Conditions for bubbles to arise under heterogeneous beliefs
CConditions for bubbles to arise under heterogeneous beliefs
Seunghyun Lee ∗ and Hyungbin Park † December 29, 2020
Abstract
This paper studies the equilibrium price of a continuous time asset traded in a marketwith heterogeneous investors. We consider a positive mean reverting asset and two groups ofinvestors who have different beliefs on the speed of mean reversion and the mean level. Weprovide an equivalent condition for bubbles to exist and show that price bubbles may not formeven though there are heterogeneous beliefs. This condition is directly related to the drift termof the asset. In addition, we characterize the minimal equilibrium price as a unique C solutionof a differential equation and express it using confluent hypergeometric functions. The financial market cannot be explained without bubbles, as price bubbles may have a seriousimpact on the economy such as over-investment, uncontrolled trading, and even financial crisisas mentioned in Xiong [2013]. In this study, we propose a simple model and show that themagnitude of investor belief difference plays an important role in the formation of bubbles. Ourmain argument is that bubbles may or may not exist, depending on the model parameters. Thisclaim differs from other research that focused on a permanent bubble. Much previous researchincluding Chen and Kohn [2011] and Scheinkman and Xiong [2003] explain that bubbles existfor all possible parameters and focus on calculating the bubble size. In contrast, we focus onthe conditions for bubbles to exist provide equivalent conditions guaranteeing a bubble. Aninteresting observation is that this condition is closely related to the drift term of the dividendrate evolution equation.Our proposed model is based on the Chen–Kohn model, with a few modifications. First, weuse a positive mean reverting model by considering a Cox–Ingersoll–Ross (CIR) process. As theChen–Kohn model has drawbacks of permitting negative dividend rates and negative prices, werestrict the dividend rate to be positive. Additionally, we consider two groups of investors whoagree on the volatility of the asset, but disagree on both the mean reversion rate and the meanlevel.We build our results based on the approaches of Harrison and Kreps [1978] and Chen andKohn [2011]. The paper of Harrison and Kreps was the first to explain price bubbles by het-erogeneous beliefs, through analysis of an asset with finitely many dividend values. Their mainargument is that investors buy the asset at a price exceeding their intrinsic value, due to the ∗ Department of Statistics, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul, Republic of Korea. Email:[email protected] † Department of Mathematical Sciences, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul, Republic ofKorea. Email: [email protected], [email protected] a r X i v : . [ q -f i n . M F ] D ec ossibility of selling it to a more optimistic buyer in the future. Even though their model ef-fectively explained the concept of bubbles, it lacked reality since they assumed a discrete timehorizon and only finitely many dividends. There has been extensive research to modify theHarrison and Kreps model to continuous time, including Scheinkman and Xiong [2003], andChen and Kohn [2011]. Chen and Kohn extended this model to continuous time by consideringan asset that follows the mean reverting Ornstein–Uhlenbeck process. They especially focus onthe influence of different mean reversion rates and proved that different beliefs always lead to aspeculative bubble. On the other hand, Scheinkman and Xiong consider a different perspectiveof explaining heterogeneous beliefs due to overconfidence in Scheinkman and Xiong [2003] andScheinkman and Xiong [2004]. Even though Scheinkman and Xiong discuss a continuous timemodel based on the work of Harrison and Kreps, they assume that the current dividend is anoisy observation with two signal streams whereas the Chen and Kohn model supposed that thedividend is observable. They also consider a trading cost and use a different definition of theequilibrium price.Recently, there have been approaches to explain bubbles through the concept of holding costsand liquidity costs (see Muhle-Karbe et al. [2020]). This research shows that the costs influencefuture trading opportunities by adding a penalization in terms of the expected return. In ourresearch, we focus on the framework in Harrison and Kreps [1978] and Chen and Kohn [2011].We consider a model without any trading costs. We also assume that our investors disagreeabout the dividend stream distribution explicitly and do not introduce the concept of signals.This paper is organized as follows. In section 2, we present our model and give the necessarydefinitions. In section 3, we characterize the minimal equilibrium price by closely followingChen and Kohn [2011]. Section 4 provides our main results on the conditions when bubblesexist. In the case where bubbles exist, we express its size in terms of confluent hypergeometricfunctions through a differential equation approach. Finally, in section 5, we conclude the paperby visualizing the bubble for a few sample cases. As mentioned in the introduction, we explore a positive mean reverting model with heteroge-neous investors. We consider two groups of investors in a continuous-time market, parallel tothe Chen–Kohn framework. We consider only one asset, a mean reverting dividend stream. TheChen–Kohn model assumed that the i th group believes that the dividend rate D t would followthe Ornstein–Uhlenbeck process dD t = κ i ( θ − D t ) dt + σdB t where i = 1 , κ > κ >
0. However, this model has a drawback that the dividend ratesmay become negative. This can cause the intrinsic value, and even the equilibrium value of theasset to be negative. In order to justify a negative price, we need to additionally assume thatthe asset must be possessed by one of the two groups, and is unrealistic. We resolve this issue byconsidering a price process that follows the CIR model. We assume that the i th group believesthat the dividend rate of the asset satisfies dD t = κ i ( θ i − D t ) dt + σ (cid:112) D t dB t . (2.1)Here, κ i , θ i , σ are positive real values. Without losing generality, we will assume that κ > κ > θ i ’s) for each group in order to observe he combined effect of the mean-reversion and mean level. This is an important issue as wecan find out when does bubbles exist. Chen and Kohn [2011] shows that bubbles always existwhen beliefs on the mean-reversion parameter differ. However, we can easily check that thediscrepancy in the mean level doesn’t lead to a bubble. This leads to a natural question: whatrelationship between the parameters creates a bubble?In this paper, we assume that there are only two groups of investors whose beliefs aredifferent. We assume that the belief of each group is constant throughout time, and cannot bealtered. Furthermore, we suppose that our investors are risk-neutral, they calculate the presentvalue of an asset by calculating the expectation of the discounted dividend, and there is notransaction cost.We also assume that κ i θ i σ ≥ i = 1 ,
2. This assumption is common for the CIRmodel, as it almost surely precludes the dividend rate to become 0. In order to discuss theconcept of bubbles, we need to define both the theoretical price and the market price. First, wedefine the “intrinsic value” as a theoretical concept determined without any transactions. Forsimplicity, we assume that our risk-neutral investors share a constant discount rate λ . Hence,group i would evaluate the value of the asset whose dividend rate is D at time t by E Q i (cid:20)(cid:90) ∞ t e − λ ( s − t ) D s ds | D t = D (cid:21) . (2.2)Here, Q i is the measure associated with our dividend stream defined in (2.1). Note that D t isa stationary process, and hence the above expectation is independent of t . In other words, wecan simplify (2.2) as E Q i (cid:20)(cid:90) ∞ e − λs D s ds | D = D (cid:21) . Now, we are ready to define the intrinsic value.
Definition 2.1.
An intrinsic value of asset is defined as a function I : R + → R + where R + isthe set of positive real numbers and I ( D ) = max i =1 , E Q i (cid:20)(cid:90) ∞ e − λs D s ds | D = D (cid:21) This definition is natural because the asset would be possessed by the group that believes highly.Since the conditional expectation of D t has an explicit formula E Q i ( D t | D ) = D e − κ i t + θ i (cid:0) − e − κ i t (cid:1) (e.g. see Jafari and Abbasian [2017]), we can easily simplify I ( D ). Since D t is positive almostsurely, we can change the expectation and integral and I ( D ) = max i =1 , (cid:90) ∞ e − λs E Q i [ D s ds | D = D ] = max i =1 , (cid:90) ∞ e − λs De − κ i t + θ i (cid:0) − e − κ i t (cid:1) = max i =1 , (cid:110) θ i λ + 1 λ + κ i ( D − θ i ) (cid:111) . (2.3)An interesting fact is that this formula is exactly the same as the intrinsic value from theOrnstein–Uhlenbeck process. In other words, the difference in the diffusion term does effect theintrinsic value. (2.3) can be simplified by maximizing over i = 1 ,
2. Since we assumed that κ ≥ κ , we calculate I ( D ) when κ = κ and κ > κ separately. First, when κ = κ , I ( D ) = Dλ + κ + max( θ , θ ) κ λ ( λ + κ ) . e can clearly see that this value is positive. Next, when κ > κ , I ( D ) = (cid:40) Dλ + κ + θ κ λ ( λ + κ ) , if D ≤ ¯ D Dλ + κ + θ κ λ ( λ + κ ) , if D ≥ ¯ D Here, ¯ D := κ κ ( θ − θ )+ λ ( κ θ − κ θ ) λ ( κ − κ ) , the D value that satisfies Dλ + κ + θ κ λ ( λ + κ ) = Dλ + κ + θ κ λ ( λ + κ ) . Note that ¯ D can be negative if κ θ < κ θ . If so, I ( D ) can be simplified as Dλ + κ + θ κ λ ( λ + κ ) . Now we consider transactions between investors, and define a market equilibrium price.Suppose that the dividend rate at time t is D t and the price of the asset is P ( D t , t ). Assumingthat group i holds the asset, they can sell it at any stopping time τ ≥ t and the time t value ofthe asset would be sup τ ≥ t E Q i (cid:20)(cid:90) τt e − λ ( s − t ) D s ds + e − λ ( τ − t ) P ( D τ , τ ) | D t (cid:21) . At the equilibrium, the present value of the asset must be equal to its price and we can formalizethis notion to the following definition.
Definition 2.2.
An equilibrium price is a function P : R + × R + → R + such that P ( D, t ) ≥ I ( D ) and P ( D, t ) = max i =1 , sup τ ≥ t E Q i (cid:20)(cid:90) τt e − λ ( s − t ) D s ds + e − λ ( τ − t ) P ( D τ , τ ) | D t (cid:21) . Since we defined the equilibrium price to be at least the intrinsic value, its positiveness isguaranteed. Due to its implicit structure, it is impossible to characterize every equilibriumprice. In this paper, we focus on the minimal equilibrium price and whether it is strictly largercompared to the intrinsic value.
In this section we focus on the specific equilibrium price, the minimal equilibrium price. As thename suggests, the minimal equilibrium price is defined as the infimum of all possible equilibriumprices. In the next theorem, we provide an iterative construction of the minimal equilibriumprice and check that it is analogous with our intuitions.
Theorem 3.1.
A minimal equilibrium price exists, is time-independent and continuous.Proof.
First, we provide a construction of the minimal equilibrium price. This is basically thesame result as Theorem 3.1. of Chen and Kohn [2011] and we just briefly discuss the formation.We define P ( D, t ) = I ( D ) and P k ( D, t ) = max i =1 , sup τ ≥ t E Q i (cid:20)(cid:90) τt e − λ ( s − t ) D s ds + e − λ ( τ − t ) P k − ( D τ , τ ) | D t = D (cid:21) (3.1)for k ≥
1. This sequence of functions is nondecreasing in k , and P ∗ ( D ) = P ∗ ( D, t ) := lim k →∞ P k ( D, t ) s time-independent and is the minimal equilibrium price. Note that we can check that P ∗ isindeed an equilibrium price. A more detailed proof is discussed in Theorem 3.1. of Chen andKohn [2011].Now, we prove that P ∗ is continuous by showing P ∗ is lower semicontinuous and uppersemicontinuous separately. First, we show that P ∗ ( D ) ≤ lim inf j →∞ P ∗ ( D j ) for any D j → D .We fix any sequence D j → D and i = 1 ,
2. We define a stopping time τ j as τ j := inf { t ≥ | D t = D } where D t is a dividend stream with D = D j that follows the belief of the i th investor. Then,since P ∗ is an equilibrium price, P ∗ ( D j ) = max i =1 , sup τ ≥ E Q i (cid:20)(cid:90) τ e − λt D t dt + e − λτ P ∗ ( D τ ) (cid:21) ≥ E Q i (cid:20)(cid:90) τ j e − λt D t dt + e − λτ j P ∗ ( D τ j ) (cid:21) = E Q i (cid:20)(cid:90) τ j e − λt D t dt (cid:21) + P ∗ ( D ) E Q i (cid:104) e − λτ j (cid:105) . Thus, for any j ≥ P ∗ ( D ) ≤ P ∗ ( D j ) − E Q i (cid:2)(cid:82) τ j e − λt D t dt (cid:3) E Q i (cid:2) e − λτ j (cid:3) (3.2)Since D j → D as j → ∞ , τ j → j →∞ E Q i (cid:104) e − λτ j (cid:105) = 1 and lim j →∞ E Q i (cid:20)(cid:90) τ j e − λt D t dt (cid:21) = 0Therefore, passing the limit in (3.2) proves that P ∗ ( D ) ≤ lim inf j →∞ P ∗ ( D j )and P ∗ is lower semicontinuous.Now, we show that P ∗ ( D ) ≥ lim sup j →∞ P ∗ ( D j ) for any D j → D . We fix any sequence D j → D , i = 1 ,
2, and consider a dividend stream D t with D = D such that follows the beliefsof the i th investor. We define a stopping time τ (cid:48) j as τ (cid:48) j := inf { t ≥ | D t = D j } . Similarly as above, P ∗ ( D ) = max i =1 , sup τ ≥ E Q i (cid:20)(cid:90) τ e − λt D t dt + e − λτ P ∗ ( D τ ) (cid:21) ≥ E Q i (cid:34)(cid:90) τ (cid:48) j e − λt D t dt + e − λτ (cid:48) j P ∗ ( D τ (cid:48) j ) (cid:35) = E Q i (cid:34)(cid:90) τ (cid:48) j e − λt D t dt (cid:35) + P ∗ ( D j ) E Q i (cid:104) e − λτ (cid:48) j (cid:105) As we take j → ∞ , τ (cid:48) j → P ∗ ( D ) ≥ lim sup j →∞ P ∗ ( D j ) . Therefore P ∗ is upper semicontinuous and our proof is complete.We end this section by defining the bubble size as the discrepancy between the minimalequilibrium price and the intrinsic price. In the next section, we investigate conditions forbubbles to exist and calculate its size. Definition 3.1.
