Like most of modern
financial theory, the systematic valuation of
options is a surprisingly new field. In the 1960s,
traded options only existed on a few of the bigger
stocks. Prices were largely set by negotiations,
and rules of thumb prevailed. Meanwhile, the question
of option pricing was heating up among economic
academics.
INSERT BLACK-SCHOLES HISTORY FROM CAPITAL
IDEAS HERE.
Thus, Black and Scholes argued the appropriate
price for an option depends on a relatively small
number of variables:
- the current price of the underlying asset
- the strike price of the option
- the time remaining until the option's expiration
- the risk-free rate of interest
- the underlying asset's volatility.
Absent from the list are predictions of future
market prices or volatility, which are captured
by the asset's volatility compared to the market
and the market-set risk-free interest rate.
Most options are written with stock as the underlying
asset, and for call options the formula for the
Black-Scholes Option Pricing Model is
| Co
= N(d1)Vs - N(d2)E/ert |
Co |
= |
Value of call option |
N(d) |
= |
cumulative normal probability density function |
Vs |
= |
Current value of stock |
E |
= |
Exercise price |
e |
= |
base, natural log (2.71828) |
r |
= |
short-term annual interest rate (continuously
compounded) |
t |
= |
length of time to option expiration (in
years) |
d1 |
= |
[ln(Vs/E) + t(r + (s2/2)]
/ (s2t)1/2 |
d2 |
= |
[ln(Vs/E) + t(r + (s2/2)]
/ (s2t)1/2 |
ln |
= |
natural log |
s |
= |
sigma [standard deviation of annual rate
of return on stock (continuously compounded)] |
The formula seems daunting, and the theory behind
it is complicated. However, we will see that use
of the formula boils down to plugging in variables
and solving for Co.
All of the variables are observable in the Wall
Street Journal with the exception of the variance,
which will be the variance of the percentage change
in daily stock prices over some previous period.
(A note of caution: The variance is usually determined
using the recent past volatility of the stock.
We should not place total confidence in this method
as past volatility is not always the best predictor
of future volatility).
There are websites where investors can input
these variables to price options: WallStreetCom
The proper risk-free rate will be the rate corresponding
to the treasury bill maturing closest to the maturity
of the option being valued. Use the attached
table to find the values for N(d1)
and N(d2). For a d1 or d2
that is positive or above the mean of 0, determining
N(d1) or N(d2) will require
a minor adjustment [1-N(d1 or 2)].
For a negative d1 or d2
, no adjustment is necessary, and N(d1)
or N(d2) will simply be the number
from the table.
Let's quickly examine the formula in a little
more depth. N(d1) represents the ratio
of shares of stock to options needed to maintain
a fully hedged, or riskless, position. If we can
consider the option holder as a levered investor
and he or she borrows an amount equal to the exercise
price at a rate of r, the second term will then
represent this loan, the present value of the
exercise price multiplied by the adjustment factor,
N(d2). Ultimately, the price of an
option will be driven by the short-term interest
rate, time to expiration, and volatility of the
underlying stock. An increase in any of these
will translate into a higher price, but an increase
in the risk-free rate will usually have the least
impact.
Let's see the formula in action and observe its
sophisticated simplicity.
|
Example
Suppose you are contemplating the purchase of
a call option on the stock of Commander Cody Enterprises.
The option has a strike price of $18 and expires
in three months. Commander Cody's stock is currently
trading at $23, and you determine that the expected
standard deviation of its continuously compounded
return is .5. The current rate on a t-bill with
similar maturity is 6%. Would you pay $5.40 for
this option?
Answer: Using the Black-Scholes
Option Pricing Model, we can approximate the appropriate
price for this option. We begin by solving for
d1 using the formula:
= |
ln(Vs/E) + (r
+ (s2 /2)) * t
square root of (s2*t) |
= |
ln(23/18) + (.06 + (.52 /2))
* (91/366)
square root of (.52 * (91/366)) |
= |
.2451 + .046
.2493 |
= |
1.1677 |
We then
look on the probability table for our z
of 1.1677.
The table tells us that the N(d1
) is 0.8790. |
Next, we solve for d2 using the formula
from above:
= |
d1 - square root
of (s2 * t) |
= |
1.1677 - square root of (.52
* (91/366)) |
= |
1.1677 - .2493 |
= |
.9184 |
| We
then look on the probability table for our
z of .9184.
The table tells us that the N(d2
) is .8212. |
We are almost finished. We now solve for the
price of the call option by using the formula
from above:
= |
S[N(d1)] - E * e-rf*t
[N(d2)] |
= |
23(.8790) - 18 * (2.7183-.06*(91/366))
(.8212) |
= |
20.217 - 14.563 |
= |
$5.65 |
Thus, as we have put the price of the option
at $5.65, you would purchase these underpriced
options at $5.40.
What if the strike price was $20 instead of $18?
What if the option expired in one year instead of
three months? Use the attached
spreadsheet to explore these scenarios. This
spreadsheet can also be used to check your answers
on future problems. |
Through use of the Black-Scholes
Option Pricing Model, we can also value puts.
First, you must determine the value of a call
with the same terms as the put to be valued. Once
the price of the call is found, finding the price
of the put is easy:
| P0
= C0 - S0 + E * e-rt |
C0 |
Previously valued call |
S0 |
Stock Price |
E |
Exercise or strike price |
e |
Exponential function = approximately 2.71828 |
r |
Continuously compounded annual risk free
rate |
t |
Time to expiration on an annual basis |
Using the Black-Scholes
spreadsheet, you can consider the effects
of changing variables in the formula. |
|
4.4.1 Option Fundamentals