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Binomial
Trees
A useful and very popular technique for pricing an option involves constructing a
binomial tree.
This is a diagram representing different possible paths that might be
followed by the stock price over the life of an option. The underlying assumption is that
the stock price follows a
random walle
In each time step, it has a certain probability of
moving up by a certain percentage amount and a certain probability of moving down by
a certain percentage amount. In the limit, as the time step becomes smaller, this model
leads to the lognormal assumption for stock prices that underlies the BlackScholes
model we will be discussing in Chapter 13.
In this chapter we will take a first look at binomial trees and their relationship to an
important principle known as riskneutral valuation. The general approach adopted
here is similar to that in an important paper published by Cox, Ross, and Rubinstein in
1979. More details on numerical procedures involving binomial and trinomial trees are
given in Chapter 17.
11.1
A
ONESTEP
BINOMIAL MODEL
We start by considering a very simple situation. A stock price is currently $20, and it is
known that at the end of 3 months it will be either $22 or $18. We are interested in
valuing a European call option to buy the stock for $21 in 3 months. This option will
have one of two values at the end of the 3 months. If the stock price turns out to be $22,
the value of the option will be $1; if the stock price turns out to be $18, the value of the
option will be zero. The situation is illustrated in Figure 11.1.
It
turns out that a relatively simple argument can be used to price the option in this
example. The only assumption needed is that arbitrage opportunities do not exist. We
set up a portfolio of the stock and the option in such a way that there is no uncertainty
about the value of the portfolio at the end of the 3 months. We then argue that, because
the portfolio has no risk, the return it earns must equal the riskfree interest rate. This
enables us to work out the cost of setting up the portfolio and therefore the option's
price. Because there are two securities (the stock and the stock option) and only two
possible outcomes, it is always possible to set up the riskless portfolio.
Consider a portfolio consisting of a long position in
tl
shares of the stock and a short
position in one call option. We calculate the value of
that makes the portfolio riskless.
241
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CHAPTER 11
Figure 11.1
Stock price movements for numerical example in Section 11.1.
Stock price
=
$22
Option price
=
$1
Stock price
=
$20
Stock price
=
$18
Option price
=
$0
Ifthe stock price moves up from $20 to $22, the value of the shares is 22.6. and the value
 of the option is 1, so that the total value of the portfolio is 22.6. 
1. If the stock price
moves down from $20 to $18, the value of the shares is 18.6. and the value of the option
is zero, so that the total value of the portfolio is 18.6.
. The portfolio is riskless if the
value of .6. is chosen so that the final value of the portfolio is the same for both
alternatives. This means that
22.6.  1
=
18.6.
or
.6.
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 Spring '11
 DonBlasius
 Math, Binomial

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