until there was a widely accepted approach to determining the value of
options. This goal was achieved with the introduction ofthe Black-Scholes
pricing model in 1972-1973. As one of the first pricing models, BlackScholes
is now considered inferior to more recent pricing models (such as
Cox-Rubinstein's binomial option pricing model). The Black-Scholes
model, although faster on a computer due to less-complex calculations,
did not take into consideration American-style options (that is, options
that enable early exercise). You can find an option-pricing calculator
online at www.marketcompass.com. or you can purchase them separately
as software from a variety of sources.
We return now to the function of our models. What we refer to as the
theoretical value of an option is the value determined by our particular
pricing model by using the six factors previously mentioned:
1. The price of the underlying stock
2. The option's strike or exercise price
3. The time until expiration of the option
4. The applicable interest rate
5. The anticipated volatility of the price movement of the underlying
security
6. Dividends (where applicable)
Entering values for the six (five if the company does not issue dividends)
required inputs into a pricing model will generate a theoretical
value for an option. A detailed discussion ofhow the various pricing models
work is beyond the scope of this book, and in the opinion of the
authors, this discussion is unnecessary for all but the most hard-core
market professionals. What is useful is a general discussion of how the
pricing formulas determine theoretical, or fair value and how the six factors
affect an options price. Furthermore, because each variable (except
for the strike price) is susceptible to change, you must be able to interpret
the values generated by the pricing model in order to understand how an
option price might react.
Although pricing models differ somewhat in the way in which they
assess data, they all essentially work the same way: Pricing models propose
a series ofpossible prices for the underlying security; assign a probability to
each price, and use this information to calculate the expected return
(expected value) as measured at expiration of an option that is purchased
with a particular exercise price. From this point, the model adjusts for any
applicable carrying costs (interest rate related) and determines a theoretically
fair value for the option. Your job, then, is to input the information
that you gather into the pricing model, acquire the probability-generated
fair value of the option, and find a bid/offer in the market place that will
enable you to establish an edge on that fair value.
Consider, for example, the odds that are associated with a game of
roulette. In roulette, the player attempts to pick one of the 38 slots on
which the ball will land. If the player chooses the correct slot, he or she
will win $36. For this opportunity; the casino charges $1. Each one of the
38 slots has an equal probability of hitting. The expected return on such
a bet played time after time is calculated by dividing the amount that is
capable of being won ($36) by the amount of probable outcomes (38). In
other words, we have 36/38 = .95. The resulting, expected return of the
bet is $.95; therefore, the fair value is $.95. In other words, a player who
pays $.95 to play will break. even over time. Hence, a player who pays less
than $.95 to play is getting a good deal-one that should produce a profit
over time. Paying more than $.95 to play is overpaying, which invariably
will result in losses over time. Casinos charge $1 to play because they
understand the mathematics of expected return.
Pricing Model Variables
Having reviewed the logic ofa pricing model and the procedures that are
necessary for using one, we now tum to the specific variables that you
will need to identify in order to set the pricing model in motion.
Underlying Stock Price. Obviously, the price of the underlying stock
is important for establishing the value of an option. Understanding how
future changes in the price might affect the value of an option will be a
significant factor in determining whether a particular option is an appropriate
investment choice, given the expectations that you might have for
the performance ofthe underlying security: We will discuss this aspect of
the pricing models in detail in our discussion of market risk. For now,
however, the Figure 4-4 indicates how a $1 change in the price of the
underlying security-with all other variables remaInIng constantaffects
the theoretical value of the 45 calls and 45 puts.
Strikel Exercise Price.
The exercise price is fixed throughout the life
ofthe option and will not change. Only in the case of a stock split would a
change to this value occur, and even so, this change would have no effect
on the theoretical value of an option.
Time until Expiration.
An option's price is directly related to the
amount of time until the option's expiration. When trading options, time
equals opportunity. Therefore, the more time that is attached to an
option, the greater its chance of finishing ITM. Consequently, a buyer is
willing to pay more for the added opportunity afforded by time on the
option. Consequently, the option seller will demand more for the added
risk that additional time requires him or her to assume. All else being
equal, then, an option that has more time is more valuable to an investor
and will therefore trade at a premium (as opposed to an option that has
less time remaining). Time until expiration is an important factor in
determining the next two factors affecting time premium: interest rates
and volatility.
For now, remember the following:
1. An option's expiration date is fIXed for the life of the option and will
not change.
2. Options that have a distant expiration date trade at a premium relative
to those that are approaching expiration.
3. As each day passes, the time to expiration decreases and the theoretical
value ofthe option erodes, thereby giving the option its status
as a wasting asset.
Read More: Pricing Models: An Overview
