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Incomplete Continuous-time Securities Markets with Stochastic Income Volatility

In an incomplete continuous-time securities market with uncertainty generated by Brownian motions, we derive closed-form solutions for the equilibrium interest rate and market price of risk processes. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously on a finite time-interval in a money market account and a single risky security. Besides establishing the existence of an equilibrium, our main result shows that if the investors' unspanned income has stochastic countercyclical volatility, the resulting equilibrium can display both lower interest rates and higher risk premia compared to the Pareto efficient equilibrium in an otherwise identical complete market.


Introduction
We consider an incomplete continuous-time securities market with uncertainty generated by Brownian motions, which allows us to derive closed-form solutions for the equilibrium interest rate and market price of risk processes. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income. The closed-form solutions facilitate a direct investigation of the impact of stochastic income volatility and preference heterogeneity on the resulting equilibrium interest rates and risk premia in a setting with unspanned income risk both at the individual and aggregate level.
The investors can trade continuously on a finite time-interval in a money market account and a single risky security, and they maximize expected time-additive exponential utility of continuous consumption. We show that the impact of unspanned income risk on the risk premium compared to the Pareto efficient analogue depends on how risk premia are measured (instantaneously or discretely). We show, in a model-free manner, that unspanned income risk can never affect the equilibrium instantaneous market price of risk process in a setting with exponential utility investors and continuous income processes governed by Brownian motions. On the other hand, if risk premia are measured over finite time-intervals (as in empirical asset pricing studies), we show that unspanned income with countercyclical income volatility can increase the discrete market price of risk process compared to the Pareto efficient equilibrium in an otherwise identical complete market setting.
It is well-known that individual unspanned income risks lower the equilibrium interest rate compared to the Pareto efficient equilibrium in an otherwise identical complete market setting (see, e.g., Wang 2003, Krueger and Lustig 2010, and Christensen, Larsen, and Munk 2011. This is due to the inefficient sharing of these risks and the resulting increased demand for precautionary savings. The key contribution of our paper is to demonstrate the non-trivial effects stochastic volatility of unspanned income can have on the resulting incomplete market equilibrium compared to the complete market analogue. While the equilibrium interest rate is always reduced, we show that the impact of unspanned income risk on the discrete market price of risk depends on whether the stochastic income volatility co-varies negatively or positively with the aggregate spanned income risk. The empirical evidence reported in, for example, Bloom (2009) and Bloom, Floetotto, and Jaimovich (2010) suggests that unspanned income risks are countercyclical, i.e., investors view their future income prospects as more uncertain in economic downturns. We show that unspanned income risk can increase (decrease) the discrete market price of risk process compared to the complete market setting if the stochastic income volatility is countercyclical (procyclical). Furthermore, we show in numerical examples that these impacts can be substantial. Christensen et al. (2011) show that unspanned income risk has no impact on risk premia for aggregate spanned income risk (measured instantaneously or discretely) in settings with deterministic income volatility and exponential utility investors. As noted above, the introduction of stochastic income volatility does not change this result as long as returns and risk premia are measured instantaneously. This result follows from the fact that even when income volatility is stochastic, unspanned income risk is deterministic instantaneously.
On the other hand, stochastic income volatility affects the return distributions over finite time-intervals and, thus, there can be an impact of unspanned income risk on the discrete market price of risk process. Increases in unspanned income risk affect an investor's expected utility negatively and, thus, if unspanned income volatility co-varies negatively with the aggregate spanned income risk, then the discrete market price of spanned income risk increases-negative shocks to spanned aggregate income do not only reduce contemporaneous consumption but these shocks also increase the volatility of subsequent unspanned income.
The questions of existence and characterization of complete market equilibria in continuous time and state models are well-studied. 2 The most common technique applied is based on the martingale method from Karatzas, Lehoczky, and Shreve (1987) and Cox and Huang (1989), which in complete market settings provides an explicit characterization of the investor's optimizer. By using the so-called representative agent method, the search for a complete market equilibrium can be reduced to a finite-dimensional fixed-point problem. To the best of our knowledge, only Cuoco and He (1994), Basak and Cuoco (1988),Žitkovic (2010), and Christensen et al. (2011) consider the existence and characterization of a non-Pareto efficient equilibrium in a continuous-time trading setting.
Our setting is similar to that of Christensen et al. (2011), who derive closed-form solutions for all the equilibrium quantities in an economy with a finite number of heterogeneous exponential utility investors, and dividends and unspanned income governed by arithmetic Brownian motions. The crucial difference between the model in Christensen et al. (2011) and our model is that we allow for stochastic income volatility and, still, we provide a tractable incomplete markets model for which the equilibrium price processes can be computed explicitly. Consequently, we can quantify the impact of the market incompleteness in the more realistic setting of stochastic income volatility supported by the empirical evidence (see op. cit. Bloom 2009 andBloom et al. 2010). The stochastic income volatility is a necessary ingredient in order to obtain an impact of unspanned income on the discrete market price of risk process. We incorporate a stochastic volatilityà la Heston's model into the income and equilibrium risky security price dynamics. We derive explicit expressions for the equi-2 See, e.g., Chapter 4 in Karatzas and Shreve (1998b) and Chapter 10 in Duffie (2001) for an overview of this literature.
More recent references on complete market equilibria in-cludeŽitkovic (2006), Cvitanić, Jouini, Malamud, and Napp (2009), Anderson and Raimondo (2008), and Hugonnier, Malamud, and Trubowitz (2009). librium instantaneous and zero-coupon interest rates as well as for the instantaneous and discrete market price of risk processes in terms of the individual income dynamics and the absolute risk aversion coefficients. The resulting type of the equilibrium market price of risk process has been widely used in various optimal consumption-portfolio models (see, e.g., Chacko andViceira 2005 andKraft 2005), whereas the resulting equilibrium interest rate is similar to the celebrated CIR term structure model. Translation invariant utility models (such as the exponential utility model we consider) allow consumption to be negative (see, e.g., the discussion in the textbook Skiadas 2009). Schroder and Skiadas (2005) show that this class of models is fairly tractable even when income is unspanned. We first conjecture the form of the equilibrium market price of risk process, and then we use the idea in Cuoco and He (1994) to re-write the individual investors' consumption-portfolio problems as problems with spanned income but heterogeneous beliefs.
In certain affine settings with a deterministic interest rate, the exponential investor's value function is available in closed-form (see, e.g., Henderson 2005, Wang 2004, Wang 2006, and Christensen et al. 2011. However, the incorporation of stochastic income volatility necessarily produces a stochastic equilibrium interest rate preventing the corresponding HJB-equation from having the usual exponential affine form. Therefore, the individual investor's value function is not available in closed-form in our setting. However, by using martingale methods, we obtain tractable expressions for the individually optimal consumption policies, which in turn are sufficient to produce the incomplete market equilibrium price processes. In a discrete infinite time horizon model with a continuum of identical exponential utility investors, Wang (2003) illustrates the negative impact unspanned income risk can have on the equilibrium interest rate. Similarly, in a discrete-time setting, Krueger and Lustig (2010) provide sufficient conditions in a setting with a continuum of identical power utility investors under which unspanned idiosyncratic income risk will lower the equilibrium interest rate, but not affect the risk premium. Christensen et al. (2011) present a continuous-time model with a finite number of exponential utility investors exhibiting the same interest rate phenomena, but also with no impact of unspanned income on the instantaneous risk premium. We extend these results by showing that as long as the income/consumption dynamics are continuous over time and the uncertainty is governed by Brownian motions, any equilibrium based on exponential preferences produces the same instantaneous risk premium as the standard Pareto efficient analogue. On the other hand, as noted above, we also prove that the discrete market price of risk process can be increased due to unspanned income risk if there is stochastic countercyclical income volatility. Constantinides and Duffie (1996), and various extensions including Storesletten, Telmer, and Yaron (2007 produce similar equilibrium implications for the impact of unspanned income risk on interest rates and risk premia. They rely on a discrete-time analysis and a continuum of identi-cal power utility investors with idiosyncratic income risks which wash-out at the aggregate level using a law of large numbers. Given virtually any pattern of risky securities and bond prices, Constantinides and Duffie (1996) show that individual income processes can be derived so that the (no-trade) equilibrium is consistent with these prices. In particular, if the cross-sectional volatility of the individual investors' income growth is countercyclical and sufficiently large, the model can produce equilibrium prices consistent with the high observed equity premium. In Chapter 21 in Cochrane (2005) it is argued that cross-sectional income data do not show such large dispersion. In contrast to Constantinides and Duffie (1996), we consider a finite number of heterogeneous investors such that there is unspanned income risk both at the individual and at the aggregate level. Importantly, while the countercyclical income volatility in Constantinides and Duffie (1996) pertains to the cross-sectional income distribution, the countercyclical income volatility in our model pertains to the individual investors' unspanned income risk.
Models based on a continuum of agents, such as Constantinides and Duffie (1996) and Krueger and Lustig (2010), rely on market clearing conditions defined by reference to a law of large numbers. Judd (1985) and Uhlig (1996) discuss both technical and interpretation issues related to using such averaging market clearing conditions. Our model uses a finite number of investors and our market clearing conditions are required to hold pointwise, i.e., the realized aggregate demands are required to equal the aggregate supplies in equilibrium.
The paper is organized as follows. The next section introduces the structure of the economy in terms of the exogenously given quantities, and in terms of conjectures for the equilibrium price processes. Section 3 presents the investors' consumption-portfolio problems in which the investors take the conjectured price processes as given. Our main Section 4 first defines and then shows the existence of an equilibrium consistent with the conjectured price processes and, secondly, it examines the impact of market incompleteness on the equilibrium interest rates and risk premia. Section 5 re-states the investors' consumption-portfolio problems in the form of an equivalent complete market setting with heterogeneous beliefs. This allows us to derive the investors' equilibrium consumption processes explicitly, which is a key ingredient in the proof of the main equilibrium theorem stated in Section 4. The concluding Section 6 discusses variations of the model, and all proofs are in the appendix.

