On increasing risk, inequality and poverty measures: Peacocks, lyrebirds and exotic options

Journal of Economic Dynamics & Control 59 (2015) 22–36

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Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

On increasing risk, inequality and poverty measures: Peacocks, lyrebirds and exotic options Christian-Oliver Ewald a,b,n, Marc Yor c,1 a b c

Department of Economics, University of Glasgow, Adam Smith Building, Glasgow G12 8QQ, United Kingdom School of Mathematics and Statistics, University of Sydney, Sydney, Australia University Pierre and Marie Curie, Paris, France

a r t i c l e in f o

abstract

Article history: Received 16 February 2015 Received in revised form 13 July 2015 Accepted 14 July 2015 Available online 20 July 2015

We extend the Rothschild and Stiglitz (1970) notion of increasing risk to families of random variables and in this way link their approach to the concept of stochastic processes which are increasing in the convex order. These processes have been introduced in seminal work by Strassen (1965), Doob (1968) and Kellerer (1972), who showed that such processes have the same marginals as a martingale. In fact, we demonstrate that their results include the results of Rothschild and Stiglitz as a special case. Further, we show that it makes sense to look at a larger class of processes, which we refer to as lyrebirds. We also show how these processes link up with the concept of second order stochastic dominance and are helpful in studying the dynamics of inequality and poverty measures. Further applications discussed include geometric and hyperbolic discounting, exotic derivatives and real options. & 2015 Elsevier B.V. All rights reserved.

JEL classification: C61 D63 G12 G13 Keywords: Increasing risk Inequality Poverty measures Peacocks

1. Introduction It is without doubt that random variables and their associated distributions play a fundamental role in economic modeling. For decision making and policy guidance it is tremendously important to be able to classify the degree of variability of random variables and in particular what it means that one random variable, say Y, is more variable than another random variable, say X. The realization of this variability can be multitude and be manifested in say the returns of a financial asset, in which case we would link variability to risk, or the income or wealth of an arbitrarily chosen individual among a certain population, in which case we would link variability to inequality. Rothschild and Stiglitz (1970, 1971) contributed in a fundamental way to the characterization of variability in the context of risk. They formalized in a mathematical rigorous way what it means for one random variable to be more risky than another and

n

Corresponding author. E-mail address: [email protected] (C.-O. Ewald). 1 Deceased.

http://dx.doi.org/10.1016/j.jedc.2015.07.004 0165-1889/& 2015 Elsevier B.V. All rights reserved.

C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

23

showed the following: Let X and Y be two sufficiently integrable random variables,2 then the two statements below are equivalent: law

A1 : Y ¼ X þ Z (equal in law) where Z is a random variable with the property that EðZjXÞ ¼ 0

ð1Þ

A2 : For all concave (utility) functions UðÞ s.t. EðUðXÞÞ and EðUðYÞÞ are both finite EðUðXÞÞ Z EðUðYÞÞ:

ð2Þ

Intuitively, the first statement says that the distribution of the random variable Y is like the distribution of X with additional risk which is unaffected by X; hence Y is riskier than X. The intuition of the second statement is that every risk averse decision maker prefers X to Y. In the context of inequality measures (as well as in the context of risk) the concept of first and second order stochastic dominance has always played a major role, compare (Levy, 1992). Denoting with ρX and ρY the cumulative distribution functions of X and Y, then X dominates Y by second order stochastic dominance, if and only if Z x Z x ρX ðtÞ dt r ρY ðtÞ dt ð3Þ 1

1

for all x A R, provided that the two integrals exist and are finite. A straightforward integration by parts gives Z x Z x   ρX ðtÞ dt ¼ ðx tÞ dρX ðtÞ ¼ E ðx  XÞ þ ; 1

ð4Þ

1

where ðx  XÞ þ ¼ maxðx X; 0Þ. If we assume that EðXÞ ¼ EðYÞ, then ðx XÞ þ  x ¼ ðX  xÞÞ þ  X implies that second order stochastic dominance of X over Y is equivalent to A3 : For all x A R we have     E ðX  xÞÞ þ rE ðY  xÞ þ :

ð5Þ

In fact under the condition EðXÞ ¼ EðYÞ it can be shown that the statement A3 is equivalent to A2. Without imposing equality in expectations of X and Y condition A3 becomes equivalent to the relaxation of condition A2, when it is only imposed upon increasing concave functions, a condition which we will later refer to as D2. This can be found for example in Atkinson (1970). Further, if the random variables X and Y above are positive, and condition (5) in A3 is valid for all x Z0, then it is automatically satisfied for all x A R, a fact that we will use later when we will be looking at various inequality measures. The linkage between Rothschild and Stiglitz's (1970, 1971) work and the work following Atkinson (1970) with the theory of derivatives and option pricing becomes evident in Eq. (5), where the expressions on both side represent the prices of European call options written on the underlying X resp. Y with strike price x. Since its inception in the early 1970s3 option pricing theory has developed sophisticated methods to evaluate expressions such as those in Eq. (5) and in particular study their dependence on certain parameters within the model, which is related to the determination of the so-called Greeks of an option. It is therefore surprising that this approach so far has not been adopted more consistently in the literature. In this paper we demonstrate how methodology that has classically appeared in the context of option pricing as well as more or less abstract probability theory can be helpful in the context of Rothschild and Stiglitz (1970, 1971) approach to classify risk as well as within the large literature on inequality and poverty, dating back to Atkinson (1970, 1987). More specifically, we extend the Rothschild and Stiglitz (1970, 1971) notion of increasing risk to families of random variables ðX t Þ and in this way link their approach to the concept of stochastic processes which are increasing in the convex order. These processes, now referred to as peacocks, have recently received increased attention in the derivatives pricing literature, compare (Hirsch et al., 2011). Originally these processes were introduced in seminal work by Strassen (1965); Doob (1968) and Kellerer (1972), who showed that such processes have the same marginals as a martingale. As we show, their results in fact include the results of Rothschild and Stiglitz as a special case. Further, we demonstrate that it makes sense to look at a larger class of processes, which we refer to as lyrebirds and which nicely link to Atkinson (1970) work. We show how these processes link up with the concept of second order stochastic dominance, which opens up the gate for exciting applications 2 Rothschild and Stiglitz (1970, 1971) only consider random variables which map into the compact interval ½0; 1. In this way they circumvent many problems that occur in the general case. A quote from their paper says “The extension (and modification) of the results to c.d.f.s defined on the whole real line is an open question whose resolution requires the solution of a host of delicate convergence problems of little economic interest.” Contrary to their statement however, most applications in Economics and Finance involve random variables with arbitrary and unbounded domains. Fortunately, the delicate convergence problems have been solved by Strassen (1965), Doob (1968) and Kellerer (1972) to which we revert in the next section. 3 Interestingly this is at about the same time when Rothschild and Stiglitz as well as Atkinson started their work.

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in the study of inequality and poverty measures, in particular on issues such as convergence and stability. We also present a number of other examples from Economics and Finance, including geometric and hyperbolic discounting as well as exotic derivatives. The remainder of the article is organized as follows. In Section 2 we demonstrate how the Rothschild and Stiglitz (1970, 1971) as well as the Atkinson (1970, 1987) framework can be expressed in terms of martingales and peacocks. In this section we also introduce a new class of processes which we call lyrebirds. This class extends the class of peacocks and fits the Rothschild and Stiglitz (1970, 1971) as well as the Atkinson (1970, 1987) in cases where expectations vary. In Section 3 we apply the theory developed in the previous section to study inequality and poverty measures. In Section 4 we consider the riskiness/inequality of solutions of stochastic differential equations, which naturally occur in Economics and Finance. These are then applied to Asian and Australian options, classical and hyperbolic discounting as well as real option theory. The main results are summarized in the conclusions. 2. Peacocks, lyrebirds and martingale marginals In this section we place the work of Rothschild and Stiglitz (1970, 1971) into the context of earlier work in probability theory, e.g. Strassen (1965) etc. and later work on the so-called peacocks which developed at the interface of probability theory and derivatives pricing. Let us first note that the condition in statement A2 includes all concave functions, whether increasing, decreasing or changing monotonic behavior. Therefore, since UðxÞ ¼ x and UðxÞ ¼  x are both concave, statement A2 implies that X and Y must have the same expectation. From the perspective of utility theory it is quite uncommon to use concave functions which are not increasing. However, it can be shown that statement A2 is equivalent4 to the following statement B2 : EðXÞ ¼ EðYÞ and for all increasing and concave utility functions UðÞ s.t. EðUðXÞÞ and EðUðYÞÞ are both finite EðUðXÞÞ Z EðUðYÞÞ:

ð6Þ

To set the framework for the following sections, we propose a different formulation of statement A1, using the terminology of martingales which is common in probability theory and finances: B1 : Consider the stochastic process ðX t Þt ¼ 0;1;2 defined by X 0 ¼ EðXÞ, X 1 ¼ X and X 2 ¼ Y. Then there exists a martingale ðM t Þt ¼ 0;1;2 which has the same marginals as ðX t Þ, i.e. X t  M t for t ¼ 0; 1; 2. It is easy to see that A1 implies B1. Simply note that the process ðM t Þt ¼ 0;1;2 defined by M 0 ¼ EðXÞ, M 1 ¼ X, M 2 ¼ X þZ is a martingale, as according to (1) EðM 2 jM 1 Þ ¼ EðX þ ZjXÞ ¼ X þ EðZjXÞ ¼ X ¼ M 1 : It is not a priori clear though, that B1 also implies A1. Note that Y  X can in general have a different distribution than M 2  M1 as statement B1 provides no information about the joint distributions. A proof that B1 indeed implies A1 can in fact be obtained from the Rothschild and Stiglitz (1970) result itself.5 Assume that UðÞ is a concave utility function. Then by application of Jensen's inequality, equality in distributions, the law of the iterated conditional expectation and the martingale property of ðM t Þ, we have EðUðYÞÞ ¼ EðEðUðM 2 ÞjM 1 ÞÞ r EðUðEðM 2 jM 1 ÞÞ ¼ EðUðXÞÞ: Hence B1 implies A2 and therefore by Rothschild and Stiglitz (1970) also A1. The advantage of the formulation B1 is that it can be easily extended to cover an arbitrary number of random variables for which the dynamics of the associated martingale implies a natural order of increasing risk. By using the equivalent characterization B1 of increasing risk identified in the previous section, a good knowledge of the mathematical literature then quickly reveals that the result presented by Rothschild and Stiglitz (1970) is in fact a special case of earlier results in probability theory. The three classical contributions in this context are Strassen (1965), Doob (1968) and Kellerer (1972). Some of these authors work with convex rather than concave functions, however as the negative of a convex function is a concave function and vice versa, the only effect of this is a reversal of the inequalities. The following two notions are central to this literature: Definition 1. C1 : The process ðX t Þ is said to have the marginals of a (sub/super)-martingale if there exists a (sub/super)-martingale ðM t Þ, with the property law

X t ¼ Mt

for all t Z 0;

ð7Þ

4 For the equivalence of A2 and B2 note that for an arbitrary not necessarily increasing concave function U(x) with bounded derivatives, there exists an a 4 0 s.t. U~ ðxÞ≔ax þ UðxÞ is increasing and concave. 5 We will consider a more general context in the following.

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C2 : The integrable process ðX t Þ is said to be increasing in convex order, or equivalently, ðX t Þ is said to be a peacock,6 if the following property holds for any convex function f ðÞ: Eðf ðX s ÞÞ r Eðf ðX t ÞÞ

for every choice of 0 rs o t o1:

ð8Þ 7

If the above property is satisfied only for increasing convex functions f ðÞ, then ðX t Þ is said to be a lyrebird.

Using the same identification as in statement B1 shows that statements C1 and C2 represent more general dynamic versions of properties A1 and A2 in Rothschild and Stiglitz (1970). Note that the equality in (7) refers to each t individually. In general the distributions of ðX t Þ and ðM t Þ as stochastic processes (on the path space) are not equal, as the latter would have the consequence that ðX t Þ itself is a martingale. Strassen (1965) showed for discrete stochastic processes, that a process is increasing in convex order, i.e. a peacock, if and only if it has the marginals of a martingale. This directly implies the result of Rothschild and Stiglitz (1970) via the identification made above. Doob (1968) extended Strassen's result for the case of continuous stochastic processes. Kellerer (1972) showed that in fact the martingale can be chosen to be a Markov process.8 The results by Strassen (1965) and Doob (1968) and Kellerer (1972) may not have been well known beyond the mathematical literature, but found a recent revival in the Derivatives Pricing and Financial Mathematics literature to which we refer in Section 4. As indicated earlier, the use of all concave (in the Rothschild and Stiglitz case) resp. all convex (Strassen, 1965, etc.) functions restricts the two random variables X and Y to have the same expectation. However much of the Finance and Economics literature has at its center a potential trade-off between expectation and riskiness, which is manifested in the certainty equivalent. In the context of inequality, Atkinson (1970) does not make a similar assumption, but restricts the utility functions to be increasing. In line with this we therefore propose to adjust Rothschild and Stiglitz's (1970) property A2 as follows: D2 : For all increasing concave (utility) functions UðÞ s.t. both EðUðXÞÞ and EðUðYÞÞ are finite EðUðXÞÞ Z EðUðYÞÞ:

ð9Þ

Let us now identify the class of stochastic processes that correspond to the adjusted condition D2 . Assume for now now, that hðÞ is an increasing convex function and ðX t Þ has the marginals of a sub-martingale. Then it follows from the Jensen inequality that for s r t EðhðX t ÞÞ ¼ EðEðhðM t ÞjF s ÞÞÞ Z EðhðEðM t jF s ÞÞÞ ZEðhðX s ÞÞ;

ð10Þ

assuming both EðhðX t ÞÞ and EðhðX s ÞÞ are finite. Here we used that EðM t jF s Þ ZM s and that hðÞ is increasing. As we will be more interested in concave functions and utilities, we formulate the corresponding result for concave functions: Proposition 1. Let ðX t Þ be a stochastic process with the marginals of a super-martingale, and U an increasing concave utility function. Then for all 0 r s rt s.t. both EðUðX s ÞÞ and EðUðX t ÞÞ are finite, we have EðUðX s ÞÞ Z EðUðX t ÞÞ:

ð11Þ

The interpretation of Proposition 1 is that a process which admits the marginals of a super-martingale is increasing in risk or in other words decreasing in desirability. The question whether the reverse holds, i.e. whether (11) for all 0 rs r t and all increasing concave utility function implies that ðX t Þ has the marginals of a super-martingale can in fact be concluded from a theorem of Kellerer (1972), which we adapt for our purposes. Here we cover both cases, convex and concave functions. Theorem 1. Let ðX t Þ be a stochastic process. 1. If EðhðX s ÞÞ rEðhðX t ÞÞ for all s r t and increasing convex functions h(x) s.t. both expectations are finite, then ðX t Þ has the marginals of a sub-martingale. 6 This name is derived from the French expression for increasing in convex order – Processus Croissant pour l'Ordre Convexe, in short PCOC – which is a homophone of the word peacock, compare (Hirsch et al., 2011). Anecdotal motivation for this name comes from the fact that the peacock is an Asian bird and the prime example, sometimes also referred to as the guiding peacock, is that of the underlying of an Asian option, see Carr et al. (2008). 7 The name lyrebird takes its motivation from the following two facts. As much as the guiding peacock is the underlying of an Asian option, the underlying of an Australian option, discussed in more detail in Section 4, represents perhaps the most prominent example of a lyrebird. Lyrebirds are native to Australia and while not genetically related to peacocks they are often referred to as peacock-wrens. The male lyrebird can fan his tail in similar, but not as striking, fashion as the common peacock. However, lyrebirds are far better known for their superb ability to mimic natural and artificial sounds. 8 Rothschild and Stiglitz consider a third equivalent notion of increasing risk based on tail weights of the corresponding random variables, and introduce the so-called mean-preserving spreads, which they use in proving their equivalence. These spreads are related to but not the same as techniques used by Strassen (1965), Doob (1968) and Kellerer (1972).

