8: OptionsWhy are the probabilities of no concern to us? There are two answers to this question. The first is
that the market sets the current price of the shares at a level that reflects the probabilities of future up-
and-down movements. In other words, the probabilities are embedded in the share price, even though
no one can see them directly.
The second answer is that the binomial model prices an option through the principle of ‘no arbitrage’.
Because it is always possible to combine a share with options (either calls or puts) into a risk-free
portfolio, the binomial model says that the value of that portfolio must be the same as the value of a risk-
free bond. Otherwise, an arbitrage opportunity would exist, because identical assets would be selling at
different prices. Hence, because the portfolio containing shares and options offers a risk-free payoff, the
probabilities of up and down movements in the share price do not enter the calculations. An investor
who holds the hedged portfolio doesn’t need to worry about movements in the shares, because they do
not affect the portfolio’s payoffs.
Almost all students object to the binomial model’s assumption that the price of a share can take just
two values in the future. Fair enough. It is certainly true that one year from today, the price of Financial
Engineers may be $70, $40 or almost any other value. However, it turns out that more complex versions
of the binomial do not require analysts to specify just two final prices for the shares. The binomial model
can accommodate a wide range of final prices. To see how this works, consider a slight modification to
our original problem.
Rather than presume that Financial Engineers shares will rise or fall by $15 each over a year’s time,
suppose they may rise or fall by $7.50 every six months. That’s still a big assumption; but if we make it,
we find that the list of potential prices of Financial Engineers shares one year from today has grown from
two values to three. Figure 8.8 proves this claim. After one year, the price of the shares may be $40, $55
or $70. Now let’s modify the assumption one more time. Suppose the price of the shares can move up or
down $3.75 every three months. Figure 8.8 shows that in this case, the number of possible share prices
one year in the future grows to five.
Given a tree with many branches like the one in Figure 8.8, it is possible to solve for the value of a
call or put option by applying the same steps we followed to value options using the simple two-step tree.
Now imagine a much larger tree, one in which the share price moves up or down every few minutes, or
even every few seconds. Each change in the share price is very small, perhaps a cent or two, but as the
tree unfolds and time passes, the number of branches rapidly expands, as does the number of possible
values of the shares at the option’s expiration date. If you imagine what this tree would look like, you can
see that when the option expires in a year, the price of Financial Engineers shares can take any one of
hundreds, or even thousands, of different values. Therefore, the complaint about the binomial model’s
artificial assumption of just two possible share prices no longer applies. Though extremely tedious,
solving for the call value involves working all the way through the tree, applying the same steps over and
over again.
The binomial model is an incredibly powerful and flexible tool that analysts can use to price
all sorts of options, from ordinary puts and calls to complex real options that are embedded in
capital investment projects. The genius of the model is in its recognition of the opportunity to use
shares and options to mimic the payoffs of risk-free bonds, the easiest of all securities to price.
That insight is also central to the second option pricing model that we discuss, the Black–Scholes
Model.
See the concept explained
step by step on the
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CONCEPTS