108156.pdf

3. Risky Assets 57

d)n−iat timen.Thereare

(n

i

)

such scenarios, the probability of each equal to
pi(1−p)n−i. As a result,

S(n)=S(0)(1 +u)i(1 +d)n−iwith probability

(

n

i

)

pi(1−p)n−i (3.3)

fori=0, 1 ,...,n.ThestockpriceS(n) at timenis a discrete random variable
withn+ 1 different values. The distribution ofS(n) as given by (3.3) is shown
in Figure 3.2 forn= 10,p=0.5,S(0) = 1,u=0.1andd=− 0. 1.

Figure 3.2 Distribution ofS(10)

The numberiof upward price movements is a random variable with a
binomial distribution. The same is true for the numbern−iof downward
movements. We therefore say that the price process follows abinomial tree.In
ann-step binomial three the setΩof all scenarios, that is,n-step paths moving
up or down at each step has 2nelements. An example of a two-step binomial
tree of stock prices is shown in Figure 3.3 and a three-step tree in Figure 3.4.

Figure 3.3 Two-step binomial tree of stock prices

In both figuresS(0) = 1 for simplicity.