The bubble B ( D ) is defined as P ∗ ( D ) − I ( D ) . We say that a bubble exists if B ( D ) > . When does bubbles exist? A differential equationapproach
We now illustrate our main results. First, we state a theorem that enables us to verify that afunction is indeed an equilibrium price. Next, we provide equivalent conditions for bubbles toexist and calculate its size in subsections 4.1 and 4.2.
Theorem 4.1.
Consider the differential equation max { κ ( θ − D ) , κ ( θ − D ) } φ (cid:48) + 12 σ Dφ (cid:48)(cid:48) − λφ + D = 0 . (4.1) Any positive C solution φ ( D ) such that ≤ φ (cid:48) ( D ) ≤ K for a positive constant K is anequilibrium price.Proof. Suppose φ ( D ) is a C solution such that 0 ≤ φ (cid:48) ( D ) ≤ K for a positive constant K . Weshow that φ ( D ) is an equilibrium price. Fix any i = 1 , D t follows the beliefs of investor i . By the Ito formula, d (cid:16) e − λt φ ( D t ) (cid:17) = − λe − λt φ ( D t ) + e − λt φ (cid:48) ( D t ) dD t + 12 e − λt φ (cid:48)(cid:48) ( D t ) (cid:104) dD (cid:105) t = e − λt (cid:20) κ i ( θ i − D t ) φ (cid:48) ( D t ) + 12 σ D t φ (cid:48)(cid:48) ( D t ) − λφ ( D t ) (cid:21) dt + σe − λt (cid:112) D t φ (cid:48) ( D t ) dB t ≤ − e − λt D t dt + σe − λt (cid:112) D t φ (cid:48) ( D t ) dB t . (4.2)Thus, for any stopping time τ ≥
0, integrating (4.2) gives E Q i (cid:104) e − λτ φ ( D τ ) (cid:105) ≤ φ ( D ) − E Q i (cid:20)(cid:90) τ e − λt D t dt (cid:21) + E Q i (cid:20)(cid:90) τ σe − λt (cid:112) D t φ (cid:48) ( D t ) dB t (cid:21) . Note that σe − λt √ D t φ (cid:48) ( D t ) ∈ H because we assumed that φ (cid:48) is bounded, and the expectationof the stochastic integral is 0. Hence, E Q i (cid:2) e − λτ φ ( D τ ) (cid:3) ≤ φ ( D ) − E Q i (cid:2)(cid:82) τ e − λt D t dt (cid:3) holds forany i = 1 , τ ≥
0. We can rewrite this as φ ( D ) ≥ max i =1 , sup τ ≥ E Q i (cid:20)(cid:90) τ e − λt D t dt + e − λτ φ ( D τ ) (cid:21) . Note that the above inequality is actually an equality by taking τ = 0. Therefore, φ satisfiesthe second condition of the equilibrium price.Now, we check that φ ( D ) ≥ I ( D ). Because φ is positive, for i = 1 , n , φ ( D ) ≥ E Q i (cid:20)(cid:90) n e − λt D t dt (cid:21) . By the monotone convergence theorem, and the definition of I ( D ), φ ( D ) ≥ max i =1 , E Q i (cid:20)(cid:90) ∞ e − λt D t dt (cid:21) = I ( D ) . Therefore, φ ( D ) is an equilibrium price. .1 Case when bubbles do not exist In this section, we consider the case where κ = κ or κ θ ≤ κ θ and prove that there is nobubble. Note that we are working under the assumption that κ ≥ κ . Our main intuition isthat the drift term in the stochastic differential equation (2.1) is uniformly dominated by onegroup and the intrinsic value becomes an equilibrium price. Before we begin the detailed proof,we define ˜ D := κ θ − κ θ κ − κ , under the assumption that κ (cid:54) = κ . This notation will be universally used throughout thissection. Note that ˜ D is the value where κ ( θ − D ) = κ ( θ − D ) and thatmax { κ ( θ − D ) , κ ( θ − D ) } = (cid:40) κ ( θ − D ) if D < ˜ Dκ ( θ − D ) if D > ˜ D .
