The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 433 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 433

It can be verified that, under the previous assumptions and provided

prices are positive, the above system admits infinite solutions. In fact, as

N + 1 < 2N, the number of equations is larger than the number of

unknowns. Therefore, if the system is solvable it admits infinite solu-

tions. To verify that the system is indeed solvable, let’s choose the first

asset and partition the set of states into two events corresponding to the

movement up or down of the same asset. Assign to these events proba-

bilities as in the binomial model

1

q^11 – r + dt^1

t = ----------------------- 1 1 - and 1 – qt

ut – dt

Choose a second asset and partition each of the previous events into

two events corresponding to the movements up or down of the second

asset. We can now assign the following probabilities to each of the fol-

lowing four events:

q1 2^2 1221

t qt , qt

(^1) ( 1 – q
t ), (^1 – qt )qt , (^1 – qt )(^1 – qt )
It can be verified that these numbers sum to one. The same process
can be repeated for each additional asset. We obtain a set of positive
numbers that sum to one and that satisfy the system by construction.
There are infinite other possible constructions. In fact, at each step, we
could multiply probabilities by “correlation factors” (i.e., numbers that
form a 2 × 2 correlation matrix) and still obtain solutions to the system.
We can therefore conclude that a system of positive binomial prices
such as the one above plus a risk-free asset is arbitrage-free and forms
an incomplete market. Recall from Chapter 8 that if we let the number
of states tend to infinity, the binomial distribution converges to a nor-
mal distribution. We have therefore demonstrated heuristically that a
multivariate normal distribution plus a risk-free asset forms an incom-
plete and arbitrage-free market. Note that the presence of correlations
does not change this conclusion.
Let’s now see under what conditions this conclusion can be changed.
Go back to the multiple binomial model, assuming, as before, that there
are N assets and T time steps. There is no logical reason to impose that
the number of states be 2NT. As we can consider each time step sepa-
rately, suppose that there is only one time step and that there are a num-
ber of states less than or equal to the number of assets plus 1: M ≤ N + 1.
In this case, the martingale condition that determines risk-neutral proba-
bilities becomes: