Pricing under dynamic risk measures

Jun Zhao; Emmanuel Lépinette; Peibiao Zhao

doi:10.1515/math-2019-0070

Artikel Open Access

Pricing under dynamic risk measures

Jun Zhao , Emmanuel Lépinette und Peibiao Zhao

Veröffentlicht/Copyright: 8. August 2019

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 17 Heft 1

Abstract

In this paper, we study the discrete-time super-replication problem of contingent claims with respect to an acceptable terminal discounted cash flow. Based on the concept of Immediate Profit, i.e., a negative price which super-replicates the zero contingent claim, we establish a weak version of the fundamental theorem of asset pricing. Moreover, time consistency is discussed and we obtain a representation formula for the minimal super-hedging prices of bounded contingent claims.

Keywords: Super-hedging; Dynamic risk measures; Time consistency; Absence of immediate profit; Pricing

MSC 2010: 49J53; 60D05; 91G20; 91G80

1 Introduction

In mathematical finance, it is very classical to solve the problem of super-replicating a contingent claim under a no-arbitrage condition (NA). In particular, in frictionless markets, the so-called fundamental theorem of asset pricing (FTAP) characterising NA condition has been studied by numerous authors, see [1, 2, 3] in discrete time and [4, 5] in continuous time. It states that NA condition holds if and only if there exist equivalent martingale measures (EMM). In complete markets, such a martingale measure Q ∼ P is unique and the (replicating) price of a derivative is uniquely computed as the expectation of the discounted payoff under Q. However, in incomplete markets, there exists an infinite number of EMM and the (minimal) super-hedging price is difficult to compute in practice. Indeed, this is a supremum of the expected discounted payoff over all probability measures (see [6] and [7, Theorem 2.1.11]).

A new pricing technique called No Good Deal (NGD) pricing has been proposed in [8, 9]. A good deal is a trade with an unusually high profit/loss or Sharpe ratio. Cherny [10] introduced the concept of good deal with respect to a risk measure as a trade with negative risk. Contrarily to the classical approach where super-replication holds almost surely, Cherny assumes that the agent seller accepts some non null risk for its portfolio not to super-hedge the payoff. In the setting of coherent risk measures, Cherny [10] provides a version of the FTAP under absence of NGD.

Risk measures are more studied and known on the space L^∞, i.e. the space of essentially bounded random variables. And the space L^p, p ∈ [1, ∞) is a natural extension, see [11, 12]. Actually, working on the restricted subspaces of L⁰, such as L^∞ and L^p, is mainly motivated by the robust representation of risk measures. However, the space L⁰, equipped with the topology of convergence in probability, is more adapted for some classical financial and actuarial problems such as hedging, pricing, portfolio choice, equilibrium and optimal reinsurance with respect to risk measures.

Delbaen in [13, 14] extends the coherent risk measure to the space L⁰ by enlarging its range to R ∪ {+∞} as there is no real-valued coherent risk measure on L⁰ when the probability space (Ω, 𝓕, P) is atomless [14, Theorem 5.1]. A robust representation with respect to a set of probability measures is then given [14, Theorem 5.4]. As the space L⁰ contains non integrable random variables, Delbaen in [14] truncates the random variables from above, i.e. only considers possible future wealth up to some threshold. It is then possible to compute the risk measures as in L^∞ and then make n tend to infinity [14, Definition 5.3]). Therefore, the robust representation on L^∞ appears to be the key point to extend coherent risk measure to L⁰, see ([10, Definition 2.2] and [15]), which allow to formulate a FTAP with respect to NGD and solve super-replication problems. In this approach, coherent risk measures remain characterised through families of probability measures which are not necessarily easy to handle in practice, see e.g. the explicit representation of this family for the Weighted VaR risk measure [16, 17].

In this paper, we define risk measures on the space L⁰ with values in R = [−∞, +∞]. They are naturally defined through the concept of acceptable set, i.e. a risk measure is seen as the minimal capital requirement added to the position for it to be acceptable. Under some natural assumptions satisfied by the acceptable set, we show that a risk measure is lower semi-continuous. This allows to compute ω-wise risk measure using similar new results on conditional essential supremum [18]. Inspired by [10], the aim of this paper is to reconsider the super-replication problem in discrete-time with respect to a risk measure without using a dual representation. The minimal super-hedging prices of a contingent claim are recursively defined in the spirit of [18].

Based on the concept of immediate profit, introduced in [18], we establish a weak version of FTAP to equivalently characterise the condition of absence of immediate profit (AIP). Moreover, we show that for bounded non-negative contingent claims, the minimal super-hedging price may be computed through a conditional (dynamic) coherent risk measure derived from the underlying risk measure. At last, we discuss the time consistency, i.e. coherent evaluations of risk in time, since it is a very important concept developed in the literatures for dynamic risk measures, see [19, 20].

The paper is organized as follows. Section 2 gives the definition of risk measures and some important properties for these risk measures are showed. Section 3 introduces the model of super-replication with respect to acceptable sets. We simplify the problem of minimal super-hedging price involving the essential infimum into a classic minimization problem just with infimum. In Section 4, a weak version of fundamental theorem of asset pricing is proved. Section 5 gives a price representation for the bounded non-negative contingent claims.

2 Dynamic risk measure

Notations:

L⁰(R, 𝓕) is the metric space of all R-valued random variables which are 𝓕-measurable;

L^p(R, 𝓕, P), p ∈ [1, ∞) (resp. p = ∞), is the normed space of all R-valued random variables which are 𝓕-measurable and admit a moment of order p under the probability P (resp. bounded). Without any confusions, we omit the notation P and just denote L^p(R, 𝓕);

L^p(R₊, 𝓕) := {X ∈ L^p(R, 𝓕)|X ≥ 0}, L^p(R₊₊, 𝓕) := {X ∈ L^p(R, 𝓕)|X > 0} and L^p(R₋, 𝓕) := {X ∈ L^p(R, 𝓕)|X ≤ 0};

In the following, we consider a complete discrete-time stochastic basis (Ω, 𝓕 := (𝓕_t)_t=0,⋯,T, P) where 𝓕_t represents the available information of the market at time t;

𝔼_P and 𝔼_Q are the expectations of any integrable random variable with respect to the probability measure P and Q. In general, we denote 𝔼_P as 𝔼 without of any confusions. All equalities and inequalities of random variables are understood up to a negligible set.

