82 Mathematics for Finance
following three cases:
1)a= 0 (a trivial portfolio consisting of no cash and no stock). ThenV(1) = 0
identically.
2)a>0 (a cash loan invested in stock). ThenV(1) =a(d−r)<0 if the price
of stock goes down.
3)a<0 (a long position in bonds financed by shorting stock). In this case
V(1) =a(u−r)<0 if stock goes up.Arbitrage is clearly impossible whend<r<u.
The above argument shows thatd<r<uif and only if there is no arbitrage
in the one-step case.
Several steps.Letd<r<uand suppose there is an arbitrage strategy.
The tree of stock prices can be considered as a collection of one-step subtrees,
as in Figure 4.1. By taking the smallestnfor whichV(n)= 0, we can find a
one-step subtree withV(n−1) = 0 at its root andV(n)≥0ateachnode
growing out of this root, withV(n)>0 at one or more of these nodes. By the
one-step case this is impossible ifd<r<u, leading to a contradiction.
Figure 4.1 One-step subtrees in a two-step binomial modelConversely, suppose that there is no arbitrage in the binomial tree model
with several steps. Then for any strategy such thatV(0) = 0 it follows that
V(n) = 0 for anynand, in particular,V(1) = 0. This implies thatd<r<u
by the above argument in the one-step case.
We shall conclude this chapter with a brief discussion of a fundamental re-
lationship between the risk-neutral probability and the No-Arbitrage Principle.
First, we observe that the lack of arbitrage is equivalent to the existence of a
risk-neutral probability in the binomial tree model.
Proposition 4.3
The binomial tree model admits no arbitrage if and only if there exists a risk-
neutral probabilityp∗such that 0<p∗<1.