Here's the collection of charts which we need:
and a collection of rules:
 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 nonzero 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 
This strategy is called a Bull Spread and we've seen it before (in the collection of charts, above).
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 "gainsslope") so, at B, we must
write (uncovered) TWO contracts ...
one to cancel the positivelysloped 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 
This strategy is called a 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 
This strategy is called a Condor .
A bird of prey ... or a bird of pray?
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 outofthemoney 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 inthemoney) 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 outofpocket 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).
Well ... not too exciting. We could lose $600 or, in the best case, make $400
... and that requires the stock to practically stand still ... but then expiry is just
around the corner so it may not move much ... but maybe we should consider
longer terms to expiration.
Let's try next March ... I mean mar 40 and mar 50 and mar 60 options:
Now were in the chips if the stock price stays in the range $42 to $58, and the maximum
gain is now $800 and maximum loss is only $200. Neato!
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
NorthEast then SouthEast then due East again, bending at each strike price. If we start
at a higher number we end up higher (meaning more gain).
Here's the butterfly, again:
 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 BlackScholes 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).
Okay, what are the chances that this stock will, in 50 days, lie outside this range?
To reach $65 in 50 days, requires an annualized increase of (65/60)^{(365/50)} = 1.79
or an annualized gain of (about) 79%.
To drop to $55 in 50 days requires an annualized increase of (55/60)^{(365/50)} = 0.53
or an annualized gain of (about) 47%.
If we think this is unlikely, for a Blue Chip stock (for example), then it seems a
reasonable strategy yielding a possible gain of $446 (see chart, above), hence a percentage
gain of 446/53 = 8.42 or a gain of 842% (in 50 days). Whooee!
Of course, we could also lose all of our initial $53 investment ...
a 100% loss!
Okay, on to something else.
We need to make the shape of the gains
chart easier to construct.
 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.).


See how easy it is to catch a Butterfly?
The sequence is B(1) S(2) B(1).
For the Bull Spread the sequence is
B(1) S(1) and for the Condor it's
B(1) S(1) S(1) B(1). Piece o' cake, eh?
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
?
Go To PART 4
buying
writing covered
writing naked
gains
