Mathematical Modeling in Finance with Stochastic Processes

2.2. MULTIPERIOD BINOMIAL TREE MODELS 83

(n+ 1,j) and (n+ 1,j+ 1) at the next trading time, with the “up” change
corresponding to (n+ 1,j+ 1) and the “down” change corresponding to
(n+ 1,j). The indexjcounts the number of up changes to that time, so
n−jis the number of down changes. Several paths lead to node (n,j), in
fact

(n

j

)

of them. The price of the risky underlying asset at trading timetn

is thenSUjDn−j. The probability of going from priceSto priceSUjDn−jis

pn,j=

(

n

j

)

pj(1−p)n−j.

To value a derivative with payoutf(SN), the key idea is that of dynamic
programming — extending the replicating portfolio and corresponding port-
folio values back one period at a time from the claim values to the starting
time.
An example will make this clear. Consider a binomial tree on the times
t 0 ,t 1 ,t 2. AssumeU= 1.05,D= 0.95, and exp(r∆ti) = 1.02, so the effective
interest rate on each time interval is 2%. We takeS 0 = 100. We value a
European call option with strike priceK= 100. Using the formula derived
in the previous section

π=

1. 02 − 0. 95

1. 05 − 0. 95

= 0. 7

and 1−π= 0.3. Then concentrating on the single period binomial branch in
the large square box, the value of the option at node (1,1) is $7.03. Likewise,
the value of the option at node (1,0) is $0. Then we work back one step and
value a derivative with potential payouts $7.03 and $0 on the single period
binomial branch at (0,0). This uses the same arithmetic to obtain the value
$4.82 at time 0. In the figure, the values of the security at each node are in
the circles, the value of the option at each node is in the small box beside
the circle.
As another example, consider a European put on the same security. The
strike price is again 100. All of the other parameters are the same. We work
backward again through the tree to obtain the value at time 0 as $0.944. In
the figure, the values of the security at each node are in the circles, the value
of the option at each node is in the small box beside the circle.
The multiperiod binomial model for pricing derivatives of a risky security
is also called theCox-Ross-Rubenstein modelorCRR modelfor short,
after those who introduced it in 1979.