call will have zero slope and that means buy and write (uncovered) calls. At A the gains graph rises, so A must be the strike price of a buy. At B the gains graph has negative slope, so B must be the strike price of a write (uncovered). AHA! A single write (uncovered) contract would have a negative slope which would just cancel the earlier buy (giving a horizontal "gains-slope") so, at B, we must write (uncovered) TWO contracts ... one to cancel the positively-sloped buy graph and a second one to take the slope of our gains graph negative. Finally, at C, in order to cancel the negative slope, we again buy a call ('cause it has a positive slope, right?). Conclusion? BUY a call and WRITE TWO uncovered calls at a higher strike price and finally BUY another call at an even higher strike price.

Example 2: Butterfly

Now an even more complcated example. There are now FOUR calls involved, with strike prices of A, B, C and D. Whooeee! For tiny stock prices (to the left of A) our gains chart has zero slope so each individual call will have zero slope and that means buy and write (uncovered) calls. At A the gains graph rises, so A must be the strike price of a buy. At B the gains graph has zero slope, so B must be the strike price of a write (uncovered) whose negative slope is just enuff to cancel the buy slope. At C, in order to acquire a negative slope, we again write (uncovered) to give a negative slope. Finally, at D, yet another buy to cancel the negative slope. Conclusion? BUY a call and WRITE an uncovered call at a higher strike, then WRITE yet another uncovered call at an even higher strike (!) and finally another BUY at an even higher strike price!! Mamma mia!

Example 3: Condor

Now for a real live example: Here's a list of october call options for AT & T with strike prices ranging from $40 to $60 and with associated option premiums from $6 5/8 to $1/16. (I've taken the price of the last trade for convenience ... my convenience.) The current stock price is $46 9/16. Note that the out-of-the-money options (like oct 60) can be bought for a song! Note, too, that in every case the strike price + the premium exceeds the current stock price of $46 9/16. (Remember Magic Formula 0 in part 1?) Of course, for the oct 40 option, the sum of $40 + $6 5/8 just barely exceeds the current price ... but then it's October 6 and the option expires in maybe a half dozen trading days! (This observation - that strike + premium is just a schnitzel greater than the current stock price - will be useful, later, in generating strategies.) Okay, suppose we buy an oct 40 contract for 100(6 5/8) = $662.50 (that's in-the-money) and write (uncovered) TWO oct 50s (receiving 2x100(1/4) = $50 for this sale (we're trying this Butterfly thingy) then we buy an oct 60 contract for 100(1/16) = $6.25 (so that our total out-of-pocket expense is $662.50 - $50 + 6.25 = $618.75).

Will we make any money in the next week or so?

Yes, if the stock price stays in the range $46 to $54 (about).

The difference here is that a longer term means higher option premiums and we get more $$ in the sale of our two options (although we pay more for the buy). This situation may or may not always occur. Just remember that our original cost (for buys and two uncovered writes) is also the starting point of the gains chart. From there this chart moves due East ... then North-East then South-East then due East again, bending at each strike price. If we start at a higher number we end up higher (meaning more gain).

• The stock is trading at $60.00
• We buy 2 calls with strikes prices above and below the current stock price, namely $55 and $65.
• They cost us $6.06 and $1.13 respectively. (Here, we use Black-Scholes to estimate the call premiums: see Part 6.)
• So far it has cost use $7.19, right?
• We write two calls with strike price = stock price, namely $60.
• That brings in 2x $3.33 = $6.66 so our net investment is now $7.19 - $6.66 = $0.53, or, for one (100 share) contract, $53.
• All calls expire in, say, 50 days. (I'm inventing this scenario, ya know!)
• We will make money unless the stock lies outside the range $55 to $65 (roughly ... stare at the chart).

Consider (at first) only buy and sell (uncovered). We could call it sell (naked) or write (short); we'll just say sell. Remember that the buy chart has a slope of +1 or +2 if we're buying two contracts or +3 if we're buying three ... Remember that the sell chart has a slope of -1 or -2 if we're selling two contracts or -3 if we're selling three ...

Step #1 Identify the Buy and Sell strike prices and their slopes (+1, +2, -1, -2, etc.). Step #2 Move due East then change slope by the amount indicated (+1, +2, -1, -2, etc.).

Indeed, we don't really need the S() and B(). We can just say, "I did an November Pfizer(1,-1,-1,1) today. How 'bout you?"

In fact, every possible sequence of integers (positiver or negative) - each associated with a strike price - will generate a strategy ... some good, most lousy.

Aah, but what we really want is to go the other way:
Sketch a gains chart then determine what buying and selling will yield that chart.

How about the sequence for ?