Financial economics without probabilistic prior assumptions

Abstract

The treatment of uncertainty in general equilibrium theory in the style of Arrow and Debreu does not require a prior probability on the state space. Finance models nevertheless treat payoffs as random variables, implicitly or explicitly using a known probability distribution. In the light of Knightian uncertainty, we might challenge such an assumption on the probabilistic sophistication of our market model. The present paper shows that one can still develop a sound model of arbitrage pricing under complete Knightian uncertainty as long as certain continuity conditions are met. The pricing functional given by an arbitrage-free market can be identified with a full support martingale measure (instead of equivalent martingale measure). We relate the no-arbitrage theory to economic equilibrium by establishing a variant of the Harrison–Kreps theorem on viability and no arbitrage. Finally, we consider (super) hedging of contingent claims and embed it in a classical infinite-dimensional linear programming problem.

1 Introduction

The recent years have seen a surge of new approaches to Knightian uncertainty, as opposed to pure risk, and deservedly so, as certain aspects of the financial crisis of the first decade of the millennium show. While the basic probabilistic assumptions implicit in modeling financial securities are justified in normal times and for well-established assets, we have to be careful when stretching the theory to new fields like credit risk, e.g. Here, the implicit assumption that probability distributions of returns or credit events are known is quite sensitive.

The new paradigm leads to some interesting questions on the foundations of asset pricing. Almost every asset pricing model starts with a probability space, and thus implicitly fixes a prior, or “empirical,” distribution, according to interpretation. In particular, one implicitly assumes that the relevant agents know all events of measure zero. From a Knightian perspective, or the point of view of model uncertainty, such an implicit assumption on probabilistic sophistication of agents might be questionable. It is thus natural to ask whether one can develop an interesting theory of asset pricing without such a probabilistic prior assumption.

There would be several ways to relax the assumption of a prior probability. We go here the extreme way and work without any probability measure at all, ex ante. One could as well imagine intermediate approaches; for example, starting with a set of (possibly singular) priors as proposed by the typical Gilboa–Schmeidler models, or a capacity.¹ Our point is that even without any such family of priors or capacities, one can still lay the basis for a non-trivial theory of asset pricing.

In finite state spaces, the basic hedging and pricing results of Mathematical Finance can be derived with simple algebra. In the binomial tree model of Cox et al. (1979), simple linear equations suffice to derive the conditions for no arbitrage, the equivalent martingale measure, the hedging portfolio, and, in turn, the unique no-arbitrage price. More generally, the fundamental theorem of asset pricing is easily developed by linear algebra even in incomplete market models.

With a finite number of states, the fundamental theorem shows the equivalence of absence of arbitrage and the existence of positive state (or Arrow-) prices. One can normalize these state prices to a probability measure to obtain a martingale measure under which the price of traded assets is equal to their discounted expected future payoff. Note that this martingale measure has full support on the state space.

As soon as one moves to more complex, infinite models, or continuous time, models usually start by fixing a prior probability $P$ on the state space. The fundamental theorem states then that the absence of arbitrage portfolios can be characterized by the existence of a continuous, positive linear pricing rule on the space of contingent claims or, equivalently, by the existence of an equivalent martingale measure. Under this measure, the price of all traded assets is equal to their properly discounted future expected value. Moreover, the measure has to be equivalent to the original prior probability $P$: It assigns positive probability exactly to those events that have positive probability under the prior² $P$.

Fixing a prior probability $P$ seems natural from a modeling point of view if one has the picture of an asset price as a random variable in mind. It is also very useful because it allows us to use the powerful methods of probability theory. From a conceptional point of view, one might nevertheless ask what is the role of the prior probability $P$, and do we really need it to develop Mathematical Finance?

Beyond its intrinsic theoretic interest, the question enters center stage of economics in the light of the recent discussions on Knightian uncertainty, as explained above. If the modeler, or the investor, cannot be certain about the choice of the prior, this can result in sensitive and unreliable prescriptions for pricing and hedging of derivatives.

On the other hand, one might reply that the measure $P$ does not really matter for pricing and hedging as the focus is on pricing or martingale measures only. These martingale measures have to be equivalent to the original $P$, however, and thus, the choice of $P$ plays an implicit role by determining the sets of measure zero, or, to put it into a fancier form, the black swan events that get price zero (even if they should not). The economic role of $P$ has also been pointed out in Harrison and Kreps (1979); these authors relate the notions of no arbitrage and viability with economic equilibrium. They point out that the prior probability $P$ determines the space of contingent claims to the extent that integrability is required. On the side of agents, it has an influence on the continuity requirement for preferences relations. Last but not least, it determines the null sets which are crucial for the notion of strict monotonicity of pricing functionals.

Recently, the attempt to model uncertain volatility has led to very interesting models where it is impossible to fix a reference or prior probability distribution ex ante. When volatility is unknown in continuous time, one has to work with mutually singular measures, and one cannot fix the null sets in advance (see Peng (2007) for the calculus, Vorbrink (2010) for no-arbitrage considerations, and Epstein and Ji (2013) and Epstein and Ji (2014) for the basic economics and asset pricing consequences).

We therefore think that it is useful to develop the foundations of Finance in a prior-free model.

In this note, we develop the fundamental theorem of asset pricing without any probabilistic assumptions on the general measurable state space. Instead, a topological continuity requirement is used: The asset payoffs are continuous in the state variable. For modeling purposes, this is merely a technical assumption that is usually satisfied (as one can construct models where the asset payoffs are projections on some product space, and hence continuous). Imposing measurability, to recall, is relatively close to imposing continuity because by Lusin’s theorem measurable functions (on nice metric spaces) are continuous on large subsets of the space.

As one cannot speak of equivalent martingale measures in this context, another notion of pricing measure is needed. It turns out that the right concept is full support martingale measure. The pricing functional has to assign positive prices to all open sets in order to avoid arbitrage opportunities. Hence, probabilistic models do enter again, as a consequence of the no-arbitrage assumption, but without imposing any probabilistic assumptions ex ante.

With the help of the new version of the fundamental theorem, one can easily characterize all arbitrage-free prices for derivatives as expectations under some full support martingale measure.

As it is important to understand the relations between Finance and Economics, we also give a variant of the Harrison–Kreps theorem which relates the economic viability of asset prices to the possibility to extend the price functional from the marketed space to the whole space of possible contingent claims while keeping strict positivity of the pricing functional.

As an application of the new setup, I show how to embed the superhedging problem in incomplete markets into the language of linear programming in infinite-dimensional spaces. The fact that superhedging and the upper bound on all no-arbitrage prices are “in some sense” dual is well known, of course. In the probabilistic setting, it is not straightforward to embed the problems directly into the language of linear programming. This is why usually a combination of stochastic and optimization techniques is used in proving the duality. In our setup, the relation to linear programming is direct and obvious. To the best of my knowledge, this embedding and treatment of the superhedging problem have not been done before.

I am not aware³ of probability-free approaches to the foundations of Finance in general, in particular for the fundamental theorem. Students of Hans Föllmer⁴ know that the basic hedging argument in complete markets can be carried out without imposing probabilistic prior assumptions, even in continuous time; one uses Föllmer’s pathwise approach to Itô integration (Föllmer 1981) to develop the Black–Scholes-like hedging argument in a purely analytic way, compare (Bick and Willinger 1994). Föllmer (1999) develops these ideas and their relations to economics masterfully; see also Föllmer (2001) for a similar account in English, and the textbook by Sondermann (2006). Here, our focus is on the fundamental theorem and the relation to economic equilibrium in the spirit of Harrison and Kreps. The notion of full support plays a role in Finance when one considers non-semimartingale models for security prices, see Bender et al. (2008) and Coviello et al. (2011). In these models, a prior probability is still fixed, however. The focus is on developing finance in a probabilistic framework with non-semimartingale prices and a restricted class of trading strategies. It turns out that a full support property for the security prices ensures no arbitrage in these models.

The paper is set up as follows. The next section develops the prior-free model of finance and proves the fundamental theorem of asset pricing. We also provide a Harrison–Kreps-like theorem on the relation between no arbitrage and economic viability. Sect. 3 describes our treatment of the superhedging problem as a linear program in infinite dimensions.

2 No arbitrage and equilibrium in a probability-free model

The usual approach to a no-arbitrage-based theory of asset pricing starts with a probability space $\left( \Omega , {\mathcal {F}}, P\right)$; future asset prices are taken to be nonnegative random variables $S_d: \Omega \rightarrow \mathbb R_+$ that can be bought at prices $f_d \ge 0$ today. In particular, one implicitly assumes that the distribution function of the future asset prices is known. A fortiori, it is presumed that the null sets, or prices that we can possibly never observe, are known.

The fundamental theorem of asset pricing states that under ($P$-almost surely) no arbitrage, there exists an equivalent martingale measure $P^*$; this measure has the same null sets as $P$ and satisfies the pricing or martingale relation

$$\begin{aligned} E^{P^*} S_d = f_d\qquad (d=0,\ldots ,D). \end{aligned}$$

Note that the imposition of a probability measure corresponds to an implicit assumption that all agents in the economy are probabilistically sophisticated [in the sense of Machina and Schmeidler (1992)]. I present here an alternative approach that is, ex ante, probability-free. Instead, I work on a measurable space with a nice topological structure.

2.1 The topological model of finance

Let $\Omega \not =\emptyset$ be the set of states of the world. Let $\tau$ be a Hausdorff topology on $\Omega$. Let ${\mathcal {F}}$ be the Borel $\sigma$-field on $\Omega$, generated by the open subsets. We assume that there exist strictly positive Radon measures $\mu$ on the measurable space $(\Omega , {\mathcal {F}})$.

Remark 2.1

• 1. There is no hope to develop a meaningful theory of arbitrage pricing when there are no probability measures with full support as we explain below.

• 2. The topological spaces that admit strictly positive Radon measures have been characterized by Casteren (1994), see also Casteren (1979). One has to be able to choose for every nonempty open set $U$ a compact set $K(U)\subset U$ such that for all countable families of open sets $(U_n)$ with

  $$\begin{aligned} \mathrm{int } \,\bigcap U_n \not =\emptyset \end{aligned}$$

  one has

  $$\begin{aligned} \bigcap _{n\in A} K(U_n)\not =\emptyset \end{aligned}$$

  for a set $A \subset \mathbb N$ of nonzero density.⁵

• 3. The assumption is satisfied for most spaces that occur in economic models. In particular, it holds for Polish spaces $(\Omega ,d)$, where $d$ is a complete metric on $\Omega$ that allows for a countable dense subset $\mathbb Q :=\left\{ (q_n) : n=1,2,\ldots ,\right\} \subset \Omega$. To see this, take the dense sequence $(q_n)_{n=1,2,\ldots }$ and define a probability measure

  $$\begin{aligned} \mu _0:=\sum _{n=1}^\infty 2^{-n} \delta _{q_n} \end{aligned}$$

  that puts weight $2^{-n}$ on $q_n$. As the sequence is dense, every nonempty open subset $U$ contains at least one $q_n$; hence $\mu _0(U)>0$. A fortiori, the assumption is satisfied for finite and countable $\Omega$. In such a discrete models, it is actually easy to develop the fundamental theorem and the subsequent hedging analysis without any probabilistic assumption.

There are $D+1$ financial assets traded at time $0$ at prices $f_d \ge 0, d=0,1,\ldots , D$ with an uncertain payoff at time $1$. As usual in Finance, we assume that there is a riskless asset $S_0(\omega )=1$ for all $\omega \in \Omega$ with price $f_0=1$ (so the interest rate is normalized to $0$). The uncertain assets have a payoff $S_d(\omega )$ that is continuous in $\omega$.

Remark 2.2

Continuity in $\omega$ is stronger than mere measurability, of course, but it comes for free in most models. Note that we speak of continuity in the state variable, not in time, here. Usually, one can construct the model in such a way that $S_d$ is a projection mapping on some product space. These mappings are automatically continuous in the associated topology. For example, with $D$ uncertain assets, you can model $\Omega$ as a product space of the form $\Omega _1 \times \cdots \times \Omega _D$; if you think that all positive prices are possible for asset $d$, then you simply take $\Omega _d = (0,\infty )$ and $F_d : \Omega \rightarrow \mathbb R$ is the projection $F_d(\omega _1,\ldots ,\omega _D)=\omega _d$.

Definition 2.3

A portfolio is a vector $\pi \in \mathbb R^{D+1}$. A portfolio $\pi$ is called an arbitrage if we have

$$\begin{aligned} \displaystyle \pi \cdot f = \sum _{d=0}^D \pi _d f_d \le 0 \\ \displaystyle \pi \cdot S(\omega ) \ge 0 \qquad \text{ for } \text{ all }\,\omega \in \Omega \\ \displaystyle \pi \cdot S(\omega ) > 0 \qquad \text{ for } \text{ some }\,\omega \in \Omega . \end{aligned}$$

We say that the market $(f,S)$ is arbitrage-free if there exists no arbitrage $\pi$.

Note how we adapt the notion of arbitrage. Instead of assuming that a riskless gain is possible $P$-almost surely, we require a riskless gain for all states $\omega$, and a strictly positive gain for some state. By continuity of asset payoffs in states, this leads to a strictly positive payoff on an open subset $U$ of the state space.

2.2 The fundamental theorem

The classic fundamental theorem of asset pricing says that markets are arbitrage-free if and only if one has a pricing or martingale measure $P^*$ that is equivalent to the prior $P$. The equivalence is important as the pricing measure must assign a positive price to all events that have positive probability under the prior $P$. In our version of the fundamental theorem, the equivalence to $P_0$ is replaced by the requirement of full support:⁶ The pricing measure needs to assign positive prices to all open sets $U \subset \Omega$.

We fix our language with the following definition.

Definition 2.4

A probability measure $P$ on $(\Omega ,{\mathcal {F}})$ is called a martingale measure (for the market $(f,S)$) if it satisfies $\int S_d \, {\hbox {d}}P=f_d, d=1,\ldots ,D$. If $P$ has also full support, we call it full support martingale measure.

We are now able to characterize the absence of arbitrage in our model.

Theorem 2.5

The market $(f,S)$ is arbitrage-free if and only if there exists a full support martingale measure.

The proof of the theorem is not much different from the usual proof in the probabilistic setting. My point here is not to deliver new techniques; rather, the aim is to discuss the conceptual foundations underlying any Financial Economics model. So we shall see that the topological model with its continuity assumption on security payoffs is able to serve as a prior-free foundation for Financial Economics.⁷

So let us see how our assumptions are used in the easy “if” part of the proof. Suppose that we have a full support martingale measure $P^*$. Let $\pi$ be an arbitrage. By continuity of $S$ in $\omega$, there exists an open set $U\subset \Omega$ such that

$$\begin{aligned} \pi \cdot S(\omega ) > 0 \end{aligned}$$

holds true for all $\omega \in U$. As $P^*$ has full support, it puts positive measure on $U$. The quantity $\pi \cdot S$ is nonnegative and positive on a set of positive $P^*$-measure, so we have $E^{P^*} \pi \cdot S = \int \pi \cdot S \, {\hbox {d}}P^* > 0$. The contradiction

$$\begin{aligned} 0 \ge f \cdot \pi = \pi \cdot E^{P^*} S =E^{P^*} \left[ \pi \cdot S\right] > 0 \end{aligned}$$

follows. This proves the easy part of the theorem. Note how we use continuity to get a strictly positive payoff on an open set and full support to obtain a contradiction. In the probabilistic proof, one has a positive payoff with positive probability under the prior probability and equivalence to the prior for the martingale measure.

The more demanding direction of the fundamental theorem requires some preparation. After that preparation, we can proceed as usual by a separation argument.

By assumption, Radon probability measures with full support exist on our measurable space $(\Omega , {\mathcal {F}})$. We can now go on and find even probability measures with full support under which all payoffs are integrable (note that we did not assume that the payoffs $S_d$ are bounded functions). Here the argument is classical and does not use our new hypotheses. Just let $M(\omega ):=\max _{d=1,\ldots ,D} S_d(\omega )$ and define a new measure $\mu _1$ via

$$\begin{aligned} \mu _1(A) = \int _A \frac{c}{1+M} {\hbox {d}}\mu _0 \end{aligned}$$

for Borel sets $A$ and the normalizing constant $c=\left( \int _\Omega \frac{1}{1+M} {\hbox {d}}\mu _0\right) ^{-1}$. As the density is strictly positive, $\mu _1$ has full support. Every $S_d$ is $\mu _1$-integrable because of

$$\begin{aligned} \frac{S_d}{1+M} \le 1. \end{aligned}$$

We conclude that the set $C$ of Radon probability measures on $(\Omega ,{\mathcal {F}})$ with full support that make each $S_d$ integrable is non-empty. It is clear that $C$ is convex.

After this preparation, we can now begin with the typical separation argument [and we follow the exposition in Föllmer and Schied (2002, Chapter 1.2 here)]. Let $R_d:=S_d-f_d, d=1,\ldots ,D$ be the returns of the uncertain assets. Define the nonempty convex subset of $\mathbb R^D$

$$\begin{aligned} K:=\left\{ \begin{pmatrix} \int R_1 {\hbox {d}}\mu \\ \vdots \\ \int R_D {\hbox {d}}\mu \\ \end{pmatrix} : \mu \in C \right\} . \end{aligned}$$

A full support martingale measure exists if and only if $0\in K$. So let us assume, to achieve a contradiction, that $0\notin K$.

By the separation theorem in $\mathbb R^D$, there exists a vector $0\not = \phi \in \mathbb R^D$ such that $\phi \cdot k \ge 0$ for all $k\in K$ and $\phi \cdot k >0$ for some $k\in K$. As we have

$$\begin{aligned} \int _\Omega \phi \cdot R(\omega ) \mu (d\omega ) \ge 0 \end{aligned}$$

for all $\mu \in C$, we must have

$$\begin{aligned} \phi \cdot R(\omega ) \ge 0 \end{aligned}$$

for all $\omega \in \Omega$. To see this, note that the set $C$ is dense in the set $\hat{C}$ of all Radon probability measures that make $S$ integrable. One can just take $\mu ^*\in C$ and approximate $\hat{\mu }\in \hat{C}$ by $\mu _n = (1-1/n) \hat{\mu }+1/n \mu ^*$ in $C$. As $S$ has finite values, each Dirac measure $\delta _\omega$ is in $\hat{C}$. Therefore, we conclude that $R(\omega )=\int R d\delta _\omega \ge 0$.

The additional strict inequality

$$\begin{aligned} \int _\Omega \phi \cdot R(\omega ) \mu (d\omega ) > 0 \end{aligned}$$

for some $\mu \in C$ implies that there must exist $\omega _0 \in \Omega$ with $\phi \cdot R(\omega _0) >0$.

Now define a portfolio $\pi$ by setting $\pi _0=-\sum _{d=1}^D \phi _d f_d$ and $\pi _d=\phi _d, d=1,\ldots ,D$. Then $\pi \cdot f =0$, and

$$\begin{aligned} \pi \cdot S(\omega )= \pi _0 + \sum _{d=1}^D \pi _d S_d(\omega ) = \sum _{d=1}^D \pi _d Y_d(\omega ) \ge 0, \end{aligned}$$

as well as $\pi \cdot S(\omega _0)>0$. Hence, $\pi$ is an arbitrage, a contradiction. We conclude that $0 \in K$. This proves the fundamental theorem of asset pricing in our topological context.

Remark 2.6

An alternative proof of the theorem goes as follows. Due to our assumptions on the measurable space, we know that $(\Omega ,{\mathcal {F}})$ admits probability measures with full support. Pick one such measure $P_0$ and consider the probabilistic model $(\Omega ,{\mathcal {F}},P_0)$ with the classical probabilistic arbitrage definition: $\pi$ is an arbitrage if $\pi \cdot S \ge 0$ holds true $P_0$-almost surely and $P_0[\pi \cdot S >0]>0$. Then, we have the following relation between the probability-free and the probabilistic model:

Lemma 2.7

The market $(f,S)$ is arbitrage-free (in the topological sense) if and only for every full support probability measure $P_0$ on $(\Omega ,{\mathcal {F}})$, the market $(f,S)$ is arbitrage-free in the probabilistic sense.

Proof

Let $P_0$ be a probability measure with full support on $(\Omega ,{\mathcal {F}})$. If $\pi$ is an arbitrage in the topological model, then $\pi \cdot S(\omega )\ge 0$ for all $\omega$, thus $P_0$-a.s. Moreover, the set $U:=\{\pi \cdot S>0\}$ is nonempty and open (due to our continuity assumption on $S$). As $P_0$ has full support, we have $P_0(U)>0$. Hence, $\pi$ is an arbitrage in the probabilistic sense.

Now, let $\pi$ be an arbitrage in the probabilistic sense under $P_0$. So $\pi \cdot S \ge 0$ $P_0$-a.s. and $P_0[\pi \cdot S>0]>0$. Then, we clearly have $\pi \cdot S(\omega )>0$ for some $\omega$. We need to show that we also have $\pi \cdot S(\omega ) \ge 0$ for all $\omega$. Here, we need again continuity and the full support property. If $\pi \cdot S(\omega _0)<0$ for some $\omega _0$, then, by continuity, $V:=\{\pi \cdot S<0\}$ is an open, nonempty set. As $P_0$ has full support, $P_0(V)>0$, a contradiction. $\square$

Combining the preceding lemma with the classic probabilistic fundamental theorem of asset pricing, we obtain an alternative proof of Theorem 2.5.

2.3 The Harrison–Kreps theorem

After the appearance of the Black–Scholes–Merton theory, a certain confusion about the magic of the formula and its supposed independence of any assumptions on preferences reigned. Harrison and Kreps clarified the situation in their seminal paper by relating the concept of no arbitrage and economic viability. We show how to obtain an analog to their result in our prior-free model.

We assume in this section that $(\Omega ,d)$ is locally compact. Our space of contingent claims is

$$\begin{aligned} {\mathcal {X}}:=C_S(\Omega ,d)=\left\{ f \in C(\Omega ): \frac{f}{\max \{1,||S||\}} \text{ is } \text{ bounded }\right\} , \end{aligned}$$

the space of continuous functions that are bounded by multiples of the assets’ payoffs. We endow this weighted space with the norm

$$\begin{aligned} \Vert f \Vert = \inf \left\{ K \ge 0 : \frac{|f|}{\max \{1,||S||\}}\le K \right\} . \end{aligned}$$

With this norm, ${\mathcal {X}}$ is a Banach lattice. Our financial market $(f,S)$ generates a marketed subspace

$$\begin{aligned} M:=\langle S_0,S_1,\ldots ,S_D\rangle = \left\{ \pi \cdot S ; \pi \in \mathbb R^{D+1}\right\} . \end{aligned}$$

Under no arbitrage, two portfolios $\pi$ and $\psi$ that lead to the same payoff $\pi \cdot S = \psi \cdot S$ must have the same value at time $0$, or $\pi \cdot f = \psi \cdot f$. The linear mapping (the price functional)

$$\begin{aligned} \phi : M \rightarrow \mathbb R \end{aligned}$$

given by $\phi \left( \pi \cdot S\right) =\pi \cdot f$ is then well-defined.

An economic agent is given by a complete and transitive relation $\succeq$ on ${\mathcal {X}}$. In addition, we assume that $\succeq$ is continuous in the sense that for all $X\in {\mathcal {X}}$ the better-than- and worse-than sets

$$\begin{aligned} \left\{ Z\in {\mathcal {X}}: Z \succeq X\right\} , \left\{ Z\in {\mathcal {X}}: X \succeq Z \right\} \end{aligned}$$

are closed (in the norm topology on ${\mathcal {X}}$). Finally, the agent’s preferences are strictly monotone: If $X \in {\mathcal {X}}$ satisfies $X \ge 0$ and $X\not =0$, then for all $Z\in {\mathcal {X}}$, we have $Z+X \succ Z$ (with the obvious definition of $\succ$).

We say that the market $(f,S)$ is viable if there exists an agent $\succeq$ such that no trade is optimal given the budget $0$: For all $\pi \in \mathbb R^{D+1}$ with $\pi \cdot f \le 0$ we have $0 \succeq \pi \cdot S$.

Theorem 2.8

The market $(f,S)$ is viable if and only if there exists a strictly positive linear functional $\Phi : {\mathcal {X}}\rightarrow \mathbb R$ such that $\Phi (X)=\phi (X)$ for $X\in M$.

Proof

A viable market $(f,S)$ must not admit arbitrage opportunities, as the agent would improve upon any portfolio by adding the arbitrage. If $(f,S)$ is arbitrage-free, Theorem 2.5 allows us to pick a full support martingale measure $\mu$. The theorem also yields that the assets $S$ are $\mu$-integrable. As $X \in {\mathcal {X}}$ satisfies $|X| \le \Vert X\Vert \max \{1,||S||\}$, we obtain that $X$ is $\mu$-integrable as well. We set $\Phi (X)=\int _\Omega X {\hbox {d}}\mu$ and obtain our desired extension of $\phi$. The extension is strictly positive as $\mu$ has full support.

On the other hand, if $\Phi$ is a strictly positive extension of $\phi$, the linear preference relation

$$\begin{aligned} X \succeq Y \quad \text{ if } \text{ and } \text{ only } \text{ if }\quad \Phi (X)\ge \Phi (Y) \end{aligned}$$

defines an agent for whom no trade is optimal given initial budget $0$. Note that positive linear functionals on the Banach lattice $C(\Omega ,d)$ are continuous. Thus, the preference relation is also continuous. $\square$

3 Contingent claims: no arbitrage prices and hedging

In the absence of complete markets, the benchmark for imperfect hedging procedures is given by super- resp. subhedging a given claim where one aims to find the cheapest portfolio that stays above resp. the most expensive portfolio that stays below the claim. We develop here a purely non-probabilistic approach to superhedging by embedding the problem fully into the language of (infinite-dimensional) linear programming. To the best of our knowledge, this has not been done before.

In our setting, superhedging a claim is a linear problem where the variable, the portfolio, is finite-dimensional, but where the constraint, staying on the safe side, is infinite-dimensional, in the space of continuous (bounded) functions. As the dual of this space consists of countably additive measures, the dual program has thus infinite-dimensional variable space. We show that the constraints imply that the dual program is to maximize the expectation of the claim over all martingale measures.

It is noteworthy that the dual constraints do not require that we look at full support martingale measures. From the point of view of optimization, it is more natural to look at the set of martingale measures only as this set is closed, and we can thus find an optimal solution to our problem.

From a Finance point of view, one wants to establish the result that the cheapest superhedge is equal to the least upper bound for all no-arbitrage prices. No-arbitrage prices, however, are determined by full support martingale measures. The two numbers coincide because the set of all full support martingale measures is dense in the set of martingale measures. The maximizer of the dual problem is not attained by such a full support martingale measure; a superhedging portfolio would be an arbitrage if its value was a no-arbitrage price. Hence, the value cannot be attained by a full support martingale measure.

3.1 Claims and no arbitrage prices

Definition 3.1

A derivative or (contingent) claim is a continuous mapping $H:\Omega \rightarrow \mathbb R_+$. Any number $h \ge 0$ is called a no arbitrage price for $H$ if the extended market with $D+2$ assets and $f_{D+1}=h$ and $S_{D+1}=H$ admits no arbitrage opportunities.

As usual, we can now apply the fundamental theorem to obtain a characterization of no arbitrage prices. If we take the expected value of the claim under a full support martingale measure, the new, extended market does have such a full support martingale measure as well, and the expected value is a no-arbitrage price. If, on the other hand, $h$ is a no arbitrage price, the other direction of the fundamental theorem implies that it is the expected value under some full support martingale measure for the extended market. This measure has to be a full support martingale measure for the old market as well.

Corollary 3.2

$h$ is a no arbitrage price for a claim $H$ if and only if

$$\begin{aligned} h=\int H \,{\mathrm{d}}P \end{aligned}$$

for a full support martingale measure $P$.

3.2 Superhedging and linear programming

We show now how to tackle the problem of superhedging in a probability-free way. We assume again that that $\left( \Omega ,d\right)$ is locally compact. For a well-defined superhedging problem, there must exist portfolios that dominate a claim. We thus work with claims in ${\mathcal {X}}=C_S(\Omega ,d)$.

Definition 3.3

A portfolio $\pi$ is called a superhedge for the claim $H\in {\mathcal {X}}$ if $\pi \cdot S (\Omega ) \ge H(\omega )$ holds true for all $\omega \in \Omega$. A portfolio $\pi$ is called a subhedge for the claim $H$ if $\pi \cdot S (\Omega ) \le H(\omega )$ holds true for all $\omega \in \Omega$.

A seller of the claim $H$ naturally asks for the cheapest superhedge. This leads to the following linear optimization problem.

Problem 3.4

(Problem SH) Find the cheapest superhedge for the claim $H$; minimize $\pi \cdot f$ over $\pi \in \mathbb R^{D+1}$ subject to $\pi \cdot S (\Omega ) \ge H(\omega )$ for all $\omega \in \Omega$.

There is a natural subhedging companion to the above problem; it can be treated by the same methods as the superhedging problem. We leave the details to the reader.

Let us now formulate the dual program for the superhedging problem SH.

It is usual to work with dual pairs of vector spaces in linear programming. Here, we have the (finite-dimensional) space of portfolios $\mathbb R^{D+1}$ whose dual is the space itself, of course. The bilinear form is the usual scalar product that we denote by $x\cdot y$ for $x,y \in \mathbb R^{D+1}$.

The linear superhedging constraint can be described by the claim $H$ and the linear mapping

$$\begin{aligned} B: \mathbb R^{D+1} \rightarrow C_S(\Omega ,d) \end{aligned}$$

with $B\pi =\pi \cdot S$ which maps portfolios to their payoffs in the space of continuous functions.

The dual space of $C_S(\Omega ,d)$ consists of functionals of the form

$$\begin{aligned} L(X)= \int _S X {\hbox {d}}\mu \end{aligned}$$

for some finite $\sigma$-additive measure $\mu$ such that the asset payoffs $S$ are $\mu$-integrable, see Summers (1970), e.g. The bilinear form on these spaces is thus given by

$$\begin{aligned} {\langle X, \mu \rangle }=\int _\Omega X {\hbox {d}}\mu ,\qquad \left( X\in C_S(\Omega ,d), \mu \in C_S^*\right) . \end{aligned}$$

In order to formulate the dual program, we need the adjoint mapping to $B$. In our context, the adjoint mapping is $B^* : C_S^* \rightarrow \mathbb R^{D+1}$ given by

$$\begin{aligned} B^* \mu = \int _\Omega S {\hbox {d}}\mu . \end{aligned}$$

It maps measures to the candidate security prices (if that measure was taken as a pricing measure).

As one easily checks, $B^*$ satisfies the defining condition for an adjoint mapping

$$\begin{aligned} {\langle B \pi , \mu \rangle } = {\langle \pi , B^* \mu \rangle } \qquad \left( \pi \in \mathbb R^{D+1}, \mu \in C_S^*\right) . \end{aligned}$$

As the superhedging problem has no sign constraints on the portfolio, the dual program has to have an equality constraint of the form $B^* \mu = f$. The inequality $B \pi \ge H$ leads to the nonnegativity constraint $\mu \ge 0$ (i.e., $\mu$ is a nonnegative measure) in the dual program. In the dual program, we thus maximize over all positive measures $\mu$ that satisfy the integral constraints

$$\begin{aligned} \int S_d {\hbox {d}}\mu = f_d, \quad d=0,\ldots , D. \end{aligned}$$

As $S_0=f_0=1$, such a $\mu$ is a probability measure. The other constraints say that $\mu$ is a martingale measure.

Problem 3.5

(DSH) Minimize the prices $\int _\Omega H {\hbox {d}}\mu$ over all martingale measures $\mu \in C_S^*$.

We are now ready to state the main duality result on probability-free superhedging.

Theorem 3.6

• 1. The linear programs SH and DSH are dual to each other. Both problems have the same value.

• 2. Both programs have optimal solutions; in particular, there exists a superhedge $\pi ^*\in \mathbb R^{D+1}$ and a martingale measure $P^*$ such that

  $$\begin{aligned} \pi ^*\cdot f = \int H {\hbox {d}}P^*. \end{aligned}$$

The details of the proof are postponed to the next subsection. Here, we want to point out that our (semi)-infinite-dimensional setup requires some care as the strong duality of the finite-dimensional linear programming theory carries over to infinite dimensions only under additional technical assumption. For our setup, it is sufficient to show that the cone

$$\begin{aligned} C:= \left\{ \left( \int S {\hbox {d}}\mu , \int H {\hbox {d}}\mu \right) : \mu C_S^*, \mu \ge 0\right\} \subset \mathbb R^{D+2} \end{aligned}$$

is closed. This follows from the fact that the cone of all positive measures in $ca\left( \Omega ,{\mathcal {F}}\right)$ is metrizable; a converging sequence in $C$ needs to stay bounded because of $S_0=1$. As such sequences are relatively weak*-compact, we find a converging subsequence of measures; see the next subsection for more details.

From a Finance point of view, one would like to go a step further and show that the superhedging value is also the least upper bound for all no-arbitrage prices. But this is immediate from the fact that the set of full support martingale measures is dense in the set of martingale measures. However, a maximizer cannot be in the set of full support martingale measures as the superhedging portfolio leads to an arbitrage.

Corollary 3.7

• 1. The price of the cheapest superhedge for a claim $H$ is equal to the least upper bound on all no-arbitrage prices for $H$.

• 2. The price of the most expensive subhedge for a claim $H$ is equal to the greatest lower bound on all no-arbitrage prices for $H$.

3.2.1 Details on embedding the superhedging problem into the language of linear programming in infinite-dimensional spaces

We now show how to embed the above superhedging problem and its dual into the language of linear programming in infinite-dimensional spaces. We will then obtain a dual characterization of the value functions via the infinite-dimensional strong duality theorem. We use the language of the textbook by Anderson and Nash (1987). Note, however, that we have swapped the primal and the dual program in our formulation.

Before we start our analysis, let us note that both problems are consistent in the sense that portfolios satisfying the constraints exist: As there exists a riskless asset and $H$ is bounded, superhedges exist. By no arbitrage, there exist martingale measures.

