1 Introduction

Radner equilibrium in incomplete Lévy models111The first author has been supported by the National Science Foundation under Grant No. DMS-1411809 (2014-2017). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Kasper Larsen

Department of Mathematical Sciences,

Carnegie Mellon University,

Pittsburgh, PA 15213, USA

email: kasperl@andrew.cmu.edu

Tanawit Sae Sue

Department of Mathematical Sciences,

Carnegie Mellon University,

Pittsburgh, PA 15213, USA

email: tsaesue@andrew.cmu.edu

July 24, 2019

Abstract: We construct continuous-time equilibrium models based on a finite number of exponential utility investors. The investors’ income rates as well as the stock’s dividend rate are governed by discontinuous Lévy processes. Our main result provides the equilibrium (i.e., bond and stock price dynamics) in closed-form. As an application, we show that the equilibrium Sharpe ratio can be increased and the equilibrium interest rate can be decreased (simultaneously) when the investors’ income streams cannot be traded.

## 1 Introduction

We construct equilibrium models in which a finite number of heterogeneous exponential investors cannot fully trade their future income streams. We show that the framework of continuous-time Lévy processes produces the Radner equilibrium in closed-form (i.e., optimal strategies, interest rates, drifts, and volatility structures are available in closed-form). Besides allowing for more model flexibility, we show that by going beyond models based on Brownian motions we can produce the following empirically desirable feature: The class of pure jump Lévy models can simultaneously lower the equilibrium interest rate and increase the equilibrium Sharpe ratio due to investors’ income streams being unspanned (i.e., due to model incompleteness).

The first construction of an incomplete continuous-time model which allows for an explicit description of the Radner equilibrium was given in [CLM12]. As an application of this model, [CLM12] show that model incompleteness can significantly lower the equilibrium interest rate. However, the (instantaneous) Sharpe ratio is unaffected by the model’s incompleteness.222Theorem 4.1 in [CL12] shows that no model based on exponential utilities, continuous consumption, and a filtration generated by Brownian motions can ever produce an incompleteness impact on the Sharpe ratio when this ratio is measured instantaneously. Besides being of mathematical interest, our motivation behind extending the Brownian framework in [CLM12] to the more general Lévy framework is to produce simultaneously a negative impact on interest rate and a positive impact on the Sharpe ratio while still maintaining a closed-form equilibrium model. Our desire to construct an incomplete equilibrium model with these features is of course due to Weil’s celebrated risk-free rate puzzle (see [Wei1989]) as well as Mehra and Prescott’s equity premium puzzle (see [MP1985]). These and other asset pricing puzzles are also discussed in detail in the survey [Cam00].

The literature on continuous-time Radner equilibrium theory in models where the investors’ income streams are spanned (i.e., complete models) is comprehensive and we refer to the recent references on endogenous completeness [AR08], [HMT12], [HR13], and [Kra2015] for more information. On the other hand, models with continuous-time trading and unspanned income streams (i.e., incomplete models) are much less developed and only in recent years has progress been made. The papers [Zit12], [Zha12], [CL15], and [KXZ15] consider models with exponential utilities, no dividends (i.e., only financial assets), and discrete-time consumption.333By restricting the investors to only consume at maturity, the economy’s interest rate cannot be determined. Furthermore, by only considering financial assets, the assets’ volatility structures also remain undetermined. Therefore, the interest rate and the volatility parameters are taken as exogenously specified model input in such models. These papers differ in how general the underlying state-processes describing the investors’ discrete-time income streams can be: [Zit12] considers a Brownian motion and an independent indicator process. [Zha12] and [CL15] consider multiple Brownian motions ([CL15] also allow for processes with mean reversion) whereas the recent paper [KXZ15] allows for a non-Markovian Brownian setting. The current paper is more related to [CLM12] and [CL12] who - in Brownian settings - consider both financial and real assets in the case of exponential investors with continuous-time consumption. Indeed, the current paper can be seen as a direct extension of [CLM12] to the setting of discontinuous Lévy processes.

The paper is organized as follows: The next section describes the underlying Lévy framework. Section 3 provides the solution to the individual investors’ problems. Section 4 contains our main result which provides the equilibrium parameters in closed-form. The last section illustrates numerically the equilibrium impacts due to incompleteness in the Gaussian compound Poisson case. The appendices contain all the proofs.

