64 Mathematics for Finance
3.3.1 Trinomial Tree Model...............................
A natural generalisation of the binomial tree model extends the range of possi-
ble values of the one-step returnsK(n) to three. The idea is to allow the price
not only to move up or down, but also to take an intermediate value at any
given step.
Condition 3.3
The one-step returnsK(n) are independent random variables of the form
K(n)=
u with probabilityp,
n with probabilityq,
d with probability 1−p−q,whered<n<uand 0< p,q,p+q<1.
This means thatuanddrepresent upward and downward price movements,
as before, whereasnstands for the middle price movement, typically a neutral
one,n=0.
Condition 3.4
The one-step returnron a risk-free investment is the same at each time step
and
d<r<u.
SinceS(1)/S(0) = 1 +K(1),Condition 3.3 implies thatS(1) takes three
different values,
S(1) =
S(0)(1 +u)
S(0)(1 +n)
S(0)(1 +d)with probabilityp,
with probabilityq,
with probability 1−p−q.Exercise 3.20
How many different values does the random variableS(2) take? What
are these values and the corresponding probabilities?The conditionE∗(K(n)) =r for risk-neutral probabilitiesp∗,q∗can be
written as
p∗(u−r)+q∗(n−r)+(1−p∗−q∗)(d−r)=0. (3.6)