There's a magic formula for determining the Fair Price that one should pay for a
call option.
>Formula?
Yes. See BlackScholes formula
Indeed, if:
 S is the current stock price (example: S = $25)
 V is the volatility (example: V = 0.25 meaning 25%)
 R is the riskfree rate (example: R = 0.04 meaning 4%)
 T is the time to expiry (example: T = 0.5 years)
 K is the strike price (example: K = $22)
then, if we use an Excel spreadsheet, the option call premium C (according to
BlackScholes) is given by:
C =
S*NORMSDIST((LN(S/K)+(R+V^2/2)*T)/(V*SQRT(T)))
 K*EXP(R*T)*NORMSDIST((LN(S/K)+(R+V^2/2)*T)/(V*SQRT(T))V*SQRT(T))
>That looks terrible! I mean ...
Pay attention. In order to make money on the purchase of the option:
 the current stock price, $S, must increase to $K + $C before the expiry date.
 That means that the stock must increase by a factor of (K+C)/S in T years.
 That means that the stock must increase at an annual rate of {(K+C)/S}^{1/T}  1
 If BlackScholes is an good estimate of the call premium, then, if we set
k = K/S we get
{(K+C)/S}^{1/T}  1 = (k + C/S)^{1/T}  1 and if we then use the BS
formula to determine C/S, we get an expression for the Required Annualized Return
(in order that the option expires with a stock price where you'd make money):
{
k +
NORMSDIST((LN(1/k)+(R+V^2/2)*T)/(V*SQRT(T)))
 (1/k)*EXP(R*T)*NORMSDIST((LN(1/k)+(R+V^2/2)*T)/(V*SQRT(T))V*SQRT(T))
}^{1/T}  1
If we assume, for example, that V = 0.25 and R = 0.04 then we can investigate how the
Required Annualized Return depends upon T (the time to expiry) and k (the ratio
StrikePrice/StockPrice).
For example, Figure 1 shows what happens when k increases (for various values
of T = time to expiry)
It appears that, if we buy an option with a StrikePrice well above the StockPrice
(that's outofthemoney) then we're expecting huge stock price increases.
If we look more closely at the graph when (StrikePrice)/(StockPrice) is under 100%, we get
Figure 2 which suggests that we should buy inthemoney call options ... where the
StrikePrice is, say 75%  80% of the current StockPrice.
Suppose we decide to buy an option with StrikePrice = 75% x StockPrice.
We must then decide upon T, the Time to expiry.
The chart below is then quite intesting.
Figure 3

Figure 1
Figure 2

>Are you saying I should buy options with a small strike price?
Isn't that a theoretical suggestion? Do call options actually obey the BlackScholes ...?
Okay. Let's look at
Microsoft,
today (July 9/02) and consider options which expire in October
 about 100 days  and we see the following:
Let's look at options with Strike Prices of $25, $35, $45 and $55.
If, for example, we buy a $25 option and pay $29, then the stock must increase to $25+$29
= $54.00
by expiry (in 101 days or 101/365 = 0.277 years). That means an equivalent ANNUAL increase of
(54.00/53.93)^{1/0.277}  1 = 0.0047 or 0.47%
On the other hand, if we go with the $55 option, paying $4.50, then the stock must increase
to $55+$4.50 = $59.50 which means an ANNUALized increase of
(59.50/53.93)^{1/0.277}  1 = 0.426 or 42.6%.
On the other hand, if we ...
>And what does BLACKSCHOLES say?
If we choose an appropriate Volatility (45%) and Riskfree Rate (4%) we get ANNUALized
returns pretty close to the actual ...
>Picture?
Yes.
>So, what are the chances that the stock price will be in the money?
At expiry? Good question.
Suppose:
 The stock has a (historical) Mean Annual Return of 9% and the
Standard Deviation (of Annual Returns) is 25%.
 The Annualized Return is then (approximately) SQRT[1.09^{2}  0.25^{2}]
 For an option which expires in 100 days, or 100/365 = 0.274 years, the expected Return
isn't 9%, of course,
but (1.09)^{0.274}  1 = 0.024 or 2.4%. Further, the SD for 100 days (or 0.274 years)
isn't 25% but 25*SQRT(0.274) = 13%.