Endowment and price processes
We consider an endowment economy with a single non-storable consumption good which also serves as the numéraire, i.e., prices are quoted in terms of this good. The economy is populated by I < ∞ consumer-investors all living on the time interval [0, T ], T < ∞.
(Ω, F, P) denotes the probability space on which all stochastic quantities are defined. (W, Z) denotes an 1 + I dimensional Brownian motion, where W is scalar valued and Z = (Z i ) I i=1 is a vector of investor-specific Brownian motions. All Brownian motions (W, Z 1 , ..., Z I ) are independent and the corresponding standard augmented Brownian filtration is denoted by F t , t ∈ [0, T ]. We consider F := F T and we will often write E t [·] instead of E P [·|F t ]. L p denotes the space of measurable and adapted processes f such that T 0 |f u | p du < ∞, P-almost surely, p ∈ {1, 2}.

Exogenously specified quantities
The investors have time-additive negative exponential utility of consumption with possible different degrees of absolute risk tolerance τ i > 0, i = 1, ..., I. For simplicity, we assume that their time-preference rates are all equal to zero. Investor i's utility function over consumption is therefore The following process v will be used to model stochastic income volatility. We define v as the Feller process where κ v , µ v , σ v are constants such that v remains strictly positive on [0, T ]. Positivity is ensured by the first part of following assumption (Feller's condition).
Assumption 2.1. The following two conditions are satisfied: Investor i's income is determined by the process .., I, are constants. The Brownian motion W affects all investors' income processes, whereas the Brownian motion Z i models investor i's idiosyncratic income risk. The income process Y i consists of the dividends from the investor's endowed portfolio of traded and non-traded assets with exogenous dividends plus the investor's stream of labor income.
It is not immediate how to adjust our approach to cover the mean-reverting income models used in Wang (2004) and Wang (2006). The affine optimal investment models used in Wang (2004) and Wang (2006) are based on an exogenously specified deterministic interest rate. However, the corresponding equilibrium interest rate cannot be deterministic or even independent of the investors' idiosyncratic income risk processes in these affine settings. 3 Unlike the power utility investor, stochastic interest rates complicate the exponential utility investor's optimal investment problem tremendously. As we shall see, the income processes (2.2) produce a stochastic equilibrium interest rate, which is adapted to the filtration generated by W , and for which the individual exponential utility investor's optimal investment problem remains partially tractable.
The aggregate income process E t := I i=1 Y it has the dynamics where we have defined the constants In order to make the following discussions and interpretations unambiguous, we assume that .., I. The cross quadratic variation between the aggregate income process and the stochastic income volatility, i.e., d E · , v · t = σ v σ E v t dt, is controlled by the parameter σ v . In what follows σ v plays an important role, and we allow for both countercyclical (σ v < 0) and procyclical (σ v > 0) stochastic income volatility.
As we noted in the Introduction, Bloom et al. (2010) demonstrate empirically that income uncertainty is strongly countercyclical both at the aggregate, the firm, and the individual level and, hence, σ v < 0. Moreover, Bloom (2009) and Bloom et al. (2010) demonstrate that income growth is negatively impacted by increases in the income volatility, for example, due to a "higher value of waiting to invest" with non-convex capital adjustment costs and, hence, κ E < 0. In turn, this implies that the constant part of expected aggregate income growth must be positive, i.e., µ E > 0, in order to have positive expected aggregate income growth (on average). In addition, Bloom (2009) demonstrates that expected income growth rebounds following positive shocks to volatility. This is in our model captured by assuming that the volatility process is mean-reverting, i.e., κ v < 0. Therefore, in the following, the "empirically relevant setting" refers to the parameter configuration:

Endogenously determined quantities
The investors can trade continuously on the time interval [0, T ] in a money market account with price process S (0) and a single risky security with price process S. We begin with the money market account.
Conjecture 2.2. The equilibrium price of the money market account has the dynamics where the F W t := σ(W u ) u∈[0,t] -adapted process r is defined by For concreteness, we let the single risky security be an annuity paying out a unit dividend rate over [0, T ]. We make the following conjecture.
Conjecture 2.3. There exists an F W t := σ(W u ) u∈[0,t] -adapted process σ S ∈ L 2 with σ St = 0 for t ∈ [0, T ) such that the equilibrium price of the risky security has the dynamics where r is defined by (2.6), and the constant µ S is defined by The idiosyncratic Brownian motions (Z i ) I i=1 do neither appear directly in the risky security price dynamics (2.7) nor in the spot interest rate dynamics (2.6). Nevertheless, a key point of this paper is to explicitly quantify the significant impact the presence of the idiosyncratic unspanned risks (Z i ) I i=1 can have on (S, S (0) ). As will become clear, this impact is due to the coefficient 1 2 I i=1 β 2 Y i /τ i in the interest rate dynamics (2.6). In order to state the third and final property regarding (S, S (0) ), we need the standard concept of state-price densities (see, e.g., Section 6F in Duffie 2001). For clarity, we define these processes explicitly.
Definition 2.4. A local state-price density ξ ν has the form where ν ∈ L 2 , W ⊥ is a W -independent Brownian motion, and If, in addition, E[M ν T ] = 1, we call ξ ν a state-price density.
The main property of local state-price densities is that both ξ t S (0) t and ξ t S t are driftless under P. For ν ∈ L 2 , M ν is always a supermartingale with E[M ν T ] ≤ 1. However, for ξ ν to be a state-price density, we require ν ∈ L 2 to produce the martingale property of M ν .
We will need the minimal state-price density ξ min for which ν := 0, i.e., The corresponding minimal martingale measure Q min is defined via the Radon-Nikodym derivative on F T as (see, e.g., the survey Föllmer and Schweizer 2010) Since v is a Feller process and µ S is a constant, we see that Novikov's condition is satisfied, which in turn ensures that M min is a martingale. 4 Therefore, ξ min is indeed a state-price density and not just a local state-price density. Consequently, Girsanov's theorem ensures is a Brownian motion under Q min which is independent of (Z 1 , ..., Z I ).
Definition 2.5. The instantaneous market price of risk process for the Brownian motion W is defined to be µ S √ v t for t ∈ [0, T ] with µ S defined by (2.8). Since we see from Definition 2.5 that the instantaneous market price of risk process is countercyclical if, and only if, the income volatility is countercyclical, i.e., if, and only if, σ v < 0.
The following conjecture identifies the risky security by identifying the volatility process σ S appearing in price dynamics (2.7).
Conjecture 2.6. The equilibrium price of the risky security has the representation (note that S T = 0) At a first glance, it may seem restrictive to take the single risky security to be an annuity.
However, we can let the risky security be any security paying out dividends at rate δ t as long as the process δ satisfies the following two properties: 5 1. δ t is an Itô-process adapted to the filtration 2. The following process is well-defined The second requirement is related to endogenous dynamic market completeness. Duffie and Huang (1985), Anderson and Raimondo (2008) and Hugonnier, Malamud, and Trubowitz (2009) provide conditions on the primitives of the economy under which an Arrow-Debreu equilibrium can be implemented by dynamic trading. In our setting, these conditions amount to ensuring that the dW -coefficient does not vanish in the above conditional expectation involving the F W tadapted dividends δ t .
In the continuous-time securities market (S (0) , S) with σ St = 0 for t ∈ [0, T ) (cf. Conjecture 2.3), all European claims written on the risky security, i.e., claims paying out g(S T ) at time T for some bounded payoff function g, are replicable. 6 Hence, the assumption of only a single traded risky security with F W t -adapted dividends is not restrictive. The key incompleteness property is that the individual investor's income process Y i cannot be fully hedged due to the presence of Z i in the dynamics of Y i . Therefore, (S (0) , S) constitutes an incomplete continuous-time securities market. Consequently, the standard method of describing the equilibrium by an representative agent cannot be applied. 5 One advantage of choosing the annuity as the risky security is that its stochastic return is only affected by changes in the stochastic volatility v, and not by changes in aggregate income E (recall that the aggregate income shocks are fully persistent). It is this property of the annuity which allows us to demonstrate that the equilibrium σSt is non-zero on [0, T ).
6 The same also holds if g is a bounded path functional of (Wt) t∈[0,T ] .

