60 Mathematics for Finance
in Chapter 5, where a precise definition of risk will be developed. The border
case of a market in whichE(K(1)) =ris referred to as risk-neutral.
It proves convenient to introduce a special symbolp∗for the probability as
well asE∗for the corresponding expectation satisfying the condition
E∗(K(1)) =p∗u+(1−p∗)d=r (3.4)for risk-neutrality, which implies that
p∗=r−d
u−d.
We shall callp∗therisk-neutral probabilityandE∗therisk-neutral expecta-
tion. It is important to understand thatp∗is an abstract mathematical object,
which may or may not be equal to the actual market probabilityp.Onlyina
risk-neutral market do we havep=p∗. Even though the risk-neutral probabil-
ityp∗may have no relation to the actual probabilityp, it turns out that for
the purpose of valuation of derivative securities the relevant probability isp∗,
rather thanp. This application of the risk-neutral probability, which is of great
practical importance, will be discussed in detail in Chapter 8.
Exercise 3.17
Letu=2/10 andr=1/10. Investigate the properties ofp∗as a function
ofd.Exercise 3.18
Show thatd<r<uif and only if 0<p∗<1.Condition (3.4) implies that
p∗(u−r)+(1−p∗)(d−r)=0.Geometrically, this means that the pair (p∗, 1 −p∗) regarded as a vector on
the planeR^2 is orthogonal to the vector with coordinates (u−r, d−r), which
represents the possible one-step gains (or losses) of an investor holding a single
share of stock, the purchase of which was financed by a cash loan attracting
interest at a rater, see Figure 3.5. The line joining the points (1,0) and (0,1)
consists of all points with coordinates (p, 1 −p), where 0<p<1. One of these
points corresponds to the actual market probability and one to the risk-neutral
probability.
Another interpretation of condition (3.4) for the risk-neutral probability is
illustrated in Figure 3.6. If massesp∗and 1−p∗are attached at the points
with coordinatesuanddon the real axis, then the centre of mass will be atr.