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2. If EðUðX s ÞÞ ZEðUðX t ÞÞ for all s r t and increasing concave functions U(x) s.t. both expectations are finite, then ðX t Þ has the marginals of a super-martingale. Proof. The first statement follows directly from Proposition 3 in Kellerer (1972). Note that the class of functions used by Kellerer, i.e. convex functions which are of limiting behavior Oðmaxðx; 0Þ þ 1Þ for x-7 1 slightly differs from the class of functions we use, however in his proof Kellerer (1972) shows that it is sufficient to restrict attention to increasing convex functions of type hðxÞ ¼ maxðx; aÞ for all a A R, which covers our case. For the second part, note that the transformation hðxÞ≔  Uð  xÞ converts increasing concave functions considered in part 2 of the proposition into increasing convex functions considered in part 1 of the proposition and that EðUðX s ÞÞ Z EðUðX t ÞÞ 3 Eðhð  X s ÞÞ rEðhð  X t ÞÞ: Hence part 2 follows from part 1, recognizing that the negative of a sub-martingale is a super-martingale. □ 3. Income and wealth distributions: increasing inequality and poverty The Rothschild and Stiglitz (1970) notion of increasing risk can be applied to distributions in the same way as it is applied to random variables. Our presentation so far has taken the point of view of families of random variables, but equivalently we could have considered families of distribution functions ρt, as in fact Strassen (1965) and Kellerer (1972) did. The latter is also more common in the study of inequality and poverty. In fact, as pointed out by Atkinson (1970, 1987) the study of inequality and poverty measures is strongly related to the study of risk. From an economic perspective, income and wealth distribution functions are of particular interest, and the question whether a series of distributions over time becomes more or less equal is fundamental in many aspects, specifically in political economy where high levels of inequality are typically associated with political and social unrest. The most popular measure for inequality in income and wealth distributions is the so-called Gini coefficient, which is defined by Z 1 G ¼ 1 2 LðxÞ dx; ð12Þ 0

where L(x) denotes the Lorentz curve, which relates the properties of wealth held by a fraction of the population to the size of that fraction, compare Lubrano (2013) for an excellent introduction into the subject. To study Gini coefficients of families of wealth distributions in the framework of this paper, it is useful to consider it as a functional of random variables. Hence, for this purpose let X denote a random variable, which represents the distribution ρX. For simplicity we assume that X takes positive values only, i.e. suppðρX Þ  ½0; 1Þ. The fraction of the population with income or wealth below a certain x A ½0; 1Þ is then given by   ρX ðxÞ ¼ PðX rxÞ ¼ E 1fX r xg ; ð13Þ where 1fX r xg denotes the indicator function, which is 1 if X r x and 0 else. On the other hand, the fraction of wealth held by this fraction of the population is   E X  1fX r xg : ð14Þ F ðxÞ ¼ EðXÞ The points on the Lorentz curve are then given by the pairs ðρX ðxÞ; FðxÞÞ Now, let Y be uniformly distributed in ½0; 1. Then law X ¼ ρX 1 ðYÞ where ρX 1 ðxÞ ¼ inffy A RjρX ðyÞ Z xg and   R 1 1 E ρX 1 ðYÞ1fρ  1 ðYÞ r xg 0 ρX ðyÞ1fρX 1 ðyÞ r xg dy X  1  ¼ : F ðxÞ ¼ R 1 1 E ρX ðYÞ 0 ρX ðyÞ dy We then obtain the Lorentz curve as R 1 1 0 ρX ðyÞ1fρX 1 ðyÞ r ρX 1 ðxÞg dy LðxÞ ¼ R 1 1 0 ρX ðyÞ dy and the integral in (12) can now be written as R 1 R 1 1 Z 1 0 0 ρX ðyÞ1fρX 1 ðyÞ r ρX 1 ðxÞg dy dx LðxÞdx ¼ R 1 1 0 0 ρX ðyÞdy 0R 1 1 1 0 ρX ðyÞ1fρX 1 ðyÞ r Xg dy @ A ¼ EðF ðX ÞÞ: ¼E R 1 1 0 ρX ðyÞ dy Using (12), (14) and (16) we can then represent the Gini coefficient as      E X  1fX r Yg ; G ρX ¼ 1  2  EðXÞ

ð15Þ

ð16Þ

ð17Þ

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27

law

where now Y ¼ X denotes an independent identical copy of X. This expression can then be further manipulated to fit into the framework of this paper, but also to highlight an alternative intuition of the Gini coefficient. In order to see this, we write         E X  1fX r Yg E ðY XÞ  1fX r Yg ¼ ; ð18Þ G ρX ¼ 1 2  EðXÞ EðXÞ where we used EðXÞ ¼ EðYÞ and that symmetry implies       E Y  1fX Z Yg E X  1fY Z Xg ¼ EðXÞ EðYÞ It now follows from symmetry in (18) that     1 EðjY  XjÞ : G ρX ¼ 2 EðXÞ

ð19Þ

Now, let ρt be a family of distributions parametrized in time. An important task is then to find appropriate results which characterize the dynamics of the corresponding measures of inequality. The following proposition provides a first hint that the notion of peacocks and their characterization can be very useful in this context. Proposition 2. If the family of distributions Gini coefficients Gt ¼ Gðρt Þ is increasing in t.

ρt for t A ½0; 1Þ is increasing in convex order, then the corresponding sequence of

Proof. As ρt is increasing in the convex order, there exists a martingale ðM t Þ with the same marginals as independent copy of of M(t). Then according to (19) we have that 3 2  ~   1 E jM t M t j 5: G ρt ¼ 4 2 EðM t Þ

~ ρt. Let MðtÞ be an

Now, the process ðZ t Þ defined by Z t ¼ M~ t  M t is obviously a martingale too, and as such a peacock, hence increasing in the  convex order. Therefore the expression E jM~ t M t j is increasing in time and as EðMt Þ is constant, the Gini coefficient Gðρt Þ is increasing in time.□ Two different interpretations of the Gini coefficient become immediately apparent, when writing the final expression in (18) as     E ðX  Y Þ þ G ρX ¼ ; ð20Þ EðXÞ using the notation x þ ¼ maxðx; 0Þ and the symmetry in X and Y. The first connects the Gini coefficient to the option pricing literature. Denoting with C X ðkÞ the price of a European call on X with strike k, we have that Z   1  C X ðkÞ dρX ðkÞ; G ρX ¼ EðXÞ i.e. the Gini coefficient is a weighted average over all strikes of European call options on X. The second interpretation is more in line with the classical literature on inequality and emphasizes the intuition of the Gini coefficient as a measure of inequality: to measure wealth inequality we consider a randomly chosen member of the population with wealth Y and tell this member the wealth X of another randomly chosen member of the population. If X is higher than Y this will cause dissatisfaction or jealousy and a perception of inequality. If on the other hand X is smaller than Y, no such feelings will occur. Taking a large sample in this way and averaging over the sample will then lead to the expression presented in (20). An interesting question is, whether there is a family of inequality measures, such that if inequality of ρt as measured by this family increases over time, ρt (or at least some functional of ρt) has the marginals of a martingale, i.e. presents a peacock. In order to answer this question we expand on the second interpretation above. In reality, it would make sense to assume that people are not concerned about small differences in wealth. If one's neighbor is a little bit richer, one would likely not mind. Only if she or he is richer by a substantial margin, then a perception of inequality arises. In order to account for this one can introduce a modified Gini coefficient     E ðX  Y  k Þ þ Gk ρ X ≔ : ð21Þ EðXÞ When inequality is measured by Gk ðρÞ, one would disregard wealth differences by a margin of k and the inequality measure would therefore be concentrated on sufficiently large wealth differences. We define the Gini coefficient Gk for all k A R via (21). For negative k, the intuition is not about jealousy, but rather of concern about one's neighbors poor income or wealth, which is accumulated across the population to arrive at a measure of inequality. Obviously G0 corresponds to the classical Gini coefficient. We can now conclude with the following:

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Proposition 3. Assume that the family of distributions ρt is increasing in inequality as measured by Gk ðρt Þ for all k A R and has constant mean. Denote by ρ t the distribution function defined via ρ t ðxÞ≔1  ρt ð  xÞ. Then the family of convoluted distributions ρt ⋆ρ t has the marginals of a martingale. Proof. We choose stochastic processes ðX t Þ and ðY t Þ such that X t ¼ ρt ¼ Y t law

law

and Xt and Yt are independent. This is always possible and we can further assume without loss of generality that EðX t Þ ¼ 1. Now consider the stochastic process Zt ¼ Xt  Y t :

ð22Þ

Using (21), we conclude from our main assumption that for all k A R the function  þ Gk ðρt Þ ¼ E ðZ t  kÞ

ð23Þ

is increasing in t. It then follows from Theorem 3.1 in Hirsch et al. (2012) that ðZ t Þ is a peacock and hence admits the marginals of a martingale. The distribution of Z t ¼ X t  Y t is by standard results in elementary probability the convolution of the distributions of Xt and Y t , i.e. ρt ⋆ρ t .□ It would be much nicer of course to be able to conclude that the family of distributions ρt rather than ρt ⋆ρ t has the marginals of a martingale. This however is in general not true as the following example shows9: it is easier to formulate the example in discrete time, a continuous time version can be easily derived from this. Consider a random variable L with law law EðLÞ ¼ 0 which maps into ½ 1; 1, such that Z is not symmetric, i.e. L a  L and hence also 1 þ L a 1 L. As the distribution is characterized by higher moments, the latter implies that there is at least one convex function h(x) s.t. Eðhð1 þ LÞÞa Eðhð1  LÞÞ:

ð24Þ

Now define the stochastic process X n ¼ 1 þ ð  1Þ L, or alternatively the corresponding sequence of distributions ρn. It follows from (24) that the associated sequence EðhðX n ÞÞ is alternating between two distinct values and so cannot be monotonic. Therefore the process ðX n Þ cannot be a peacock and cannot have the marginals of a martingale. However, with an independent copy M  L and Y n ¼ 1 þð 1Þn M we have that n

Z n ≔X n  Y n  Y n  X n ¼ X n þ 1 Y n þ 1 ¼ Z n þ 1

ð25Þ

and hence EðhðZ n ÞÞ is constant in n for all convex functions. Therefore Zn is a peacock and thus has the marginals of a martingale. In fact a suitable martingale can be easily identified as i.i.d. copies of L  M. Obviously the Gini coefficients associated to Zn and Z n þ 1 are also identical. For other appropriately generalized coefficients of inequality it may well be possible to conclude that the sequence of distributions is in fact a peacock. If the coefficient of variation, see for example Forster et al. (2014), is extended by a sufficiently rich family of convex functions, we could well establish the peacock property from increasing inequality. The same holds for appropriate families of the Pietra index. Rather than pursuing this, we consider below a number of well known classical poverty measures, which are discussed in Atkinson (1987). These are the normalized deficit, Watts’ measure and the Clark measure of poverty. It is remarkable how well these measures fit into the framework of peacocks and general option pricing theory, and it is surprising that this has not been pursued before. Let us first look at the normalized deficit. Let X be a random variable representing the income or wealth distribution ρ  ρX . Let z Z 0 be the poverty threshold, then Z z    x 1  1  dρX ðxÞ ¼ E ðz  XÞ þ D ρX ¼ ð26Þ z z 0 is called the normalized deficit. This expression measures how far people falling below the poverty threshold differ from the poverty threshold, on average. The value z of the poverty threshold causes a lot of controversy and is usually determined only within a range. However, we can make the following statement: Proposition 4. Let ðρt Þ be a sequence of wealth distributions with suppðρt Þ  ½0; 1Þ and Eðρt Þ constant, such that the normalized deficit Dðρt Þ is increasing for all z Z0, then ðρt Þ represents a peacock and hence has the marginals of a martingale. The sequence of distributions ðρt Þ converges weakly towards a limit distribution ρ1 ðxÞ Proof. With the identification in (26) as well as the statements from Section 1, this follows directly from Theorem 3.1 in Hirsch et al. (2012). The second part follows from Doob's martingale convergence theorem (see Oksendal, 2003, Theorems C.5 and C.6, p. 302) noticing that under the assumptions made the sequence ðρt Þ is automatically uniformly integrable.□ The convergence in Proposition 4 is actually stronger than stated. The convergence of the martingale associated to the sequence of distributions is in fact in the L1 norm. However, the proposition above is restrictive in the way that it requires the expectations of the series of distributions to be constant. This disadvantage can be overcome by using the more general concept of Lyrebirds: 9

We are thankful to Francis Hirsch and Bernard Roynette for suggestions and comments that lead to this example.

C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

29

Proposition 5. Let ðρt Þ be a sequence of wealth distributions with suppðρt Þ  ½0; 1Þ and assume there exists a constant C 4 0 such that Eðρt Þ rC for all t. Assume that the normalized deficit Dðρt Þ is monotonically increasing for all z Z 0. Then ðρt Þ has the marginals of a super-martingale and the sequence of distributions converges weakly towards a distribution ρ1 . proof. That the sequence ðρt Þ admits the marginals of a super-martingale follows from Theorem 3. Note that under the assumption of Dðρt Þ monotonically increasing for all z Z 0, we have that EðUðX t ÞÞ is decreasing for all increasing concave utility function of the form UðxÞ ¼ a  b  ðz xÞ þ with b 4 0 and that these functions build a sufficiently rich basis of all increasing concave utility function. The convergence follows from Doob's super-martingale convergence theorem, noticing again that the distribution is concentrated on R þ and that the sequence of expectations is bounded.□ Martingale convergence theorems have been used in the Economics literature before, perhaps most notably by Thomas and Worrall (1990). However they consider a functional of a state variable (the marginal utility) within a dynamic programming context.10 The setup above does not require any of this and simply makes use of properties of the sequence of distributions. It is possible to use other measures of inequality and poverty to make analogous statements. These include Watts' (1968) measure Z z  þ W¼ logðx=zÞ dρX ðxÞ ¼ E ðz  logðXÞÞ ; 0

with z ¼ logðzÞ, which leads to logðX t Þ being a peacock or a lyrebird, or the measure proposed by Clark (1981) Z þ  1 z xα 1  1 dρX ðxÞ ¼ E z X α C¼ z α 0 zα α

with z ¼ zα and 0 o α o 1 which leads to Xt being a peacock, since x↦x1=α is a convex function and Xt is a peacock. 4. Increasing risk and inequality for solutions of stochastic differential equations Continuous time models in which economic variables are described as the solution of certain stochastic differential equations are becoming more and more popular. It is therefore important to understand how risk and inequality measures attached to such variables change over time. In the context of risk the importance of stochastic differential equations as a modeling tool is documented in manifold. In the context of inequality stochastic differential equations have played a less dominant role, however recent interesting applications include (Benhabib et al., 2015) as well as (Forster et al., 2014). Let us therefore consider the case, where ðX t Þ is given as the solution of a stochastic differential equation dX t ¼ αðt; X t Þ dt þ σ ðt; X t Þ dW t ;

X 0 ¼ x;

ð27Þ

where αðt; xÞ and σ ðt; xÞ are sufficiently smooth functions on ½0; 1Þ  U with U  R an open set and X t A U for all t. Trivially, a sub-martingale has the marginals of a sub-martingale and is hence a lyrebird. Further, any integrable Itô process with positive drift term is a sub-martingale. Hence if αðt; xÞ Z 0 for all ðt; xÞ A ½0; 1Þ  U, then ðX t Þ is a sub-martingale and hence a lyrebird. The following result goes beyond this. Proposition 6. Let ðX t Þ be given by (27) with σ ðt; xÞ linear and αðt; xÞ convex in x, s.t. for all ðs; xÞ A ½0; 1Þ  U

αs ðs; xÞs þ αðs; xÞ Z 0 and 2σ ðs; xÞσ s ðs; xÞs þ 12 σ 2 ðs; xÞ Z0;

ð28Þ

then Xt is a Lyrebird, i.e. for every convex and increasing function h(x) the functional t↦EðhðX t ÞÞ is increasing in t. Proof. This result follows by combination of Propositions A1–A3 in the appendix.□ Note that Proposition A1 in the Appendix can in fact lead to stronger results, when specific functions h(x) are under consideration, but without further restriction Proposition 6 naturally emerges from Proposition A1. The criterion derived in Proposition 6 is merely sufficient but not necessary. In particular criterion (28) applied to a time homogeneous diffusion is equivalent to the condition that its drift term is positive. In this case, the solution is in fact a submartingale, hence trivially a lyrebird. This raises the following question. Is a lyrebird obtained from the solution of a timehomogeneous SDE perhaps automatically a sub-martingale? The answer to this question is no, as the example of X t ¼ sinhðBt Þ for Bt a Brownian motion shows, compare Theorem 7.20 in Hirsch et al. More generally, it can be shown that the process X t ¼ sinhðBt þxÞ for all x 4 0 is a lyrebird, but not a peacock. These examples show that condition (28) is not necessary.

10

See for example the proof of Proposition 3 in Thomas and Worrall (1990).

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C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

Interesting applications of Proposition 6 lie in the field of exotic options, in particular Asian and Australian options, as we now demonstrate. Let St ¼ eðμ  ð1=2Þσ

2

Þt þ σ W t

denote the price of a share, then an Asian option is a call on the underlying Z 1 t ¼ Su du X Asian t t 0 while an Australian option is a call on the underlying R 1 t t 0 Su du X Aus ¼ : t St

ð29Þ

ð30Þ

ð31Þ

The latter expression is in fact related to what is referred to risk elasticity in the work of Hansen and Scheinkman (2009) as well as Borovička et al. (2011). Expressions (29)–(31) can be considered under the real world measure or the risk neutral measure. In the context of option pricing the latter is relevant, and for such purposes we simply identify μ ¼ r as the risk-less rate. It was unknown for some time, whether Asian option prices are increasing in volatility and maturity. Carr et al. (2008) were the first to prove that the price of an Asian option is indeed increasing in the volatility of the asset price and if μ ¼0, the Asian same is true with respect to maturity. A direct consequence of this is that under the assumption of μ ¼ 0 the process Xt is a peacock. In fact this example is now considered as the “guiding” peacock in the literature. The case μ a 0 has however been discarded so far. In this case, it is obvious that ðX t Þ cannot have the marginals of a martingale, as its expectation is not constant. However, as we will show below, in the case of μ 4 0, the processes ðX Asian Þ and t ðX Aus Þ are still lyrebirds and hence increasing in convex order for increasing convex functions. t Proposition 7. Assume that μ o σ 2 . Then the Australian underlying ðX Aus t Þ defined in (31) is a lyrebird. Proof. It is not difficult to verify that the dynamics of the process ðX Aus t Þ defined by (31) is given as  1 Aus σ 2  μ  X Aus þ1 dt  σ X Aus dW t : dX t ¼ t t t In terms of the general dynamics (27) we therefore have that  1 αðs; xÞ ¼ σ 2  μ  x þ 1 s σ ðs; xÞ ¼  σ x:

ð32Þ

ð33Þ ð34Þ

Obviously σ ðs; xÞ is linear and independent of s. Furthermore, we have that

αs ðs; xÞ ¼

x s2

ð35Þ

and therefore the left hand sides of (28) can be computed as

αs ðs; xÞs þ αðs; xÞ ¼ ðσ 2  μÞx þ1 2σ ðs; xÞσ s ðs; xÞ þ 12 σ 2 ðs; xÞ ¼ 12 σ 2 ðs; xÞ

ð36Þ

As αðs; xÞ is obviously convex in x, the result then follows directly from Proposition 6 and (36), noticing that Xt takes only positive values.□ The following corollary can be obtained from exploiting a general relationship between the Australian and Asian underlying, that was uncovered in Ewald et al. (2013). Corollary 1. In the case of μ 4 0 the Asian underlying ðX Asian Þ is a lyrebird. t Proof. It follows from Proposition 2.1 in Ewald et al. (2013) that for every t 40 the Asian underlying Xt with μ 4 0 and X 0 ¼ 1 has the same distribution as the Australian underlying with μ replaced by σ 2  μ. The latter is a Lyrebird according to Proposition 7 as long as σ 2  μ o σ 2 , which is equivalent to μ 4 0.□ The relationship between time and volatility can be used to derive results on the volatility dependence of Asian and Australian options, we omit this here. Another interesting case is where the stochastic dynamics actually describes a whole family of dynamics parametrized by a parameter ϵ, i.e. ϵ

dX t ¼ αðt; X ϵt ; ϵÞ dt þ σ ðt; X ϵt ; ϵÞ dW t :

ð37Þ

This situation is often encountered in economic and financial modeling, see for example Hansen and Scheinkman (2010) and

C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

31

Borovička et al. (2011, 2014)).11 In this case we are interested in conditions that guarantee that d   ϵ  E h Xt Z 0 dϵ

ð38Þ

for all t Z 0. Such conditions can be derived by analogy to the previous case. We have the following proposition. Proposition 8. Assume that the partial derivatives of the coefficient functions satisfy

αϵ ðs; x; ϵÞ Z 0;

ð39Þ

σ ðs; x; ϵÞσ ϵ ðs; x; ϵÞ Z0;

ð40Þ ϵ

and further that αðs; x; ϵÞ is convex in x and σ ðs; x; ϵÞ linear in x. Then ðd=dϵÞEðhðX t ÞÞ Z 0 for all convex and increasing functions h ϵ (x), i.e. Xt is a lyrebird in ϵ for all t. Proof. See Appendix A.□ 5. Time integral, average and hyperbolic discounting of sub- and super-martingales The case of an Asian option considered in the previous section in fact presents an example, where a time average of a sub- resp. super-martingale is considered. Instead of the average one can also consider a simple time integral with the possibility of certain discount factors. The following proposition presents a generalization of Hirsch et al. (2011). Proposition 9. Let ðMt Þ be a sub- resp. super martingale and α: ½0; 1Þ-½0; 1Þ be an increasing continuous function such that

αð0Þ ¼ 0. Then the processes ðX t Þ and ðX~ t Þ defined via 1 αðtÞ

Z

Xt ≔

X~ t ≔

Z

t 0

t

0

Ms dαðsÞ

ð41Þ

ðM s  M0 Þ dαðsÞ

ð42Þ

have the marginals of a sub- resp. super-martingale

Proof. The proof of these results follows the ideas presented in Hirsch et al. (2011) and we only point out the differences in the following. First, note that the results for super-martingales can be obtained from those for sub-martingales from the simple fact that the negative of a sub-martingale is a super-martingale and vice versa. We therefore consider the submartingale case only. In order to prove the first point, Hirsch et al. (2011) use integration by parts to represent the process ðX t Þ as X t ¼ Mt  with M αt ¼

Rt 0

1

αðtÞ

M αt

ð43Þ

αðsÞdMs and obtain that

dX t ¼ M αt

dαðtÞ : α2 ðtÞ

ð44Þ

As ðM t Þ is no longer a martingale, the proof in Hirsch et al. (2011) now needs to be adjusted in the following way: note that α as the integral of a positive integrand with respect to a sub-martingale, Mt is also a sub-martingale. It then follows from (44) that

Z t  dαðuÞ

E X t jF s Þ ¼ X s þE M αu 2

F s α ðuÞ s Z t dαðuÞ ZX s þ M αs 2 s α ðuÞ  1 1  ; ¼ X s þM αs αðsÞ αðtÞ α

ðuÞ where we used in the second line that Mu is a sub-martingale and that dαα2 ðuÞ is deterministic and positive. Now note that in the light of Section 2 in difference to Hirsch et al. (2011) we only need to consider increasing convex functions ψ ðxÞ (using

11 Hansen and his coauthors studied a variation of the derivative in expression (38) applied to cash flows and stochastic discount factors as a way of quantifying the exposure of cash flows to risk and the prices of risk associated with these exposures. They have also been interested in the dependence of these exposures and prices on the time horizon t as a way of capturing the “term-structure” of risk.

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C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

the same notation as in Hirsch et al., 2011). This guarantees that Z s ψ 0 ðX u ÞdMαu Z 0; E ϵ

which allows us to complete the proof of the first point exactly as in Hirsch et al. (2011). For the second point the only relevant issues are that Z t EðM u jF s Þ dαðuÞ ZðαðtÞ  αðsÞÞM s s

and since we can restrict ourselves to increasing convex functions ψ ðxÞ that Eðψ ðEðX~ t jF s ÞÞÞ Z Eðψ ðX~ s þðαðtÞ  αðsÞÞM s ÞÞ;

ð45Þ

at which point the proof can be completed exactly in the same way as in Hirsch et al. (2011).□ Proposition 9 has a very useful application in the presence of discounting, both geometric and hyperbolic. Let us start with the case of hyperbolic discounting which has recently drawn a lot of attention within the economic literature, see for example Noor (2009), Masatlioglu (2007) and Dasgupta and Maskin (2005). To see this, we make the following choice: 

1



αðsÞ≔ log 1 þ ρs : ρ We then have dαðsÞ ¼

1 ds 1 þ ρs

and for any sub-martingale Mt can conclude from Proposition 9 that the process Z t 1 ðM s  M0 Þ ds X~ t ¼ 0 1 þ ρs

ð46Þ

is a lyrebird and hence increasing in convex order. To state this in more explicit terms and assuming for simplicity that M 0 ¼ 0, we have that for any increasing and convex function h(x), the function  Z t 1 Ms ds ð47Þ t↦E h 0 1 þ ρs is increasing in t. The previous expression takes explicit account of separation between time-preference risk and uncertainty, compare Masatlioglu (2007) and Noor (2009). This applies, whenever the underlying economic variable ðM t Þ is a sub-martingale. On the other hand, if ðM t Þ is a super martingale, we conclude from Proposition 9, that the corresponding process X~ t has in fact the marginals of a super-martingale. Replacing h(x) from above with an increasing concave function then shows that the analog of expression (47) in this case is decreasing in time. Perhaps more important in this case is the following convergence result: Proposition 10. Let ðM t Þ be a positive super-martingale, then the process corresponding to (46) converges in distribution for t-1. Proof. This follows from the Doob martingale convergence theorem for positive super-martingales, as under these assumptions ðX~ t Þ as defined in (46) has the same marginals as a positive super-martingale.□ The case of more conventional geometric discounting can also be recovered from Proposition 9, by choosing

αðsÞ ¼

 1 1  e  ρs ;

ρ

leading to dαðsÞ ¼ e  ρs ds and a functional of the type  Z t e  ρs M s ds t↦E h

ð48Þ

0

for which similar results to those for hyperbolic discounting can be obtained. 6. Application to real options The underlying for the Australian option discussed in Section 4 is strongly related to a process commonly referred to as geometric mean reversion. In fact let ðX t Þ be given by (31), then as shown in Ewald et al. (2013), the process ðY t Þ given as Yt ¼ R t 0

St Su du

¼

1 t  Xt

ð49Þ

C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

defined for t 4 0 satisfies   dY t ¼ Y t μ  Y t dt þ σ dW t :

33

ð50Þ

This process is commonly referred to as a geometric mean reversion process, compare Dixit and Pindyck (1994), and often considered in the framework of real options. In the biological context it is interpreted as a stochastic version of the logistic growth equation, and is often used in the context of natural resource harvesting, compare (Ewald and Wang (2010) and Ewald (2013)). Note that the drift term in (50) is concave rather than convex, which will put this process in a slightly different category than those considered in Section 4. The following proposition concerns the evolution of riskiness of the process ðY t Þ in the framework of Rothschild and Stiglitz (1970). Proposition 11. Consider the geometric mean reversion process ðY t Þ given in (50). Assume μ o σ 2 and let g(x) be a utility function with Arrow–Pratt risk aversion coefficient 

g ″ ðxÞx Z2 g 0 ðxÞ

for all x 40:

ð51Þ

Then for all 0 o s ot we have EðgðY t ÞÞ r EðgðY s ÞÞ: Proof. Let us first show that the function  1 hðxÞ≔ g x is increasing and convex. In fact we have   1 1 0   2 h ðx Þ ¼  g 0 x x   

  1 1 1 2 ″  4 þg 0  : h ðxÞ ¼  g ″ x x x x3

ð52Þ

ð53Þ

ð54Þ

ð55Þ

As gðÞ is increasing by assumption, (54) implies that hðÞ is also increasing. The function hðÞ is convex if and only if the expression in (55) is positive. The latter is the case if     1 1 2 0 1   3 for all x 40; g″ r  g x x x4 x which is equivalent to (51). It follows from Proposition 7 that ðX t Þ has the marginals of a sub-martingale, and as the product of a sub-martingale and an increasing function is also a sub-martingale, that the process ðt  X t Þ has the marginals of a submartingale too, hence it is a lyrebird. Therefore EðhðtX t ÞÞ ZEðhðsX s ÞÞ:

ð56Þ

Using (49) and (53) the latter is equivalent to EðgðY t ÞÞ r EðgðY s ÞÞ:&

ð57Þ

In the real option framework, if (50) describes the value of a project and an investor has to chose a deterministic time t to invest in it, then as long as the investor is risk averse, she/he would invest as soon as possible. If on the other hand the investor has some managerial flexibility, and can invest according to an optimal stopping time, which requires that the process ðY t Þ is observable at any time, then the result is different and is discussed in Dixit and Pindyck (1994) for example. In that case the decision to invest will occur at a specified investment threshold. 7. Conclusions We have demonstrated how the Rothschild and Stiglitz (1970, 1971) notion of increasing risk can be extended to families of random variables ðX t Þ and linked their concept to the concept of stochastic processes which are increasing in the convex order. By doing this we showed how Rothschild and Stiglitz' results follow from earlier results in the mathematical literature, e.g. Strassen (1965), Doob (1968) and Kellerer (1972). We also showed how the more recently developed literature around the topic of the so-called peacocks leads to very interesting and useful results on risk and inequality. Going far beyond Rothschild and Stiglitz (1970, 1971), the observation that increasing risk is related to processes with the marginals of a martingale uncovers important issues in the dynamics of risk, inequality and poverty measures, given the richness of the theory of martingales in the mathematical literature. We further demonstrated that it makes sense to look at a larger class of processes, which we refer to as lyrebirds. These processes provide greater flexibility than peacocks and such is required in particular in the study of inequality and poverty measures. Specific inequality and poverty measures that we have included

34

C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

in our study involved the Gini coefficient and the normalized deficit. In addition we looked at other applications in Economics and Finance such as geometric and hyperbolic discounting as well as exotic options and real options.

Acknowledgements The suggestions and advice of two anonymous referees as well as an associate editor helped greatly to improve the manuscript and have been highly appreciated. Appendix 0

Proposition A 1. Let ðX t Þ be given by (27) and h(x) be a given function, s.t. h ðX t Þ Z 0 for all t a.s. and not vanishing identically. Let vðs; t; xÞ ¼ EðhðX t ÞjX s ¼ xÞ:

ð58Þ

The functional t↦EðhðX t ÞÞ is increasing in t if the following condition holds: For all ðs; xÞ A ½0; 1Þ  U and t Zs   2½ðαs ðs; xÞs þ αðs; xÞÞvx ðs; t; xÞ þ 2σ ðs; xÞσ s ðs; xÞs þ 12 σ 2 ðs; xÞ vxx ðs; t; xÞ Z 0:

ð59Þ

Proof. It follows from the Feynman–Kac theorem that vðs; t; xÞ satisfies the following PDE and boundary condition vs þ αðs; xÞvx þ 12 σ 2 ðs; xÞvxx ¼ 0

ð60Þ

vðt; t; xÞ ¼ hðxÞ:

ð61Þ

The following is a generalization of the approach pioneered by Carr et al. (2008) which is based on the Pontryagin maximum principle for parabolic PDEs. For this let ϵ 40 and consider the time changed stochastic process ðX ϵ2 t Þ with corresponding filtration ðF ϵ2 t Þ. For Δ 4 0 we have that Z ϵ2 ðt þ ΔÞ X ϵ2 ðt þ ΔÞ  X ϵ2 t ¼ αðs; X s Þ dsþ σ ðs; X s Þ dW s Z

ϵ2 t

¼ t

tþΔ

αðϵ2 u; X ϵ2 u Þϵ2 du þ σ ðϵ2 u; X ϵ2 u Þ dW ϵ2 u ;

ð62Þ

where we made use of the substitution s ¼ ϵ2 u. Using that ðBt Þ defined via 1 Bt ≔ W ϵ2 t

ð63Þ

ϵ

~ϵ

is a Brownian motion adapted to the filtration ðF~ t Þ with F~ t ¼ F ϵ2 t , we conclude from ð62Þ that the process X t ¼ X ϵ2 t satisfies ϵ ϵ ϵ dX~ t ¼ αðϵ2 t; X~ t Þϵ2 dt þ σ ðϵ2 t; X~ t Þϵ dBt :

ð64Þ

It now follows from (58) and the Feynman–Kac theorem applied to (64) that ϵ ϵ ~ t; x; ϵÞ vðϵ2 s; ϵ2 t; xÞ ¼ EðhðX~ t ÞjX~ s ¼ xÞ≕vðs;

for all s; t; x; ϵ;

where v~ satisfies     v~ s þ α ϵ2 s; x ϵ2 v~ x þ 12 σ 2 ϵ2 s; x ϵv~ xx ¼ 0 ~ t; x; ϵÞ ¼ hðxÞ: vðt; ~ t; x; ϵÞ and hence We now consider the case s ¼0, for which vð0; ϵ2 t; xÞ ¼ vð0;

  d d 1 d

EðhðX t ÞÞ ¼ vð0; t; xÞ ¼ v 0; ϵ2 t; x dt dt 2t dϵ ϵ ¼ 1

1 d

1 v~ ð0; t; x; ϵÞ ¼ v~ ϵ ð0; t; x; 1Þ: ¼ 2t dϵ ϵ ¼ 1 2t

ð65Þ

ð66Þ

ð67Þ

d EðhðX t ÞÞ is positive hence depends only on the positivity of the derivative v~ ϵ ð0; x; t; ϵÞ. In order to The question whether dt investigate this expression further we differentiate the PDE (66) with respect to ϵ and obtain       0 ¼ v~ ϵs þ 2αs ϵ2 s; x sϵ3 v~ x þ 2α ϵ2 s; x ϵv~ x þ α ϵ2 s; x ϵ2 v~ ϵx  2   2  2     þ 2σ ϵ s; x σ s ϵ s; x sϵ v~ xx þ 12 σ 2 ϵ2 s; x v~ xx þ 12 σ 2 ϵ2 s; x ϵv~ ϵxx : ð68Þ

Evaluation of (68) at ϵ ¼ 1, using the notation wðs; t; xÞ≔v~ ϵ ðs; t; x; 1Þ and the fact that the boundary condition in (66) does not depend on ϵ, we obtain ws þ αðs; xÞwx þ 12 σ 2 ðs; xÞwxx ¼  ½2αs ðs; xÞs þ 2αðs; xÞÞvx

C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

  þ 2σ ðs; xÞσ s ðs; xÞs þ 12 σ 2 ðs; xÞ vxx

35

ð69Þ

wðt; t; xÞ ¼ 0:

ð70Þ

Here we also used that evaluated at ϵ ¼ 1 we have v~ x ¼ vx and v~ xx ¼ vxx . Eqs. (69) and (70) represent a parabolic PDE to which the Pontryagin maximum principle in the version of Stroock and Varadhan (1979) can be applied. This principle implies that as long as the right hand side of (69) is negative for all (s,x), we have wðs; t; xÞ Z0 for all ðs; t; xÞ and hence positivity in (67). The latter is implied by Eq. (59).□ Proposition A 2. Assume ðX t Þ satisfies (27). Then vðs; x; tÞ as defined in (58) is increasing in x, i.e. vx 4 0. Proof. Without loss of generality we assume that h(x) is two times continuously differentiable. Using results on the first variation process ðZ t Þ of ðX t Þ, according to Protter (2004) and Fournié et al. (1999), we have vx ðs; x; t Þ ¼ with

 0 ∂   E h X t jX s ¼ xÞ ¼ E h ðX t ÞZ t jX s ¼ xÞ ∂x

Z Z t ¼ exp

t

s

αx ðu; X u Þ  12 σ 2x ðu; X u Þ ds þ

Z

t s

ð71Þ

σ x ðu; X u Þ dW u :