The individual investor's problem
Investor i chooses trading strategies (θ (0) , θ) as well as some consumption rate process c in excess of the income Y i . θ t denotes the number of units held of the risky security in addition to the endowed portfolio of this asset (the endowed portfolio has dividends included in Y i ). Since the money market account has endogenous dividends paid at time T , the dividends generated by the endowed portfolio θ i0− of this asset are not included in the investor's income process Y i . Therefore, θ (0) t denotes the total number of units held of the money market account at time since the risky security is an annuity paying a unit dividend stream.
In order to ensure well-posedness of the individual investor's optimization problem we need to impose conditions which ensure that the measurable and adapted processes (θ, c) are such that the wealth dynamics (3.1) are well-defined. Moreover, in order to rule out arbitrage, we need additional constraints on the possible choices. First, the investor is required to leave no obligations behind after the finite horizon: admissible if additionally the process is a supermartingale for all state-price densities ξ ν (see Definition 2.4). In this case, we write (c, θ) ∈ A. This supermartingale condition ensures that there are no arbitrage opportunities in the admissible set A. In order to verify this claim, we let τ be a stopping time valued in By using this inequality with c := 0, we see that there are no arbitrage opportunities on [0, T ] in the admissible set A.
Investor i maximizes time-additive expected utility stemming from consumption in addition to the investor's income, i.e., investor i seeks In Section 5 we show how to re-phrase (3.4) in terms of heterogeneous beliefs and spanned income. This ultimately allows us to solve explicitly for the optimal consumption strategieŝ c i (see Theorem 5.1 below), while only providing the abstract existence ofθ i via the martingale representation theorem.

Equilibrium
Before stating the following equilibrium definition (of the Radner-type), we recall that consumption c i is measured in excess of the income rates Y i , and that the trading strategies θ i denote the units held of the risky security in addition to the investors' endowed portfolios of this asset. On the other hand, the trading strategies θ i denote the total number of units held of the money market account. Since the money market account has endogenous dividends determined by the spot interest rates, this asset must be in zero net-supply in order to ensure that aggregate consumption is exogenous. Of course, this also implies that the endowments of the money market account must satisfy the clearing condition Definition 4.1. An equilibrium is a set of security price processes (S (0) , S), characterized by (r, µ S ), and a set of investor strategies (ĉ i ,θ i ) ∈ A such that given (r, µ S ), the processes (ĉ i ,θ i ) are optimal for investor i, i = 1, 2..., I, and such that all markets clear, i.e., In order to state our main equilibrium existence theorem, we need the following assumption on the exogenous model parameters.
are such that the following two restrictions hold: The first restriction in (4.2) trivially holds if When (4.3) holds, the spot interest rate process r t defined by (2.6) is bounded from below by µ E /τ Σ , i.e., r t is bounded from below by the constant part of risk-adjusted expected aggregate consumption growth per capita (recall we assume that the investors' time-preference rates are all equal to zero). Empirically, real interest rates can be negative and from Bloom (2009) the constant κ E is likely to be negative. Therefore, we will use the weaker condition (4.2) in the following analysis. On the other hand, in the empirically relevant setting in which (4.3) fails, the spot interest rate process r t becomes unbounded from below and, consequently, zerocoupon bond prices may explode in finite time. Condition (4.2) ensures finite zero-coupon bond prices for all maturities which is all we need to prove our main equilibrium existence theorem.
The proof of the following main theorem shows that clearing in the good's market, i.e., I i=1ĉ it = 0, ensures market clearing for both the risky security and the money market. Theorem 4.3. Under Assumptions 2.1 and 4.2, the security price processes (S (0) , S) characterized by r and µ S defined in (2.6) and (2.8), respectively, with the resulting individually optimal strategies (ĉ i ,θ i ) ∈ A, i = 1, 2..., I, constitute an equilibrium for which Conjectures 2.2, 2.3, and 2.6 hold.
The following lemma shows that our equilibrium produces exponential-affine zero-coupon bond prices, and this property constitutes an important ingredient in the proof of Theorem 4.3. We refer to the appendix in Kim and Omberg (1996) for a detailed description of Riccati equations, see, in particular, the discussion on normal Riccati solutions.
has unique non-exploding solutions satisfying b(s) = 0 for s ∈ (0, ∞). Furthermore, for µ S defined by (2.8), we have for t ∈ [0, U ] that the zero-coupon bond prices are given by Depending on whether (4.3) holds, the second restriction in (4.2) ensures that (4.4) has a positive or negative solution b(s) for s ∈ [0, ∞). In the empirically relevant setting in which (4.3) fails, the solution to (4.4) is positive. Therefore, the zero-coupon bond prices are increasing in the volatility v t , which is consistent with increasing incentives for precautionary savings when income risk increases (see, e.g., the discussion in Christensen et al. 2011).
The proof of Theorem 4.3 shows that Conjecture 2.6 holds with the volatility coefficient which is non-zero on the interval [0, T ) under Assumption 4.2. The sign of σ S is determined by the sign of σ v and the sign of the function b. In the empirically relevant setting in which the income volatility is countercyclical (σ v < 0), and in which increasing income volatility reduces the expected aggregate income growth (κ E < 0, implying that (4.3) fails), the function b is positive. This implies that the volatility process (4.7) is negative. Therefore, the instantaneous risk premium for the annuity, i.e., σ St µ S √ v t /S t , is also negative.
Finally, we mention that Theorem 4.3 does not make any uniqueness statement regarding the equilibrium. In other words, we are not claiming that S (0) defined by (2.5) and S defined by (2.7) is the only equilibrium possible in our pure exchange economy.

