- Risky Assets 61
Figure 3.5 Geometric interpretation of risk-neutral probabilityp∗Figure 3.6 Barycentric interpretation of risk-neutral probabilityp∗3.2.2 Martingale Property
By Proposition 3.4 the expectation ofS(n) with respect to the risk-neutral
probabilityp∗is
E∗(S(n)) =S(0)(1 +r)n, (3.5)
sincer=E∗(K(1)).
Example 3.6
Consider a two-step binomial tree model such thatS(0) = 100 dollars,u=0.2,
d=− 0 .1andr=0.1. Thenp∗=2/3 is the risk-neutral probability, and the
expected stock price after two steps is
E∗(S(2)) =S(0)(1 +r)^2 = 121dollars. After one time step, once it becomes known whether the stock price has
gone up or down, we shall need to recompute the expectation ofS(2). Suppose
that the stock price has gone up to $120 after the first step. In these circum-
stances the set of possible scenarios reduces to those for whichS(1) = 120
dollars, and the tree of stock prices reduces to the subtree in Figure 3.7. Given
thatS(1) = 120 dollars, the risk-neutral expectation ofS(2) will therefore be