The dynamic risk measure X ↦ (ρ_t(X))_t=0,⋯,T we consider is defined on L⁰. It is constructed from its acceptance sets defined as follows:

Definition 2.1

A dynamic acceptable set is a family (𝓐_t)_t=0,⋯,T of subsets of L⁰(R, 𝓕_T) satisfying the following conditions:

X + Y ∈ 𝓐_t for all X, Y ∈ 𝓐_t;
Y ∈ 𝓐_t whenever Y ≥ X for some X ∈ 𝓐_t;
𝓐_t ∩ L⁰(R, 𝓕_t) = L⁰(R₊, 𝓕_t);
k_tX ∈ 𝓐_t for any X ∈ 𝓐_t and k_t ∈ L⁰(R₊, 𝓕_t).

Any element of 𝓐_t is said acceptable at time t. For any X ∈ L⁰(R, 𝓕_T), we denote by AtX the set of all Y ∈ L⁰(R, 𝓕_t) such that X + Y ∈ 𝓐_t.

Definition 2.2

Let (𝓐_t)_t=0,⋯,T be a dynamic acceptance set. The risk measure associated to (𝓐_t)_t=0,⋯,T is, at time t, the mapping ρ_t : L⁰(R, 𝓕_T) → L⁰(R ∪ {−∞, +∞}, 𝓕_t) defined as

ρt(X):=essinfAtX (2.1)

up to a negligible set.

Observe that ρ_t(X) is the the minimal capital requirement we add to the position X for it to be acceptable at time t. The effective domain of ρ_t is denoted as

domρt:={X∈L0(R,FT)|ρt(X)<+∞}.

In this paper, we just consider the positions whose risk measures are not infinite at any time t. In other words, we assume that ρ_t(X) < +∞ for any X ∈ L⁰(R, 𝓕_T).

Lemma 2.3

For any X ∈ L⁰(R, 𝓕_T), there exists a sequence Y_n ∈ AtX such that ρ_t(X) = lim_n→∞ ↓ Y_n a.s.

Proof

We first observe that the set AtX is 𝓕_t-decomposable, i.e. if Λ_t ∈ 𝓕_t and Y₁, Y₂ ∈ AtX , then Y₁ 1_{Λ_t} + Y₂ 1_{Ω∖Λ_t} ∈ AtX . To see it, we use conditions 1) and 4) of Definition 2.1. We then deduce that AtX is directed downward, i.e. if Y₁, Y₂ ∈ AtX , then Y₁ ∧ Y₂ ∈ AtX . Indeed, Y₁ ∧ Y₂ = Y₁ 1_{{Y₁≤Y₂}} + Y₂ 1_{Y₁>Y₂} with {Y₁ ≤ Y₂} ∈ 𝓕_t. Therefore, there exists a sequence Y_n ∈ AtX such that ρ_t(X) = lim_n→∞ ↓ Y_n a.s., see [7, Section 5.3.1.]

The following proposition is straightforward due to the definition. The proofs are showed in the Appendix C.

Proposition 2.4

The risk measure ρ_t defined as (2.1) satisfies the following properties:

Normalization: ρ_t(0) = 0;

Monotonicity: X ≤ X′ means ρ_t(X) ≥ ρ_t(X′);

Cash invariance: for all m_t ∈ L⁰(R, 𝓕_t), ρ_t(X + m_t) = ρ_t(X) − m_t;

Subadditivity: for all X, X′ ∈ L⁰(R, 𝓕_T), ρ_t(X + X′) ≤ ρ_t(X) + ρ_t(X′);

Positive homogeneity: for all k ∈ L⁰(R₊, 𝓕_t), ρ_t(kX) = kρ_t(X).

Moreover, if acceptable set 𝓐_t is closed, then ρ_t is lower semi-continuous a.s. with the constraint ρ_t(X) > -∞ a.s. for all X ∈ L⁰(R, 𝓕_T) and 𝓐_t can be represented by ρ_t:

At={X∈L0(R,FT)|ρt(X)≤0}. (2.2)

Definition 2.5

A system (ρ_t)_0≤t≤T is called dynamic risk measure if ρ_t is a risk measure function defined as (2.1) for each 0 ≤ t ≤ T.

3 Minimal super-hedging prices

In the discrete-time model, let (S_t)_0≤t≤T be the discounted price process of asset where S_t ∈ L⁰(R₊, 𝓕_t). And (ρ_t)_0≤t≤T is dynamic risk measure defined in Definition 2.5. A contingent claim at time T is denoted by a real-valued 𝓕_T-measurable random variable h_T. The question is to find a self-financing strategy process (θ_t)_0≤t≤T to super-replicate the contingent claim h_T. Here we use the concept of super-replication in the sense of acceptable set, that is the resulting risk is negative, instead of super-hedging almost surely as the most literatures did. In fact, super-replication almost surely usually can not be realized in a real market.

First let us start with the one step model, that is to super-replicate the contingent claim h_T at time T − 1. And the acceptable set 𝓐_T−1 is assumed to be closed in this section. An notion of super-hedging with respect to the acceptable set is given as follows. In this paper, we just consider the contingent claims which can be super-hedged in the sense of the following definition.

Definition 3.1

Contingent claim h_T is said to be super-hedged at time T − 1 if there exists some P_T−1 ∈ L⁰(R, 𝓕_T−1) and strategy θ_T−1 ∈ L⁰(R, 𝓕_T−1) such that P_T−1 + θ_T−1 Δ S_T − h_T ∈ 𝓐_T−1. And P_T−1 are called the super-hedging prices of the contingent claim h_T at time T − 1.

We show that h_T can be super-hedged if it satisfies the condition h_T ≤ a_T−1 S_T + b_T−1 where a_T−1, b_T−1 ∈ L⁰(R, 𝓕_T−1). In detail, take θ_T−1 = a_T−1 and P_T−1 = a_T−1S_T−1 + b_T−1, then P_T−1 + θ_T−1 Δ S_T − h_T ≥ 0 such that P_T−1 + θ_T−1 Δ S_T − h_T ∈ 𝓐_T−1 since 𝓐_t ∩ L⁰(R, 𝓕_t) = L⁰(R₊, 𝓕_t).

The set 𝓟_T−1(h_T) consists of all super-hedging prices at time T − 1, that is

PT−1(hT):={PT−1∈L0(R,FT−1)|∃θT−1∈L0(R,FT−1)s.t.PT−1+θT−1ΔST−hT∈AT−1}.

Since we assume that the contingent claims of consideration can be super-hedged, that is to say, we may suppose that 𝓟_T−1(h_T) ≠ ∅. According to (2.2) and the cash invariance property of ρ_T−1, P_T−1 + θ_T−1 Δ S_T − h_T ∈ 𝓐_T−1 if and only if P_T−1 ≥ θ_T−1 S_T−1 + ρ_T−1(θ_T−1 S_T − h_T). Then the set 𝓟_T−1(h_T) can be equivalently written as

PT−1(hT)={θT−1ST−1+ρT−1(θT−1ST−hT):θT−1∈L0(R,FT−1)}+L0(R+,FT−1).

Let

g(ω,x):=xST−1+ρT−1(xST−hT),

then the set of super-hedging prices can be expressed as

PT−1(hT)={g(θT−1):θT−1∈L0(R,FT−1)}+L0(R+,FT−1). (3.3)

Actually, we may construct a jointly measurable version of the random function g(ω, x) such that g(θ_T−1) = θ_T−1 S_T−1 + ρ_T−1(θ_T−1 S_T − h_T). And we can prove that g(ω, x) is convex and lower semi-continuous in x for almost all ω under the assumption that the acceptable set 𝓐_T−1 is closed.

Lemma 3.2

Let 𝓖_T−1 = {(X, Y) ∈ L⁰(R², 𝓕_T−1)|Y ≥ X S_T−1 + ρ_T−1(X S_T − h_T)}. Then, 𝓖_T−1 is a non-empty closed convex subset of L⁰(R², 𝓕_T−1). Moreover, 𝓖_T−1 is 𝓕_T−1-decomposable such that 𝓖_T−1 = L⁰(G_T−1, 𝓕_T−1) for some non-empty 𝓕_T−1-measurable random closed convex set G_T−1.

Proof

Trivially 𝓖_T−1 is closed and convex since 𝓐_T−1 is supposed to be closed and a convex cone. And 𝓖_T−1 ≠ ∅ since 𝓟_T−1(h_T) ≠ ∅ from the assumption. Moreover, 𝓐_T−1 is 𝓕_T−1-decomposable and so 𝓖_T−1 is. Thus we can deduce that 𝓖_T−1 = L⁰(G_T−1, 𝓕_T−1) for some 𝓕_T−1-measurable random closed set G_T−1, see [7, Proposition 5.4.3]. As 𝓖_T−1 is not empty, we deduce that G_T−1 ≠ ∅ a.s. Moreover, there exists a Castaing representation of G_T−1 such that G_T−1(ω) = cl {Zⁿ(ω) : n ≥ 1} for every ω ∈ Ω, where (Zⁿ)_n≥1 is a countable family of 𝓖_T−1, see [21, Proposition 2.7]. Then, by a contradiction argument and using a measurable selection argument, we may show that G_T−1 is convex as 𝓖_T−1. □

Proposition 3.3

There exists a 𝓕_T−1 × 𝓑(R)-measurable function g_T−1 such that G_T−1 = {(x, y) : y ≥ g(ω, x)} and Y ≥ X S_T−1 + ρ_T−1(X S_T − h_T) if and only if Y ≥ g_T−1(X) where X, Y ∈ L⁰(R, 𝓕_T−1). Moreover, x ↦ g(ω, x) is a.s. convex and lower semi-continuous.

Proof

Define the following random function

g(ω,x):=inf{α∈R:(x,α)∈GT−1(ω)}. (3.4)

We first show that g is 𝓕_T−1 × 𝓑(R)-measurable. To see it, since the x-sections of G_T−1 are upper sets, we get that g(ω, x) := inf{α ∈ Q : (x, α) ∈ G_T−1(ω)} where Q is the set of all rational numbers of R. Let us define the 𝓕_T−1 × 𝓑(R)-measurable function I(ω, x) = 1 if (ω, x) ∈ G_T−1 and I(ω, x) = +∞ if (ω, x) ∉ G_T−1. Then, define, for each α ∈ Q, θ^α(ω, x) = α I(ω, x) with the convention R ×(+∞) = +∞. As θ^α is 𝓕_T−1 × 𝓑(R)-measurable, we deduce that g(ω, x) = infα∈Q θ^α(ω, x) is also 𝓕_T−1 × 𝓑(R)-measurable.

Since G_T−1 is closed, it is clear that (x, g(ω, x)) ∈ G_T−1(ω) a.s. when g(ω, x) < ∞ and, moreover, g(ω, x) > -∞ by Proposition 2.4. Therefore, G_T−1(ω) is the epigraph of the random function x ↦ g(ω, x). As Y ≥ X S_T−1 + ρ_T−1(X S_T − h_T) if and only if (X, Y) ∈ 𝓖_T−1, or equivalently (X, Y) ∈ G_T−1 a.s., we deduce that it is equivalent to Y ≥ g(X).

Moreover, as G_T−1 is convex, we deduce that x ↦ g(ω, x) is a.s. convex. Let us show that x ↦ g(ω, x) is a.s. lower-semi continuous. Consider a sequence xⁿ ∈ R which converges to x₀ ∈ R. Let us denote βⁿ := g(xⁿ). We have (xⁿ, βⁿ) ∈ G_T−1 from the above discussion. In the case where inf_n βⁿ = −∞, g(ω, x) − 1 > β_n for n large enough (up to a subsequence) hence (xⁿ, g(ω, x) − 1) ∈ G_T−1(ω) since the xⁿ-sections of G_T−1 are upper sets. As n → ∞, we deduce that (x, g(ω, x) − 1) ∈ G_T−1(ω). This contradicts the definition of g. Moreover, the inequality g(x) ≤ lim inf_n βⁿ is trivial when the right hand side is +∞. Otherwise, β^∞ := lim inf_n βⁿ < ∞ and (x₀, β^∞) ∈ G_T−1 as G_T−1 is closed. It follows by definition of g that g(x₀) ≤ lim inf_n g(xⁿ), i.e. g is lower-semi continuous. □

Corollary 3.4

We have g(X) = X S_T−1 + ρ_T−1(X S_T − h_T) a.s. whatever X ∈ L⁰(R, 𝓕_T−1).

Proof

Consider a measurable selection (x_T−1, y_T−1) ∈ 𝓖_T−1 ≠ ∅. We have y_T−1 ≥ g(x_T−1) by definition hence g(x_T−1) < ∞ a.s. Let us define X_T−1 = x_T−11_{g(X)=∞} + X 1_{g(X)<∞}. Since we have

g(XT−1)=g(xT−1)1{g(X)=∞}+g(X)1{g(X)<∞},

is a.s. finite, (X_T−1, g(X_T−1)) ∈ G_T−1 a.s. We deduce that

g(XT−1)≥XT−1ST−1+ρT−1(XT−1ST−hT)

as 𝓖_T−1 = L⁰(G_T−1, 𝓕_T−1). Therefore, g(X) ≥ X S_T−1 + ρ_T−1(X S_T − h_T) on the set {g(X) < ∞}. Moreover, the inequality trivially holds when g(X) = +∞. Similarly, let us define

YT−1=XST−1+ρT−1(XST−hT)1{XST−1+ρT−1(XST−hT)<∞}+yT−11{XST−1+ρT−1(XST−hT)=+∞}.

We have (X_T−1, Y_T−1) ∈ 𝓖_T−1 a.s. hence, by definition of g, g(X_T−1) ≤ Y_T−1. Then, g(X) ≤ X S_T−1 + ρ_T−1(X S_T − h_T) on {X S_T−1 + ρ_T−1(X S_T − h_T) < ∞}. The inequality being trivial on the complementary set, we finally conclude that the equality holds a.s. □

The minimal super-hedging price is given in the sense of (conditional) essential infimum. A generalized concept and existence of conditional essential supremum (resp. conditional essential infimum) of a family of vector-valued random variables with respect to a random partial order are discussed in [22, 23]. Here we use the classical case with a natural partial order for a family of real-valued random variables (see Appendix A).

Definition 3.5

The minimal super-hedging price of the contingent claim h_T at time T − 1 is defined as

PT−1∗:=essinfθT−1∈L0(R,FT−1)PT−1(hT). (3.5)

Omit L⁰(R₊, 𝓕_T−1) and denote PT−1′(hT) := {g(θ_T−1) : θ_T−1 ∈ L⁰(R, 𝓕_T−1)}, then

PT−1∗=essinfθT−1∈L0(R,FT−1)PT−1(hT)=essinfθT−1∈L0(R,FT−1)PT−1′(hT).

Lemma 3.6

The set PT−1′(hT) is directed downward.

Proof

For any θT−11, θT−12 ∈ L⁰(R, 𝓕_T−1), define

θT−1:=θT−111{g(θT−11)≤g(θT−12)}+θT−121{g(θT−11)>g(θT−12)}∈L0(R,FT−1).

Due to the convexity of g, it holds

g(θT−1)≤g(θT−11)1{g(θT−11)≤g(θT−12)}+g(θT−12)1{g(θT−11)>g(θT−12)}=g(θT−11)∧g(θT−12).

That implies that there exists θ_T−1 ∈ L⁰(R, 𝓕_T−1) such that g(θ_T−1) ∈ PT−1′(hT) where g(θ_T−1) is the lower bound of any pair g(θT−11) and g(θT−12) from the set PT−1′(hT) . □

Theorem 3.7

PT−1∗=essinfθT−1∈L0(R,FT−1)⁡g(θT−1)=limn↓g(θT−1n) (3.6)

for some sequence θT−1n ∈ L⁰(R, 𝓕_T−1). Moreover, it holds

essinfθT−1∈L0(R,FT−1)⁡g(θT−1)=infx∈Rg(x). (3.7)

Proof

The first equality (3.6) is a direct consequence of Lemma 3.6. In order to obtain (3.7), we first prove that infx∈Rg(x) g(x) is 𝓕_T−1-measurable. Define

Domg(ω):={x∈R: g(ω,x)<∞}={x∈R: ρT−1(xST−hT)<∞}.

Observe that Dom g is an upper set, i.e. an interval. Since 𝓟_T−1(h_T) ≠ ∅, there exists a strategy a_T−1 ∈ Dom g hence Dom g contains the interval [a_T−1, ∞). Thus we can say that Dom g_T−1 admits a non empty interior on which g_T−1 is convex hence continuous. It follows that

infx∈Rg(x)=infx∈Domgg(x)=infx∈intDomgg(x)=infx∈Q∩intDomgg(x).