0

ð72Þ 0

The result follows from h ðxÞ Z 0 and Z t 4 0 and the assumption that h ðxÞ does not vanish identically.□ In order for ðX t Þ to qualify as a Lyrebird, we need that t↦EðhðX t ÞÞ is increasing for all increasing and convex functions h. The specific choice of the function h(x) obviously affects the values of vx and vxx. The previous lemma shows that vx is always positive, as long as h(x) is increasing and convex. However, unlike the gamma of a European call option in the Black–Scholes model, for general αðs; xÞ and σ ðs; xÞ the function vxx can be negative. Geometric mean reversion, which we discussed in Section 6, is one example for this. The following lemma holds: Proposition A 3. Let ðF t Þ denote the stochastic process defined by Z t Z t ðαxx ðu; X u Þ  σ x ðu; X u Þσ xx ðu; X u ÞÞZ u du þ σ xx ðu; X u ÞZ u dW u ; Ft ¼ Zt  s

ð73Þ

s

then ″

0

vxx ðs; x; tÞ ¼ Eðh ðX t ÞZ 2t jX s ¼ xÞ þ Eðh ðX t ÞF t jX s ¼ xÞ

ð74Þ

Proof. Note that from Eq. (71) it follows that vxx ðs; x; t Þ ¼

∂  0 E h ðX t ÞZ t jX s ¼ xÞ ∂x

ð75Þ

and that Zt depends on x through Xu's dependence on x in Eq. (72). In fact differentiation of (72) with respect to x yields (73), which is identified as the second variation process of ðX t Þ, compare Protter (2004, Chapter 5.7). Executing the differentiation in (75) inside the expectation we obtain (74).□ Proof of Proposition 8. We set vðs; t; x; ϵÞ ¼ EðhðX ϵt ÞjX ϵs ¼ xÞ: Then vs þ αðs; x; ϵÞvx þ 12 σ 2 ðs; x; ϵÞvxx ¼ 0 vðt; t; x; ϵÞ ¼ hðxÞ:

ð76Þ

Further let v~ ðs; t; x; ϵÞ ¼

 d d   

vðs; t; x; ϵÞ ¼ E h X ϵt X ϵs ¼ x dϵ dϵ

then by differentiation of (76) we have that for all v~ s þ αðs; x; ϵÞv~ x þ αϵ ðs; x; ϵ ~ t; x; ϵÞ ¼ 0; vðt;

Þvx þ 12

ð77Þ

ϵ

σ ðs; x; ϵÞv~ xx þ σ ðs; x; ϵÞσ ϵ ðs; x; ϵÞvxx ¼ 0 2

which we can rearrange to v~ s þ αðs; x; ϵÞv~ x þ 12 σ 2 ðs; x; ϵÞv~ xx ¼  ðαϵ ðs; x; ϵÞvx þ σ ðs; x; ϵÞσ ϵ ðs; x; ϵÞvxx Þ ~ t; x; ϵÞ ¼ 0: vðt; Under the assumptions Proposition 8, the right hand side of the parabolic PDE above is negative (compare Propositions A1 and A2), and hence the maximum principle implies that v~ ¼ vϵ is positive, from which the statement of the proposition follows.□

36

C.-O. Ewald, M. Yor / Journal of Economic Dynamics & Control 59 (2015) 22–36

References Atkinson, A., 1970. On the measurement of inequality. J. Econ. Theory 2. Atkinson, A., 1987. On the measurement of poverty. Econometrica 55 (4). Benhabib, J., Bisin, A., Zhu, S., 2015. The distribution of wealth in the Blanchard–Yaari model. Macroecon. Dyn., in press, http://dx.doi.org/10.1017/ S1365100514000066. Borovička, J., Hansen, L.P., Hendricks, M., Scheinkman, J., 2011. Risk-price dynamics. J. Financ. Econom. 9 (1), 3–65. Borovička, J., Hansen, L.P., Scheinkman, J., 2014. Shock elasticities and impulse responses. Math. Financ. Econ. 8 (4), 333–354. Carr, P., Ewald, C.-O., Xiao, Y., 2008. On the qualitative effect of volatility and duration on prices of asian. Finance Res. Lett. 5 (3). Clark, S., Hemming, R., Ulph, D., 1981. On indices for the measurement of poverty. Econ. J. 91, 515–526. Dasgupta, P., Maskin, E., 2005. Uncertainty and hyperbolic discounting. Am. Econ. Rev. 95 (4). Dixit, A.K., Pindyck, R.S., 1994. Investment under Uncertainty. Princeton University Press, Princeton, New Jersey. Doob, J.L., 1968. Generalized sweeping-out and probability. J. Funct. Anal. 2, 207–225. Ewald, C.-O., Menkens, O., Ting, S.H.M., 2013. Asian and Australian options: a common perspective. J. Econ. Dyn. Control 37 (5). Ewald, C.-O., 2013. Derivatives on nonstorable renewable resources: fish futures and options, not so fishy after all. Nat. Resour. Model. 26 (2), 215–236. Ewald, C.-O., Wang, W.-K., 2010. Sustainable yields in fisheries: uncertainty, risk-aversion and mean–variance analysis. Nat. Resour. Model. 23 (3), 303–323. Forster, M., la Torre, D., Lambert, P., 2014. Optimal control of inequality under uncertainty. Math. Soc. Sci. 68. Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L., Touzi, N., 1999. Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3, 391–412. Hansen, L.P., Scheinkman, J.A., 2009. Long-term risk: an operator approach. Econometrica 77 (1), 177–234. Hansen, L.P., Scheinkman, J., 2010. Pricing growth-rate risk. Finance Stoch. 16 (1), 1–15. Hirsch, F., Profeta, C., Roynette, B., Yor, M., 2011. Peacocks and Associated Martingales with Explicit Constructions. Springer. ISBN-10: 8847019079. Hirsch, F., Roynette, B., Yor, M., 2012. Kellerer's Theorem Revisited. Working Paper. Available at: 〈http://www.maths.univ-evry.fr/prepubli/361.pdf〉. Kellerer, H.G., 1972. Markov komposition und eine anwendung auf martingale. Math. Ann. 198. Levy, H., 1992. Stochastic dominance and expected utility: survey and analysis. Manag. Sci. 38 (4). Lubrano, M., 2013. The Econometrics of Inequality and Poverty Lecture 4: Lorentz Curves, the Gini Coefficient and Parametric Distributions. Available at: 〈http://www.vcharite.univ-mrs.fr/PP/lubrano/cours/Lecture-4.pdf〉. Noor, J., 2009. Hyperbolic discounting and the standard model: eliciting discount functions. J. Econ. Theory 144 (2009), 2077–2083. Protter, P., 2004. Stochastic Integration and Differential Equations, 2nd edition. Springer. ISBN-10: 3540003134. Oksendal, B., 2003. Stochastic Differential Equations: An Introduction with Applications 5th edition. Springer. Ok, E., Masatlioglu, Y., 2007. A theory of (relative) discounting. J. Econ. Theory 137, 214–245. Rothschild, M., Stiglitz, J., 1970. Increasing risk. I. A definition. J. Econ. Theory 2, 225–243. Rothschild, M., Stiglitz, J., 1971. Increasing risk. II. Its economic consequences. J. Econ. Theory 3, 66–84. Strassen, V., 1965. The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439. Thomas, J., Worrall, T., 1990. Income fluctuation and asymmetric information: an example of a repeated principal-agent problem. J. Econ. Theory 51, 367–390. Watts, H.W., 1968. An economic definition of poverty. In: Moynihan, D.P. (Ed.), On Understanding Poverty, Basic Books, New York.