We can easily check that ¯ D and ˜ D are closely related by the formula ¯ D = ˜ D + κ κ ( θ − θ ) λ ( κ − κ ) . Corollary 4.2.
Suppose that κ = κ or κ θ ≤ κ θ . Then, I ( D ) is an equilibrium price andhence the minimal equilibrium price P ∗ ( D ) .Proof. First, we consider the case when κ = κ . Without losing generality, we can additionallyassume that θ > θ . Here, we omit the case when θ = θ because it would mean that thereare only 1 group of investors. Under these assumptions,max { κ ( θ − D ) , κ ( θ − D ) } = κ ( θ − D ) , and the equation (4.1) can be written as κ ( θ − D ) φ (cid:48) + 12 σ Dφ (cid:48)(cid:48) − λφ + D = 0 . Recall that we calculated I ( D ) in section 2 as I ( D ) = Dλ + κ + θ κ λ ( λ + κ ) . It is clear that I ( D ) is a C solution of (4.1) with a constant derivative, and I ( D ) is an equilib-rium price by Theorem 4.1.Next, we consider the case when κ θ ≤ κ θ . Since the case κ = κ was discussed in theprevious paragraph, we can assume that κ > κ . Under these assumptions, can check that˜ D ≤ { κ ( θ − D ) , κ ( θ − D ) } = κ ( θ − D )for all positive D . Then (4.1) becomes κ ( θ − D ) φ (cid:48) + 12 σ Dφ (cid:48)(cid:48) − λφ + D = 0and I ( D ) = Dλ + κ + κ θ λ ( λ + κ ) is clearly a solution satisfying the conditions in Theorem 4.1 (Theformula of I ( D ) follows from (2.3) and the fact that ¯ D < I ( D ) is anequilibrium price and our proof is complete.This simple corollary shows that if the long term mean level of the two investors differsignificantly, there is no price bubble. The case when κ θ ≤ κ θ can be explained intuitivelyby thinking that the drift term of the 2nd group is high enough to ignore the possibility oftransactions. The 2nd group would always price the intrinsic value higher and consistently holdthe asset, so there would be no bubble. .2 Case when bubbles exist In the introduction, we mentioned that bubbles exist when θ = θ and disappear when κ = κ .A natural question would be to determine the boundary conditions where bubbles are created.In this section, we prove that bubbles exist if κ > κ and κ θ > κ θ . Note that this is exactlythe opposite of the conditions in section 4.1, and hence the equivalent condition for bubbles toexist. We consider two sub-cases and prove that bubbles actually exist for both of them. Forthe first case (which is more significant), we explicitly calculate the bubble size using confluenthypergeometric functions. For the second case, we provide a lower bound of the bubble size. θ ≤ θ First, we consider the case when κ > κ , θ ≤ θ , κ θ ≥ κ θ . The latter two conditions canbe rewritten as κ κ θ ≤ θ ≤ θ . We use the concept of viscosity solutions to explicitly find theminimal equilibrium price and show that bubbles exist. The following theorem characterizesthe minimal equilibrium price as a viscosity supersolution of a differential equation. Note thatthe sign of the equation has changed from (4.1). Theorem 4.3.
The minimal equilibrium price P ∗ is a viscosity supersolution of the differentialequation − max { κ ( θ − D ) , κ ( θ − D ) } φ (cid:48) − σ Dφ (cid:48)(cid:48) + λφ − D = 0 . (4.3) Proof.