We deduce that infx∈Rg(x)≥infx∈Qg(x) so that the equality holds and finally infx∈Rg(x) is 𝓕_T−1-measurable.

As g(θ_T−1) ≥ infx∈Rg(x) for any θ_T−1 ∈ R, then essinfθT−1∈L0(R,FT−1)⁡g(θT−1)≥infx∈Rg(x) from the measurability of infx∈Rg(x) . For the reverse, take x_n ∈ R, of course x_n ∈ L⁰(R, 𝓕_T−1) (basically x_n is a constant), then g(xn)≥essinfθT−1∈L0(R,FT−1)⁡g(θT−1) such that infx∈Rg(x)=infn∈Ng(xn)≥essinfθT−1∈L0(R,FT−1)⁡g(θT−1). Finally, the equality (3.7) holds. □

Actually, it is not very clear how to solve the optimization problem with the essential infimum. Now it has been transferred into a classical one just with infimum according to Theorem 3.7 so that we can know how to deal with it. Before characterizing the optimal solutions and studying the existence of optimal strategies, we first recall the concept of immediate profit (IP) as introduced in [18] and give a weak version of fundamental theorem of asset pricing to build a basic principle for the hedging and pricing.

4 Weak fundamental theorem of asset pricing

Let us extend the acceptable set 𝓐_t to 𝓐_t,t+s ⊆ L⁰(R, 𝓕_t+s) by the same axiomatic conditions in Definition 2.1. In what follows, all acceptable sets are supposed to be closed. The risk measure ρ_t is defined on L⁰(R, 𝓕_t+s) for some s ≥ 0 instead of L⁰(R, 𝓕_T), the risk measure function is

ρt(X)=essinf{Y∈L0(R,Ft)|X+Y∈At,t+s}

and the corresponding acceptable set is

At,t+s={X∈L0(R,Ft+s)|ρt(X)≤0}.

First we consider the general one-step model from t to t + 1, super-hedging the contingent claim h_t+1 at time t means that there exists some P_t ∈ L⁰(R, 𝓕_t) and strategy θ_t ∈ L⁰(R, 𝓕_t) such that P_t + θ_t Δ S_t+1 − h_t+1 is acceptable with respect to the acceptable set 𝓐_t,t+1. Similarly we can express the set of all super-hedging prices as

Pt(ht+1)={θtSt+ρt(θtSt+1−ht+1):θt∈L0(R,Ft)}+L0(R+,Ft).

The minimal super-hedging price at time t for this one-step model is

Pt∗:=essinfθt∈L0(R,Ft)Pt(ht+1). (4.8)

For the contingent claim h_T we define recursively

PT∗=hTandPt∗:=essinfθt∈L0(R,Ft)Pt(Pt+1∗)

where Pt+1∗ can be regarded as the contingent claim h_t+1.

Let us recall the concept of immediate profit as introduced in [18], which means that it is possible to super-replicate contingent claim zero with a negative price.

Definition 4.1

Absence of Immediate Profit (AIP) holds if

Pt(0)∩L0(R−,Ft)={0} (4.9)

for any 0 ≤ t ≤ T.

It is obvious that (AIP) property automatically holds at time T since 𝓟_T(0) = L⁰(R₊, 𝓕_T). Next we characterize (AIP) for general model with t ≤ T − 1.

Theorem 4.2

(Weak Fundamental theorem of asset pricing) (AIP) property holds if and only if

−ρt(St+1)≤St≤ρt(−St+1) (4.10)

for all 0 ≤ t ≤ T − 1.

Proof

For the backward recursion starting from PT∗ = h_T = 0, the set of super-hedging prices for contingent claim zero at time T − 1 is

PT−1(0)={θT−1ST−1+ρT−1(θT−1ST):θT−1∈L0(R,FT−1)}+L0(R+,FT−1)

and the minimal super-hedging price is PT−1∗=essinfθT−1∈L0(R,FT−1)PT−1(0). From Theorem 3.7 we know

PT−1∗=essinfθT−1∈L0(R,FT−1)⁡g(θT−1)=infx∈Rg(x)

where g(x) = x S_T−1 + ρ_T−1(x S_T) for the case h_T = 0. Now it is easy to see that

g(x)=x[ST−1+ρT−1(ST)]1x≥0+x[ST−1−ρT−1(−ST)]1x<0.

Denote Λ_T−1 := {−ρ_T−1(S_T) ≤ S_T−1 ≤ ρ_T−1(−S_T)}, then we can deduce that

PT−1∗=(0)1ΛT−1+(−∞)1Ω∖ΛT−1.

Now (AIP) at time T − 1 implies that the set Ω ∖ Λ_T−1 is empty, that is

−ρT−1(ST)≤ST−1≤ρT−1(−ST) (4.11)

holds almost surely. By repeating the procedure for time T − 2, T − 3, … we can get the conclusion. □

Example 4.3

For the classical one-step super-hedging problem, i.e., a contingent claim h_T can be super-replicated at time T − 1 means that there exist some P_T−1 ∈ L⁰(R, 𝓕_T−1) and strategy θ_T−1 ∈ L⁰(R, 𝓕_T−1) such that P_T−1 + θ_T−1Δ S_T − h_T ≥ 0 almost surely. In this case the acceptable set 𝓐_T−1 is as follows:

AT−1={X∈LT0|X≥0} ={X∈LT0|essinfFT−1X≥0} ={X∈LT0|−essinfFT−1X≤0}.

This also implies that ρ_T−1(X) = −ess inf_{𝓕_T−1} X. Then from Theorem 4.2 AIP property can be expressed as the same equivalent condition:

essinfFT−1ST≤ST−1≤esssupFT−1ST. (4.12)

Indeed, S_T−1 ≥ −ρ_T−1(S_T) = ess inf_{𝓕_T−1} S_T and S_T−1 ≤ ρ_T−1(−S_T) = −ess inf_{𝓕_T−1}(−S_T) = ess sup_{𝓕_T−1}S_T. Thus the second equivalent condition of (AIP) in [18, Theorem 3.4] is one of the special cases in our paper when taking the worst-case risk measure ρ_T−1(X) = −ess inf_{𝓕_T−1} X.

Remark 4.4

The condition (4.11) implies (4.12) trivially. Actually, the risk at time T − 1 of position X ∈ L⁰(R, 𝓕_T) given by ρ_T−1(X) = −ess inf_{𝓕_T−1} X is the worst-case (maximum) one. Indeed, from (2.1), we can easily see that

ρT−1(X)≤esssupFT−1(−X)=−essinfFT−1X

since X + ess sup_{𝓕_T−1}(−X) ∈ 𝓐_T−1 and ess sup_{𝓕_T−1}(−X) is 𝓕_T−1-measurable. By considering −X it holds that

ρT−1(−X)≤esssupFT−1(X)

such that we can get by taking X = S_T that

essinfFT−1ST≤−ρT−1(ST)≤ST−1≤ρT−1(−ST)≤esssupFT−1ST.

5 Price representation

In this section, the study is restricted to bounded non-negative contingent claims. The main purpose is to give the specific expression of minimal super-hedging prices in the sense of risk management.

Notice that the risk measure ρ_t is based on the space L⁰ and its dual representation is not used in the previous content. Next we give a new risk measure defined on the space L^∞ under which the minimal super-hedging price of a bounded contingent claim is just the risk of its opposite payoff.

Let us recall the general axiomatic definition of conditional coherent risk measure ρ_t : L^∞(R, 𝓕_T) → L⁰(R, 𝓕_t) (see Definition 1, 2 and 3 in [24]):

Definition 5.1

([24]) A map ρ_t : L^∞(R, 𝓕_T) → L⁰(R, 𝓕_t) is said to be a conditional coherent risk measure if it satisfies the following properties:

Normalization: ρ_t(0) = 0;
Conditional translation invariance: for all X ∈ L^∞(R, 𝓕_T) and m_t ∈ L^∞(R, 𝓕_t),

ρt(X+mt)=ρt(X)−mt;
Monotonicity: for all X, X ∈ L^∞(R, 𝓕_T), X ≤ X′ means ρ_t(X) ≥ ρ_t(X′);
Subadditivity: for all X, X′ ∈ L^∞(R, 𝓕_T), ρ_t(X + X′) ≤ ρ_t(X)+ρ_t(X′);
Conditional positive homogeneity: for all k ∈ L^∞(R₊, 𝓕_t), ρ_t(kX) = kρ_t(X).

Let us define recursively (ρ̃_t)_0≤t≤T for some bounded position Y ∈ L^∞(R, 𝓕_T) based on the given dynamic risk measure (ρ_t)_0≤t≤T as

ρ~T(Y)=−Yandρ~t(Y)=infx∈Rρt(xΔSt+1−ρ~t+1(Y)).

Actually, it can be proved that ρ̃_t are conditional coherent risk measures defined in Definition 5.1 for all 0 ≤ t ≤ T and (ρ̃_t)_t is time-consistent, that is for all X, Y ∈ L^∞(R, 𝓕_T) and 0 ≤ t ≤ T, ρ̃_t+1(X) = ρ̃_t+1(Y) implies ρ̃_t(X) = ρ̃_t(Y) (see Section 5 in [24]). Then the pricing problem is naturally equivalent to measure the risk of contingent claim under the conditional coherent risk measure ρ̃_t, that is

Pt∗=ρ~t(−hT) (5.13)

which is the time-consistent price process.

Lemma 5.2

Assume the condition (AIP) holds, then ρ̃_t are conditional coherent risk measures for all 0 ≤ t ≤ T on L^∞. Moreover, (ρ̃_t)_t is time-consistent whenever the underlying dynamic risk measure (ρ_t)_t is or not.

Proof

Indeed, ρ̃_T(⋅) trivially satisfies the conditions in the Definition 5.1 such that ρ̃_T(⋅) is a conditional coherent risk measure. And all the other properties except normalization for ρ̃_t with 0 ≤ t ≤ T − 1 are easy to be inherited from ρ_t by the induction. Here we just need to prove the normalization. Assume ρ̃_t+1(0) = 0, then

ρ~t(0)=infx∈Rρt(xΔSt+1)=infx∈R[xρt(ΔSt+1)1x≥0−xρt(−ΔSt+1)1x<0]=0

as (AIP) implies that ρ_t(Δ S_t+1) and ρ_t(−Δ S_t+1) are both non-negative. The time-consistency can be easily deduced from the definition of (ρ̃_t)_t. □

Next we can give the expression of Pt∗ in the sense of robust representation for conditional coherent risk measure ρ̃_t. First let us give the following sets of probability measures for all 0 ≤ t ≤ T as:

Qt:={Qis a probability measure|Q≪PandQ=P|Ft}. (5.14)

Theorem 5.3

Assume (AIP) property holds, then the minimal super-hedging price can be represented as

Pt∗=esssupQ∈Qt∗EQ(hT|Ft)

where

Qt∗:={Q∈Qt|EQ(Y|Ft)≥−ρ~t(Y),∀Y∈L∞(R,FT)}. (5.15)

Proof

From Lemma 5.2 ρ̃_t is a conditional coherent risk measure. And the lower semi-continuity of ρ̃_t is inherited from the underlying risk measure ρ_t. Thus the following robust representation (see [24]) can be obtained

ρ~t(Y)=esssupQ∈Qt∗{−EQ(Y|Ft)}

where 𝓠_t and Qt∗ are defined as (5.14) and (5.15). Then let Y = −h_T it is easy to conclude from (5.13). □

Appendix A. Conditional essential supremum/infimum

Given a measurable probability space (Ω, 𝓕, P) and 𝓗 is a sub-σ-algebra of 𝓕. Recall the concept of generalized conditional essential supremum (see [22], Definition 3.1) in L⁰(R^d) as well as the existence and uniqueness for the case where d = 1 (see [22], Lemma 3.9). A similar result holds for the conditional essential infimum.

Lemma 3.9

([22]) Let Γ ≠ ∅ be a subset of L⁰(R ∪ {+∞}, 𝓕). Then there exist a unique 𝓗-measurable random variable γ̂ ∈ L⁰(R ∪ {+∞}, 𝓗) denoted as ess sup_𝓗 Γ such that the following conditions hold:

γ̂ ≥ γ a.s. for any γ ∈ Γ;
if γ̃ ∈ L⁰(R ∪ {+∞}, 𝓗) satisfies γ̃ ≥ γ a.s. for any γ ∈ Γ, then γ̃ ≥ γ̂ a.s.

B. Measurable subsequences

First, let us recall the existence of convergent subsequences of the random sequence from L⁰(R^d), see [7, Lemma 2.1.2]. The technical constructions of these convergent subsequences can be found in the proof of this lemma.

Lemma 2.1.2

([7]) Let ηⁿ ∈ L⁰(R^d) be such that η := lim inf|ηⁿ| < ∞. Then there are η̃^k ∈ L⁰(R^d) such that for all ω the sequence of η̃^k(ω) is a convergent subsequence of the sequence of ηⁿ(ω).

It is worth noting that the subsequence η̃^k is random due to the fact that

η~k(ω)=ηnk(ω)(ω)=∑p≥k∞ηp(ω)1nk=p.

The more detailed results about the random convergent subsequence can be found in [25, Section 6.3]. Let (𝓚, d) be a compact metric space and ℕ be the set of all natural numbers.

Definition 6.3.1

([25]) An ℕ-valued, 𝓕-measurable function is called a random time. A strictly increasing sequence (τk)k=1∞ of random times is called a measurably parameterised subsequence or simply a measurable subsequence.

Lemma 6.3.2

([25]) Let (fn)n=1∞ be a sequence of 𝓕-measurable function f_n : Ω → 𝓚. Let τ : Ω → {1, 2, 3, ⋯} be an 𝓕-measurable random time, then g(ω) = f_τ(ω) (ω) is 𝓕-measurable.

Proposition 6.3.3

([25]) For a sequence (fn)n=1∞ ∈ L⁰(Ω, 𝓕, P; 𝓚) we may find a measurably parameterised subsequence (τk)k=1∞ such that (fτk)k=1∞ converges for all ω ∈ Ω.

Proposition 6.3.4

([25]) Under the assumptions of Proposition 6.3.3 we have in addition:

Let x₀ ∈ 𝓚 and define

B={ω∈Ω: x0 is an accumulation point of (fn(ω))n=1∞}.

Then the sequence (τk)k=1∞ in Proposition 6.3.3 may be chosen such that

limkfτk(ω)(ω)=x0, for each ω∈B.
Let f₀ ∈ L⁰(Ω, 𝓕, P;𝓚) and define

C={ω∈Ω: f0 is not the limit of (fn(ω))n=1∞},

where the above means that either the limit does not exist or, if it exists, it is different from f₀(ω). Then the sequence (τk)k=1∞ in Proposition 6.3.3 may be chosen such that

limkfτk(ω)(ω)≠f0(ω), for each ω∈C.

C. Proof of Proposition 2.4

The first five conventional properties are directly deduced from the definition of ρ_t in (2.1). Let us prove the “Moreover” part under the assumption that the acceptable set 𝓐_t is closed.

First, we can prove that ρ_t(X) > yq∞ a.s for all X ∈ L⁰(R, 𝓕_T). Indeed, by Lemma 2.3, for any X ∈ L⁰(R, 𝓕_T) there exists a sequence Y_n ∈ AtX , i.e. Y_n ∈ L⁰(R, 𝓕_t) satisfying X + Y_n ∈ 𝓐_t such that ρ_t(X) = lim_n ↓ Y_n. Suppose that P{ρ_t(X) = −∞} > 0. Denote the 𝓕_t-measurable set Λ_t := {ω ∈ Ω : ρ_t(X) = −∞} = {ω ∈ Ω : lim_n ↓ Y_n = −∞}. Let us consider it by the following steps:

