The risk-neutral non-additive probability with market frictions

The fundamental theory of asset pricing has been developed under the two main assumptions that markets are frictionless and have no arbitrage opportunities. In this case the market enforces that replicable assets are valued by a linear function of their payoffs, or as the discounted expectation with respect to the so-called risk-neutral probability. Important evidence of the presence of frictions in financial markets has led to study market pricing rules in such a framework. Recently, Cerreia-Vioglio et al. (J Econ Theory 157:730–762, 2015) have extended the Fundamental Theorem of Finance by showing that, with markets frictions, requiring the put–call parity to hold, together with the mild assumption of translation invariance, is equivalent to the market pricing rule being represented as a discounted Choquet expectation with respect to a risk-neutral nonadditive probability. This paper continues this study by characterizing important properties of the (unique) risk-neutral nonadditive probability vf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_f$$\end{document} associated with a Choquet pricing rule f, when it is not assumed to be subadditive. First, we show that the observed violation of the call–put parity, a condition considered by Chateauneuf et al. (Math Financ 6:323–330, 1996) similar to the put–call parity in Cerreia-Vioglio et al. (2015), is consistent with the existence of bid-ask spreads. Second, the balancedness of vf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_f$$\end{document}—or equivalently the non-vacuity of its core—is characterized by an arbitrage-free condition that eliminates all the arbitrage opportunities that can be obtained by splitting payoffs in parts; moreover the (nonempty) core of vf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_f$$\end{document} consists of additive probabilities below vf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_f$$\end{document} whose associated (standard) expectations are all below the Choquet pricing rule f. Third, by strengthening again the previous arbitrage-free condition, we show the existence of a strictly positive risk-neutral probability below vf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_f$$\end{document}, which allows to recover the standard formulation of the Fundamental Theorem of Finance for frictionless markets.


Introduction
This paper 1 considers a two-date stochastic model, where today (t = 0) is known and tomorrow (t = 1) is uncertain; the uncertainty is represented by a set Ω of states of nature, one (and only one) of which will be disclosed tomorrow. The set Ω will be finite in the whole paper. A (stream of) payoff is a random variable x : Ω → R or a vector x ∈ R Ω , where x(ω) is the payoff (money) at t = 1 if state ω prevails. We adopt the convention that if x(ω) < 0 (resp. > 0), then |x(ω)| is paid (resp. gained). A financial market is defined as a family of securities that allows investors to generate payoffs by choosing adequately the quantities of the securities that are bought or sold. The hedging price associated with this market is then a function f : R Ω → R that associates to every payoff x ∈ R Ω the price (or the cost) f (x) to be paid today for the delivery of x at t = 1; here, by convention, | f (x)| is paid (resp. gained) if f (x) > 0 (resp. < 0), in other words, a negative cost is a gain. We refer to Cerreia-Vioglio et al. (2015a) and Chateauneuf and Cornet (2022) for the detailed presentation of a market and the way to derive its hedging price f : R Ω → R from underlying given securities. Here, the hedging price f , taken as the primitive concept, is supposed to be exogenously given.
The study of markets with frictions has led to consider Choquet pricing rules that are subadditive (instead of linear in the frictionless case) as in Chateauneuf et al. (1996) and Chateauneuf and Cornet (2022). In this paper pricing rules are not assumed to be subadditive and we recall the following important generalization of the Fundamental Theorem of Finance.
(i) f satisfies: [ Monotonicity, Translation Invariance, and Put-Call Parity; (iii) there exists a nonadditive probability v and a risk-free rate r > −1 such that The first two assumptions in (i) are quite standard in the literature. The last one of Constant Modularity, introduced in financial markets by Cerreia-Vioglio et al. (2015a), can be equivalently formulated in financial terms by the put-call parity; see the PCP Condition defined and discussed hereafter. Further generalizations are provided by Cerreia-Vioglio et al. (2015b) to deal with pricing rules that may not be monotone or may not be defined on the whole space R Ω , a framework that goes beyond the scope of our paper.
We recall that a nonadditive probability is a set function v : for the definition and properties of the Choquet integral, we refer to Denneberg (1994) and Marinacci and Montrucchio (2004). Let f : R Ω → R be monotone, and translation invariant (hence in particular when f is a monotone Choquet pricing rule), one checks that f (1 Ω ) > 0 if and only if f = 0. Thus under the assumptions of Theorem 1, we define r f ∈ R and the set function v f : 2 Ω → R by: As noticed in Cerreia-Vioglio et al. (2015a), the interest rate r and the nonadditive probability v, associated with f by Theorem 1 are uniquely defined and coincide respectively with r f and v f (as defined above), i.e., r = r f and v = v f . Hereafter, r f will be called the risk-free interest rate associated with f , and v f the risk-neutral nonadditive probability associated with f .
We can now present the main results of our paper, which studies the properties of the risk-neutral nonadditive probability v f , when f is a Choquet pricing rule. First, we notice that the converse of the above Assertion (b) of Theorem 1 may not be true (unless f is assumed to be subadditive) as shown by Example 1 in the Appendix. Second, Theorem 2, characterizes the Positive Bid-Ask Spread Condition, i.e., f (−x)+ f (x) ≥ 0 for all x, by a weak form of the Call-Put Parity Condition introduced by Chateauneuf et al. (1996). The Positive Bid-Ask Spread Condition with x ≥ 0 also implies the absence of "buy and sell" arbitrage opportunities, that is, there is no payoff , thus the cost f (x) ≥ 0 of buying x is smaller than the gain − f (−x) ≥ 0 of selling the same payoff x. Equivalently, for K = 2, there is no x k ∈ R Ω + ∪−R Ω + (k = 1, . . . , K ) such that x 1 +· · ·+x K ≥ 0 and f (x 1 )+· · ·+ f (x K ) < 0. Then Theorem 3, shows that the elimination of all "buy and sell" arbitrage opportunities at any order K ∈ N-as defined above-characterizes the nonemptyness of the core of v f . Note that, under the additional assumption that f is subadditive, then v f is submodular, hence the core of v f is nonempty (Shapley 1971) but under the assumption made by Cerreia-Vioglio et al. (2015a), v f may have an empty core as shown by Example 1.
Moreover, Theorem 3 characterizes the existence of a strictly positive (additive) probability P in the core of v f , and thus extend the Fundamental Theorem of Asset Pricing in the standard frictionless case. For this purpose, more arbitrage opportunities than before need to be eliminating, by assuming that, for all integer K , there is no Finally, Theorem 3 also characterizes the existence of a strictly positive probability P in the core of v f (resp. the nonemptyness of the core) by the strict positivity (resp. nonnegativity) of the exact cover v e f of v f . This condition cannot be weakened by only assuming v f > 0 (resp. v f ≥ 0) as shown by Example 1.
The main results discussed previously are formally presented in the next Sect. 2, the proofs are given in Sect. 3, and the Appendix provides Example 1, together with some conclusions.

Characterizing the absence of buy and sell arbitrage opportunities
We first recall the notion of Call-Put Parity introduced by Chateauneuf et al. (1996): and we notice that, for a frictionless market, that is f is linear, it is equivalent to the notion of Put-Call Parity introduced by Cerreia-Vioglio et al. (2015a): Our first result states that, for Choquet pricing rules, the Call-Put Parity Condition CPP is equivalent to the No-Spread Condition, i.e., f (x) + f (−x) = 0 for all x ≥ 0 (or for all x). Moreover, a weaker version of the Call-Put Parity (in which the equality is replaced by an inequality) is equivalent to the Positive Bid-Ask Spread Condition or the absence of buy and sell arbitrage opportunities of order 2. It is worth pointing out that the weaker form of CPP has been confirmed by empirical research, see e.g. Gould and Galai (1974), Klemkosky and Resnick (1979), and Sternberg (1994).
Theorem 2 Let f : R Ω → R be a nonzero, monotone, Choquet pricing rule. (a) The following assertions are equivalent: The following assertions are equivalent: The proof of Theorem 2 is given in Sect. 3.1. We point out that the Call-Put Parity Condition CPP (or its equivalent form of No-Spread) does not guarantee in general that v f has a nonempty core, as shown in Example 1 in the Appendix. As proved hereafter in Theorem 3, the elimination of arbitrage opportunities at any order K ≥ 1 will be required for v f to have a nonempty core.
We end the section with the following remark.
Remark 1 (CPP Choquet pricing rules with a nonempty core) When the Choquet pricing rule f satisfies the Call-Put Parity CPP and core

Absence of "split" arbitrage opportunities
We have defined previously the following Arbitrage-free Conditions: that strengthen standard notions encountered in the finance literature, by ruling out the standard (order 1) arbitrage opportunities, the order 2 buy and sell arbitrage opportunities together with the arbitrage opportunities at any order K . See Example 1 in the Appendix for the importance of eliminating arbitrage opportunities of order K > 2.
The next proposition shows that AF ++ is stronger than AF + and, whenever f is subadditive there is no need to eliminate arbitrage opportunities of order K > 1.
The proofs of the remaining assertions are straightforward.
The next result shows that the Arbitrage-free Conditions AF ++ and AF + (that consider multiple buy and sell strategies with either x k ≥ 0 or x k ≤ 0 for every k) have equivalent formulations also of interest, when x k can be taken in the whole space R Ω for all k. We assume hereafter that f only satisfies f ([x] for all x in the whole space R Ω , a property that holds when f is a Choquet pricing rule since it is constant modular by Theorem 1 (take t = 0). Note that this assumption has a clear financial meaning, that is, splitting Proposition 2 Let f : R Ω → R be monotone and satisfy Then the three following assertions are equivalent: (ii ++ ) For all integer K , for all x k ∈ R Ω (k = 1, . . . , K ) Similarly, the three following assertions are equivalent: The proof is similar to the previous one.

Characterizing arbitrage-free Choquet pricing rules
The core of a nonzero monotone Choquet pricing rule f : R Ω → R is defined as the core of its associated risk-neutral nonadditive probability v f , that is: . We let P(Ω) := {P ∈ R Ω + : P · 1 Ω = 1} be the set of (additive) probability on Ω, and we notice that core ( f ) ⊆ R Ω + since f is monotone and all elements of the core are additive probabilities, thus one has: Theorem 3 Let f : R Ω → R be a nonzero, monotone, Choquet pricing rule. Then the following assertions are equivalent: (resp. and P ∈ R Ω ++ ).
The proof of Theorem 3 is given in Sect. 3.2. We end the section with a remark.
Remark 2 (Subadditive Pricing Rules) When f is additionally subadditive, the Choquet pricing rule f is submodular, thus v f is also submodular, hence exact, that is, v e f = v f . Consequently, f satisfies AF ++ (resp. AF + ) if and only if v f is strictly positive (resp. nonnegative). We point out that this equivalence may not hold when f is no longer assumed to be subadditive as shown in Example 1 in the Appendix.

Proof of Theorem 2
Let f : R Ω → R be a nonzero, monotone, Choquet pricing rule, we show that the following four conditions are equivalent.

Proof of Theorem 3
Let f be a monotone, nonzero, Choquet pricing rule, without any loss of generality we assume hereafter that f (1 Ω ) = 1, thus r f = 0. We define: We consider the following assertions: (1) f satisfies AF + (resp. AF ++ ) and we prove the following implications: using the fact that f is positively homogeneous and is the value of a linear programming problem whose dual is defined by: We claim that 0 ≤ c * (x) < +∞ for all x ≥ 0. Indeed, first we have: [c(x) > 0 for all x > 0] Let x > 0. We have proved previously that c(x) = c * (x) and is finite, thus the dual problem has a solution, by the Strong Duality Theorem of Linear Programming. Hence, there exist  (4), we deduce that v e f ({ω}) > 0 for all ω ∈ Ω. Thus, there exists μ ω ∈ core ( f ), which is nonempty and compact, such that:
We end the section with a remark.
Remark 3 (Arbitrage-free and Balancedness Conditions) In the proof of Theorem 3 we have also shown that the following Arbitrage-free and Balancedness Conditions are equivalent: • f satisfies the Arbitrage-free Condition AF + ; The two Balancedness Conditions, which are equivalent since f is monotone, characterize the nonemptyness of core (v f ) from Bondareva (1963), and Shapley (1967). and they can be interpreted as follows. If the bond 1 Ω is sold as a whole and bought as parts that are fractions of event payoffs, θ A 1 A , then the aggregate cost of buying the parts cannot be smaller than the gain from selling 1 Ω , that is, Similarly, the two following conditions are also equivalent: • f satisfies the Arbitrage-free Condition AF ++ ;

Conclusion and appendix
This paper studies the properties of the risk-neutral nonadditive probability v f associated with a Choquet pricing rule f following Cerreia-Vioglio et al. (2015a) (Theorem 1) in the absence of subadditivity assumption on the pricing rule. In this framework, the usual conditions of absence of arbitrage opportunities do not guarantee the nonemptyness of the core of the risk-neutral nonadditive probability v f associated with the pricing rule f as shown in Example 1 hereafter. Second, Theorem 2, characterizes the Positive Bid-Ask Spread Condition, i.e., f (−x) + f (x) ≥ 0 for all x, by a weak form of the Call-Put Parity Condition introduced by Chateauneuf et al. (1996). The Positive Bid-Ask Spread Condition can be re-interpreted as the absence of "buy and sell" arbitrage opportunities of order K = 2, that is, there is no x k ∈ R Ω + ∪−R Ω + (k = 1, . . . , K ) such that x 1 +· · ·+x K ≥ 0 and f (x 1 ) + · · · + f (x K ) < 0.
Then Theorem 3, shows that the elimination of all "buy and sell" arbitrage opportunities for any order K ∈ N-as defined previously-characterizes the nonemptyness of the core of v f . Note that, under the additional assumption that the Choquet pricing rule f is subadditive, then v f is submodular, hence the core of v f is nonempty (Shapley 1971) but under the assumption made by Cerreia-Vioglio et al. (2015a), v f may have an empty core as shown by Example 1.
We then define a risk-neutral probability P as an element of the core of v f whenever it is nonempty. Moreover, the (standard) expectation of every payoff x with respect to the risk-neutral probability P is below its non-additive expectation with respect to v f , hence 1 1 + r f Ω xd P ≤ 1 1 + r f Ω xdv f = f (x) for all x, and the inequality is an equality when f is linear. Moreover, Theorem 3 characterizes Condition AF ++ , by the existence of a strictly positive risk-neutral (additive) probability P in the core of f . Clearly this latter condition implies that v f is strictly positive (i.e., positive for all nonempty events) but the converse assertion may not be true (unless f is subadditive), as shown by the following Example 1 below. Finally, Theorem 3 also characterizes Condition AF ++ (resp. AF + ) by the strict positivity (resp. nonnegativity) of the exact cover v e f of v f . In particular, if f is subadditive, then the Choquet pricing rule f is submodular, thus v f is also submodular, hence exact, that is, v e f = v f ; consequently, f satisfies AF ++ (resp. AF + ) if and only if v f is strictly positive (resp. nonnegative).
Finally the following Example 1 exhibits a monotone Choquet pricing rule f such that core( f ) = ∅, while f satisfies all the Arbitrage-free Conditions of order 1 and 2, together with the Call-Put Parity Condition CPP, and the Put-Call Parity Condition PCP.