DYNAMIC RISK MEASURES AND G-EXPECTATION

Abstract

A standard deviation has been a starting point for a mathematical definition of risk. As a remedy for drawbacks such as subadditivity property discouraging the diversification, coherent and convex risk measures are introduced in an axiomatic approach. Choquet expectation and g-expectations, which generalize mathematical expectations, are widely used in hedging and pricing contingent claims in incomplete markets. The each risk measure or expectation give rise to its own pricing rules. In this paper we investigate relationships among dynamic risk measures, Choquet expectation and dynamic g-expectations in the framework of the continuous-time asset pricing.

1. INTRODUCTION

Various kinds of risk measures have been proposed and discussed to measure or quantify the market risks in theoretical and practical perspectives. A starting point for a mathematical definition of risk is simply as standard deviation. Markowitz [19] used the standard deviation to measure the market risk in his portfolio theory but his method doesn’t tell the difference between the positive and the negative deviation. Artzner et al. [2, 3] proposed a coherent risk measure in an axiomatic approach, and formulated the representation theorems. Frittelli [12] proposed sublinear risk measures to weaken coherent axioms. Heath [16] firstly studied the convex risk measures and Föllmer & Schied [9, 10, 11] and Frittelli & Rosazza Gianin [13] extended them to general probability spaces. They had weakened the conditions of positive homogeneity and subadditivity by replacing them with convexity.

There exist stochastic phenomena like Allais paradox [1] and Ellsberg paradox [8] which can not be dealt with linear mathematical expectation in economics. Choquet [6] introduced a nonlinear expectation called Choquet expectation which applied to many areas such as statistics, economics and finance. But Choquet expectation has a difficulty in defining a conditional expectation. Peng [21] introduced a nonlinear expectation, 𝑔-expectation which is a solution of a nonlinear backward stochastic differential equation. It’s easy to define conditional expectation with Peng’s 𝑔-expectation. In this paper, we show that Choquet expectation is equal to 𝑔-expectation under some conditions via {Ƒt}t∈[0, T]-consistent expectation Ɛ satisfying Ɛμ-domination and translability condition.

The coherent (or convex) risk measure which is a static risk measures is defined in section 2. Peng’s 𝑔-expectation, Choquet expectation and dynamic risk measure are introduced in section 3. The relationships between Choquet expectation and 𝑔-expectation are given as in the literature in section 4. It is shown that Choquet expectation is equal to 𝑔-expectation under some conditions via {Ƒt}t∈[0, T]-consistent expectation Ɛ in section 5.

 

2. STATIC RISK MEASURES

Risk measures are introduced to measure or quantify investors’ risky positions such as financial contracts or contingent claims. Let ( Ω, Ƒ , P) be a probability space and T be a fixed horizon time. Assume that 𝒳 = Lp( Ω, Ƒ, P), with 1 ≤ p ≤ + ∞ is the space of financial positions to be quantified or measured. Lp( Ω, Ƒ, P) is endowed with its norm topology for p ∈ (1, + ∞) and with the weak topology σ(L∞, + L1) for p = + ∞.

Definition 2.1. A coherent risk measure ρ : 𝒳 → ℝ is a mapping satisfying for X, Y ∈ 𝒳

(1) ρ(X) ≥ ρ(Y) if X ≤ Y (monotonicity), (2) ρ(X + m) = ρ(X) − m for m ∈ ℝ (translation invariance), (3) ρ(X + Y) ≤ ρ(X) + ρ(Y) (subadditivity), (4) ρ(λX) = λρ(X) for λ ≥ 0 (positive homogeneity).

The subadditivity and the positive homogeneity can be relaxed to a weaker quantity, i.e. convexity

ρ(λX + (1 − λ)Y) ≤ λρ(X) + (1 − λ)ρ(Y) ∀λ ∈ [0,1],

which means diversification should not increase the risk.

 

3. PENG’S 𝑔-EXPECTATION AND CHOQUET EXPECTATION

Let (Wt)t≥0 a standard d−dimensional Brownian motion and (Ƒt)t≥ 0 the augmented filtration associated with the one generated by (Wt)t≥0. Let be the space of the adapted processes (ξt)t∈[0, T] such that

where ∥·∥ represents the Euclidean norm on ℝn.

Suppose that for t ∈ [0, T], L2(Ƒt) := L2(­Ω, Ƒt, P) is the space of real-valued, Ƒt-measurable and square integrable random variables endowed with the L2-norm ∥·∥2 topology.

Let 𝑔 : ­ Ω × [0, T] × ℝ × ℝn → ℝ a function that 𝑔 ⟼ 𝑔(t, y, z) is measurable for each (y, z) ∈ ℝ × ℝn and satisfy the following conditions

Theorem 3.1 ([20]). For every terminal condition X ∈ L2(ƑT) the following backward stochastic differential equation

has a unique solution

Definition 3.2. For each X ∈ L2(ƑT) and for each t ∈ [0, T] 𝑔 -expectation of X and the conditional 𝑔-expectation of X under Ƒt is respectively defined by

Ɛ𝑔[X] := y0, Ɛ𝑔[X|Ƒt] := yt,

where yt is the solution of the BSDE (3.2).

Since 𝑔-expectation and conditional 𝑔-expectation can be considered as the extension of classic mathematical expectation and conditional mathematical expectation, they preserve most properties of classic mathematical expectation and conditional mathematical expectation except the linearity.

Definition 3.3. A real-valued set function c : Ƒ → [0, 1] is called capacity if it satisfies (1) c(A) ≤ c(B) for A ⊂ B, (2) c(∅) = 0 and c(Ω­) = 1.

Definition 3.4. A capacity is called submodular or 2-alternating if

c(A∪B) + c(A∩B) ≤ c(A) + c(B).

Definition 3.5. Two measurable functions X and Y on (Ω, Ƒ) are called comonotone if there exists a measurable function Z on (Ω, Ƒ) and increasing functions f and 𝑔 on ℝ such that

X = f(Z) and Y = 𝑔(Z).

A risk measure ρ on Lp(ƑT ) is called comonotonic if

ρ(X + Y) = ρ(X) + ρ(Y)

whenever X and Y are comonotonic.

Define the Choquet integral of the loss as

Then ρ : 𝒳 → ℝ satisfies monotonicity, translation invariance and positive homogeneity, and other properties according to the given conditions.

(1) (Constant preserving) ∫ λdc = λ for constant λ. (2) (Monotonicity) If X ≤ Y, then ∫(−X)dc ≥ ∫(−Y)dc. (3) (Positive homogeneity) For λ ≥ 0, ∫ λ(−X)dc = λ∫(−X)dc. (4) (Translation invariance) ∫(−X + m)dc = ∫(−X)dc + m, m ∈ ℝ. (5) (Comonotone additivity) If X and Y are comonotone functions, then

(6) (Subadditivity) If c is submodular or concave function, then

The static risk measures do not account for payoffs or new information according to the time evolution(refer to [25, 26]).

Definition 3.6. A dynamic risk measures are defined as the mappings (ρt)t∈[0, T] satisfying

(1) ρt : Lp(ƑT) → L0(­Ω, Ƒt, P), for all t ∈ [0, T],(2) ρ0 is a static risk measure,(3) ρT (X) = −X   P−a.s., for all X ∈ Lp(ƑT).

 

4. NONLINEAR EXPECTATIONS AND NONLINEAR PRICING

To quantify riskiness of financial positions, coherent (or convex) risk measures, Choquet expectation and 𝑔-expectation are widely used. It depends on practitioner’s appropriate choices. The paper [5] shows that the pricing with the coherent risk measure is less than one with the Choquet expectation.

Denote the Choquet expectation 𝒞(·) as 𝒞𝑔(·) with respect to the capacity V𝑔 defined as

V𝑔(A) := Ɛ𝑔[IA] ∀A ∈ ƑT.

Theorem 4.1 ([5]). If Ɛ𝑔[·] is a coherent risk measure, then Ɛ𝑔[·] is bounded by the Choquet expectation 𝒞𝑔(·), that is

Ɛ𝑔[X] ≤ 𝒞𝑔(X), X ∈ L2(Ω, Ƒ, P)

But if Ɛ𝑔[·] is a convex risk measure, then the above inequality does not hold generally.

Theorem 4.2 ([15]). Let 𝑔 be convex function with respect to z, independent of y and deterministic. Let 𝑔 also satisfy (3.1). Then ρ𝑔(X) ≤ 𝒞𝑔[−X] for X ∈ L2(ƑT) if and only if ρ𝑔 is a coherent risk measure. Here ρ𝑔(X) is defined as ρ𝑔(X) := Ɛ𝑔[−X] for X ∈ L2(ƑT).

Note that ρ𝑔 : L2(ƑT) ⟼ ℝ is a coherent (or convex) risk measure if and only if 𝑔 is independent of y and is positively homogeneous and subadditive (or convex) with respect to z (see [23, 14, 22]).

The positive homogeneity and comonotonic additivity hold in the Choquet expectation. The time consistency holds in the 𝑔-expectation.

E[ξ + η] = E[ξ] + E[η] ∀ξ, η ∈ L2(Ω, Ƒ, P).

The above equality holds for the Choquet expectation if ξ and η are comonotonic. But if 𝑔 is nonlinear, the above equality does not hold for the 𝑔-expectation even if ξ and η are comonotonic. These facts means that 𝑔-expectation is more nonlinear than the Choquet expectation on L2(Ω, Ƒ, P) [15].

The following Lemmas (4.3) and (4.6), Proposition (4.4), and Theorem (4.5) are from the paper [5].

Lemma 4.3. For any X ∈ L2(Ω, ƑT, P), there exists unique η ∈ L2(Ω, Ƒt, P) such that

Ɛ𝑔[IAX] = Ɛ𝑔[IAη] ∀A ∈ Ƒt.

The η is called the conditional 𝑔-expectation of X and it is written as Ɛ𝑔[X|Ƒt]. This Ɛ𝑔[X|Ƒt] is exactly the yt which is the solution of BSDE (3.2).

Proposition 4.4. Let μ = {μt}t∈[0, T] be a continuous functions. Suppose that g(t, y, z) = μt|zt| and the process (zt}t∈[0, T) is one dimensional. Then for any ξ ∈ L2(Ω, Ƒ, P), the conditional 𝑔-expectation satisfies

Ɛ𝑔[ξ|Ƒt] = ess supQ∈QEQ[ξ|Ƒt] for μ > 0

where Q is a set of probability measures defined as

Theorem 4.5 ([5]). Suppose that 𝑔 satisfies the given Hypotheses. Then there exists a Choquet expectation whose restriction to L2(Ω, Ƒ, P) is equal to a 𝑔-expectation if and only if 𝑔 is independent of y and is linear in z, i.e. there exists a continuous function v(t) such that

𝑔(y, z, t) = v(t)z.

Lemma 4.6. Suppose that 𝑔 is a convex (or concave) function. If Ɛ𝑔 [·] is comonotonic additive on , then Ɛ𝑔 [·|Ƒt] is also comonotonic additive on for any t ∈ [0, T).

Corollary 4.7. Suppose that 𝑔 is a convex (or concave) function. If Ɛ𝑔[·] is a Choquet expectation on then Ɛ𝑔 [·|Ƒt] is also a Choquet expectation on for any t ∈ [0; T).

 

5. Ƒt-CONSISTENT EXPECTATION

In this section, an {Ƒt}t∈[0, T]-consistent expectation Ɛ is defined as a nonlinear functional on L2(ƑT ). We’ll show that Choquet expectation is an {Ƒt}t∈[0, T]-consistent expectation Ɛ under some conditions.

Definition 5.1. A nonlinear expectation is defined as a functional Ɛ : L2(ƑT) → ℝ satisfying

(1) (Monotonicity) If X ≥ Y P-a.s., then Ɛ(X) ¸ Ɛ(Y). Moreover, under the inequality X ≥ Y , Ɛ(X) = Ɛ(Y ) if and only if X = Y P-a.s.. (2) (Constancy) Ɛ(c) = c ∀c ∈ ℝ.

Definition 5.2. An {Ƒt}t∈[0, T]-consistent expectation is defined as the nonlinear expectation Ɛ such that if for any X ∈ L2(ƑT ) and any t ∈ [0; T] there exists η ∈ L2(Ƒt) satisfying

The η satisfying (5.1) is called conditional {Ƒt}t∈[0, T]-consistent expectation of X under Ƒt and denoted by Ɛ[X|Ƒt].

Definition 5.3. It is called that {Ƒt}t∈[0, T]-consistent expectation Ɛ is dominated by Ɛu (u > 0) if

Ɛ[X + Y ] − Ɛ[X] ≤ Ɛu[Y] ∀X, Y ∈ L2(ƑT )

where Ɛu is 𝑔-expectation with 𝑔(t, y, z) = u|z|.

An {Ƒt}t∈[0, T]-consistent expectation Ɛ is called to satisfy the translability condition if

The following theorem tells us the relationships between conditional 𝑔-expectation and {Ƒt}t∈[0, T]-consistent expectation.

Theorem 5.4 ([7]). Let Ɛ : L2(ƑT ) → ℝ be a {Ƒt}t∈[0, T]-consistent expectation. If Ɛ is Ɛu-dominated for some u > 0 and if it satisfies translability condition (5.2), then there exists a unique 𝑔 which is independent of y, satisfies the assumptions (3.1) and |𝑔(t,z)| ≤ u|z| such that

Ɛ[X] = Ɛ𝑔[X] and Ɛ[X|Ƒt] = Ɛ𝑔[X|Ƒt] ∀X ∈ L2(ƑT ).

Theorem 5.5 ([11]). For the Choquet integral with respect to a capacity c, the following are equivalent.

(1) ρ0(X) := ∫(−X) dc is a convex risk measure on L2(ƑT ). (2) ρ0(X) := ∫(−X) dc is a coherent risk measure on L2(ƑT ). (3) For Qc :={Q ∈ M1,f|Q[A] ≤ c(A) ∀A∈ ƑT},

(4) The set function c is submodular. In this case, Qc = Qmax.

The set ℳ1,f = ℳ1,f(Ω, Ƒ) in Theorem (5.3) is the one of all finitely additive set functions Q : Ƒ → [0; 1] which is normalized to Q[­Ω] = 1. The Qmax is defined as

where Aρ is defined as

Aρ := {X ∈ L2(ƑT) ρ(X) ≤ 0 }.

From the viewpoint of Proposition (4.4) and Theorem (4.5), the set Qc of (5.3) is unnecessarily too large so that it could be reduced to a suitable set of probability measures for consistency, i.e.

It can be shown that Qc is indeed the set of equivalent martingale measures by the following Proposition (5.6).

Proposition 5.6 ([11]). If Q << P on Ƒ, then Q is equivalent to P if and only if P-a.s.

Assume that the capacity c is submodular. Under the new set Qc as in (5.4), we define a nonlinear expectation as

We will show that the above Ɛ[X] satisfies all the assumptions of Theorem (5.4). It is easy to show that Ɛ[X] satisfies the monotonicity and constancy in the Defini- tion (5.1) but if X ≥ Y , Ɛ[X] = Ɛ[Y] if and only if X = Y P-a.s.. Suppose that X ≥ Y and Ɛ[X] = Ɛ[Y]. We prove it contrapositively. Suppose X = Y P-a.s. does not hold. Let A = {w ∈ Ω | X ≠ Y} ∈ Ƒ. Then EQ[1AX] > EQ[1AY] for each Q ∈ Qc and there exists a such that EQ[1AX] > r > EQ[1AY]. By taking supremum on the left hand side first, we have ess sup Q∈Qc EQ[1AY], > r > EQ[1AY] and so ess supQ∈Qc EQ[1AX], > r ≥ ess supEQ[1AY], it’s a contradiction.