For simplicity, define F ( D, ψ, p, γ ) as F ( D, d, p, γ ) := − max { κ ( θ − D ) , κ ( θ − D ) } p − σ Dγ + λd − D Suppose P ∗ is not a viscosity supersolution. Then, we can take a C function ψ and a positive real D such that P ∗ ( D ) ≥ ψ ( D ) for D ∈ R + , P ∗ ( D ) = ψ ( D ), and F ( D , P ∗ ( D ) , ψ (cid:48) ( D ) , ψ (cid:48)(cid:48) ( D )) ≤− δ < δ . Note that we have used the fact that P ∗ is lower semicontinuoushere.First, we consider the case when D < ˜ D . Because ψ is a C function, there exists (cid:15) > F ( D, ψ ( D ) , ψ (cid:48) ( D ) , ψ (cid:48)(cid:48) ( D )) ≤ − δ D ∈ [ D − (cid:15), D + (cid:15) ] where D > (cid:15) and D + (cid:15) < ˜ D . Now we define a dividend stream D t such that D is the initial dividend rate and i = 1. Define a stopping time τ := inf { s > | D s − D | ≥ (cid:15) } > . Then, by integrating (4.2) we get E Q (cid:104) e − λτ ψ ( D τ ) (cid:105) = ψ ( D ) + E Q (cid:20)(cid:90) τ e − λt (cid:18) κ ( θ − D t ) ψ (cid:48) + 12 σ D t ψ (cid:48)(cid:48) − λψ (cid:19) dt (cid:21) ≥ ψ ( D ) + E Q (cid:20)(cid:90) τ e − λt (cid:18) δ − D t (cid:19) dt (cid:21) (4.4)Note that the expectation of the stochastic integration is 0 because D t is bounded in 0 ≤ t ≤ τ .Meanwhile, using our definition of D and the fact that P ∗ is an equilibrium price, ψ ( D ) = P ∗ ( D ) ≥ E Q (cid:20)(cid:90) τ e − λt D t dt + e − λτ P ∗ ( D τ ) (cid:21) ≥ E Q (cid:20)(cid:90) τ e − λt D t dt + e − λτ ψ ( D τ ) (cid:21) . (4.5)Combining (4.4) and (4.5) gives 0 ≥ δ E Q (cid:20)(cid:90) τ e − λt D t ds (cid:21) nd this is clearly a contradiction since δ > τ > D > ˜ D can be done almost identically and we omit the details.Finally, we consider the case when D = ˜ D . The outline of the proof is similar but we needto be more cautious with the details. We begin by defining a constant X = κ ( θ − ˜ D ) ψ (cid:48) ( ˜ D ).We can easily check that X = κ ( θ − ˜ D ) ψ (cid:48) ( ˜ D ) as well using the definition of ˜ D .Now, we define positive constants D < ˜ D < D such that F ( d, ψ ( d ) , ψ (cid:48) ( d ) , ψ (cid:48)(cid:48) ( d )) ≤ − δ (cid:12)(cid:12) κ i ( θ i − d ) ψ (cid:48) ( d ) − X (cid:12)(cid:12) ≤ δ d ∈ [ D , D ], i = 1 ,
2. This choice is possible because F ( ˜ D, P ∗ ( ˜ D ) , ψ (cid:48) ( ˜ D ) , ψ (cid:48)(cid:48) ( ˜ D )) ≤ − δ and κ i ( θ i − d ) ψ (cid:48) ( d ) is a continuous function that has value X at d = ˜ D . The rest of the proof issimilar to the case D < ˜ D . We fix a group i and define a dividend stream D t with the initialvalue D that follows the beliefs of group i . Next, we define a stopping time τ := inf { s > | D s = D or D } . Note that our assumptions on D and D forces κ i ( θ i − D t ) ψ (cid:48) ( D t ) ≥ X − δ σ D t ψ (cid:48)(cid:48) ( D t ) − λψ ( D t ) ≥ δ − (cid:18) X + δ (cid:19) − D t = 23 δ − X − D t for 0 ≤ t ≤ τ . Finally, we integrate (4.2) from 0 to τ to get E Q i (cid:104) e − λτ ψ ( D τ ) (cid:105) = ψ ( D ) + E Q i (cid:20)(cid:90) τ e − λt (cid:18) κ i ( θ i − D t ) ψ (cid:48) + 12 σ D t ψ (cid:48)(cid:48) − λψ (cid:19) dt (cid:21) ≥ ψ ( D ) + E Q i (cid:20)(cid:90) τ e − λt (cid:18) X − δ δ − X − D t (cid:19) dt (cid:21) = ψ ( D ) + E Q i (cid:20)(cid:90) τ e − λt (cid:18) δ − D t (cid:19) dt (cid:21) (4.6)Since (4.5) also holds under the beliefs of group i , we can prove that ψ ( D ) ≥ E Q i (cid:20)(cid:90) τ e − λt D t dt + e − λτ ψ ( D τ ) (cid:21) . Comparing this inequality with (4.6) leads to a contradiction. In conclusion, we can find acontradiction for all three cases and P ∗ must be a viscosity supersolution of (4.3). Remark 4.1.
The proof of the case when D (cid:54) = ˜ D can be done without assuming the converse. Now we find classical solutions of (4.3). In particular, we identify a specific solution thatcoincides with our intuition about asset prices. Trivially, a price of an asset whose current divi-dend is 0 must have a finite value, and it is plausible that the price has an asymptotically linearrelationship as the current dividend rate diverges. We set these intuitive beliefs as boundaryconditions for the differential equation and prove that there is an unique solution.
