Call Option Strategies a continuation of
Part II

Here's what we'll do:
 Assume a Return R = 0.10 (meaning 10%) and Standard Deviation S = 0.25
(meaning 25%) for some stock.
 Assume we're interested in some strategy (Bull Spread? Butterfly? Condor?) involving
Call Options
which expire in N = 180 days.
 We calculate the losing range of stock prices: the range of stock prices
where we lose money.
 With these three parameters (R, S and N)  and some assumption
concerning the distribution of returns (normal? lognormal?) 
we create a probability distribution for the stock prices after N days (at expiry).
 From this distribution of prices (at expiry of our Option) we determine the probability that
the
stock price lies in the losing range.
 We change the strategy (Bull Spread? Butterfly? Condor?)
and keep our eye on the Probability of Losing.
>Example, please.
Okay. Suppose the distribution of prices, at expiry, is as shown in Fig. 1 and
suppose, further, that stock prices below $27 will cause us to lose money.
>And the probability of losing, eh?
Yes, and in particular ...
>Go for it!
Okay, here's a picture (Fig. 2) with a Bull Spread:
 The stock currently trades at $100.
 We Buy a Call with a Strike Price of $95 and pay $11.65.
 We Write an (uncovered) Call with Strike at $105 and receive $7.10.
So far we've paid an initial cost of $11.65  $7.10 = $4.55.
 If the stock price is $99.55 we exercise our option (receiving a $99.55 stock for $95)
thereby making $99.55  $95.00 = $4.55 which exactly covers our initial cost.
 We stare at the pertinent cumulative distribution and note that this Break Even price will
be attained 38% of the time.
 We conclude that, 38% of the time, we will lose money.

Figure 2

>You invented the option premiums, the $11.65 and $7.10?
Well, I'm assuming a 10% Annual Return, a 20% Standard Deviation, a 4% Riskfree Rate,
200 days to expiration and I've used BlackScholes and ...
>And the distribution?
I've assumed a Normal distribution and ...
>Is that valid? I mean, doesn't BlackScholes assume a Lognormal distribution?
Well ... yes, but I'm trying to demonstrate the idea of superimposing the Gain/Loss chart
(for a Bull Spread, for example) with some Cumulative Distribution so that we can
identify some probability of a loss.
>Why not assume a Lognormal distribution?
Yes, we could do that, but remember that everybody stares at BlackScholes when trading in
options so we accept BlackScholes when calculating the option premium but we can assume any
distribution we like for the
future distribution of stock prices. It could be Normal or Lognormal or something else.
>And the probability is 38% ... for a Bull Spread?
Of course, not! It depends upon so many parameters. For example, if the Standard Deviation
is increased to 25% and the timetoexpiry decreased to 100 days (and we keep the Normal distribution)
we'd get a higher probability of loss ... like so:
>I assume those "Gains" are for 100share contracts?
Yes and ...
>Do you have a spreadsheet to do all this?
Uh ... give me some time ... but the spreadsheet will probably look like
this
with an explanation which looks like
this.
>Can I try it?
Sure ... such as it is!
Just
RIGHTclick here and select
"Save Target".
See also more on Call Options and
BlackScholes.