We need the stability property of a set Qc to show that Ɛ[X] is a {Ƒt}t∈[0, T] - consistent expectation. In the following definitions, the stopping times σ and τ can be replaced by t ∈ [0, T] without any loss.

Definition 5.7. Let Q1 and Q2 be two equivalent probability measures and σ be a stopping time. The probability measure

[A]:=EQ1[Q2[A|Ƒσ, A ∈ ƑT

is called the pasting of Q1 and Q2 in σ.

Note that by the monotone convergence theorem for conditional expectation is a probability measure and

Definition 5.8. A set 𝒬 of equivalent probability measures on (Ω, Ƒ) is called stable if, for any Q1,Q2 ∈ 𝒬 and the stopping time σ, also their pasting in σ is contained in 𝒬.

Proposition 5.9 ([11]). The set 𝒬c of equivalent martingale measures is stable.

Theorem 5.10 ([11]). Let 𝒬 be a set of equivalent probability measures. If 𝒬 is stable, then the following holds for X ∈ L2(ƑT )

From the Theorem (5.10), we can easily see that Ɛ[X] is a {}t ∈ [0,T]-consistent expectation condition (5.1), Ɛ[1AX] = Ɛ[X|]] ∀A ∈ .

Let us show that {}t ∈ [0,T]-consistent expectation Ɛ is dominated by Ɛμ (μ>0). Since Ɛ[X+Y] − Ɛ[X] ≤ ess sup Q∈𝒬c EQ[Y ] and there exists 𝑔-expectation Ɛμ with 𝑔(t, y, z) = μz satisfying Ɛμ[X] = ess sup Q∈𝒬c EQ[Y ] by Theorem (4.5), Ɛ is dominated by Ɛμ. Note that Ɛμ-dominated nonlinear expectation Ɛ implies that Ɛ is lower semi-continuous [7].

Finally we show that {}t ∈ [0,T]-consistent expectation Ɛ satisfies the translability condition. Let X ∈ L2() and β ∈ L2(). Then by the de¯nition of Ɛ we have

Therefore, the nonlinear expectation Ɛ defined as (5.5) satisfies the all the conditions of Theorem (5.4). Thus the results so far can be summarized in the following Theorem (5.11).

Theorem 5.11. Let the nonlinear expectation Ɛ be defined as (5.5). Then there exists a unique 𝑔 which is independent of y, satisfies the assumptions (3.1) and |𝑔(t; z)| ≤ μ|z| such that

Note that the generator 𝑔 in Theorem (5.11) should be the form of 𝑔(t; y; z) = μtz which is linear in z and so Ɛ𝑔 = Ɛμ to be consistent to the results of Theorem (4.5).

In fact, for 𝑔(t, y, z) = μtz, let us consider the BSDE

The above differential equation (5.6) is reduced to

By Girsanov’s Theorem, (t)0≤t≤T is a Q-Brownian motion under Q defined as

Therefore we have the relations

Ɛ𝑔[X] = EQ[X], Ɛ𝑔[X|Ƒt] = EQ[X|Ƒt]

which means that 𝑔-expectation is a classical mathematical expectation.

Proposition 5.12 ([23]). Let the risk measure be defined as

where 𝑔 satisfies the conditions (3.1). Moreover, if 𝑔 is sublinear in (y; z), i.e. positively homogeneous in (y; z) and subadditive in (y; z), then is a dynamic coherent and time-consistent risk measure.

Note that if 𝑔 satisfies both positive homogeneity and subadditivity, 𝑔 is indepen- dent of y. The proposition (5.12) and Theorem (4.2) tells us that for Theorem (5.11) to hold the linearity of 𝑔 is necessary.