Lemma 4.4. Φ( D ) defined as Φ( D ) := (cid:40) Dλ + κ + θ κ λ ( λ + κ ) + Em ( D ) if D < ˜ D Dλ + κ + θ κ λ ( λ + κ ) + F u ( D ) if D > ˜ D s the unique positive C solution of (4.1) such that has a finite value at 0 and has a lineargrowth as D → ∞ .Here m ( D ) := M ( a , b , x ) , u ( D ) := U ( a , b , x ) where a i = λκ i , b i = κ i θ i σ , x i = κ i Dσ for i = 1 , , and M and U are confluent hypergeometric functions of the first and second typedefined in Daalhuis [2010].E and F are positive constants defined in the appendix.Proof. First, we solve the differential equation (4.1). We begin by considering the case when D ≤ ˜ D . The maximum in (4.1) is achieved when i = 1 and the differential equation can berewritten as κ ( θ − D ) φ (cid:48) + 12 σ Dφ (cid:48)(cid:48) − λφ + D = 0 . Define ψ ( D ) := φ ( D ) − Dλ + κ − θ κ λ ( λ + κ ) . Then ψ ( D ) satisfies κ ( θ − D ) ψ (cid:48) + 12 σ Dψ (cid:48)(cid:48) − λψ = 0 . (4.7)Let x = κ Dσ and write ψ ( D ) = ξ ( x ). Then (4.7) can be rewritten as xξ (cid:48)(cid:48) + ( b − x ) ξ (cid:48) − a ξ = 0 . (4.8)where a = λκ and b = κ θ σ . (4.8) is a differential equation also called the Kummer’s equation(see Daalhuis [2010] or Olver et al. [2010]) which is known to have an explicit solution formula ξ ( x ) = C M ( a , b , x ) + C U ( a , b , x ) . Hence, the solution of (4.1) can be written as φ ( D ) = ξ (cid:18) κ Dσ (cid:19) + Dλ + κ + θ κ λ ( λ + κ )= C M (cid:18) a , b , (cid:18) κ Dσ (cid:19)(cid:19) + C U (cid:18) a , b , (cid:18) κ Dσ (cid:19)(cid:19) + Dλ + κ + θ κ λ ( λ + κ )for D ≤ ˜ D . Similarly, the solution when D > ˜ D is φ ( D ) = C M (cid:18) a , b , (cid:18) κ Dσ (cid:19)(cid:19) + C U (cid:18) a , b , (cid:18) κ Dσ (cid:19)(cid:19) + Dλ + κ + θ κ λ ( λ + κ ) . Now we find the appropriate constants C , C , C , C that constrains φ ( D ) to have a finitevalue at 0 and is linear near infinity. First, observe that M ( a, b, x ) = 1 + O ( x ) and U ( a, b, x ) = Γ(1 − b )Γ( a +1 − b ) x − b + O ( x − b ) as x → b = κ θ σ ≥ U (cid:0) a , b , (cid:0) κ Dσ (cid:1)(cid:1) diverges and C must be 0.Next, note that M ( a, b, x ) = O ( e x x a − b Γ( a ) ) and U ( a, b, x ) = O ( x − a ) as x → ∞ (again, seechapter 13.2 of Olver et al. [2010]). Regardless of a and b , M (cid:0) a , b , (cid:0) κ Dσ (cid:1)(cid:1) diverges at anexponential rate, so C must be 0. Since U (cid:0) a , b , (cid:0) κ Dσ (cid:1)(cid:1) converges to 0, we can see that C U (cid:0) a , b , (cid:0) κ Dσ (cid:1)(cid:1) converges to 0 for any C , and φ ( D ) is linear at infinity. Hence, φ ( D ) canbe simplified as the following, using m ( D ) = M ( a , b , x ) and u ( D ) = U ( a , b , x ). φ ( D ) = (cid:40) φ ( D ) := Dλ + κ + θ κ λ ( λ + κ ) + C m ( D ) if D < ˜ Dφ ( D ) := Dλ + κ + θ κ λ ( λ + κ ) + C u ( D ) if D > ˜ D Finally, we uniquely determine values C and C that makes φ a C function. We need φ ( ˜ D ) = φ ( ˜ D ), φ (cid:48) ( ˜ D ) = φ (cid:48) ( ˜ D ), and φ (cid:48)(cid:48) ( ˜ D ) = φ (cid:48)(cid:48) ( ˜ D ). We focus on the first two equations.Using the fact that ∂M ( a, b, x ) ∂x = ab M ( a + 1 , b + 1 , x ) (4.9) nd ∂U ( a, b, x ) ∂x = − aU ( a + 1 , b + 1 , x ) (4.10)we can uniquely determine C and C . The linear independence of the two equations followfrom a simple observation about the sign of coefficients. We omit the details, and