Equilibrium impacts due to incompleteness
In this section we analytically show how the incomplete market equilibrium established in Theorem 4.3 can be used to simultaneously explain the risk-free interest rate puzzle and the equity premium puzzle. We compare the equilibrium characterized in Theorem 4.3 to the equilibrium in an otherwise identical complete market economy in which all risks are spanned.
In the complete market economy, there exists a representative agent, and the equilibrium is characterized by the representative agent's first-order condition. The representative agent is modeled by the utility function where γ is a Negishi-weight vector. Since each investor is modeled by a negative exponential utility function, the representative agent's utility function becomes (see, e.g., Section 5.26 in Huang and Litzenberger 1988) This expression shows that the weight γ does not matter for the representative agent's preferences (Gorman aggregation). The first-order condition for the representative agent produces the proportionality requirement where the aggregate income process E t is defined by (2.3), and ξ rep t is the unique state-price density in the representative agent setting. By computing the dynamics of both sides of (4.8) and matching the coefficients we find the spot interest rate based on the representative agent economy to be Since τ Σ := I i=1 τ i , we have that τ Σ ≥ τ i for all i, which produces the key inequality In an economy with unspanned idiosyncratic risks Z i , this inequality combined with Theorem 4.3 produces the interest rate reduction Similarly, from the dynamics of (4.8) we find that the instantaneous market price of risk process based on the representative agent is identical to the market price of risk process This is also an analogue of the result presented in Christensen et al. (2011) (compare to their equation (27)). This equilibrium implication is not limited to our particular income model (2.1)-(2.2). Theorem 4.6 below shows that this result holds true in any model based on exponential investors and continuous income rates based on Itô-processes driven by Brownian motions.
We will next establish that unspanned income risk with stochastic volatility can affect the risk premium measured over finite time-intervals, even though there is no impact on the instantaneous market price of risk process as demonstrated above. Our motivation is that empirical studies of asset pricing properties, such as the risk-free rate and the equity premium puzzles, necessarily must measure returns, spot interest rates, and risk premia over finite time-intervals, where the length U > 0 of the time-intervals is determined by the sampling frequency. In order to precisely quantify risk premia measured over [0, U ], we need to introduce the minimal forward measure Q U . Since the equilibrium spot interest rate derived in Theorem 4.3 is stochastic, the minimal martingale measure Q min and the minimal forward measure Q U differ. The probability measure Q U is defined by the Radon-Nikodym Lemma 4.4 provides an explicit representation for equilibrium zero-coupon bond prices B(t, U ).
Based on this lemma, Girsanov's theorem ensures that is a Q U -Brownian motion, where the deterministic function b is defined by the Riccati equation (4.4). We can then make the following definition.
Definition 4.5. Under Assumption 4.2: The discrete market price of risk process measured over the finite time-interval [0, U ], U ∈ (0, T ], is defined by µ dis Our reasoning behind Definition 4.5 is the following. Let σ X ∈ L 2 be a bounded process and consider a traded security with price process (use the wealth dynamics (3.1) with c := 0) The main characterizing property of Q U is that all prices of traded securities denominated in terms of the price of the zero-coupon bond maturing at time U have no drift under Q U .
Since σ X is bounded, we therefore have where the last equality follows from B(U, U ) = 1. This identity implies that the expected return over the interval [0, U ] under the minimal forward measure Q U is equal to the zerocoupon rate for this interval, i.e., In other words, the process µ dis S (t) √ v t is the drift-correction in the W -dynamics needed to produce the riskless return as the expected return under Q U of X over the interval [0, U ].
We focus on the discrete market price of risk process, since the W -drift correction µ dis is universal across all traded securities. Alternatively, we could consider the risk premium over the interval [0, U ] for a security with the price process X t . This premium is defined by where the equality follows from (4.12). The Radon-Nikodym derivative dQ U dP is completely determined by the discrete market price of risk process µ dis From this we see that the impact on the risk premium over the interval [0, U ] due to market incompleteness depends on the security. In other words, unlike the discrete market price of risk process, the significance of the impact on the risk premium over [0, U ] depends on the security's volatility process σ X . Moreover, contrary to when returns and risk premia are measured instantaneously, normalizing the risk premium over the interval [0, U ] by the standard deviation of the security's return 1 X 0 Var[X U ] to produce the discrete "Sharpe ratio" does not remove the dependence on the security's volatility process σ X .
Similarly to the probability measures Q min and Q U , we can introduce Q min rep and Q U rep corresponding to the representative agent based on the spot interest rate r rep t defined by (4.9). This interest rate r rep t produces the zero-cupon bond prices for 0 ≤ t ≤ U ≤ T : By using the inequality (4.10), we see that Assumption 4.2 ensures that the Riccati equation describing b rep has a unique non-exploding solution on [0, ∞). Therefore, for U ∈ (0, T ], the is a Brownian motion under the representative agent's minimal forward measure Q U rep . The discrete market price of risk process measured over [0, U ] corresponding to the representative agent is defined similarly to Definition 4.5 as µ dis,rep By comparing the coefficients for the two Riccati equations describing b and b rep and using the inequality (4.10), we see that b rep (t) ≤ b(t) for all t ∈ [0, T ). Consequently, provided that σ v = 0 as in the second part of Assumption 2.1, we obtain an impact on the equilibrium discrete market price of risk process measured over [0, U ]. In particular, if the stochastic income volatility is countercyclical (σ v < 0), the equilibrium discrete market price of risk process measured over finite time-intervals is higher than in an otherwise identical complete market.
Similarly to the derivation of σ S in (4.7) presented in the proof of Theorem 4.3, we can show that the annuity's volatility coefficient in the representative agent setting is In the empirically relevant setting in which σ v < 0 and κ E < 0, and, hence, also a(t) ≤ a rep (t), for all t ∈ [0, T ). It therefore follows from (4.6) and (4.13) that B rep (t, U ) ≤ B(t, U ) which produces the inequality In other words, the drift-correction σ rep St µ dis,rep S (t) √ v t in the S-dynamics needed to produce the riskless return as the expected return of the annuity over the interval [0, U ] in the complete market setting is larger (less negative) than in the incomplete market setting.
This section has explicitly illustrated the equilibrium impacts due to market incompleteness, which we summarize as follows: 1. The equilibrium spot interest rate is impacted negatively.
2. The equilibrium instantaneous market price of risk process is unaffected. Theorem 4.6 below shows that this feature carries over to any model based on exponential utility investors and continuous income processes driven by Brownian motions.
3. The equilibrium discrete market price of risk process measured over a finite interval

Numerical illustrations
This section serves to illustrate that the impact on the equilibrium interest rate and the discrete market price of risk stemming from investors receiving partially unspanned income with stochastic volatility can be significant. The numerical values reported in this section only serve to illustrate the potential impact.
The impact on the interest rate and on the discrete market price of risk is determined by (4.10): We consider first a homogeneous investor setting in which all investors have the same risk tolerance τ i := τ as well as the same unspanned income risk parameter β Y i := β Y , i = 1, ..., I.
In this setting, τ Σ = Iτ , and we find that as I → ∞. For given parameter values, Table 1    . We see that the impact on the discrete market price of risk is highest when the less risk tolerant investors face the largest unspanned income risk.  Table 2: Increase in the discrete market price of risk µ dis S (0) − µ dis,rep S (0) in the limiting case (I → ∞) for various weights w and various risk tolerance parameters (τ A , τ B ). The numbers are based on β Y A := 0.1, β Y B := 0.4, whereas the remaining exogenous parameters are as in Table 1.

No impact on the instantaneous market price of risk process
In this section we show that the instantaneous market price of risk process based on the representative agent is always identical to the equilibrium instantaneous market price of risk process in a setting based on exponential investors and continuous income processes governed by Brownian motions. We consider the following model for t ∈ [0, T ]: (4.14) for some (δ, r) ∈ L 1 , (σ ′ S , λ ′ ) ∈ L 2 , σ ′ S = 0 and a Brownian motion B. In the following theorem we refer to Definition 2.4 for the notion of a local state-price density ξ ν .
Theorem 4.6. For t ∈ [0, T ] we consider the income dynamics Assume that (4.14) constitutes an equilibrium in which each investor's optimal consumption processĉ it satisfies the following first-order condition (4.15) whereξ i is an investor-specific local state-price density. Then the equilibrium instantaneous market price of risk process λ ′ satisfies In the setting of this theorem, let E ′ t := I i=1 Y ′ it denote the aggregate endowment. By computing the dynamics of the representative agent's state-price density (proportional to , we see that the instantaneous market price of risk process based on the representative agent agrees with λ ′ stated in Theorem 4.6. In other words, Theorem 4.6 shows that any model based on exponential utility investors and continuous income processes governed by Brownian motions produces the same instantaneous market price of risk process as suggested by the standard representative agent model. There is no loss of generality in assuming the above form for (Y ′ i ) I i=1 and S. Indeed, by assuming that an equilibrium risky security price S exists, we can use Lévy's characterization for Brownian motion as well as the martingale representation theorem for F t := σ(W u , Z u1 , ..., Z uI ) u∈[0,t] to write the martingale component of dS as σ ′ St dB t for some Brownian motion B and some process σ ′ S ∈ L 2 . Subsequently, we can decompose the martingale part of Y ′ i into its projection onto B and some residual orthogonal martingale component (possibly depending on i) which produces the above form for dY ′ i for i = 1, ..., I. Finally, we discuss the first-order condition (4.15). In the case of utility functions defined on the positive semi-axis, Cvitanić, Schachermayer, and Wang (2001) show that the introduction of unspanned endowments may require finite additive measures in the dual space, in which case (4.15) makes no sense. However, Owen andŽitkovic (2009) show that for utility functions defined over R-such as our setting-the dual optimizer is always a (countable additive) measure and (4.15) holds. Both papers Cvitanić et al. (2001) and

Heterogeneous beliefs
This section contains the key ingredient required to prove our main Theorem 4.3. We introduce a technique which allows us to partially solve the individual investor's consumptionportfolio problem (3.4). Because the interest rate r t is stochastic, the PDE produced by the HJB-approach does not have the usual exponential affine solution that Henderson (2005) and Christensen et al. (2011) rely on. Inspired by Cuoco and He (1994), we instead convert the optimization problem into an equivalent problem with spanned income but heterogeneous beliefs (see also Section 5 in Christensen et al. 2011). We define the P-equivalent probability measures P i , i = 1, ..., I, via the Radon-Nikodym derivative dP i dP := π iT > 0 on F T , where By Novikov's condition and Girsanov's theorem, we know that under each P i , i = 1, ..., I, the √ v u du are independent Brownian motions. We will need the processesỸ i defined byỸ i0 := Y i0 and for t ∈ [0, T ]. By using the processes (π i ,Ỹ i ), we can re-write the objective in (3.4) as where the last equality follows from the martingale property of π i and iterated conditional expectations. Problem (3.4) can then be re-stated as sup (c,θ)∈A which can be seen as a complete market consumption-portfolio optimization problem with the spanned income rate processỸ i and heterogeneous beliefs P i . As detailed in the proof section, the following result follows from a variation of the martingale method for complete markets.  (5.4) and the consumption process has the dynamics Furthermore, there exists an investment strategyθ i such that the pair (ĉ i ,θ i ) ∈ A is optimal for investor i, i = 1, ..., I.
The proof of this result produces the optimal investment strategyθ i using the martingale representation theorem (see equation (A.5)) via the relation However, a tractable expression for the optimal investment strategyθ i is not available because the interest rate r t is stochastic. Fortunately, our equilibrium approach only requires the abstract existence ofθ i . The proof of Theorem 5.1 shows that the optimal strategies (θ i ,ĉ i ) are such that the process (3.3) is a martingale for all state-price densities ξ ν .
Finally, let us find the individual investors' optimal state-price densities. From the proof of Theorem 5.1 (see equation (A.10)) we get the relation where the Lagrange multiplier α i is defined by (5.3). Therefore, investor i's state-price density isξ it := π it ξ min t , which implies that the ratios between the investors' marginals are non-constant. By the second welfare theorem, we therefore do not expect Pareto efficiency of the equilibrium allocations, and this was indeed confirmed in Section 4.1.

Adjusted aggregate endowment
In order to put our equilibrium into a different perspective, let us re-consider the heterogenous beliefs formulation (5.2). It follows from Bayes' rule that W remains a Brownian motion under each P i . Because the adjusted income processes (Ỹ i ) I i=1 defined by (5.1) as well as the wealth dynamics dX θ,c t -including r t -are driven solely by W , we have for i = 1, 2..., I and (θ, c) ∈ A. Hence, if we defineẼ t := I i=1Ỹ it with the dynamics as the economy's "aggregate endowment" rate, we can reduce the search for an equilibrium to a complete market equilibrium with aggregate endowmentẼ t . In other words, by replacing (4.8) with the following adjusted first-order condition in the representative agent's problem we recover the correct incomplete securities markets equilibrium derived in Theorem 4.3.

Model variations
The model used for the income processes (2.1)-(2.2) is chosen for its mathematical simplicity.
In this section, we briefly point to a number of variations of the model, some of which are needed in order for the model to produce realistic equilibrium predictions. First of all, it is straightforward to replace the constants (µ

Gaussian models
We can modify our setting to produce an equilibrium in which the absolute income volatility process follows a Gaussian process. This is inspired by Stein and Stein's stochastic volatility model Stein and Stein (1991), where the relative volatility process is the Gaussian process Instead of the income dynamics (2.2), we consider In this setting, the equilibrium interest rate as well as the equilibrium instantaneous market price of risk processes will be affine functions of v, i.e., Gaussian processes. The resulting interest rate is Vasiček's famous term structure model, whereas this type of instantaneous market price of risk process was originally used in Kim and Omberg (1996). Gaussian based instantaneous market price of risk models have been widely used in the finance literature (see, e.g., Wachter 2002, Munk and Sørensen 2004and Benth and Karlsen 2005.

Terminal consumption
Instead of running consumption, we can consider terminal consumption only. As we shall see in the next result, we need to allow µ S in (2.7) to be a continuous function on [0, T ]. In this setting, the optimization problem (3.4) is replaced by The wealth process X θ i is defined by setting c := 0 in (3.1), i.e., We define the admissible strategies A term to be those measurable and adapted processes θ for which X θ t is well-defined and X θ ξ ν is a supermartingale for all state-price densities ξ ν . The analogue of Theorem 4.3 is the following result.
Theorem 6.1. Under Assumptions 2.1 and 4.2, there exists an equilibrium for which Conjectures 2.2,2.3,and 2.6 hold with r t := 0 in (2.5) and µ S in (2.7) replaced by the deterministic function where b is defined by the Riccati equation (4.4).
In this setting of consumption at time T only, the interest rate cannot be determined in equilibrium, and we choose r t := 0 for simplicity. Consequently, the minimum martingale measure Q min and the minimum forward measure Q U are identical, and the instantaneous market price of risk process is identical to the discrete market price of risk process measured over finite time-intervals [0, U ], U ∈ [0, T ].
Contrary to Theorem 4.6, the setting of terminal consumption only produces an impact on the instantaneous market price of risk process due to income incompleteness. In order to see this, we proceed as in Section 4.1 except that the first-order-condition (4.8) is only required to hold at t = T . To compute the instantaneous market price of risk process corresponding to the representative agent, we need the dynamics of the martingale ξ rep Similarly to the proof of Lemma 4.4 we have where b rep is defined in Section 4.1. By comparing (6.2) and the dW -coefficient in (6.3) we see from Section 4.1 that the incompleteness impact on the instantaneous market price of risk process in the case of only terminal consumption at time T is identical to the impact on the discrete market price of risk process measured over the interval [0, T ] in the case of continuous consumption.
Because the equilibrium interest rate is zero, the investors value functions will be of the exponential-affine form. Consequently, the investors' optimal trading strategiesθ it can be computed explicitly using HJB-techniques.
In conclusion, this section shows that it is not the difference between the minimal martingale and forward measures due to equilibrium stochastic interest rates which produces the impact of unspanned income on the discrete market price of risk process. Instead, the key observation is that the impact of stochastic unspanned income volatility must be integrated over a time interval in order to produce an effect on the market price of risk process such as returns measured over finite time-intervals (Section 4.1) or consumption only taking place at discrete points in time.

Appendix: Proofs
We start by proving Lemma 4.4 since this result is used in the later proofs. Then we state and prove a result which is used in the proof of Theorem 5.1 for individual optimality, given Conjectures 2.2, 2.3, and 2.6 about the spot interest rate and the risky security price processes. We then prove Theorem 4.3 stating that the conjectured spot interest rate and the risky security price processes indeed clear all markets. The proof of Theorem 6.1 is similar to the proof of Theorem 4.3 and we will only outline the few differences. Finally, we prove Theorem 4.6 stating that there is no impact of incompleteness on the instantaneous market price of risk process in settings based on exponential investors and continuous income processes governed by Brownian motions.
Proof of Lemma 4.4. The discriminant corresponding to the Riccati equation (4.4) is defined as Under Assumption 4.2, q is positive. The appendix in Kim and Omberg (1996)

on normal
Riccati equations ensures that (4.4) has a non-exploding unique normal solution b with b(s) = 0 for s ∈ (0, ∞).
In order to calculate the zero-coupon bond prices, we need the dynamics of the volatility process v defined by (2.1) under the minimal martingale measure Q min : Therefore, v t is also a Feller process under the minimal measure Q min . By Itô's lemma we see that the process N t : Since v t has a non-central χ 2 -distribution and b is a bounded continuous function, Novikov's condition is satisfied locally on [0, U ]. We can then use the argument on p.233 in Liptser and Shiryayev (2001) to see that N is a martingale on [0, U ]. This martingale property and the terminal condition

and the claim follows. ♦
In the later proofs we will need the following result, where the main complication is that ν can depend on both W and W ⊥ and, hence, the random variable Proof. By the definition of a state-price density ξ ν , we can find a W -independent Brownian motion W ⊥ as well as ν ∈ L 2 such that is a martingale. We define the corresponding P-equivalent probability measure Q ν by dQ ν dP := M ν T on F T . Q min denotes the minimal martingale measure under which √ v u du is a Brownian motion. By Girsanov's theorem, W Q min is also a Q ν -Brownian motion.
For a set A t ∈ F W t the martingale representation theorem produces an F W -adapted process f ∈ L 2 such that Since 1 At is bounded, the process g v is a bounded Q min -martingale. Furthermore, since W Q min is also a Q ν -Brownian motion, g v is a local Q ν -martingale. However, by g v 's boundedness property, g v is actually a Q ν -martingale.
To conclude the proof, we let A s ∈ F s be arbitrary. Then we have The first equality follows from iterated expectations and the F s -measurability of A s . The second equality is Bayes' rule for conditional expectations. The third equality is g's martingale property under Q ν . The fourth equality is again Bayes' rule, whereas the last equality is pro- Proof of Theorem 5.1. Letĉ i be defined as in the theorem's statement. Our first task is to show that the following process is well-defined (we note thatX iT = 0) Under Assumption 4.2, we can find a constant p > 1 such that We then consider the coupled system of ODEs for s ∈ (0, ∞) The restriction (A.6) ensures a positive discriminant corresponding tob's ODE. Therefore, the appendix in Kim and Omberg (1996) on normal solutions ensures thatb and, hence, alsõ a, is a continuous function on [0, ∞). Arguing as in the proof of Lemma 4.4, we find To verify that (A.5) indeed is well-defined, we can use Tonelli's theorem to write By Hölder's inequality it therefore suffices to show that the expectations are bounded uniformly in u ∈ [t, T ]. We start with the first term. If (4.3) holds, r t defined by (2.6) is bounded from below, and the claim follows since p > 1. On the other hand, if (4.3) fails, we get the inequality for u ∈ [t, T ] Since bothã andb are continuous functions on [0, T ] and, hence, bounded, we obtain We will next provide a uniform bound (in u ∈ [0, T ]) of the second term in (A.7). In the following argument C 1 , C 2 , ... denote various irrelevant positive constants. Since v t > 0, we have the following chain of inequalities The first inequality follows from the definition ofĉ i . The second inequality uses Jensen's inequality (recall p > 1) and Tonelli's theorem on the ds-integral, whereas the estimate of the dW Q min -integral follows from the Burkholder-Davis-Gundy inequality (see, e.g., Theorem 3.28 on p.166 in Karatzas and Shreve (1998a)). The third inequality first uses u ≤ T and Jensen's inequality on the second ds-integral. The last estimate is similar. The dynamics (A.4) for v ensure that v s is non-central χ 2 -distributed under Q min and, hence, the ds-integrals are finite.
All in all, we have shown that when Assumption (4.2) holds, the process (A.5) is well-defined and finite.
We now turn to the admissibility requirement. Using the relation between Q min and ξ min andX it 's definition produce for t ∈ [0, T ] The integrand f i ∈ L 2 appearing in the second equality comes from the martingale representation theorem for F W t := σ(W u ) u∈ [0,t] after noticing that all involved quantities are F W t -adapted. The last equality follows from (5.3) and by defininĝ which is possible since we are assuming that Conjecture 2.3 holds. All in all, this shows that X it has the form (3.1) and that is a P-martingale. We will next show the supermartingale requirement (3.3) by proving the stronger martingale property. By the definition of a state-price density ξ ν t , we can find a martingale M ν such that ξ ν t = M ν t /S (0) t . We then get for 0 ≤ s ≤ t ≤ T Finally, we will verify the optimality of (θ i ,ĉ i ) for problem (5.2) and, hence, also for problem (3.4). For the case of positive wealth processes the standard argument can be found in Section 3.6 in Karatzas and Shreve (1998b). In the following, V denotes the convex conjugate of U (see Section 3.4 in Karatzas and Shreve 1998b). By Fenchel's inequality, we have U (c u +Ỹ iu ) ≤ V (α i ξ min u ) + α i ξ min u c u +Ỹ iu , u ∈ [0, T ].
Integrating with respect to du and adding the positive random variable α i ξ min T X c,θ iT , see (3.2), give us T 0 U (c u +Ỹ iu )du ≤ T 0 V (α i ξ min u ) + α i ξ min u c u +Ỹ iu du + α i ξ min T X c,θ iT .
Since π it ξ min t is a state-price density and the subjective probability measure P i is defined by dP i dP := π iT , we can use the supermartingale property (3.3) to obtain the inequality The first equality follows from the established martingale property, whereas the last equality follows from the first-order condition U ′ i (ĉ iu +Ỹ iu ) = α i ξ min u , u ∈ [0, T ], (A.10) and the relation between U and V stated in Lemma 4.3(i) in Karatzas and Shreve (1998b). In order to verify that (A.10) holds, we use (5.4) to see that (A.10) holds for u = 0. Furthermore, by using (5.5) we see that the dynamics of both sides of (A.10) are identical and, hence, (A.10) holds for all u ∈ [0, T ].

♦
Proof of Theorem 4.3. We define S (0) by (2.5) and S by (2.10). The already proven Lemma 4.4 produces the zero-coupon bond dynamics We can then use Tonelli's theorem to re-write (2.10) as follows S t = Therefore, Conjecture 2.3 holds with the volatility coefficient (4.7).
We now establish clearing in the goods market. By summing up the expressions for dĉ it , we find d I i=1ĉ it = 0, see (2.6)-(2.8) for the definitions of r t and µ S . Since I i=1 X i0 = 0, we see from (A.8) that I i=1ĉ i0 = 0 and, hence, the goods market clears. To see that the risky security market also clears, we sum over i = 1, ..., I in (5.6) to see I i=1 Xθ i ,ĉ i t = 0. By dividing both sides in this relation by S (0) , we find the Q min -dynamics The second equality follows from the definition of W Q min , and the last equality is due to clearing in the goods market. By matching the dW Q min -coefficients and using 1 S (0) σ S = 0, we obtain the clearing condition.
Finally, to show clearing in the money market, we use The first equality was established above, whereas the last equality follows from the already established clearing in the risky security market. Since S (0) > 0, the clearing condition in the money market follows. the analogue of (A.8): (A.11) In order to see that all markets clear we introduce the martingale for t ∈ [0, T ] By using Fubini's theorem for conditional expectations we find the dynamics where the second equality follows from (4.4). However, by using integration by parts together with b(0) = 0 we obtain dN t = 0 for t ∈ [0, T ].
We can then finish the proof and as in the proof of Theorem 4.3 it suffices to show I i=1 Xθ i T = 0, P-a.s., to ensure clearing in all markets. By the definition of ξ min t the requirement I i=1 Xθ i T = 0, P-a.s., is equivalent to By the definitions of µ S andỸ i this requirement is equivalent to By the definition of the martingale N , this requirement can be re-written as