INFORMES Y OTROS TIPOS DE INFORMACIÓN

In this section we will consider the positive cone of L∞(P ) excluding zero, i.e. L+∞(P )\{0}.

Note that the set

{w ∈ L∞(P ) : P (w ≥ 0) = 1 and P (w > 0) > 0},

as considered in [63], is equal to L+∞(P )\{0}.

We will interpret each element w ∈ L+_∞(P )\{0} as the time T payoff of a claim.

If we short sell this claim, then we will receive a positive amount now, but will have to pay back −w at time T . But today we could also choose an admissible trading strategy, with zero (or negative) initial cost, that might ‘hedge’ the claim w. At time t, our payoff will then be f − w, where f ∈ C.

Lemma 3.4.1. If for some w ∈ L+_∞(P )\{0} and f ∈ C f − w ≥ 0 P -a.s., then the no arbitrage condition is violated.

Proof. Select w ∈ L+_∞(P )\{0} and f ∈ C = (K−L+₀(P ))∩L∞(P ) such that f −w ≥ 0 P -a.s..

Since f ∈ (K − L+₀(P )), we can write f = k − l where k ∈ K and l ∈ L+₀(P ). Then f − w = k − (l + w),

where l + w ∈ L+₀(P ). Hence f − w ∈ (K − L+₀(P )) and thus f − w ∈ C. In other words, f − w ∈ C ∩ L+_∞(P ), which implies that arbitrage is possible.

Definition 3.4.2. The essential infimum, denoted by ess inf f , is given by ess inf f = sup{z : f ≥ z a.e.}.

Lemma 3.4.3. The following are equivalent. (i) There is FLVR.

(ii) There exist w ∈ L+∞(P )\{0} and a sequence (fn) ⊆ C such that

lim

n→∞kfn− wk∞= 0. (3.3)

(iii) There exist w ∈ L+_∞(P )\{0} and f ∈ C such that sup

f ∈C

{ess inf

Ω (f − w)} ≥ 0. (3.4)

Proof. To show (i) ⇔ (ii) is trivial.

(ii) ⇒ (iii): Suppose there exist w ∈ L+∞(P )\{0} and a sequence (fn) ⊆ C such that

lim

n→∞kfn− wk∞= 0. Let ˜fn= fn− (fn− w)

+ _{for all n ∈ N. Then ( ˜f}

n) ⊆ C and −1 n ≤ ˜fn− w ≤ (fn− w) − (fn− w) +_{≤ 0.} Hence, ess inf Ω ( ˜fn− w) ≥ − 1 n, and so sup f ∈C {ess inf Ω(f − w)} ≥ 0.

(iii) ⇒ (ii): Suppose there exist w ∈ L+∞(P )\{0} and f ∈ C such that

sup

f ∈C

{ess inf

Ω (f − w)} ≥ 0. Then, for all n ∈ N, there exists fn∈ C such that

ess inf

Ω (fn− w) ≥ −

1 n.

Let ˜fn= fn− (fn− w)+. Then ˜fn∈ C and

−1

n ≤ ˜fn− w ≤ (fn− w) − (fn− w)

+ _{≤ 0 ≤} 1

n. Hence, we have that

lim

n→∞|| ˜fn− w||∞= 0.

Let U be a certain set of utility functions u : R → [−∞, ∞]. We assume that the preference ”  ” of the investors in the market under consideration can be represented by the expected utility, i.e.

f1 f2 ⇔ EQ[ u(f1) ] ≥ EQ[ u(f2) ],

where Q ∈ P, u ∈ U and f1, f2∈ L0(P ).

Frittelli [63] introduced the notion of a market free lunch that depends on the preferences of the investors in the market. Market free lunch with respect to U is defined as follows. Definition 3.4.4. [63, Definition 3] There is a market free lunch with respect to U if for all P ∈ P and u ∈ U, there exists w ∈ L+

∞(P )\{0} such that

sup

f ∈CE

P[ u(f − w) ] ≥ u(0). (3.5)

Hence, there is no market free lunch (NMFL(U)) with respect to U if for all w ∈ L+

∞(P )\{0}

there exist P ∈ P and u ∈ U such that sup

f ∈CEP

[ u(f − w) ] < u(0).

