The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 403 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 403

mation structure It. The latter is a discrete, hierarchical organization of

partitions It with the following properties:

Ik ≡ ( {Aik }); k = 0 , , ...T; i = 1 , , ...Mk; 1 = M 1 ≤≤· Mk ≤≤· MT = M

Mk

Aik ∩Ajk = ∅if i ≠j and ∪Aik = Ω

i = 1

and, in addition, given any two sets Aik, Ajh, with h > k, either their

intersection is empty Aik ∩Ajh = ∅or Aik ⊇Ajh. In other words, the par-

titions become more refined with time.

Each security i is characterized by a payoff process dt

i

and by a

i i

price process St

i

. In this finite-state setting, dt and St are discrete vari-
ables that, given that there are M states, can be represented by M-vec-
i
ω
i
tors dt ω ω ω
i
= [dt()] and St = [St
i
()] where dt
i
() and St
i
() are,
respectively, the payoff and the price of the i-th asset at time t, 0 ≤t ≤T
and in state ω ∈ Ω. Following Chapter 6, all payoffs and prices are sto-
chastic processes adapted to the filtration ℑt. Recall from Chapter 6
i
that, given that dt
i
and St are adapted processes in a finite probability
space, they have to assume a constant value on each partition of the
information structure It. It is convenient to introduce the following
notation:

dAi jt = di t()ω , ω ∈Ajt

i

= St

i

SAjt ()ω, ω ∈Ajt

where i

i

dAi jt and SAjtrepresent the constant values that the processes dit

and St assume on the states that belong to the sets Ajt of each partition

It. There is M 0 = 1 value for dAi j 0

i

and SAi j 0 , Mt values for dAi jtand SAi jt

and MT = M values for dAi jT and SAjT. The same notation and the same

consideration can be applied to any process adapted to the filtration ℑt.

Trading Strategies

We have to define the meaning of trading strategies in this multiperiod

setting. A trading strategy is a sequence of portfolios θsuch that θt is the

portfolio held at time t after trading. To ensure that there is no anticipa-

tion of information, each trading strategy θmust be an adapted process.

The payoff dθgenerated by a trading strategy is an adapted process dθ

t

with the following time dynamics: