Coherent Price Systems and Uncertainty-Neutral Valuation Patrick Beißner∗ First Version: 2/2012

This Version: 6/2015

Abstract I consider fundamental questions of arbitrage pricing arising when the uncertainty model incorporates uncertainty about volatility. This additional ambiguity motivates a new principle of risk- and ambiguityneutral valuation as an extension of Ross (1976). In the spirit of Harrison and Kreps (1979), I establish a microeconomic foundation of viability in which ambiguity-neutrality imposes a fair pricing principle via symmetric martingales. The resulting equivalent symmetric martingale measure set exists if the uncertain volatility in asset prices is driven by an ambiguous Brownian motion. Key words: ambiguous volatility, non-linear expectations and prices, arbitrage, asset pricing, preference-free valuation, martingales JEL subject classification: G13, G14, D46, D52, C62

1

Introduction

A very basic assumption in finance models points to the quantification of the uncertain future. Usually, a single probability measure or prior is postulated. Instead, I guarantee an awareness of potential volatility misspecification which results in Knightian uncertainty and consider its implications for fair pricing. Such an approach deviates from models in which the term structures of volatilities, including stochastic volatility models such as Heston (1993), are described by another stochastic process. Doubts about the modeler’s ability to be aware of all relevant information leads to the view that knowing the true volatility regime is often impossible. As argued in Epstein and Ji (2013), I question the hypothetical confidence of a universal dependency between ∗

Email: [email protected]. Bielefeld University (IMW). I thank Larry Epstein, Simon Grant, Chiaki Hara, Peter Klibanoff, Frank Riedel, Rabee Tourky, Walter Trockel, and Nicholas Yannelis for fruitful discussions.

past and future, and avoid modeling the volatility in terms of a stochastic process whose law of motion is exactly known. However, the knowledge that the volatility of the state variable moves within a given confidence interval remains certain. A typical costless arbitrage is a positive claim, and with positive probability, is strictly positive. Such a definition clearly depends on the chosen objective prior that forms the law of the underlying asset price. As in the seminal work by Harrison and Kreps (1979), real world probabilities are assumed to be exactly known. The Fundamental Theorem of Asset Pricing (FTAP) then asserts the equivalence between the absence of arbitrage and the existence of a consistent linear price system. Risk-neutral probabilities assigns to each contingent claim an expected value. The situation changes when uncertainty is described by a set of mutually singular priors P.1 A robust form of arbitrage refers to a positive claim that is strictly positive under some prior in P. By accepting this modified notion of arbitrage as a weak dominance principle, one crucial issue points to inconsistencies between linear price systems and the present concept of arbitrage. To realize this point, recall that the easy part of an FTAP under one prior is that the existence of a risk-neutral measure implies absence of arbitrage. When seeking a FTAP under uncertain volatility, the following question serves to clarify the issue: Is the existence of a single risk neutral measure Q, equivalent to some prior in P sufficient for the absence of arbitrage? A short argument yields a negative answer2 and indicates that the robust arbitrage notion is inconsistent with a linear theory of valuation. A single pricing measure Q is incapable of containing all the information about what is possible under P. In the same vein, the idea of “no empty promises” in Willard and Dybvig (1999) points to the unpleasant but possible ignorance of payoffs in states with zero probability for only a particular prior. Section 2 begins with a space of contingent claims that consists of all payoffs with finite variance under all priors. As in the single-prior setup, linear prices on that space of claims are represented by a prior dependent stateprice density or stochastic discount factor. Typically, a risk-neutral measure Q refers to such a price system in the sense of Arrow-Debreu. However, linear prices only capture strictly positive payoffs on events that are in domain of the representing prior. They are blind to an “arbitrage event” under other Singular priors P, P0 live on disjoint supports: P(A) = 1 and P0 (A) = 0 for an event A. The argument goes as follows: Let X ∈ M be a marketed claim with price 0 = π(X). It follows that E Q X = 0 since E Q = π on M . Suppose X ∈ M is an arbitrage with P0 (X > 0) > 0. In the present setting, P0 ∈ P can be singular to P and hence, by equivalence, to Q. 1

2

2

possible priors. For this reason, I aggregate the uncertainty to a robust pricing scheme. The corresponding coherent price systems and the notion of viability then depend on prior-dependent linear price functionals and do not share this problematic feature. The first result states an equivalence between scenario-based viability and extensions to coherent prices systems. Section 3 moves to a dynamic financial market model of a risky and ambiguous asset price process (St ). A sequence of time-consistent conditional sub-linear expectations (Et ) allows for a martingale theory of a fair game against Nature. Instead of one measure representing the risk-neutral world, I suggest the concept of an equivalent symmetric martingale measure set (EsMM-set) Q. The relation to the objective world P comes from a family of state prices that creates a risk- and ambiguity- adjusted expectation E Q . Intuitively, an EsMM-set represents an uncertainty-neutral world, in which ambiguity neutrality, as a part of uncertainty neutrality, matters. The central idea follows the same intuition as the heuristics of risk-neutral valuation. The additional component, preferences for ambiguity, become neutral when we move to the uncertainty-neutral world Q. It is exactly this kind of neutrality that corresponds to the notion of symmetric martingales and means that the price process is a fair game under every pricing measure in Q. This reasoning qualifies the valuation principle to be called uncertainty neutral. The second result shows that there is a one-to-one between viability and EsMM-sets, whenever P allows for time consistent updating. The final step treats a special class of asset prices driven by a Brownian motion with uncertain volatility. This is a zero-mean and stationary process with novel N(0, [σ, σ])-normally distributed independent increments.3 The resulting process is a canonical generalization of the standard Brownian motion, in such a way that the volatility moves almost arbitrarily within [σ, σ]. A Samuelson-type model incorporates this kind of volatility uncertainty by a risky and ambiguous asset price model that follows the stochastic differential equation dSt = µt dt + Vt dBt . (1) The increment dSt in (1) is divided into a locally certain part and the locally uncertain part Vt dBt . The interpretation is h i d varPr (St ) r=t ∈ Vt · σ , Vt · σ , (2) dr where varPr (St ) refers to the conditional variance under P. In abuse of notation, the issue in (2) is displayed by vart (dSt ) = Vt2 dhBit . Under a standard 3

Such random variables are the outcome of a robust central limit theorem for a given confidence interval [σ, σ], see Chapter II in Peng (2010).

3

Brownian motion, the description (2) of (1) boils down to the textbook ded scription dr varr (St ) r=t = Vt , see Chapter 5 in Duffie (2010). The third main result states an equivalence between the existence of an EsMM-sets and absence of P-arbitrage. The state price density has a similarly explicit form as it does in the classical case. Section 4 provides a conclusion. The Appendix presents details and proofs. Related Literature Harrison and Kreps (1979) described how the arbitrage pricing principle provided an economic foundation. Kreps (1981) continued laying the foundation. Later, Delbaen and Schachermayer (1994) presented a general FTAP in continuous time. Here, the uncertainty and risk are seen as equals. Moving to a dynamic multiple prior model, the concept of time consistency becomes crucial, see Epstein and Schneider (2003). Priors are mutually equivalent and share the same null sets; the resulting drift uncertainty leaves the valuation principle almost unaffected. Cont (2006) notes that such an assumption “means that all models agree on the universe of possible scenarios[...], if P0 defines a complete market model, this hypothesis entails that there is no uncertainty on option prices!” For consideration of financial markets under volatility uncertainty, see Denis and Martini (2006) and seminal work by Peng (2006) on ambiguous Brownian motions. Epstein and Ji (2013) provide a discussion in economic terms. Jouini and Kallal (1995) consider nonlinear pricing caused by bid-ask spreads and transaction costs. In Araujo, Chateauneuf, and Faro (2012), pricing rules with finitely many states are considered. An equilibrium with super-linear prices is discussed in Aliprantis, Tourky, and Yannelis (2001).

2

Viability and Coherent Prices

To emphasize key aspects under volatility uncertainty, I start with: Classical Viability Let there be two dates 0 and T ; claims at T have a finite variance and lie in the Lebesgue space LP ≡ L2 (P). Price functionals are linear and continuous, the dual L∗P can be identified with densities in LP . A strictly positive functional Π : LP → R evaluates a non-zero positive payoff X, such that Π(X) > 0. The collection of all such is denoted by L∗P+ . Fix a linear price system π : M → R, where the marketed space M ⊂ LP contains all frictionless achievable claims. AP is the set of convex, strictly monotone and continuous preference relations on R × LP . Viability means ˆ ∈ Bπ with (ˆ ˆ % that there is an %∈ AP and an feasible bundle (ˆ x, X) x, X) (x, X) for all (x, X) ∈ Bπ = {(y, Y ) ∈ R × M : y + π(Y ) ≤ 0}. One can show that π is viable if and only if there is an extension Π ∈ L∗P+ of π to LP . 4

2.1

Uncertainty, Claims and Prices

The states of the world, Ω, build a complete separable metric space. B(Ω) = F is the Borel σ-algebra of Ω, and let Cb (Ω) denote the space of all bounded continuous real-valued functions. The uncertainty is described by a weakly compact subset of Borel probability measures P ⊂ ∆(Ω) on (Ω, F).4 The following example preludes the dynamic setting of Section 3. Example 1 The Wiener measure P0 makes the coordinate process Bt (ω) = ωt into a Brownian motion. A model for volatility uncertainty is then based upon martingale laws: Pσ ≡ P0 ◦ (B σ )−1 ,

where dBtσ = σt dBt

(3)

and (σt ) is an adapted and non-negative process. Via the construction in (3), a set D Rof volatility regimes builds P = (Pσ )σ∈D . The quadratic variation t hB σ it = 0 σs2 ds then describes the volatility process of B σ .5 For payoffs X : Ω → R define E P (X) = maxP ∈P E P X, the upper expectation, 1 and the norm k · kP on Cb (Ω) given by kXkP = E P (X 2 ) 2 . The completion of Cb (Ω) under k · kP is denoted by LP , and let LP = LP /N be the quotient space of LP by the k · kP -null elements N . A payoff X is positive if P(X ≥ 0) = 1 for every P ∈ P . This induces an order relation ≥P , denoted by ≥, on LP so that classical arguments prove that (LP , k · kP , ≥) is a Banach lattice, see Appendix A for details. Moreover, it follows that LP = L{P} . Remark 1 The order relation ≥P usually defines the arbitrage cone by deleting the zero of LP+ = {X ∈ LP : X ≥P 0}. Every R ⊂ P defines a weaker order relation ≥R and a larger arbitrage cone. The arbitrage concept in Definition 3 allows any R ⊂ P to determine what an arbitrage is. To realize the need of coherent price systems, consider the dual space of continuous and linear functionals on LP . The mutual singularity of elements in P generates a different space of contingent claims, and the dual space L∗P changes as well. As stated in Appendix B, the space of all linear and k · kP -continuous functionals on LP is given by  L∗P = Π(·) = E P [ψP ·] where P ∈ P, ψP ∈ LP . (4) Similarly to the LP -setting, an analogous interpretation for the space in (4) holds true. The state price density ψP is supported by some prior P ∈ P, but 4

As shown in Denis, Hu, and Peng (2011), the related capacity c(·) = supP ∈P P (·) is regular if and only if the set of priors is relatively compact, see Appendix B for a criterion. 5 If σ ˆ and σ only agree up to time t, then Pσ and PRσˆ can be neither mutually equivalent r nor singular. Consider the event A = {ω : hBir (ω) = 0 σt2 (ω)dt, r ∈ [0, s]} having positive mass under Pσ and Pσˆ ; but they are mutually singular when restricted to Ω \ A.

5

the stronger norm k·kP , i.e., k·k{P} ≤ k·kP , causes a richer dual space that is parametrized by P. In abuse of changing domains, this means L∗P = ∪P∈P L∗P . Each Π ∈ L∗P assigns no value to strictly positive payoffs outside the support of the prior in the representation. Under P, an insufficiency of linear prices comes into play, which is avoided by allowing sublinear prices on LP . I extend the price space, and take a cue from Aliprantis and Tourky (2002). cx(P) denotes the convex hull of P. The space of coherent price systems of Ψ : LP → R, generated by strictly positive and linear functionals, is given by n o P L~ = Ψ(·) = sup E [ψ ·], P ⊂ cx(P) and ψ ∈ L (5) P Ψ P P+ . P+ P∈PΨ

Elements in L~ P+ rely on a set of strictly positive price functionals (ΠP ), which are consolidated by combining the pointwise maximum and convex combination of linear price systems. This leads to a consolidation operation Y Γ: L∗P+ → L~ P+ P∈P

to aggregate the prior-dependent price systems. PΨ = ΓP then refers to these priors that appear in the representation of Ψ. A linear price system ΠP corresponds to a consolidation via the (second-order) point measure δ{P} ∈ ∆(P), such that ΓP = {P}. Appendix B.1 gives an example how a sublinear functional in L~ P+ can be constructed. Corollary 1 clarifies the domain where prices are linear. Proposition 1 Each Ψ ∈ L~ P+ satisfies, ∀ X, Y ∈ LP , c ∈ R and λ ≥ 0, 1. sub–linearity Ψ(λX + Y ) ≤ λΨ(X) + Ψ(Y ) 2. constant preserving Ψ(c) = c 3. strict positivity X ≥ 0 and X 6= 0 on PΨ implies Ψ(X) > 0. 4. monotonicity If ΓP is closed, X ≥ Y implies Ψ(X) ≥ Ψ(Y ). 5. continuity Xn → X in k · kP implies limn Ψ(Xn ) = Ψ(X). X 6= Y on PΨ refers to the presence of some P ∈ PΨ such that P(X 6= Y ) > 0.

2.2

Viable Price Systems

An FTAP with a sound microeconomic foundation contains a third statement about the existence of an agent (preferring more than less) and being in an optimal state. With regard to Remark 1, the notion of strict monotone preferences is subtle.6 I begin with the introduction of marketed spaces MP ⊂ 6

≥-strict monotonicity means that X ≥ Y and X = 6 Y implies X  Y . The most common preference under ambiguity, such as maxmin, variational or smooth ambiguity representations, are excluded.

6

LP , P ∈ P. Any claim in MP can be achieved frictionlessly, if P is the true prior. The deviation from Kreps (1981) where the market model is described by the quintuple (L, k · k, L+\{0}, M, π), comes from the uncertainty model P and the resulting effect of ambiguity about the distribution of payoffs. Remark 2 In accordance with Remark 1 fix the triple (LP , k · kP , LP+ ). A marketed space, call it M ⊂ LP collects all traded claims. Such claims are on a stock with price R T dynamics R TdSt = Vt dBt and spans M , via terminal outcomes X = ηT ST = 0 ηt dSt = 0 ηt Vt dBt , where η is some feasible and replicating trading strategy. The price of ηT ST is then η0 S0 . When there is also ambiguity in the price dynamics, say Bt has ambiguous volatility as in (3), then the uncertainty in S reappears in a claim on S: σ

Z

X = 0

T

ηt dStσ

Z = 0

T

ηt Vt dBtσ

Z

T

ηt Vt d

= 0

R

t σ dBs 0 s



Z =

T

ηt Vt σt dBt 0

The uncertain volatility σ results in an ambiguous payoff and leads to an uncertain marketed space and price η0 S0σ . To incorporate this ambiguity on replication, the marketed space MPσ and the price πPσ depend now on the prior. πP and πP0 may have a rich common domain, but also different evaluations, i.e. πP (X) 6= πP0 (X) with X ∈ MP ∩ MP0 . Coherence is based on sublinear price systems, as illustrated in the following example and discussed in Heath and Ku (2006). Example 2 Let P = {P, P0 }. If P is the true law, each claim in MP is priced by a linear functional πP . An agent is unable to choose a portfolio in MP0 + MP , due to ignorance about the true prior. An equality of prices on the intersection is less intuitive, since the different priors create different price structures. A robust price for X ∈ MP0 ∩ MP is max{πP0 (X), πP (X)}. The set (πP )P∈P of linear scenario-based price functionals inherits all the information of the underlying financial market. In the single prior setting, Q the market is incomplete if and only if MP 6= LP . MP MP0 denotes the Cartesian product of MP and MP0 . Definition 1 Fix a set of linear prices πP on MP . A price system [(π QP )P∈P ; Γ] consists of a consolidation Γ and the product of Γ-relevant prices P∈ΓP πP . Agents are characterized by their preference for trades on R × LP , P ∈ P. There is a single consumption good, a numeraire. Bundles (x, X) are elements in R × LP , the units at time 0 and T . AP denotes the set of rational, convex, strictly monotone and LP -continuous preference relations %P on R × LP . For a price πP : MP → R define the budget set by 7

BπP = {(y, Y ) ∈ R × MP : y + πP (Y ) ≤ 0}. An appropriate notion of viability defines a minimal consistency criterion, and can be regarded as an inverse no-trade equilibrium condition. Definition 2 A price system is scenario-based viable, if for each P ∈ ΓP there is a preference relation %P ∈ AP and a bundle (xP , XP ) ∈ BπP such that (xP , XP ) is %P -maximal on BπP . The conditions are necessary and sufficient for a classical economic equilibrium under each relevant scenario in ΓP . Note that this definition has to some degree the preference type of Bewley (2002). When MP = M for every P, scenario-based viability exactly captures the existence of an agent with Bewley preferences and a maximal consumption bundle (x, X). The next result connects viability with price systems in L~ P+ . Theorem 1 A price system is scenario-based viable if and only if there is a Ψ ∈ L~ P+ such that πP ≤ Ψ for each P ∈ ΓP . The characterization of scenario-based viability takes scenario-based marketed spaces (MP ) as given. The same holds true for the consolidation operator Γ. With this in mind, one could think that in a general equilibrium price system the locally given prices (πP ) are incorporated. Extensions refer to a Γ-regulated and coherent price system for every claim in LP . There is an intrinsic subspace of mean-ambiguity free claims in which the valuation is unique and the price system acts in a linear manner. In Section 3, the related symmetry property corresponds to a refined martingale notion. Corollary 1 Every Ψ ∈ L~ P+ is linear on the Ψ-marketed space  MΨ = X ∈ LP : E P [ψP X] is constant on P ∈ PΨ .

(6)

Elements in (6) reveal the endogenous structure of possible bilateral trade, such that the law of one price holds. This results in a trade-off between the robustness of prices and the scope of the law of one price on LP .

3 Asset Markets and Symmetric Martingales This section extends the primitives by trading dates and trading strategies on [0, T ], a riskless security, and a security such that its price dynamics has uncertain volatility. Before, let us recall the case without ambiguity.

8

Risk-Neutral Pricing Fix a filtration (Ft ) on (Ω, F, P) and an adapted risky asset price (St ). The wealth process of a strategy η will be denoted by X η . Self-financing strategies are piecewise constant adapted processes η such that dX η = ηdS, and will be denoted by AP . A P-arbitrage in AP is a strategy with no cost and 0 6= XTη ≥ 0. A marketed claim X ∈ M means that there is an η ∈ AP such that X = ηT ST , by the law of one price π(X) = η0 S0 . An equivalent martingale measure (EMM) Q is such that S is an E Q -martingale and dQ = ψdP. ψ ∈ LP+ is a state price with respect to P. Harrison and Kreps (1979) state that under no P–arbitrage, linear and strictly positive extensions Π of π : M → R to LP and EMM’s Q are in a one–to–one correspondence via Π(X) = E Q [X].

3.1

Arbitrage under Volatility Uncertainty

The following example is a refinement of Example 1. Example 3 Suppose that two volatility models σ 1 and σ 2 are consistent with a given data set. Their implications implication for a trading decision may differ considerably. As in Remark 2, the asset span depends on each resulting law Pσ1 and Pσ2 as constructed in (3). In order to address the possibilistic issue, define the universal extreme cases σ t = inf(σt1 , σt2 ) and σ t = sup(σt1 , σt2 ). When thinking about reasonable uncertainty management, no scenario between σ and σ should be ignored. These boundaries result in:  P = Pσ : σt ∈ [σ t , σ t ] for all t ∈ [0, T ] . (7) For the sake of simplicity, the price of the riskless asset will be assumed to satisfies St0 = 1, i.e., the interest rate is zero. Fix a financial market (1, S) on the filtered uncertainty space (Ω, F, P; F), where the price process of the uncertain asset (St ) satisfies St ∈ LP for each t and is F-adapted. Trading strategies are simple.7 The fraction invested in the riskless asset is denoted by ηt0 . A trading strategy η = (η 0 , η 1 ) is self-financing if ηtn−1 (1, S)tn = ηtn (1, S)tn for all dates. The value of the portfolio satisfies Xtη ∈ LP for every t ∈ [0, T ] and the set of self-financing trading strategies is denoted by A. This market (1, S), taken together with A, is denoted by MS . Remarks 1 and 2 motivate considering a robust notion of no-arbitrage. Definition 3 Let R ⊂ P. There is an R-arbitrage opportunity in MS if there exists an admissible pair η ∈ A such that η0 S0 ≤ 0, ηT ST ≥ 0 R-q.s.

and

P∗ (ηT ST > 0) > 0 for some P∗ ∈ R.

7

A simple strategy is an Ft -adapted stochastic process η = (η 0 , η 1 ) such that there is a finite of dates 0 < t0 ≤ · · · ≤ tN = T such that η i , i = 0, 1 can be written as PNsequence −1 i ηt = k=0 1[ti ,ti+1 ) (t)η i,k , with η i,k ∈ LP and Ftk -measurable.

9

The definition rests upon the following thought experiment. An arbitrage strategy is riskless for each P ∈ R, and if the prior P∗ constitutes the market from which one would gain a profit with a strictly positive probability. For instance, the P-arbitrage notion can be seen as a rather weak arbitrage opportunity with the corresponding cone LP+ \{0}. Alternatively, one could argue that no R-arbitrage is consistent with a weak dominance principle based on R. In conjunction with Remark 2, a claim X ∈ LP is P-marketed in MS at time zero under P ∈ P if there is an η ∈ A such that X = ηT ST holds only P-almost surely. In this case η hedges X ∈ MP and η0 S0 = πP (X) is the price of X in MS . Let η, η 0 ∈ AP replicate the same claim X ∈ MP , i.e. ηT ST = η T0 ST P-a.s. Under the absence of P-arbitrage η0 S0 = η 00 S0 = πP (X), ˆ A market MS is viable if although the price may differ under some P 6= P. it is ΓP -arbitrage free, and the associated price system πQ P is viable.

3.2

Equivalent Symmetric Martingale Measure Sets

Let us move to the dynamics of a continuous-time, multiple-prior uncertainty model. For each t, define Pt,P = {P0 ∈ P : P = P0 on Ft } which consists of all priors in P that agree with P in the events up to time t. Fix a contingent claim X ∈ LP . The unique existence of sublinear conditional expectations (EtP )t∈[0,T ] is provided through the following construction 0

EtP (X) = sup EtP X

P-a.s.

P0 ∈Pt,P

for all P ∈ P,

(8)

see section 2 in Epstein and Ji (2014) for details. The sequence of conditioning satisfies the crucial Law of Iterated
Expectation, i.e., for every X ∈ LP P the updating rule EtP (X) = EtP Et+s (X) holds true.8 An Ft -adapted process X = (Xt ) is an E P -martingale if EtP (Xt+s ) = Xt

for all s, t.

(9)

For an E P -martingale X, its negative, −X, is in general not again an E P martingale. If this is the case, we say that the process is a symmetric E P martingale, which is equivalent to the E P -martingale property of X for every P ∈ P. The notion of symmetric martingales lets preferences for ambiguity become neutral by choosing a proper set of risk neutral measures. Definition 4 A set Q in ∆(Ω) is called an equivalent symmetric martingale measure set (EsMM-set) if the following conditions hold: 8

This is an alternative formulation of the rectangularity of Epstein and Schneider (2003). Consistent conditioning as a type of sequential rationality allows defining a martingale. For the multiple prior case with equivalent priors, I refer to Riedel (2009).

10

1. For every Q ∈ Q there is a P ∈ cx(P) that is equivalent via Q

2. (St ) is a symmetric E -martingale and E

Q

dQ dP

∈ LP .

is the expectation under Q.

The case P = {P} boils down to the well-known risk-neutral valuation principle. The first condition states a direct relation between an element Q ∈ Q and the primitive priors P. Integrability is a technical condition to guarantee the connection to viability. The second part points to the adjusted martingale condition. The idea of a fair gamble under Q reflects the neutrality of preferences for both risk and ambiguity, i.e., uncertainty neutrality. The symmetric martingale property of S implies, as discussed in the Introduction, that the (expected) value of the claim is constant on the EsMM-set. The valuation is mean-unambiguous, i.e., preferences for ambiguity under Q are neutral. One can think of the ambiguity-neutral part in terms of risk-neutral maxmin preferences from Gilboa and Schmeidler (1989), i.e., the worst-case expected utility under Q. For a claim X on (St ), E Q [X] is constant on Q ∈ Q. Similarly to pricing under risk, where risk preferences are irrelevant, analogous reasoning should be true for preferences for ambiguity. As such, uncertainty neutrality immediately leads to the need for an the uncertaintyneutral expectation E Q . The following result justifies the discussion of the connection between uncertainty neutrality and symmetric martingales. Theorem 2 Suppose the financial market model MS is P-arbitrage free. There is a bijection between viable prices Ψ ∈ L~ P+ and EsMM sets, such that PΨ = ΓP is stable under pasting.9 Ψ = E Q holds and the EsMM set is Q = {Q ∈ ∆(Ω) : dQ = ψP dP,

P ∈ ΓP and ψP ∈ LP+ } .

(10)

EsMM-sets are ordered by inclusion. MR denotes the collection of all EsMM sets Q as in (10) so that the corresponding consolidation Γ satisfies ΓP = R ⊂SP. Theorem 2 establishes then to a one-to-one mapping between L~ P+ and R⊂cx(P) MR . Analogous to the single-prior setting, this view yields some further insights by combining Theorems 1 and 2. Corollary 2 Suppose given an R = ΓP ⊂ P that is stable under pasting. 1. MS is viable if and only if there is an EsMM-set in MR . 2. If MR is nonempty, then there exists no R-arbitrage. The expected return under the uncertainty neutral expectation E Q equals that of the riskless asset. 9

The set of priors is stable under pasting if for every P ∈ P, every (Ft )-stopping time τ, B ∈ Fτ and P, P0 ∈ Pτ,P , we have Pτ ∈ P, where Pτ (A) = E P P(A|Fτ )1B + P0 (A|Fτo )1B c , for all A ∈ Fτ . It is crucial to define a notion of conditional expectation, that will satisfy the iterated law. For instance, the set of priors in Example 4 are, by construction, stable under pasting.

11

3.3

A Samuelson Model with Ambiguous Brownian Motion

Next I will discuss asset prices that are driven by a G-Brownian motion, in which the volatility uncertainty is contained in the quadratic variation. If P = {P}, the asset price is often an Itˆo process dSt = µt dt + Vt dBt , driven by a Brownian motion (Bt ). The state price density solves dψRt = ψt θt dBt , T P 12 0 θt2 dt satisfies E [e ] < ∞. ψ0 = 1 and the market price of risk θt = µtV−r t Harrison and Kreps (1979) then observe an FTAP under P. 3.3.1

Existence of EsMM-sets

Volatility uncertainty in asset prices (St ) means that every stochastic process (σt ) taking values in [σ, σ] is possible model for the volatility of the state process. More precisely, S is driven by a G-Brownian motion B = (B σt )σt ∈[σ,σ] , see Appendix B.2. As a special case of Example 3, P is induced by [σ, σ]. The asset price is determined by the following stochastic differential equation10 dSt = µt (St )dhBit + Vt (St )dBt .

(11)

Moreover, let V take strictly positive values only. The riskless asset has an interest rate of zero. To verify the symmetric E Q -martingale property of the price process S under some E Q , a Girsanov transformation of (Bt ) guarantees the existence of a nontrivial EsMM-set Q.11 For this approach Rdefine an expectation on LP , generated by Q = {Q ∈ ∆(Ω) : A 7→ Q(A) = ψ(ω)dP(ω) and P ∈ P} via A max E Q X = E Q (X) = E P (ψX),

(12)

Q∈Q

where the state price in (12) is now an aggregated object under the uncertainty model, i.e., ψ = ψP P-a.s. for all P ∈ P. The universal state price density ψ = ψTθ turns out to be a symmetric exponential martingale dψtθ = ψtθ θt (St )dBt , with ψ0θ = 1. The process (θt ) is an integrable pricing kernel, or market price of uncertainty. The results in Appendix B.2 allow us to write ψTθ explicitly as   Z Z T 1 T 2 θ θt (St ) dhBit − θt (St )dBt . (13) ψT = exp − 2 0 0 10

Let µ : [0, T ]×Ω×R → R and V : [0, T ]×Ω×R → R+ be processes such that a unique solution of (11) exists, see part 5 in Peng (2010) for some classical Lipschitz continuity conditions on µ and V in the state variable. 11 Trivial EsMM-sets consist of mutually equivalent priors, associated to a single P ∈ P.

12

Theorem 3 justifies the choice of this shifted sublinear expectation when the asset price is restrained by a symmetric martingale under an uncertaintyneutral expectation. pricing kernel in (13) solve Vt (St )θt (St ) = µt (St ). R t Let the ∗ ∗ ∗ Define St = S0 + 0 Vr (Sr )dBr , which is a symmetric E P -martingale, by the RT
2 second result in Appendix B.2.1. I assume that E P eδ· 0 θt (St ) dhBit < ∞ for some δ > 12 . Under this moderate conditions we have the following FTAP: Theorem 3 There is an EsMM-set Q ∈ MP ⇔ P-arbitrage in MS is absent. The concept of scenario-based viability can be combined via Theorem 2. Let X ∈ LP be a contingent claim such that it is priced by P-arbitrage. Then the value is given by Ψ(X) = E P (ψX), whenever Γ is the maximum operation.12 Remark 3 Theorem 3.3 in Epstein and Ji (2013) obtains analogous state prices by using a utility-gradient approach and assuming µ(t, St ) = µt St and V (t, St ) = Vt St . The local functional form of (11) would then be appart = bt dhBit + Vt dBt . The relation between the asset ently governed by dS St
price processes, with the special pricing kernel µVtt = (θt ), is as follows Rt Rt Rt d µ dhBi = µ a ˆ ds = b ds, where a ˆs = ds hBis . s s s s 0 0 0 s

4

Conclusion

I have presented a theory of derivative security pricing in which volatility uncertainty is incorporated. The classical notion of equivalent martingale measures changes, and the valuation by means of expectations becomes nonlinear. The results of this paper establish a preliminary version of the FTAP under volatility uncertainty. The present uncertainty model is closely related to that of Epstein and Ji (2013), while the present valuation principle follows the preference-free approach by Ross (1976). The price of a claim is the expected value under an uncertainty neutral world. Expectations for the security price do not merely depend on one “risk-neutral” prior; the principle of a risk-neutral valuation is insufficient, as different mutually singular priors deliver completely different linear risk-neutral expectations. No one of them is able to assign a strictly positive price to all strictly positive payoffs. The shortcoming of linear prices is rearranged. A single prior measure, as the output of a linear-price equilibrium, can create an invisible threat ˆ One can define a new G-expectation related to the volatility uncertainty of a closed ˆ α subinterval [ˆ σ1 , σ ˆ2 ] ⊂ [σ, σ] and then identify a consolidation operator by ΓG P = {P : α ∈ [ˆ σ1 , σ ˆ2 ]}. Theorem 3 states then the equivalence between the existence of an EsMM-set ˆ QGˆ ∈ MΓGˆ and absence of ΓG P -arbitrage. 12

P

13

of convention and may lead to an illusion of security when faced with an uncertain future. Under the present type of volatility uncertainty, the focus on a single prior creates a hazard. Payoff-relevant events with a positive probability may be for free under a supporting prior of a linear price. For instance, this can result from the first order condition in a consumption-based approach.

A

Details and Proofs

A property holds quasi-surely (q.s.) if it holds outside a polar set, P-polar sets evaluated under every prior are zero or one. As shown in Theorem 25 of Denis, Hu, and Peng (2011), the space LP is characterized by n o LP = X ∈ L0 (Ω) : X has a q.c. version, lim E P (|X|2 1{|X|>n} ) = 0 . n→∞

A mapping X : Ω → R is said to be quasi-continuous if for all ε > 0 there exists an open set O with supP ∈P P (O) < ε such that X|Oc is continuous. We say that X : Ω → R has a quasi-continuous version (q.c.) if there exists a quasi-continuous function Y : Ω → R with X = Y q.s. Since, for all X, Y ∈ LP with |X| ≤ |Y | imply kXkP ≤ kY kP , we have that LP is a Banach lattice. Proof of Proposition 1 Claim 1 and 2 follow from the construction of the functionals in L~ P+ . Claim 4 directly follows from the fact that P(X ≥ Y ) = 1 for every P ∈ PΨ and the fact that each ψP is positive, i.e. Ψ(Y ) = E P [ψP Y ] ≤ E P [ψP X] ≤ Ψ(X) for some P ∈ PΨ . The first equality applies the compactness of ΓP . Claim 3 follows a similar argument by using the strict positivity of each ψP . By the second part of Proposition 1 we have Ψ(0) = 0. Since LP is a Banach lattice, Claim 5 follows from Theorem 1 in Biagini and Frittelli (2010).  For the proof of Theorem 1, we redefine the shifted preference relationship %P such that every feasible net trade is worse off than (0, 0) ∈ BπP . Obviously, an agent given by %P does not trade. Proof of Theorem 1 Fix the price system πQ P . For the easy direction fix P a Ψ ∈ L~ P+ on LP such that πP ≤ Ψ on MP = MP ∩ LP for each P ∈ ΓP . We have Ψ = maxP∈ΓP ΠP . The preference relation %0P defined by (x, X) <0P (x0 , X 0 )

if x + −ΠP (−X) ≥ x0 + −ΠP (−X 0 )

lies in AP . For each P ∈ ΓP , the bundle (xP , XP ) = (0, 0) satisfies the viability condition of Definition 2, hence πQ P is scenario-based viable. 14

In the other direction, let πQ P be scenario-based viable. The preference relation

P∈ΓP

Q Q and  are both convex sets. The Riesz space product LP = B P P Q L (see paragraph 352 K in Fremlin (2000)), is under the norm k · kP P∈ΓP P again a Banach lattice (see paragraph 354 X (b) in Fremlin Q (2000)). By the LP -continuity of each %P , the set P is k · kP -open in LP . An application of the separation theorem for a topological vector space, for each Q P ∈ ΓP , yields a non-zero linear and k · kP -continuous functional φP on R × LP with 1. φP (x, X) ≥ 0 for all (x, X) ∈P 2. φ(x, X) ≤ 0 for all (x, X) ∈ BQ P 3. {(yP , YP )}P∈ΓP = (y, Y ) with prR×LP (φP )(y, Y ) =: φP (yP , YP ) < 0, since φP is nontrivial. Note that condition 3. depends on the chosen P. Strict monotonicity of %P implies (1, 0) P (0, 0). The LP -continuity of each

0, hence φP (1 + εyP , εYP ) = −φP (1, 0) + εφP (yP , YP ) ≤ 0 and φP (1, 0) ≥ −εφP (yP , YP ) > 0. We have φP (1, 0) > 0 and after a renormalization let φP (1, 0) = 1. Moreover, we can write φP (xP , XP ) = xP + ΠP (XP ), where ΠP : LP → R can be identified as an element in the topological dual L∗P . We show strict positivity of ΠP on LP . Let X ∈ LP+\{0} we have (0, X) P (0, 0), hence (−ε, X) P (0, 0), and therefore ΠP (X) − ε ≥ 0. Moreover, ΠPLP is positive on LP , i.e. X ≥ 0 P-q.s. implies ΠPLP ≥ 0. Since LP is a Banach lattice, positivity of implies continuity and ΠP ∈ L∗P follows. Let X ∈ MPP , since (−πP (X), X), (πP (X), −X) ∈ Bπ we have 0 = φ(πP (X), X) = πP (X) − ΠP (X) and ΠP = πP on MPP follows. Γ((ΠP )P∈ΓP ) = Ψ is by construction in L~ P+ , as the strict positivity of Ψ follows from the strict positivity of each ΠP . ΨMPP ≥ πP follows from an inequality in the last part of Proposition 1 and ΠP = πP on MP .  15

I illustrate the construction of the previous proof in the following diagram: Q  / [(πP : MP → R)P∈P ; Γ]  π ; Γ P P∈ΓP _

Hahn Banach





ΠP : LP → R

P∈ΓP

,Γ 

Γ

/

Ψ : LP → R

Proof of Corollary 1 By construction every functional Ψ can be represented as the supremum of priors in cx(P). Since X ∈ MΨ , the supremum and the infimum of P 7→ E P [ψP X] coincide on ΓP . The assertion follows.  Proof of Theorem 2 We fix an EsMM-set Q. The related consolidation Γ gives us the set of relevant priors ΓP ⊂ P. From Definition 4 there is a ∈ LP , for each Q ∈ Q and a related P ∈ P. Let the associated ψP = dQ dP strictly positive Ψ ∈ L~ P+ be given. Take a marketed claim X m ∈ MPP with P ∈ ΓP and let η ∈ A be a selffinancing trading strategy that hedges X m . Since η ∈ A, by the decomposition rule for conditional E Q -expectation, see for instance Theorem 2.6 (iv) in Epstein and Ji (2014), and since S is a symmetric E Q -martingale, the following equalities EtQ (ηu Su ) = ηt+ EtQ (Su ) + ηt− EtQ (−Su ) = ηt+ St − ηt− St = ηt St hold, where η = η + − η − with ηt+ , ηt− ≥ 0 and 0 ≤ t ≤ u ≤ T . Therefore we achieve Ψ(X m ) = E0Q (ηT ST ) = η0 S0 ≥ πP (X m ),

for every P ∈ ΓP .

For the other direction, fix a strictly positive price system Ψ ∈ L~ P+ with ΨMP ≥ πP , related to a set of linear functionals (πP )P∈P and (ΠP )P ∈P , such that ΠMP = πP . This can be inferred from Ψ and the construction in the proof of the second part of Theorem 1. Now, we define Q in terms of Γ. We illustrate the possible cases which can appear. For simplicity we assume P = {P1 , P2 , P3 }. Let Pk,j = 21 Pk + 12 Pj and ψ k,j = 12 ψ k + 12 ψ j , recall that we can represent each functional ΠP (·) by E P [ψP ·]. We have  1 (Π1 + Π2 ) ∧ Π3 becomes ψ 1,2 · P1,2 , ψ3 · P3 = Q. 2 Consequently, Q = {Q : dQ = ψP dP, P ∈ ΓP , ψP ∈ LP }, where ψP with P ∈ ΓP is constructed by the procedure of Example 4 in Appendix B.1. The first condition of Definition 4 follows, since the square integrability of each ψP follows from the k · kP -continuity of linear functionals which generate Ψ.

16

We prove the symmetric Q-martingale property of the asset price process. Let B ∈ Ft and η ∈ A be a self-financing trading strategy satisfying  (  s ∈ [t, u) and ω ∈ B St , 1 s ∈ [t, u) and ω ∈ B 1 0 ηs = ηs = Su − St , s ∈ [u, T ) and ω ∈ B  0 else ,  0 else. This strategy yields a portfolio value ηT ST = (Su − St ) · 1B , the claim ηT ST is marketed at price zero. Under the uncertainty neutral conditional sublinear expectation (EtQ )t∈[0,T ] , we have with t ≤ u EtQ ((St − Su )1B ) = 0. By Theorem 4.7 Xu and Zhang (2010), it follows that St = EtQ (Su ).13 But this means that (St )t∈[0,T ] is an E Q -martingale. The same argumentation holds for −S, hence the asset price S is a symmetric E Q -martingale.  Proof of Corollary 2 1. Suppose there is a Q ∈ MP and let η ∈ A such that ηT ST ≥ 0 in LP and P0 (ηT ST > 0) > 0 for some P0 ∈ P. Since for all Q ∈ Q there is a P ∈ cx(P) such that Q ∼ P, there is a Q0 ∈ Q with Q0 (ηT ST > 0) > 0. Hence, E Q (ηT ST ) > 0 and by Theorem 2 we observe E Q (ηT ST ) = η0 S0 . This implies that no P-arbitrage exists. 2. By Theorem 2, there is a related Ψ in L~ P+ , wit ΓP = R = PΨ . Fix a costless strategy η ∈ A such that η0 S0 = 0 hence Ψ(ηT ST ) = 0. The viability of Ψ implies ηT ST = 0 R-q.s. Hence, no R-arbitrage exists. Proof of Theorem 3 ⇒ This part is the content of Corollary 2.2 with R = P. ⇐ Let Q = {Q : dQ = ρdP, P ∈ P} be an EsMM-set, where the density ρ satisfies ρ ∈ LP and E P (ρ) = −E P (−ρ). Next define the stochastic process (ρt )t∈[0,T ] by ρt = EtP (ρ) resulting in a symmetric E P -martingale to which we apply the martingale representation theorem for E P -expectation, stated in Appendix B.2. Hence, there is a γ ∈ M 2 such that we can write Z t ρt = 1 + γs dBs . 0

By the G-Itˆo formula, stated in the Appendix B.2, we have Z t Z 1 t 2 ln(ρt ) = φs dBs + φ dhBis , 2 0 s 0 13

The result is proven for the G-framework. The assertion holds true as well by the martingale representation in Proposition 4.10 of Nutz and Soner (2012).

17

for every t ∈ [0, T ] and hence ρ=

ψTφ

1 = exp − 2 

Z

T

θs2 dhBis

Z −

T



θs dBs . 0

0

With this representation of the density process and by the assumed integrability condition we can apply the Girsanov theorem, stated in Appendix B.2. Rt φ ρt Set φt = γt and consider the process Bt = Bt − 0 φs ds, t ∈ [0, T ]. We deduce that Bφ is a G-Brownian motion under E φ (·) = E P (ρ·) and S satisfies Z t Z t φ (µs + Vs φs )dhBφ is t ∈ [0, T ] Vs dBs + St = S0 + 0

0

on (Ω, LP , E φ ). Since V is a bounded process, the stochastic integral is a symmetric martingale under E φ . S is a symmetric E φ -martingale if and only if µt +Vt φt = 0 P-q.s. We have shown that ρ is a simultaneous Radon-Nikodym type density of the EsMM-set Q. Hence, there is a nontrivial EsMM-set in MP , since φt = θt P-q.s for every t ∈ [0, T ]. 

B

Required results

We state the mentioned criterion for the weak compactness of P. Let σ 1 , σ 2 : [0, T ] → R+ determine two measures with a H¨older continuous distribution function t 7→ σ i ([0, t]) = σ i (t). As introduced in (3), a measure P on (Ω, F) is a martingale probability measure if the coordinate process is a martingale. Weak compactness of P, Denis and Martini (2006): The set of probability measures P(σ 1 , σ 2 ) induced by (3) and dσt1 ≤ dhBiPt σ ≤ dσt2 is weakly compact.

B.1

The sub order dual

We discuss the mathematical preliminaries for the price space L~ P. The topological dual space: 1. Let k · kP be a capacity norm, defined in SectionR 2.2. Every continuous linear form l on LP admits a representation l(X) = Xdµ, X ∈ LP , where µ is a bounded signedmeasure F. R defined on a σ-algebra containing 2. We have L∗P = µ = ψP dP : P ∈ P and ψP ∈ L2 (P) . The first claim is stated in Proposition 11 of Feyel and de La Pradelle (1989). The second assertion can be proven via a modification of of Lemma I.28 and Theorem I.30 in Kervarec (2008). 18

For the space of Q coherent price systems L~ P+ , every consolidation operator ∗ Γ has a domain in P ∈P LP and maps to L~ P . Different operations for consolidation are possible. Let ΠP = E P [ψP ·] ∈ L∗P , with P ∈ P and µ ∈ ∆(P) has full support on P. In this context we can consider the additive case in L~ P+ , where a new prior is generated: Z µ Γ ({ΠP }P∈P ) = ψP · µ(dP) = E Pµ [ψPµ ·], P

where ψPµ is constructed as in Example 4. The Dirac measure δP is a particular example for µ where only one particular prior P ∈ P is drives the pricing scheme. The operation in question is given by (ΠP )P∈P 7→ E P [ψP ·]. The second operation in L~ P+ is a point-wise maximum: Γmax ({ΠP }P∈P ) = max E P [ψP ·]. P∈P

Combinations between the maximum and an addition operation are possible: Example 4 Let {Pn }n∈N be a partition of P and µn ∈ ∆(P). For each n, the resulting prior Pn (A) = E µn [P(A)] is given by a second-order weighting operation Γµn . Apply Γµn to the densities ψP so that ψn = E µn [ψP ]. Each group of priors Pn is consolidated to one pair (ψn , Pn ). These resulting pairs (ψn , Pn ) can then be consolidated by Γ(P) = supn∈N E Pn [ψn ·]. Representation of sublinearity, Biagini and Frittelli (2010): Let Ψ be a sublinear functional on L, then PΨ = {x∗ ∈ L∗ : x∗ (X) ≤ Ψ(X) ∀X ∈ L} is nonempty and Ψ(X) = max x∗ (X). ∗ x ∈PΨ

B.2

Stochastics under Sublinear Expectations

This section recalls some notions and results for the G-Brownian motion. Let ΩT = C0 ([0, T ]). A sublinear G-expectation E on LP = L is a functional E : L → R satisfying monotonicity, constant preserving, sub-additivity and positive homogeneity. The triple (Ω, L, E) is called a sublinear expectation space. For the construction of the G-expectation and a general overview see Part 1 and 2 in Peng (2010). In this Appendix, I will only focus on the very basic concepts. In the classic probabilistic setting, where some probability measure P (or a resulting linear expectation E P ) captures the uncertainty, the random vector X is determined by FX (A) = P(X ∈ A) = E P [1{X∈A} ] = E P [fA (X)]. 19

Here, the space of “test functions” fA consists of all the indicator functions with respect to all elements in the Borel σ-algebra B(Rn ). In comparison to the (linear) probability theory, the nonlinearity of E changes the basic notions of stochastics. Definition 5 The distribution of a random vector X = (X1 , . . . , Xn ) ∈ Ln under E is a functional FX : Cb (Rn ) → R given by φ 7→ FX [φ] := E[φ(X)]. Note that for each X, the triple (Rn , Cb (Rn ), FX ) is again a sublinear expectation space. Due to the nonlinearity of E, the (cumulative) distribution function is no longer able to capture all the ”information” about the uncertainty of X under E. From this perspective, the present type of distribution is determined by all the test functions in Cb (Rn ). In accepting this definition of a distribution, the notion of an identical distribution has an intuitive appeal. Definition 6 Two random vectors X, Y ∈ Ln are identical distributed if E[φ(X)] = E[φ(Y )] for every φ ∈ Cb (Rn ) and denoted by X ∼ Y . • We say Y is E-independent of X if for every Cb (R2 ) we have h i E[φ(X, Y )] = E E[φ(x, Y )]|x=X √ • X 0 an independent copy of X. If aX + bX 0 ∼ a + b · X 0 for every a, b ≥ 0, then X is called N (0, [σ, σ])-normal distributed The connection to G-expectation, i.e. EG = E comes from the sublinear and monotone function G : R → R G(a) =

1 sup σa = E[aX 2 ] 2 σ∈[σ,σ]

As a canonical generalization of the standard normal distribution N (0, σ) = N(0, [σ, σ]), a N(0, [σ, σ])-distributed X is characterized by a nonlinear heat PDE ∂t u√= G(uxx ), with φ = u|t=0 such the solution reads as follows u(t, x) = E[φ(x + tX)]. B.2.1

Stochastic analysis with G-Brownian motion

A process (Bt )t≥0 on (Ω, L, E) is called a G-Brownian motion with B0 = 0 if: • Bt+s − Bt is E-independent of (Bt1 , . . . , Btn ). • the increment Bt+s − Bt is N (0, [σ · s, σ · s])-distributed 20

R The Itˆo integral ηdBt for B can also be defined for the following integrands: Let H 0 be the vector space of all simple trading strategies η as in footnote 6. RT 
1/2 For η ∈ H 0 let kηkM 2 ≡ E 0 |ηs |2 ds and denote by M 2 the completion 0 of H under this norm. Itˆo-formula, Li and Peng (2011): Let Φ ∈ C 2 (R) and dXt = µt dhBit + Vt dBt , t ∈ [0, T ], µ, V ∈ M 2 are bounded processes. Then: Z t Z 1 t Φx (Xu )Vu dBu + Φx (Xu )µu + Φxx (Xu )Vu2 dhBiu . Φ(Xt ) − Φ(Xs ) = 2 s s Martingale representation, Soner, Touzi, and Zhang (2011): Let ξ ∈ LP . The E-martingale Xt ≡ E[ξ|Ft ] has the following unique representation Z t Xt = E[ξ] + zs dBs − Kt . 0

K is an increasing with K0 = 0, KT ∈ LP , z ∈ M 2 , and −K is an E-martingale. K ≡ 0 if and only if (Xt ) is a symmetric martingale. Girsanov for G-expectation, Xu, Shang, and Zhang (2011): Assume the following Novikov type condition: There is an ε > 12 such that h  Z T i 2 E exp ε · θs dhBis < ∞ 0 R t Then Bθt = Bt − 0 θs dhBis is a G-Brownian motion under the sublinear expectation E θ (·) given by, E θ (X) = E[ψTθ · X], P θ = ψTθ · P with X ∈ LθP .

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