Unformatted text preview: cient, we
¯
¯
have V ∩ D = ∅. Therefore by the separating hyperplane theorem, there exists
0 ̸= λ ∈ RI such that ⟨λ, v ⟩ ≤ ⟨λ, u + d⟩ for any v ∈ V and d ∈ RI . Letting
¯
++
di → ∞, it must be λi ≥ 0, so λ ∈ RI . Letting d → 0, we get ⟨λ, v ⟩ ≤ ⟨λ, u⟩,
¯
+
which implies
I I λi u(xi ) ≤
i=1 λi u(¯i )
x
i=1 for any feasible {xi }. Hence {xi } maximizes W over all feasible allocations.
¯ 4.2 Asset pricing Consider an economy with two periods, denoted by t = 0, 1. Suppose that at
t = 1 the state of the economy can be one of s = 1, . . . , S . There are J assets in
the economy, indexed by j = 1, . . . , J . One share of asset j trades for price qj
at time 0 and pays Vsj in state s. (It can be Vsj < 0, in which case the holder
of one share of asset j must deliver −Vsj > 0 in state s.) Let q = (q1 , . . . , qJ )
the vector of asset prices and V = (Vsj ) be the matrix of asset payoﬀs. Deﬁne
W = W (q, V ) = −q ′
V 2014W Econ 172B Operations Research (B) Alexis Akira Toda be the (1 + S ) × J matrix of net payments of one share of each asset in each
state. Here, state 0 is deﬁned by time 0 and the presence of −q = (−q1 , . . . , −qJ )
means that in order to receive Vsj in state s one must purchase one share of
asset j at time 0, thus paying qj (receiving −qj ).
Let θ ∈ RJ be a portfolio. (θj is the number of shares of asset j an investor
buys. θj < 0 corresponds to shortselling.) The net payments of the portfolio θ
is the vector
−q ′ θ
Wθ =
∈ R1+S .
Vθ
Here the investor pays q ′ θ at t = 0 for buying the portfolio θ, and receives (V θ)s
in state s at t = 1.
Let ⟨W ⟩ = W θ θ ∈ RJ be the set of payoﬀs generated by all portfolios,
called the asset span. We say that the asset span ⟨W ⟩ exhibits noarbitrage if
⟨W ⟩ ∩ R1+S = {0} .
+
That is, it is impossible to ﬁnd a portfolio that pays a nonnegative amount in
every state and a positive amount in at least one state.
Theorem 10 (Fundamental Theorem of Asset Pricing). The asset span ⟨W ⟩
exhibits noarbitrage if and only if there exists π ∈ RS such that [1, π ′ ]W = 0.
++
In this case, the asset prices age given by
S qj = πs Vsj .
s=1 πs > 0 is called the state price in state s.
Proof. Suppose that such a π exists. If 0 ̸= w = (w0 , . . . , wS ) ∈ R1+S , then
+
S [1, π ′ ]w = w0 + πs ws > 0,
s=1 so w ∈ ⟨W ...
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 Winter '08
 Foster,C
 Convex set, Convex function, Alexis Akira Toda, Akira Toda

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