**Journal of Mathematical Finance** Vol.3 No.2(2013), Article ID:31821,11 pages DOI:10.4236/jmf.2013.32028

Risk Measures and Nonlinear Expectations*

^{1}School of Mathematics, Shandong University, Jinan, China

^{2}Department of Financial Engineering, Ajou University, Suwon, Korea

^{3}Department of Mathematics, Donghua University, Shanghai, China

^{4}Department of Statistical and Actuarial Science, The University of Western Ontario London, Ontario, Canada

Email: zjchen@sdu.edu.cn

Copyright © 2013 Zengjing Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received November 20, 2012; revised May 22, 2013; accepted June 16, 2013

**Keywords:** Risk Measure; Coherent Risk; Convex Risk; Choquet Expectation; g-Expectation; Backward Stochastic Differential Equation; Converse Comparison Theorem; BSDE; Jensen’s Inequality

ABSTRACT

Coherent and convex risk measures, Choquet expectation and Peng’s g-expectation are all generalizations of mathematical expectation. All have been widely used to assess financial riskiness under uncertainty. In this paper, we investigate differences amongst these risk measures and expectations. For this purpose, we constrain our attention of coherent and convex risk measures, and Choquet expectation to the domain of g-expectation. Some differences among coherent and convex risk measures and Choquet expectations are accounted for in the framework of g-expectations. We show that in the family of convex risk measures, only coherent risk measures satisfy Jensen’s inequality. In mathematical finance, risk measures and Choquet expectations are typically used in the pricing of contingent claims over families of measures. The different risk measures will typically yield different pricing. In this paper, we show that the coherent pricing is always less than the corresponding Choquet pricing. This property and inequality fails in general when one uses pricing by convex risk measures. We also discuss the relation between static risk measure and dynamic risk measure in the framework of g-expectations. We show that if g-expectations yield coherent (convex) risk measures then the corresponding conditional g-expectations or equivalently the dynamic risk measure is also coherent (convex). To prove these results, we establish a new converse of the comparison theorem of g-expectations.

1. Introduction

The choice of financial risk measures is very important in the assessment of the riskiness of financial positions. For this reason, several classes of financial risk measures have been proposed in the literature. Among these are coherent and convex risk measures, Choquet expectations and Peng’s g-expectations. Coherent risk measures were first introduced by Artzner, Delbaen, Eber and Heath [1] and Delbaen [2]. As an extension of coherent risk measures, convex risk measures in general probability spaces were introduced by Föllmer & Schied [3] and Frittelli & Rosazza Gianin [4]. g-expectations were introduced by Peng [5] via a class of nonlinear backward stochastic differential equations (BSDEs), this class of nonlinear BSDEs being introduced earlier by Pardoux and Peng [6]. Choquet [7] extended probability measures to nonadditive probability measures (capacity), and introduced the so called Choquet expectation.

Our interest in this paper is to explore the relations among risk measures and expectations. To do so, we restrict our attention of coherent and convex risk measures and Choquet expectations to the domain of g-expectations. The distinctions between coherent risk measure and convex risk measure are accounted for intuitively in the framework of g-expectations. We show that 1) in the family of convex risk measures, only coherent risk measures satisfy Jensen’s inequality; 2) coherent risk measures are always bounded by the corresponding Choquet expectation, but such an inequality in general fails for convex risk measures. In finance, coherent and convex risk measures and Choquet expectations are often used in the pricing of a contingent claim. Result 2) implies coherent pricing is always less than Choquet pricing, but the pricing by a convex risk measure no longer has this property. We also study the relation between static and dynamic risk measures. We establish that if g-expectations are coherent (convex) risk measures, then the same is true for the corresponding conditional g-expectations or dynamic risk. In order to prove these results, we establish in Section 3, Theorem 1, a new converse comparison theorem of g-expectations. Jiang [8] studies gexpectation and shows that some cases give rise to risk measures. Here we are able to show, in the case of gexpectations, that coherent risk measures are bounded by Choquet expectation but this relation fails for convex risk measures; see Theorem 4. Also we show that convex risk measures obey Jensen’s inequality; see Theorem 3.

The paper is organized as follows. Section 2 reviews and gives the various definitions needed here. Section 3 gives the main results and proofs. Section 4 gives a summary of the results, putting them into a Table form for convenience of the various relations.

2. Expectations and Risk Measures

In this section, we briefly recall the definitions of g-expectation, Choquet expectation, coherent and convex risk measures.

2.1. g-Expectation

Peng [5] introduced g-expectation via a class of backward stochastic differential equations (BSDE). Some of the relevant definition and notation are given here.

Fix and let be a -dimensional standard Brownian motion defined on a completed probability space. Suppose is the natural filtration generated by, that is We also assume. Denote

Let satisfy

(H1) For any is a continuous progressively measurable process with

.

(H2) There exists a constant such that for any

(H3)

In Section 3, Corollary 3 we will consider a special case of with.

Under the assumptions of (H1) and (H2), Pardoux and Peng [6] showed that for any, the BSDE

(1)

has a unique pair solution

.

Using the solution of BSDE (1), which depends on, Peng [5] introduced the notion of g-expectations.

Definition 1 Assume that (H1), (H2) and (H3) hold on g and. Let be the solution of BSDE (1).

defined by is called the g-expectation of the random variable.

defined by is called the conditional g-expectation of the random variable.

Peng [5] also showed that g-expectation and conditional -expectation preserve most of basic properties of mathematical expectation, except for linearity. The basic properties are summarized in the next Lemma.

Lemma 1 (Peng) Suppose that

.

1) Preservation of constants: For any constant .

2) Monotonicity: If, then.

3) Strict monotonicity: If, and, then.

4) Consistency: For any,

.

5) If does not depend on, and is - measurable, then

In particular,.

6) Continuity: If as in, then.

The following lemma is from Briand et al. [9, Theorem 2.1]. We can rewrite it as follows.

Lemma 2 (Briand et al. ) Suppose that is of the form where is a continuous bounded process. Then

where the limit is in the sense of.

2.2. Choquet Expectation

Choquet [7] extended the notion of a probability measure to nonadditive probability (called capacity) and defined a kind of nonlinear expectation, which is now called Choquet expectation.

Definition 2

1) A real valued set function is called a capacity if a)

b), whenever and.

2) Let be a capacity. For any, the Choquet expectation is defined by

Remark 1 A property of Choquet expectation is positive homogeneity, i.e. for any constant

2.3. Risk Measures

A risk measure is a map where is interpreted as the “habitat” of the financial positions whose riskiness has to be quantified. In this paper, we shall consider.

The following modifications of coherent risk measures (Artzner et al.[1]) is from Roorda et. al. [10].

Definition 3 A risk measure is said to be coherent if it satisfies 1) Subadditivity:,;

2) Positive homogeneity: for all real number

3) Monotonicity: whenever

4) Translation invariance: for all real number.

As an extension of coherent risk measures, Föllmer and Schied [3] introduced the axiomatic setting for convex risk measures. The following modifications of convex risk measures of Föllmer and Schied [3] is from Frittelli and Rosazza Gianin [4].

Definition 4 A risk measure is said to be convex if it satisfies 1) Convexity:

,

;

2) Normality:;

3) Properties (3) and (4) in Definition 3.

A functional in Definitions 3 and 4 is usually called a static risk measure. Obviously, a coherent risk measure is a convex risk measure.

As an extension of such a functional Artzner et al. [11,12], Frittelli and Rosazza Gianin [13] introduced the notion of dynamic risk measure which is random and depends on a time parameter.

Definition 5 A dynamic risk measure

is a random functional which depends on, such that for each it is a risk measure. If satisfies for each the conditions in Definition 3, we say is a dynamic coherent risk measure. Similarly if satisfies for each the conditions in Definition 4, we say is a dynamic convex risk measure.

3. Main Results

In order to prove our main results, we establish a general converse comparison theorem of g-expectation. This theorem plays an important role in this paper.

Theorem 1 Suppose that and satisfy (H1), (H2) and (H3). Then the following conclusions are equivalent.

1) For any

2) For any

(2)

Proof: We first show that inequality (2) implies inequality 3).

Let and be the solutions of the following BSDE corresponding to the terminal value and and the generator and, respectively

(3)

Then

For fixed, consider the BSDE

(4)

It is easy to check that is the solution of the BSDE (4).

Comparing BSDEs (4) and (3) with and, assumption (2), (2) then yields

Applying the comparison theorem of BSDE in Peng [5], we have Taking, thus by the definition of -expectation, the proof of this part is complete.

We now prove that inequality (1) implies (2). We distinguish two cases: the former where does not depend on, the latter where may depend on.

Case 1, does not depend on. The proof of this case 1 is done in two steps.

Case 1, Step 1: We now show that for any, we have

Indeed, for, set

If for all, we have then we obtain our result.