## 2 Mathematical Setting

### 2.1 Underlying Lévy process

We let denote the time-horizon and we let denote the number of investors. denotes the underlying filtered probability space and we assume that . For some underlying -dimensional pure jump Lévy process we denote by the random counting measure on associated with ’s jumps. The corresponding compensated random measure is denoted where is referred to as the Lévy measure on associated with ’s jumps, see, e.g., [App2009] and [Sat1999] for more details about these objects. We assume that is the filtration generated by (right-continuous and completed).

The following regularity assumption on the Lévy measure will be made throughout the paper:

###### Assumption 2.1.

In addition to the usual properties

 ν({0})=0,∫RI+1(||z||2∧1)ν(dz)<∞, (2.1)

the Lévy measure satisfies the following three conditions:

 ∫||z||<1|z(0)|ν(dz)<∞, (2.2) ∫||z||≥1eu(0)z(0)+u(i)z(i)ν(dz)<∞ for all u(0),u(i)∈R and i=1,...,I, (2.3) ν(z(0)>0)>0 and ν(z(0)<0)>0. (2.4)

Assumption 2.1 requires a few remarks: [CLM12] consider the case of correlated Brownian motions with drift which is why we focus exclusively on the pure jump case. The requirement that (2.3) holds for all and in can be relaxed to a certain domain at the cost of more cumbersome notation (this can be seen from the proofs in Appendix B). Condition (2.2) is not implied by (2.1) because it requires that can integrate instead of on the unit ball and has a number of implications; e.g., (2.2) ensures that the process

 Jt=∫||z||<1z(0)N(dz,dt),t∈[0,T], (2.5)

is well-defined and is of finite variation.444The process will be related to the stock’s dividend process in the next section. We note that can still have infinite activity on finite intervals. The last condition (2.4) can also be relaxed to requiring that a certain explicit function is onto (see the last part of Lemma A.1 in Appendix A).

### 2.2 Exogenously specified model input

The investors are assumed to have heterogeneous exponential utilities over running consumption, i.e.,

 Ui(c)=−e−cτi,c∈R,τi>0,i=1,...,I. (2.6)

Here the investor specific constants are referred to as the risk tolerance coefficients.

We consider a pure exchange economy in the sense that bond prices, stock prices, income, and dividend processes are all quoted in terms of the model’s single consumption good. The i’th investor’s income rate process is modeled by

 dYit=μidt+σi∫RI+1z(i)~N(dt,dz),Yi0∈R, (2.7)

where and are constants for . The single stock’s dividend rate process is modeled by

 dDt=μDdt+σD∫RI+1z(0)~N(dt,dz),D0∈R, (2.8)

where and are constants. Because (2.5) is of finite variation, we see that is of finite variation too ( defined above by (2.7) might not be).

We note that the processes (2.7) and (2.8) are not independent; indeed, the quadratic cross characteristics555The brackets are also called the conditional quadratic cross variation; see, e.g., Section III.5 in [Pro04]. between and are given by

 d⟨Yi,D⟩t=∫RI+1σiσDz(0)z(i)ν(dz),t∈[0,T],i=1,...,I.

### 2.3 Endogenously determined price dynamics

We will restrict the financial market to only consist of two traded securities (one financial asset and one real asset). The financial asset is taken to be the zero net supply money market account. Its price process will be shown to have the following equilibrium dynamics

 dS(0)t=S(0)trdt,S(0)0=1, (2.9)

where is a constant. Because the interest rate is deterministic, the money market account is equivalent to zero-coupon bonds of all maturities. In the following, we will need the corresponding annuity

 A(t)=∫Tte−r(s−t)ds,t∈[0,T]. (2.10)

The real asset is a stock paying out the dividend stream (see 2.8). This security is in unit net supply and we will show that its equilibrium price dynamics are given by

 dSt+Dtdt=(rSt+μ(t))dt+σDA(t)∫RI+1z(0)~N(dt,dz),ST−=0, (2.11)

for where is a deterministic function. In order to have defined for , we explicitly define to be (i.e., there is no jump in at by definition).

The following assumption is placed on the equilibrium output parameters and needs to be verified for any candidate set of parameters.

###### Assumption 2.2.

The function is continuous on and is constant.

As discussed in the Introduction we are interested in how model incompleteness impacts the interest rate and the stock’s Sharpe ratio . The stock’s (instantaneous) Sharpe ratio is defined as the constant

 λ=μ(t)σDA(t)√∫RI+1(z(0))2ν(dz). (2.12)

The Sharpe ratio (2.12) measures the stock’s return (cleaned for interest and dividend components) relatively to the standard deviation of its noise term. Sharpe ratios have been widely studied and used in the literature and we refer to [CP2007] for an application of the Sharpe ratio (2.12) in a continuous-time jump diffusion setting.