This definition clearly depends on the set of utility functions, which we choose. The above definition only makes economical sense if our utility function is non-decreasing on R. Consider w ∈ L+

∞(P )\{0} such that Equation 3.5 holds. A market free lunch implies that

all investors in the market, who are represented by their beliefs P ∈ P and their preferences u ∈ U, regard the risk w as a free lunch, as each investor can hedge the risk g in such a way that their preferences and beliefs are not compromised.

Consider the following families of utility functions

U0 = {u : R → [−∞, ∞] : u is non-decreasing on R},

U1 = {u ∈ U0 : u is left continuous at 0 ∈ int(dom(u))} and

U2 = {u ∈ U0 : u is finite-valued and concave on R}.

Proposition 3.4.5. (i) NMFL(U0) ⇔ NA.

(ii) NMFL(U1) ⇔ NFLVR.

Proof. ([63, Proposition 5]) (i): Assume that an arbitrage opportunity exists. We need to show that there exists w ∈ L+

∞(P )\{0} such that (3.5) holds. Consider w ∈ C ∩L+∞(P )\{0}.

Since L+∞(P )\{0} ⊆ L∞(P ), we have that w ∈ C ∩ L∞(P ), which shows that w is an

arbitrage opportunity. Then, for u ∈ U0

sup

f ∈CE

P[ u(f − w) ] ≥ EP[ u(w − w) ] = u(0),

i.e. there is a MFL(U0).

Conversely, assume there exists a MFL(U0), i.e. there exists w ∈ L+∞(P )\{0} such that

sup

f ∈C

EP[ u(f − w) ] ≥ u(0) holds for all P ∈ P and u ∈ U0. Take P ∈ P and define u ∈ U0 by

u(x) = ( 0 for x ≥ 0 −∞ for x < 0. Thus, sup f ∈CEP

[ u(f − w) ] ≥ 0 and due to the definition of u, there must exist f ∈ C such that P (f − w ≥ 0) = 1, i.e. by Lemma 3.4.1, there is an arbitrage opportunity.

(ii): Let n ≥ 1. Suppose there is a free lunch with vanishing risk. Then, by Lemma 3.4.3, there exist w ∈ L+∞(P )\{0} and a sequence (fn) ⊆ C such that (3.4) is satisfied.

From the left continuity of u at 0, there exists n > 0 such that if −n < x ≤ 0, then

u(x) > u(0) −_n1.

For each n, there exists δn> 0 such that lim

n→∞kfn− wk∞> δn. Set ˜fn:= fn−(fn−w) +_.

Then ˜fn ∈ C and −δn< ( ˜fn− w) ≤ 0 P -a.s.. Hence, u( ˜fn− w) > u(0) −_n1 P -a.s.. Thus,

we can conclude that sup

f ∈C

EP[ u(f − w) ] ≥ u(0), i.e. a MFL(U1) exists.

Conversely, suppose that there exists a MFL(U1). Take P ∈ P and for all n ≥ 1, define

un by

un(x) =

(

0 for x > −_n1 −∞ for x ≤ −_n1.

Then un∈ U1. By assumption there exists w ∈ L+∞(P )\{0} such that sup f ∈C

EP[ un(f − w) ] ≥

un(0) = 0 for all n ≥ 1. Hence, there exists fn ∈ C such that P (fn− w ≤ −_n1) = 0, i.e.

such that ess inf(fn− w) ≥ −_n1 for all n ≥ 1. Thus, sup f ∈C

{ess inf(f − w)} ≥ 0, which, by Lemma 3.4.3, implies that there exists a FLVR.

This proposition shows that the difference, from an economic perspective, between NA and NFLVR is due to the differing preferences of the investors.

Under this new concept of NMFL, Bellini and Frittelli [11] proved another version of the fundamental theorem of asset pricing.

Theorem 3.4.6. For any semimartingale S, the following are equivalent: (i) S satisfies NMFL(U2),

(ii) M ∩ P 6= ∅.

We will not prove this theorem here, as we will be considering an alternative version of it in Section 3.5, which we will then prove. See [14] for a proof of Theorem 3.4.6.

If S is not locally bounded, then M cannot be reduced to a simpler form. Hence, we need to introduce a new setup, introduced by Frittelli [64], which allows us to work with unbounded processes.