We now embed our problems into the formulation of Anderson and Nash. Let $X=C_S^*$ and $Y=C_S\left( \Omega ,d\right)$. The (separating) bilinear form on $X$ and $Y$ is

$$\begin{aligned} {\langle \mu , K \rangle }=\int _\Omega K(\omega ) \mu (d\omega ) \end{aligned}$$

for $\mu \in X$ and $K\in Y$. We let $Z=W=\mathbb R^{D+1}$ with the usual bilinear form given by the scalar product. Let us set $c:=-H$. Recall that we introduced the linear mapping $B^* : X \rightarrow Y$ with $B\mu =\int S {\hbox {d}}\mu$ above. Let $A=-B^*$. The primal program in the sense of Anderson and Nash is then

Problem 3.8

(EP) minimize ${\langle \mu , c \rangle }$ over $\mu \in X$ subject to $A \mu = f$ and $\mu \ge 0$.

Note that this is exactly our problem DSH above.

The adjoint mapping to $A$ is $A^*=-B$, of course. The dual problem is now

Problem 3.9

$(\mathrm{EP}^{*})$ maximize ${\langle f, \pi \rangle }$ over $\pi \in W$ subject to $A^* \pi \le c$.

This is our superhedging problem SH.

According to Theorem 3.10 in Anderson and Nash (1987), strong duality holds true if the set

$$\begin{aligned} C:= \left\{ \left( \int S {\hbox {d}}\mu , \int c {\hbox {d}}\mu \right) : \mu \ge 0, \mu \in C_S^* \right\} \subset \mathbb R^{D+2} \end{aligned}$$

is closed. Let $(\mu _n)$ be a sequence of nonnegative finite measures such that

$$\begin{aligned} \int S {\hbox {d}}\mu _n \rightarrow z \in \mathbb R^{D+1}, \int H {\hbox {d}}\mu _n \rightarrow r \in \mathbb {R} \end{aligned}$$

as $n \rightarrow \infty$. As the zeroth asset is riskless, we conclude that $\mu _n(\Omega )=\int S_0 {\hbox {d}}\mu _n$ remains bounded. In particular, the sequence $(\mu _n)$ is relatively compact in the weak topology (given by continuous, bounded functions on $\Omega$). As this topology is metrizable (on the set of nonnegative measures, e.g., by the Prohorov distance), we can assume without the loss of generality that $(\mu _n)$ converges in the weak topology to some $\mu \ge 0$. By continuity of the integral with respect to weak convergence, we then get

$$\begin{aligned} z=\int S \,{\hbox {d}}\mu , r = \int c \,{\hbox {d}}\mu \end{aligned}$$

and this establishes that $C$ is closed.

We finally have to show that optimal solutions to both problems exist. The set of martingale measures is weakly compact because it is a closed subset of the weakly compact set of measures that are bounded by $1$ (Alaoglu’s theorem). Hence, a maximizing martingale measure for the continuous linear evaluation $\int H {\hbox {d}}\mu$ exists.

The argument for the existence of an optimal superhedge is somewhat lengthier (but follows the same route of arguments as in the probabilistic case). Let $(\pi _n)$ be a minimizing sequence of portfolios. If the sequence is bounded, there exists a converging subsequence with limit $\pi ^*$, and this is the desired optimal superhedge.

So let us assume that $(\pi _n)$ is unbounded. The sequence

$$\begin{aligned} \eta _n:=\frac{\pi _n}{\Vert \pi _n\Vert } \end{aligned}$$

is then well-defined for large $n$. Without the loss of generality, it converges to some $\eta$ with norm $1$.

We can assume that the asset payoffs $S_0,\ldots , S_D$ are linearly independent, or else we choose a maximal linearly independent market including the riskless asset. As $\pi _n$ are superhedges, and $H$ is bounded, we obtain

$$\begin{aligned} \eta ^* \cdot S = \lim \eta _n \cdot S \ge \limsup H / \Vert \pi _n\Vert = 0. \end{aligned}$$

As $\pi _n \cdot f$ converges to the finite value of the superhedging problem, we also get

$$\begin{aligned} \eta ^* \cdot f = 0. \end{aligned}$$

By no arbitrage, we conclude $\eta ^*=0$ which is a contradiction to $\Vert \eta ^*\Vert =1$.

4 Conclusion

A theory of hedging and pricing of derivative securities can be developed without referring to a prior probability measure that fixes the sets of measure zero. In this sense, one can say that the ideas of hedging and pricing (as far as it is related to and based on hedging) are independent from probability theory, from an epistemological point of view.

When one starts an economic model of uncertain markets, a financial model, one can impose different strengths of probabilistic sophistication, either implicitly or explicitly. The strongest form of the efficient market hypothesis requires that there is an “objective” probability for all events, known by the market, and reflected by returns. The weaker form of the efficient market hypothesis, as nowadays commonly used in Mathematical Finance, just requires that the market participants share the same view of all null and probability one events. Returns and derivative prices are then determined by a martingale measure that is equivalent to the prior probability. Our approach goes a step further and relaxes the requirement that agents and the market agree on null events. It is then still possible to develop a reasonable theory of derivative pricing. The no-arbitrage condition is strong enough to introduce a pricing probability even without any probabilistic prior assumption; this pricing probability has full support on the state space and thus assigns positive probability to all open sets.