The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 402 Wednesday, February 4, 2004 1:08 PM

402 The Mathematics of Financial Modeling and Investment Management

ψ 0 = ψ 1 + ψ 2 = ¹¹
₁₀

and consequently the risk-neutral probabilities:

ψψψψˆ

ψˆ 1

ψˆ 2

ψ 1 ⁄ ψ 0

ψ 2 ⁄ ψ 0

⁸
₁₁

³
₁₁

= = =

Risk-neutral probabilities sum to one while state prices do not. We can

now check if our market is complete. Write the following equations:

50 θ 1 + 30 θ 2 + 38 θ 3 = ξ 1

100 θ 1 + 120 θ 2 + 112 θ 3 = ξ 2

The rank of the coefficient matrix is clearly 2 as the determinant of the

first minor is different from zero:

50 30 = 50 × 120 – 100 × 30 = 300 ≠ 0

100 120

Our sample market is therefore complete and arbitrage-free. A portfolio

made with the first two securities can replicate any payoff and the third

security can be replicated as a portfolio of the first two.

ARBITRAGE PRICING IN A MULTIPERIOD FINITE-STATE

SETTING

The above basic results can be extended to a multiperiod finite-state set-

ting using the probabilistic concepts developed in Chapter 6. The econ-

omy is represented by a probability space (Ω,ℑ,P) where Ω is the set of

possible states, ℑ is the algebra of events (recall that we are in a finite-

state setting and therefore there are only a finite number of events), and

P is a probability function. As the number of states is finite, finite prob-

abilities P({ω}) ≡ P(ω) ≡ pω are defined for each state. There is only a

finite number of dates from 0 to T.

Propagation of Information

Recall from Chapter 6 that the propagation of information is repre-

sented by a filtration ℑt that, in the finite case, is equivalent to an infor-