By taking 𝓚 = R ∪ {−∞} and x₀ = −∞ in [25, Proposition 6.3.4 (i)], there is a 𝓕_t-measurably parameterised subsequence (τk)k=1∞ such that the subsequence (Lk)k=1∞:=(Yτk)k=1∞ diverges to −∞ on the set Λ_t of positive probability. Since (X(ω) + L_k(ω)) 1_{Λ_t} = (X(ω) + Y_{τ_k(ω)} (ω)) 1_{Λ_t} = (X(ω) + ∑p≥k∞ Y_p(ω) 1_{τ_k=p}) 1_{Λ_t} = ∑p≥k∞ (X(ω) + Y_p(ω)) 1_{τ_k=p} 1_{Λ_t} and τ_k is 𝓕_t-measurable, then we can deduce from the additivity and the positive homogeneity of 𝓐_t that X + L_k ∈ 𝓐_t on the set Λ_t.
By the normalization procedure X¯k:=X|Lk| and L¯k:=Lk|Lk|, we get that X̄_k + L̄_k ∈ 𝓐_t on the set Λ_t. Applying [25, Proposition 6.3.3] to the sequence (L¯k)k=1∞, there is a 𝓕_t-measurably parameterised subsequence (σi)i=1∞ such that the subsequence (L¯σi)i=1∞ converges to some L̄. As |L̄_k| = 1 for any k ≥ 1, we can see that |L̄| = 1. Actually, L̄ = -1 a.s. as L̄_{σ_i} < 0 for large enough i.
Next we can say that −1 is also the limit of the sequence (L¯k)k=1∞ a.s. Otherwise, the set C := {ω ∈ Ω : −1 is not the limit of (L¯k(ω))k=1∞ } has positive probability. By taking f₀ = −1 in [25, Proposition 6.3.4 (ii)], then 𝓕_t-measurably parameterised subsequence (σi)i=1∞ may be chosen such that lim_i L̄_{σ_i(ω)}(ω) ≠ −1 for each ω ∈ C. This is contradicted with the above statement L̄ = −1. Thus, we can deduce that lim_k L̄_k = −1.
On the other hand, X¯k=X|Lk| trivially converges to zero as L_k diverges to −∞. Finally, we deduce that lim_k(X_k + L̄_k) = −1 ∈ 𝓐_t on the set Λ_t if 𝓐_t is closed. This is contradicted with the third condition : 𝓐_t ∩ L⁰(R, 𝓕_t) = L⁰(R₊, 𝓕_t) in the Definition 2.1. Thus, the assumption ρ_t(X) = −∞ with a positive probability is impossible, that is ρ_t(X) > −∞ with probability one.

Since we assume that ρ_t(X) < +∞ for any X ∈ L⁰(R, 𝓕_T), then it holds ρ_t(X) ∈ L⁰(R, 𝓕_t). By Lemma 2.3, we know that ρ_t(X) = lim_n ↓ Y_n a.s. where Y_n ∈ L⁰(R, 𝓕_t) satisfying X + Y_n ∈ 𝓐_t. As the set 𝓐_t is closed, 𝓕_t-decomposable and contains 0, we deduce that X + ρ_t(X) ∈ 𝓐_t for any X ∈ L⁰(R, 𝓕_T). Now let us prove the lower semi-continuity of ρ_t. Consider a sequence X_n ∈ L⁰(R, 𝓕_T) which converges to X₀. Denote α_n := ρ_t(X_n), then X_n + α_n ∈ 𝓐_t. Our goal is to prove the inequality ρ_t(X₀) ≤ lim infα_n a.s. Let us divide it into the following three cases:

As for the case where lim inf α_n = +∞, the inequality ρ_t(X₀) ≤ lim infα_n holds trivially. Thus we may assume w.l.o.g. that lim infα_n < +∞.
Let us consider the case where lim infα_n = −∞. Suppose that the 𝓕_t-measurable set Γ_t := {ω ∈ Ω : lim infα_n = −∞} has a positive probability. Obviously, −∞ is an accumulation point of (αn)n=1∞ on the set Γ_t. For convenience, denote α_∞ := lim infα_n. Again, [25, Proposition 6.3.4 (i)] implies that there is a 𝓕_t-measurably parameterised subsequence (μk)k=1∞ such that the subsequence (βk)k=1∞:=(αμk)k=1∞ diverges to −∞ on the set Γ_t of positive probability. Let (Zk)k=1∞:=(Xμk)k=1∞ be the corresponding subsequence of the sequence X_n. Then we can see that Z_k + β_k ∈ 𝓐_t on the set Γ_t as (Z_k(ω) + β_k(ω)) 1_{Γ_t} = (X_{μ_k(ω)}(ω) + α_{μ_k(ω)} (ω)) 1_{Γ_t} = ∑p≥k∞ (X_p(ω) + α_p(ω)) 1_{μ_k=p} 1_{Γ_t}. Then, using the normalization procedure Z~k:=Zk|βk| and β~k:=βk|βk|, we get that Z̃_k + β̃_k ∈ 𝓐_t on the set Γ_t. By passing once again to a measurably parameterised subsequence, we may assume that β̃_k converges to −1 according to the similar statements in the above Step 2 and Step 3. Note that Z_k = X_{μ_k} converges to X₀ and β_k diverges to −∞ such that Z̃_k converges to zero, we finally get that lim_k(Z̃_k + β̃_k) = −1 ∈ 𝓐_t on the set Γ_t if the set 𝓐_t is closed. This contradicts with the third condition in the Definition 2.1. Thus, α_∞ = lim infα_n > −∞ with probability one.
Combining the cases a) and b), we can assume w.l.o.g. that α_∞ ∈ L⁰(R, 𝓕_t) and X₀ + α_∞ ∈ 𝓐_t. It follows that ρ_t(X₀) ≤ α_∞ = lim inf ρ_t(X_n) a.s.

At last, if the set 𝓐_t is closed, the acceptable set 𝓐_t can be represented as 𝓐_t = {X ∈ L⁰(R, 𝓕_T)|ρ_t(X) ≤ 0}. Indeed, it is clear that ρ_t(X) ≤ 0 for all X ∈ 𝓐_t. Reciprocally, if ρ_t(X) ≤ 0, we get that X = −ρ_t(X) + a_t where a_t ∈ 𝓐_t. Finally we can deduce that X ∈ 𝓐_t since 0 ≤ −ρ_t(X) ∈ 𝓐_t and 𝓐_t + 𝓐_t ⊆ 𝓐_t. □

Acknowledgements

The authors are grateful to the responsible editor and the anonymous referees for their valuable comments and suggestions. This work was supported by the [National Nature Science Foundation of China] under Grant [number 11871275] and Grant [number 11371194]; [Grant-in-Aid for Scientific Research from Nanjing University of Science and Technology] under Grant [number KN11008] and Grant [number 2011YBXM120].

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Received: 2018-08-14

Accepted: 2019-06-19

Published Online: 2019-08-08

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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https://doi.org/10.1515/math-2019-0070

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Super-hedging; Dynamic risk measures; Time consistency; Absence of immediate profit; Pricing

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