So what's a Call option worth - and what does its price depend upon?
If you buy a Call with a strike price of $45, you can then buy the stock for $45 per share.
If the stock happens to be trading for $52, then you're going to pay
Aah, now suppose the expiry date is months away. The stock could very well be worth considerably
more than $52 by then ... so, unless the expiry date is tomorrow (or sooner!), you're going to
pay MORE than that $7. How much more? It depends upon the eventual price of the stock, umpteen
days into the future ... and that, of course, depends upon the number of So far, we've got the option premium depending upon four things, but there's more.
If we go through some elaborate calculation and determine that the "expected"
Now we have the option price depending upon five things.
Okay, it's time for the magic
C = S - K) has changed!
However, at expiry, t = 0, e ^{-rt} = 1, d_{1} = d_{2} = infinity
so N(d_{1}) = N(d_{2}) = 1 and we'll get, again,
C = S - K.
Stock Price, umpteen
days into the future: it's obtained by multiplying the current
Stock Price by some magic factor which takes into account the
randomness of this price over the time until expiry
- hence involves (as well as the
Stock and Strike
Prices) the Days to Expiry, the
Risk Free Rate and the Volatility
(which appears in the magic formula in terms of the Standard Deviation,
σ.
The second term computes the
The use of the word "expected" naturally implies some kind of probability distribution, and
that's
given by the normal distribution terms, N(d Personally, I think there are too many variables ... so if we express all the dollar values, like Stock Price (S) and Call Premium (C) as a multiple of the Strike Price (K) ... by dividing the Black-Scholes formula by K, we get something neat:
For example, we can now generate charts without knowing the Strike Price, but expressing
dollar values as multiples of a Strike Price.
So, does Black-Scholes do a good job of estimating
I followed Ford options, strike = $25, expiry in Oct, 2000, starting on Aug 31, 2000 and
it looks like the following chart (where I
Did Black-Scholes do good? Of course, if everybuddy uses Black-Scholes to make decisions regarding options, it sorta guarantees that Black-Scholes will do good, eh?
Oh, one other thing: in the above chart, I used the
Enough of the mathematical details - no need to understand them fully (I don't). Remember, it was worth a Nobel Prize, so you wouldn't expect it to be simple! It's more important to know how each of the five variables affect the Option Premium ... so here are some pretty pictures, each showing the variation of the Black-Scholes
Option Price when just one of the five variables changes.
Fig. 1: Dependence upon Stock Price Options way out-of-the-money are cheep, cheep. Who wants an option with strike = $50 when the stock is trading at $35?
- Out-of-the-money options (high Strike Price compared to Stock Price) tend to be cheep, cheep but increase in price
for longer Days to Expiry; you pay a so-called
*Time Premium*: the more time, the bigger the premium. - The cheep out-of-the-money option premiums increase dramatically for increasingly volatile stocks.
- Deep in-the-money options (low Strike Price, high Stock Price) increase in price
in synch with the stock price as the stock price gets larger. That means a
**delta**very nearly**1**. (Remember**delta**, from Part 5?) - Wanna know who
**delta**is?
Note the dependence of
For example, if the stock price is about 5% above the strike price, then delta is about 0.8 (where we're assuming 25% volatility and 6% risk-free rate).
Just in case y'all wanna stick this stuff into a spreadsheet:
Oh ... by the way, there's a significant difference in the "Cumulative Probabilities"
(which run from a probability of 0.00 to a probability of 1.00),
namely N(d
in-the-moneyNote: delta = N(d_{1}) is nearly 1
etc.
... but y'all git the idea, eh?Suppose we bought a call option for $9.80 with a Strike Price of $30. At the time we bought the option the stock was selling for $39 and there were 150 days to expiry. Now, here's a neat chart: Fig. 6
When we bought the call we had (Stock Price)/(Strike Price) = $39/$30 = 1.3 or
Now let's see. When the stock is at $45, that's $45/$30 = 1.5 or
Of course, the graphs depend upon the Volatility and Risk-free Rate ... so we've picked a couple of numbers just to give y'all the general idea. One last thing: notice that, as the number of days left (to expiry) approaches zero, the curves approach a straight line through (100%,0%), with slope "1". That means that when the Stock Price increases by 10% (of the Strike Price), the Option Price increases by 10% (of the Strike Price) ... and the "3-days to expiry" curve is almost there! Okay, one more last thing: In our example above, we bought a Call option with Strike Price of $30 and we paid $9.80 for it, and the stock itself was available on the market for $39. In order to make any money, the Stock Price had better increase to $30.00 + $9.80 = $39.80 in the next 150 days. That's a gain of $39.80/$39.00 = 1.02 or 2% (in 150 days). Is that too much to expect? It's easier to compare if this gain factor is annualized: 1.02 ^{365/150} = 1.05 so it
means our stock must increase at an annual rate at least 5% (over the next 150 days) in order to
make money. Now that's not asking much, eh?
So, we ask ourselves (when buying a Call option):
The result is sometimes surprising! Here's an example: - It's Aug 25 and we look at AT&T oct/30 Calls. We can buy it for $2
^{5}/_{8}= $2.625 - From Aug 25 to Oct 21 (when the Call expires) is
**56**days. - To make money, the Stock Price must exceed $30 + $2.625 = $
**32.625** - The stock is currently trading at $
**31.00** - That means a gain factor (in
**56**days) of**32.625**/**31.00**= 1.052 hence an*annualized*gain factor of 1.052^{365/56}= 1.40 or**40%**.
So, is the moral to BUY deep in-the-money Calls?
Of course, if we WRITE a
Remember:
One-of-these-days, I'll spend a day computing, for a jillion Call options, the annualized percentage gain necessary to make money
... and consider buying the option which generates the minimum number! one last thing? I lied. Here are some "required annualized gains" bumpf for some heavily traded stocks ... using the actual numbers, as of Aug. 25/00 (no Black-Scholes estimates):
:(
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