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Arbitrage opportunities are usually considered in the risky context only. In this section we first define arbitrage opportunities when there is ambiguity and derive the analogue of the first fundamental theorem of asset pricing in our single period context with a finite set of possible future outcomes Xt+1, but allowing for ambiguity. Next, we present our hypothesis on arbitrage free pricing that will be our maintained hypothesis when dealing with the anomalies that seem to reject the standard approach.

2.3.1 On the First Fundamental Theorem of Asset Pricing

From now on X_t+1 = S_t+1+ D_t+1, denoting the vector of total payoff of the assets at time t + 1, will be defined as

X_t+1: Xt+13 x 7→ x ∈ Xt+1.

Since Ωt+1 might be unknown, we simply take X_t+1 as the random vector being the identity transformation on Xt+1, avoiding in this way a specification of Ωt+1.²² Let there be J + 1 assets available, numbered from 0 to J, with asset 0 the numéraire, whose price is normalized to one. We assume that the numéraire pays off zero dividend. A portfolio at time t is given by a vector

22CompareAhn(2008).

Section 2.3 Arbitrage Free Pricing

23

h_t ∈ R^J+1, with time t price h_t· St and with total payoff at time t + 1 given by h_t· Xt+1= ht· St+1+ D_t+1

(where a · means taking an inner product). Given the zero probability set E⁰_t+1, the payoffs that are believed possible or possibly possible are given by h_t· x, for x ∈ Xt+1\E⁰t+1. Portfolio h_t is a zero investment portfolio in case its price is equal to zero, i.e., h_t · St = 0. Given the zero probability set E⁰_t+1, a zero investment portfolio h_t is an arbitrage opportunity in case h_t· x ≥ 0, for x ∈ Xt+1\E⁰_t+1, with at least one strict inequality. Thus, an arbitrage opportunity is a portfolio payoff, requiring zero initial investment, such that it generates a nonnegative payoff, unequal to zero, but only in case of outcomes that are believed to be possible or possibly possible. This definition is the usual one in case E_t+1⁰ = ∅.

Next, we present the analogue of the first fundamental theorem of asset pricing given an econometric model Et, with set of priors Πt, yielding as belief set Bt = Bt Et, Πt

, with corresponding zero probability set E⁰_t+1 = E⁰_t+1 Bt

. The support of a probability distribution P will be denoted by supp(P). For some given P with supp(P) ⊂ Xt+1, we write the expectation of some transformation f of the vector of payoffs X_t+1 with respect to this probability distribution P as E_P f X_t+1

. Then the First Theorem of Asset Pricing becomes:

Theorem 2.1

Given S_t,Xt+1, and the zero probability set E⁰_t+1(Xt+1, there are no arbitrage opportunities if and only if there exists some probability distribution Q_t with supp Qt

= Xt+1\E⁰_t+1, such that for all portfolios h_t we have h_t· St= E_Qt h_t· Xt+1
.

Proof. See Appendix.

With a single belief, this is just the standard first fundamental theorem of asset pricing, with supp Qt

determined by the support of the single belief. As in the standard case, the probability measure Qt is a martingale probability measure in our more general case: prices are the expected values of their payoffs under the measure Qt when arbitrage opportunities are absent. But now supp Qt

is determined by the set of beliefs. Indeed, with more than one belief this martingale probability measure need not be equivalent to any individual belief,

so that according to individual beliefs there might be arbitrage opportunities.

However, different arbitrage opportunities correspond to different beliefs, and it is the presence of ambiguity about these separate beliefs that prevents the possibility to exploit the belief specific arbitrage opportunities.²³

2.3.2 Maintained Hypothesis

As maintained hypothesis we shall impose that the asset prices are set such that according to the beliefs employed by the investors, interacting on the asset markets, there are no arbitrage opportunities. In general, different investors might have different beliefs, and there is no reason to exclude, for instance, non-rational beliefs. However, it makes sense to postulate that prices are set such that there are no arbitrage opportunities given at least rational and well-performing beliefs, since such beliefs cannot be rejected on the basis of empirical evidence, and also never exclude what might be possible. To deal with the “worst case scenario” of the previous section (to achieve well-performing beliefs), we shall restrict attention to rational and well-performing single-period beliefs. Thus, we postulate that the asset prices are set such that there are no arbitrage opportunities given rational and well-performing single-period beliefs. This hypothesis substantially extends the traditional hypothesis that prices are set such that there are no arbitrage opportunities given rational expectation-based single- and multi-period beliefs.

Our version of the first fundamental theorem of asset pricing shows that the assumption that asset prices exclude arbitrage is a characteristic of the equivalent class of all couples Et, Πt

, with set of beliefs Bt = Bt Et, Πt

, sharing the same zero probability set E⁰_t+1 = E⁰_t+1 Bt

. This means that, given S_t, Xt+1, and some E⁰_t+1, if we can find a martingale probability measure Qt, with supp Qt

= Xt+1\E⁰t+1, such that for all portfolios h_t we have h_t· St = E_Qt h_t· Xt+1
, so that there are no arbitrage opportunities, then any couple

Et, Πt

, such that the resulting zero probability set is equal to the given E⁰_t+1,

23A typical case is the term structure of interest rates. For instance, the Nelson-Siegel parametrization of the term structure might allow for arbitrage opportunities, when the uncertainty is generated by a single probability distribution P, see, for example,Björk and Christensen(1999) orFilipovi´c(1999). However, when using the Nelson-Siegel specification in applied work, such asDiebold and Li(2006), estimation inaccuracy (or even allowing for model misspecification) might generate ambiguity, potentially avoiding arbitrage opportunities.