If not, then there exists such that. We will now obtain a contradiction.

For this,

That is

Taking g-expectation on both sides of the above inequality, and apply the strict monotonicity of -expectation in Lemma 1 (3), it follows

But by Lemma 1 (4) and (5),

Note that by Lemma 1(v)

Thus

This induces a contradiction, thus concluding the proof of this Step 1.

Case 1, Step 2: For any with and, let us choose . Obviously,

By Step 1,

Thus

Let applying Lemma 2, since g does not depend on we rewrite simply as thus The proof of Case 1 is complete.

Case 2, g depends on y. The proof is similar to the proof of Theorem 2.1 in Coquet et al. [14]. For each and define the stopping time

Obviously, if for each , for all then the proof is done. If it is not the case, then there exist and

such that

.

Fix and consider the following (forward) SDEs defined on the interval

and

Obviously, the above equations admit a unique solution which is progressively measurable with

Define the following stopping time

It is clear that and note that whenever thus, Hence.

Moreover, we can prove

In fact, setting

then

Thus

It follows that on, This implies

(5)

By the definition of and, the pair processes and are the solutions of the following BSDEs with terminal values and,

and

Hence,

and

Applying the strict comparison theorem of BSDE and inequality (5), by the assumptions of this Theorem, we have

This induces a contradiction. The proof is complete.

Lemma 3 Suppose that g satisfies (H1), (H2) and (H3). For any constant, let

.

Then for any,

Proof: Letting, then is the solution of BSDE

Since

the above BSDE can be rewritten as

(6)

Let, then satisfies

(7)

Comparing with BSDE (6) and BSDE (7), by the uniqueness of the solution of BSDE, we have

Let then. The conclusion of the Lemma now follows by the definition of g-expectation. This concludes the proof.

Applying Theorem 1 and Lemma 3, immediately, we obtain several relations between g-expectation and. These are given in the following Corollaries.

Corollary 1 The g-expectation is the classical mathematical expectation if and only if g does not depend on and is linear in.

Proof: Applying Theorem 1, is linear if and only if is linear in. By assumption (H3), that is for all. Thus does not depend on. The proof is complete.

Corollary 2 The -expectation is a convex risk measure if and only if does not depend on and is convex in.

Proof: Obviously, -expectation is convex risk measure if and only if for any

(8)

For a fixed, let

Applying Lemma 3,

Inequality (8) becomes

Applying Theorem 1, for any

which then implies that g is convex. By the explanation of Remark for Lemma 4.5 in Briand et al. [9], the convexity of and the assumption (H3) imply that does not depend on. The proof is complete.

The function is positively homogeneous in if for any,.

Corollary 3 The -expectation is a coherent risk measure if and only if does not depend on and it is convex and positively homogenous in. In particular, if, is of the form

with.

Proof: By Corollary 2, the -expectation is a convex risk measure if and only if does not depend on and is convex in. Applying Theorem 1 and Lemma 3 again, it is easy to check that -expectation is positively homogeneous if and only if is positively homogeneous (that is for all and if and only if for any,.

In particular, if, notice the fact that is convex and positively homogeneous on, and that does not depend on. We write it as then

(9)

Note that, , but

Thus from (9)

Defining

,.

Obviously since the convexity of yields

The proof is complete.

Remark 2 Corollaries 2 and 3 give us an intuitive explanation for the distinction between coherent and convex risk measures. In the framework of g-expectations, convex risk measures are generated by convex functions, while coherent measures are generated only by convex and positively homogenous functions. In particular, if d = 1, it is generated only by the family with. Thus the family of coherent risk measures is much smaller than the family of convex risk measures.

Jensen’s inequality for mathematical inequality is important in probability theory. Chen et al. [15] studied Jensen’s inequality for g-expectation.

We say that -expectation satisfies Jensen’s inequality if for any convex function then

(10)

Lemma 4 [Chen et al. [15] Theorem 3.1] Let be a convex function and satisfy, and. Then 1) Jensen’s inequality (10) holds for -expectations if and only if does not depend on and is positively homogeneous in;

2) If the necessary and sufficient condition for Jensen’s inequality (10) to hold is that there exist two adapted processes and such that

.

Now we can easily obtain our main results. Theorem 2 below shows the relation between static risk measures and dynamic risk measures.

Theorem 2 If -expectation is a static convex (coherent) risk measure, then the corresponding conditional g-expectation is dynamic convex (coherent) risk measure for each

Proof: This follows directly direct from Theorem 1.

Theorem 3 below shows that in the family of convex risk measure, only coherent risk measure satisfies Jensen’s inequality.

Theorem 3 Suppose that is a convex risk measure. Then is a coherent risk measure if and only if satisfies Jensen’s inequality.

Proof: If is a convex risk measure, then by Corollary 2, is convex. Applying Lemma 4, satisfies Jensen’s inequality if and only if is positively homogenous. By Corollary 2, the corresponding is coherent risk measure. The proof is complete.

Theorem 4 and Counterexample 1 below give the relation between risk measures and Choquet expectation.

Theorem 4 If is a coherent risk measure, then is bounded by the corresponding Choquet expectation, that is where. If is a convex risk measure then inequality above fails in general. By construction there exists a convex risk measure and random variables and such that

The prove this theorem uses the following lemma.

Lemma 5 Suppose that does not depend on. Suppose that the -expectation satisfies (1) (2) For any positive constant,

Then for any the -expectation is bounded by the corresponding Choquet expectation, that is

(11)

Proof: The proof is done in three steps.

Step 1. We show that if is bounded by, then inequality (11) holds.

In fact, for the fixed, denote by

Then in

Moreover, can be rewritten as

But by the assumptions (1) and (2) in this lemma, we have

(12)

Note that

and

Thus, taking limits on both sides of inequality (12), it follows that The proof of Step 1 is complete.

Step 2. We show that if is bounded by, that is, then inequality (11) holds.

Let then Applying Step 1,

(13)

But by Lemma 1(v), On the other hand,

Thus by (13)

Therefore

Step 3. For any let then. By Step 2,

Letting, it follows that

The proof is complete.

Proof of Theorem 4: If the -expectation is a coherent risk measure, then it is easy to check that the -expectation satisfies the conditions of Lemma 5.

Let . By Lemma 5 and the definition of Choquet expectation, we have The first part of this theorem is complete.

Counterexample 1 shows that this property of coherent risk measures fails in general for more general convex risk measures. This completes the proof of Theorem 4.

Counterexample 1 Suppose that is 1-dimensional Brownian motion (i.e. d = 1). Let where. Then is a convex risk measure. Let and Then

However Here the capacity in the Choquet expectation is given by

Proof of the Inequality in Counterexample 1: The convex function satisfies (H1), (H2) and(H3). Thus, by Corollary 2, -expectation is a convex risk measure. This together with the property of Choquet expectation in Remark 1 implies

Moreover, since by Corollary 3, is a convex risk measure rather than a coherent risk measure. We now prove that In fact, since

we only need to show

Let be the solution of the BSDE

(14)

First we prove that

(15)

where is Lebesgue measure on

If it is not true, then a.e. and BSDE (14) becomes

Thus

By the Markov property,

Recall that and are independent and. Thus

where is the density function of the normal distribution. Thus where is the Malliavin derivative. Thus can be greater than 1 whenever is near 0 and is near 0. Thus (15) holds, which contradicts the assumption a.e..

Secondly we prove that

Let be the solution of the BSDE

(16)

Obviously,

which means is the solution of BSDE

But Thus by the uniqueness of the solution of BSDE, On the other hand, let be the solution of the BSDE

(17)

Comparing BSDE(17) with BSDE (16), notice (15) and the fact

and

whenever z >1. By the strict comparison theorem of BSDE, we have

Setting, thus

The proof is complete.

Remark 3 In mathematical finance, coherent and convex risk measures and Choquet expectation are used in the pricing of contingent claim. Theorem 4 shows that coherent pricing is always less than Choquet pricing, while Counterexample 1 demonstrates that pricing by a convex risk measure no longer has this property. In fact the convex risk price may be greater than or less than the Choquet expectation.

4. Summary

Coherent risk measures are a generalization of mathematical expectations, while convex risk measures are a generalization of coherent risk measures. In the framework of -expectation, the summary of our results is given in Table 1. In that Table, the Choquet expectation is.

Counterexample 1 shows that convex risk may be or Choquet expectation. Only in the case of coherent

Table 1. Relations among coherent and convex risk measures, choquet expectation and Jensen’s inequality.

risk there is an inequality relation with Choquet expectation.

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NOTES

^{*}This work is supported partly by the National Natural Science Foundation of China (11231005; 11171062) and WCU(World Class University) program of the Korea Science and Engineering Foundation (R31-20007) and the NSERC grant of RK.

This work has been supported by grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada.