108156.pdf

62 Mathematics for Finance

Figure 3.7 Subtree such thatS(1) = 120 dollars

2

3 ×144 +

1
3 ×108 = 132 dollars, which is equal to 120(1 +r). Formally, this can
be written using the conditional expectation^1 ofS(2) given thatS(1) = 120,

E∗(S(2)|S(1) = 120) = 120(1 +r).

Similarly, if the stock price drops to $90 after one time step, the set of possible
scenarios will reduce to those for whichS(1) = 90 dollars, and the tree of stock
prices will reduce to the subtree in Figure 3.8. Given thatS(1) = 90 dollars,

Figure 3.8 Subtree such thatS(1) = 90 dollars

the risk-neutral expectation ofS(2) will be^23 ×108 +^13 ×81 = 99 dollars, which
is equal to 90(1 +r). This can be written as

E∗(S(2)|S(1) = 90) = 90(1 +r).

The last two formulae involving conditional expectation can be written as a
single equality, properly understood:

E∗(S(2)|S(1)) =S(1)(1 +r).

This analysis can be extended to any time step in the binomial tree model.
Suppose thatntime steps have passed and the stock price has becomeS(n).
What is the risk-neutral expectation of the priceS(n+ 1) after one more step?

(^1) The conditional expectation of a random variableξgiven an eventAsuch that
P(A)
= 0 is defined byE(ξ|A)=E(ξ (^1) A)/P(A), where 1Ais the indicator random
variable, equal to 1 onAand 0 on the complement ofA.