We end this section by introducing the set of equivalent martingale measures which will play a key role in what follows. For a process we consider sigma-martingales of the linear form

 dZψt=Zψt−∫RI+1ψt(z)~N(dt,dz),t∈[0,T],Zψ0=1. (2.13)

To ensure that a unique solution of (2.13) exists, we require that is -integrable in the sense that is a predictable flow satisfying the integrability condition

 P(∫T0∫RI+1ψt(z)2ν(dz)dt<∞)=1. (2.14)

To ensure that the solution of (2.13) is strictly positive we require

 ψt(z)>−1,P-a.s., for all z∈RI+1 and t∈[0,T]. (2.15)

Under conditions (2.14) and (2.15), the unique solution of (2.13) is a strictly positive local martingale. We additionally require that this process is a martingale. This martingale property allows us to define the associated probability measure on by the Radon-Nikodym derivative . The final requirement we place on the integrand is the sigma-martingale property under of the process , . Itô’s product rule can be used to see that this requirement is equivalent to the property666Conditions (2.1) and (2.3) of Assumption 2.1 allow us to use Cauchy-Schwartz’s inequality together with (2.14) to see that is -integrable.

 μ(t)σDA(t)+∫RI+1ψt(z)z(0)ν(dz)=0,P-a.s.,for all t∈[0,T]. (2.16)

When satisfies the above requirements we refer to the associated measure as an equivalent martingale measure. Finally, we note that because the model is incomplete, there exist infinitely many integrands satisfying the above requirements (Theorem 3.1 and Theorem 4.2 below explicitly construct such integrands).

## 3 Individual investors’ optimization problems

In this section the price dynamics (2.9) and (2.11) as well as Assumption 2.2 are taken as input and we consider the i’th investor’s utility maximization problem. Because , the investor’s initial wealth is given by where investor’s i’th initial endowments are units of the money market account and units of the stock. In the following we will let denote the consumption rate in excess of the income rate , i.e., investor i’ths cumulative consumption at time is given by .

We next describe the i’th investor’s set of admissible strategies where is some investor specific equivalent martingale measure (Theorem 3.1 below gives the specific ). The investor can choose predictable processes and to generate the self-financing gain dynamics

 dXit=(rXit−cit+θtμ(t))dt+θtσDA(t)∫RI+1z(0)~N(dt,dz),Xi0∈R, (3.1)

provided that the various integrals exist on and provided that the left limit exists. In that case, we define the terminal value in order to have defined for all . The investor is required to leave no financial obligations behind at maturity in the sense that

 XiT≥0,P-a.s.

Finally, to rule out arbitrage opportunities, we require the process is a -supermartingale for . When these requirements are satisfied we write .

The investor’s maximization problem is given by

 sup(θ,ci)∈AiE[∫T0Ui(cit+Yit)dt], (3.2)

where the exponential utility function is defined by (2.6). The rest of this section is devoted to describing the solution of (3.2). To do this, we first note that Lemma A.1 in Appendix A ensures that the function

 R∋u(0)→∫RI+1z(0)eu(0)z(0)+u(i)z(i)ν(dz),u(i)∈R, (3.3)

has a well-defined continuous inverse with domain . We can then define the constants by (here we use (2.2) and the assumption that is constant)

 θ∗i=−τiσDfi−1τiσi(−μ(t)σDA(t)+∫RI+1z(0)ν(dz)), (3.4)

where is the annuity defined by (2.10). This allows us to define the constants by

 gi=−r+1τiA(t)−1θ∗iμ(t)+1τiμi−∫RI+1(e−1τiθ∗iσDz(0)−1τiσiz(i)−1+1τiθ∗iσDz(0)+1τiσiz(i))ν(dz). (3.5)

In terms of these objects, the following result provides the explicit solution to (3.2); see Appendix B for the proof. This shows that Theorem 1 in [CLM12] carries over from the Brownian framework to the current setting of Lévy processes.

###### Theorem 3.1.

Let Assumptions 2.1 and 2.2 hold. Then the deterministic integrand defined by

 ψi(z)=e−1τiθ∗iσDz(0)−1τiσiz(i)−1,z∈RI+1, (3.6)

produces an equivalent martingale measure via the martingale density (2.13). Furthermore, for the corresponding admissible set , the processes attain the supremum in (3.2) where is defined by (3.4) and

 c∗it=A(t)−1X∗it+τiA(t)−1∫Tte−r(s−t)gi(s−t)ds,t∈[0,T]. (3.7)

In (3.7) the process denotes the gain process (3.1) produced by .

The proof of Theorem 3.1 establishes that the optimal controls produce a corresponding gain process with the property

 X∗iT=0,P-a.s.,

which means that optimally the investor leaves no wealth behind.

This section contains our main result which provides the Radner equilibrium in closed-form. We start by defining what we mean by an equilibrium in the present setting:

We call given by (2.9) and (2.11) for an equilibrium if these price processes are produced by a pair satisfying Assumption 2.2 and if:

1. There exist equivalent martingale measures and processes which attain the supremum in (3.2) for .

2. The markets clear in the sense that for all we have

 I∑i=1c∗it=Dt,I∑i=1θ∗it=1,I∑i=1θ(0)∗it=0. (4.1)

Our main existence result (the proof is in Appendix B) is stated in terms of two deterministic functions : First we define the Sharpe ratio (constant) through the requirement

 σD+I∑i=1τifi−1τiσi(−λ√∫RI+1(z(0))2ν(dz)+∫RI+1z(0)ν(dz))=0, (4.2)

where the inverse functions are defined in the previous section (see 3.3). Lemma A.1 in Appendix A ensures that (4.2) uniquely determines . Then we can define constants for by

 θ∗i=−τiσDfi−1τiσi(−λ√∫RI+1(z(0))2ν(dz)+∫RI+1z(0)ν(dz)). (4.3)

In turn, this allows us to define the constant

 r=1τΣ(μD+I∑i=1μi−∫RI+1(I∑i=1τie−1τiθ∗iσDz(0)−1τiσiz(i)−τΣ+σDz(0)+I∑i=1σiz(i))ν(dz)), (4.4)

where . Finally, we can define the annuity by (2.10) for defined by (4.4) and we can define the drift

 μ(t)=λ√∫RI+1(z(0))2ν(dz)σDA(t),t∈[0,T]. (4.5)
###### Theorem 4.2.

Let Assumption 2.1 hold, let , , and define the above functions by (4.4)-(4.5) as well as

 ψ∗t(z)=ef(−μ(t)σDA(t)+∫RI+1z(0)ν(dz))z(0)−1,z∈RI+1,t∈[0,T], (4.6)

where is the inverse of the mapping

 R∋u(0)→∫RI+1z(0)eu(0)z(0)ν(dz). (4.7)

Then with defined by (2.9) and defined by

 St=EQ∗[∫Tte−r(s−t)Dsds∣∣Ft],t∈[0,T], (4.8)

where is the equivalent martingale measure defined via , constitute an equilibrium.

We note that defined by (4.6) does not depend on time because (4.5) ensures that is constant.

## 5 Application

In this section we will compare the incomplete equilibrium of Theorem 4.2 with the corresponding complete equilibrium based on the representative agent. In the second part of this section we specify the Lévy measure to be the widely used compound Poisson process with Gaussian jumps and illustrate numerically the impacts on the resulting parameters due to model incompleteness.777The geometric form of this Lévy process was first used in finance by Merton in his classical paper [Mer76]. It is also the basis for Bates’ asset pricing model developed in [Bat1996].

### 5.1 Representative agent’s equilibrium

It is well-known that when all investors have exponential utilities, then so does the sup-convolution describing the representative agent’s preferences with risk tolerance coefficient ; see, e.g., Section 5.26 in [HL88]. We therefore define the representative agent’s utility function by

 Urep(c)=−e−c/τΣ,c∈R. (5.1)

The consumption-based capital asset pricing model developed in [Bre1979] (and extended in [GS1982] to certain incomplete models) is based on constructing price processes by applying the first-order condition for optimality in the representative agent’s problem through the proportionality requirement

 U′rep(Dt+I∑i=1Yit)∝e−rreptZrept,t∈[0,T]. (5.2)

Here is the interest rate and is the model’s (unique) martingale density. This model (i.e., and ) will serve as the basis for our comparison. Itô’s lemma produces the following dynamics of the left-hand-side of (5.2)

 dU′rep(Dt+∑Ii=1Yit)U′rep(Dt+∑Ii=1Yit)

By matching coefficients with the right-hand-side of (5.2) we find

 dZrept =Zrept−∫RN+1ψrep(z)~N(dt,dz),ψrep(z)=e−1τΣ(∑Ii=1σiz(i)+σDz(0))−1, (5.3) rrep =1τΣ(μD+I∑i=1μi) (5.4)

We denote by the measure defined by on . We find the parameters describing the stock dynamics by computing the dynamics of

 Srept=EQrep[∫Tte−rrep(s−t)Dsds∣∣Ft],t∈[0,T]. (5.5)

As in the proof of Theorem 4.2 in Appendix B, we find that the dynamics of (5.5) are of the form (2.11) but with and where

 Arep(t) =∫Tte−rrep(s−t)ds, (5.6) μrep(t) =−σDArep(t)∫RI+1ψ% rep(z)z(0)ν(dz), (5.7)

for . Finally, the Sharpe ratio based on the representative agent is defined as:

 λrep=μrep(t)σDArep(t)√∫RI+1(z(0))2ν(dz), (5.8)

which is the analogue of (2.12).

### 5.2 Incompleteness impacts in a numerical example

In this section we consider the Lévy measure corresponding to a compound Poisson process with Gaussian jumps (i.i.d. zero-mean normals with covariance matrix ) and a unit constant Poisson intensity. In other words, for a symmetric positive definite matrix with unit diagonal elements, we consider the Lévy measure

 ν(dz)=1√(2π)I+1det(Σ)e−12z′Σ−1zdz,z∈RI+1. (5.9)

This measure satisfies Assumption 2.1. Furthermore, the functions and (the inverse functions of 3.3 and 4.7) can be expressed via the Gaussian moment generating function , , and its derivatives.

Based on (4.4) and (5.4), we see that the incompleteness impact on the equilibrium interest rate is given by

 rrep−r =∫RI+1(I∑i=1τiτΣe−1τiθ∗iσDz(0)−1τiσiz(i)−e−1τΣ(σDz(0)+∑Ii=1σiz(i)))ν(dz).

Jensen’s inequality and the clearing property (see 4.5) can be used to see that this difference is always non-negative (a similar observation is made in [CLM12] and [CL12]). On the other hand, the impact on the instantaneous Sharpe ratio due to model incompleteness, i.e., , can be both positive and negative. Here is defined by (5.8) and the (instantaneous) Sharpe ratio in the incomplete equilibrium is defined by (2.12) and is found implicitly by solving

 σD+I∑i=1τifi−1τiσi(−λ)=0. (5.10)

This follows from (4.2) and the zero-mean and unit variance properties of .

To proceed with the numerics, we will use a flat correlation matrix in the sense that for and for where . The remaining parameters used to generate Figure 1 are

 σD=.2I,σi=.1,τi=τ,i=1,...,I. (5.11)

From Figure 1 we see that our model simultaneously can produce a positive impact on the equilibrium (instantaneous) Sharpe ratio and a negative impact on the equilibrium interest rate. As discussed in the Introduction, these effects are empirically desirable because they are linked to the asset pricing puzzles in [Wei1989] and [MP1985]. Finally, we note that as the resulting model approaches the representative agent’s complete model and both incompleteness impacts vanish.

## Appendix A Two auxiliary lemmas

###### Lemma A.1.

Suppose Assumptions 2.1 holds. Then the partial derivative

 φi(u(0),u(i))=∂∂u(0)∫RI+1eu(0)z(0)+u(i)z(i)ν(dz),u(0),u(i)∈R, (A.1)

is a well-defined function and satisfies the following properties:

1. The function has the representation

 φi(u(0),u(i))=∫RI+1z(0)eu(0)z(0)+u(i)z(i)ν(dz),u(0),u(i)∈R. (A.2)
2. The function is jointly continuous.

3. For fixed , the function is strictly increasing and onto . Consequently, the inverse function exists and is continuous on .

###### Proof.

For the first claim, we can use the bound

 |ehz−1||h|≤|z|e|z|,z(0)∈R,|h|≤1. (A.3)

This bound is integrable by (2.2) and (2.3) of Assumption 2.1. Therefore, the dominated convergence theorem can be used to produce the representation (A.2). The second claim follows similarly. The strict monotonicity property in the last claim follows directly from (A.2). Finally, (2.4) ensures that the map is onto .

###### Lemma A.2.

Suppose Assumption 2.1 holds. Let be a constant and define by (2.10). For any two continuous functions on there exists a unique solution of the linear SDE

 dXt=((r−A(t)−1)Xt+m(t))dt+A(t)c(t)∫RI+1z(0)~N(dt,dz),X0∈R, (A.4)

on with , -a.s., as .

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