- Risky Assets 65
The triple (p∗,q∗, 1 −p∗−q∗) regarded as a vector inR^3 is orthogonal to the
vector with coordinates (u−r, n−r, d−r) representing the possible one-step
gains (or losses) of an investor holding a singe share of stock, the purchase of
which was financed by a cash loan. This means that (p∗,q∗, 1 −p∗−q∗) lies on
the intersection of the triangle{(a, b, c):a, b, c≥ 0 ,a+b+c=1}and the plane
orthogonal to the gains vector (u−r, n−r, d−r), as in Figure 3.9. Condition 3.4
Figure 3.9 Geometric interpretation of risk-neutral probabilitiesp∗,q∗guarantees that the intersection is non-empty, since the line containing the
vector (u−r, n−r, d−r) does not pass through the positive octant. In this
case there are infinitely many risk-neutral probabilities, the intersection being
a line segment.
Another interpretation of condition (3.6) for the risk-neutral probability is
illustrated in Figure 3.10. If massesp∗,q∗and 1−p∗−q∗are attached at the
points with coordinatesu,nanddon the real axis, then the centre of mass
will be atr.
Figure 3.10 Barycentric interpretation of risk-neutral probabilitiesp∗,q∗Exercise 3.21
Letu=0.2,n=0,d=− 0 .1, andr= 0. Find all risk-neutral probabili-
ties.