The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 401 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 401

Prices can then be expressed as

M M M

S ψj

i = ∑ dijψj = ∑ pjdij----- = ∑ pjdijπj

j = 1 j = 1 pj j = 1

which demonstrates that S = E[Dππππ].

We can now specialize the above calculations in the numerical case

of the previous section. Recall that in the previous section we gave the

example of three securities with the following prices and payoffs

expressed in dollars:

S =

70

60

80

D =

50 100

30 120

38 112

We first compute the relative state prices:

50 ψ 1 + 100 ψ 2 = 70

30 ψ 1 + 120 ψ 2 = 60

38 ψ 1 + 112 ψ 2 = 80

Solving the first two equations, we obtain

ψ 1

ψ 2

⁴
₅

³
₁₀

=

However, the third equation is not satisfied by these values for the state

prices. As a consequence, there does not exist a state-price vector which

confirms that there are arbitrage opportunities as observed in the first

section.

Now suppose that the price of security C is $64 and not $80. In this

case, the third equation is satisfied and the state-price vector is the one

shown above. Risk-neutral probabilities can now be easily computed.

Here is how. First sum the two state prices: ⁴
₅ + ³⁄₁₀ = ¹¹⁄₁₀ to obtain