Options ... in the money

Once upon a time I played with call options, to answer the question: "What're the chances of making money?"

>Yes. That was here, right?
Right, so I thought ...

>You're not gonna do that again, are you?
Well, yes ... sorta. I just want to look at whether one should buy a call option with strike price below the stock price, or above and whether one should consider options that expire in a month or two or ...

>Okay, I got it. You done that already ... what else can you say?
I looked, again, at the necessary Compound Annual Growth Rate (CAGR) in order that you'd make money.
If K is the strike price and S the stock price when you buy the option and Rf the risk-free rate and V the volatility and I use the Black-Scholes magic to get the option premium C, namely:
C = S*NORMSDIST((LN(S/K)+(Rf+V^2/2)*T)/(V*SQRT(T)))-K*EXP(-Rf*T)*NORMSDIST((LN(S/K)+(Rf+V^2/2)*T)/(V*SQRT(T))-V*SQRT(T))
where T is the number of years to expiry and, if the option expires in n months, then T = n/12 years and if x = K/S we can calculate the required CAGR in order that the stock price increase from S to K + C as CAGR = [(K + C)/S]12/n = [x + C/S]12/n and we note that
C/S = NORMSDIST((-LN(x)+(Rf+V^2/2)*T)/(V*SQRT(T)))-x*EXP(-Rf*T)*NORMSDIST((-LN(x)+(Rf+V^2/2)*T)/(V*SQRT(T))-V*SQRT(T))
and ...

>A picture is worth a thousand ...
Okay, here's my picture: Figure 1

>So, unless you have inside information, you should buy a call where the stock price is higher than the strike price.
Yes, since it won't take miraculous CAGR to make money and ...

>And buy a call which expires many months from now.
Yes, because it doesn't take miraculous CAGR to make ...

>And that's it?
Well ... yes, except to note that lower volatility is good: Figure 2

>You kidding? And investors don't know that?
Well, maybe they do buy in-the-money options,
where S/K = 0.8 or thereabouts. Let's check GE call options in May, 2005 that expire in 7 months. The stock price is S = $36.
The popular choice has a strike K = $27.50 hence
K/S = 27.5/36 = 0.76

Figure 1

Figure 2

Note that some buy options out-of-the-money since the returns differ.

>How 'bout an example?
Okay, here's an example:
Suppose the stock increased at some annual rate, say r.
Then, in n months the stock would have increased from S   to   P = S(1+r)n/12
Then, if you exercised the option at expiry (in n months), you'd pay K for the stock, and it'd have cost you K + C in total (including the option premium), then you'd sell at P and your gain would be G = P / (K + C) = (1+r)n/12/ (K/S+ C/S) = (1+r)n/12/ (x + C/S)

>But that's a gain over n months, right?
Yes, annualized, the return would be G12/n - 1 = (1+r) / [x + C/S]12/n - 1

So, for our example, we assume the stock increases at the annual rate of r = 0.05 or 5%.
Then, at expiry, our annualized return, had we bought an option which expired in 1 month or 2 or 3 months and sold at expiry and ...


Figure 3
(BUT, see Figure 4, below!)

>So, an option for which K/S = 1.2 or 1.3 would provide a huge gain.
Could provide a huge gain. You take a greater risk and you get a greater reward.

Of course.

P.S. The required CAGR (in order to break even by the expiry date) has been included in the spreadsheet here.

Added, June 29, 2005  

So I get this e-mail from John W. who says about that last chart, Figure 3:
"Either it is the wrong chart or I should give up considering investing (i.e.speculating) in options."

>And you think, "What does he know", eh?
Uh ... no. At my age, creepin' senility gets in the way and ...

>Yeah, yeah, so what's the real chart look like?
Okay, look at Figure 1. See the black dot? It says that, in order to make money, the stock must increase at an annual rate of about 45%.
Aah, but that last chart assumes the stock increases at just 5% ... so you'd lose money.
But Figure 3 (incorrectly) says you'd make a bundle, something like a 50% annualized return, so ...

>So Figure 3 is wrong! I could've told you that!
Anyway, while Figure 1 plots   CAGR = [x + C/S]12/n   Figure 3 is supposed to plot   (1+r) / CAGR - 1, so ...

>So when one goes up the other goes down, right?
Uh ... yes. Instead of (1+r) / CAGR - 1, I actually plotted (1+r) CAGR - 1 so Figures 1 and 3 naturally look the same and ...

>Yeah, yeah, so what's the real chart look like?

Figure 4

>I could've told you that!