the solutions C , C are defined as E, F in the appendix. In the appendix we show that E and F are bothpositive. The relationship φ (cid:48)(cid:48) ( ˜ D ) = φ (cid:48)(cid:48) ( ˜ D ) follows from our assumptions that φ ( ˜ D ) = φ ( ˜ D )and φ (cid:48) ( ˜ D ) = φ (cid:48) ( ˜ D ) by taking limits D → ˜ D + and D → ˜ D − in (4.1). Therefore, E, F are theunique constants that makes φ a C function and our proof is complete.One may question whether Φ( D ) is increasing. This property must hold in order to beanalogous with our intuitions about the price process. The following lemma confirms this. Lemma 4.5. Φ( D ) is convex and λ + κ < Φ (cid:48) ( D ) < λ + κ .Proof. Recall that we defined Φ( D ) asΦ( D ) = (cid:40) Dλ + κ + θ κ λ ( λ + κ ) + Em ( D ) if D < ˜ D Dλ + κ + θ κ λ ( λ + κ ) + F u ( D ) if D > ˜ D We can easily check that m ( D ) and u ( D ) are convex by the differential formula given in (4.9)and (4.10). Therefore, Φ( D ) is also convex and Φ (cid:48) ( D ) is increasing in D . Note thatΦ (cid:48) ( D ) = (cid:40) λ + κ + κ σ ab M (cid:0) a + 1 , b + 1 , κ Dσ (cid:1) E if D < ˜ D λ + κ − κ σ aU (cid:0) a + 1 , b + 1 , κ Dσ (cid:1) F if D > ˜ D and lim D → M (cid:18) a + 1 , b + 1 , κ Dσ (cid:19) = 1 , lim D →∞ U (cid:18) a + 1 , b + 1 , κ Dσ (cid:19) = 0 . (Here, we used the asymptotic properties discussed in the proof of the previous lemma) Hence,lim D → Φ (cid:48) ( D ) = 1 λ + κ + 2 κ σ ab , lim D →∞ Φ (cid:48) ( D ) = 1 λ + κ and λ + κ < Φ (cid:48) ( D ) < λ + κ follows from the convexity of Φ.Finally, we use theorems 4.1 and 4.3 to prove that Φ( D ) is actually the minimal equilibriumprice. This simple corollary links the complex and implicit concept “minimal equilibrium price”with the differential equation (4.1). Corollary 4.6. Φ( D ) defined in lemma 4.4. is the minimal equilibrium price, i.e. Φ( D ) = P ∗ ( D ) .Proof. By theorem 4.1, Φ is an equilibrium price. By the minimality of P ∗ ( D ), we know thatΦ( D ) ≥ P ∗ ( D ). We prove Φ( D ) ≤ P ∗ ( D ) by considering the minimization probleminf D ≥ { P ∗ ( D ) − Φ( D ) } (4.11)mentioned in the erratum Chen and Kohn [2013].First, we consider the case when the infimum is achieved as D → ∞ . Note that 0 ≤ Φ( D ) − P ∗ ( D ) ≤ Φ( D ) − I ( D ) = F U ( D ) for a large enough D and U ( D ) → D → ∞ .Hence, Φ( D ) − P ∗ ( D ) → D → ∞ and the infimum of (4.11) is 0.Next, we consider the case when the infimum of (4.11) is achieved at a finite value. Let D be the value achieving the infimum. We show that P ∗ ( D ) ≥ Φ( D ), which completes the proof. he definition of D makes it a global minimum of P ∗ − Φ. Since P ∗ is a viscosity supersolutionof (4.3), F ( D , P ∗ ( D ) , Φ (cid:48) ( D ) , Φ (cid:48)(cid:48) ( D )) ≥
0. In other words, − max i =1 , { κ ( θ − D ) , κ ( θ − D ) } Φ (cid:48) − σ D Φ (cid:48)(cid:48) + λP ∗ ( D ) − D ≥ . (4.12)However, since Φ is a classical solution of (4.3), − max i =1 , { κ ( θ − D ) , κ ( θ − D ) } Φ (cid:48) − σ D Φ (cid:48)(cid:48) + λ Φ( D ) − D = 0 . (4.13)Comparing (4.12) and (4.13) clearly shows that P ∗ ( D ) ≥ Φ( D ) and inf D ≥ { P ∗ ( D ) − Φ( D ) } ≥ P ∗ ( D ) ≥ Φ( D ) and our proof is complete.We end this section by calculating the bubble size B ( D ) = P ∗ ( D ) − I ( D ). Using the formulaof P ∗ ( D ) = Φ( D ) and I ( D ), B ( D ) can be written as B ( D ) = Em ( D ) if D ≤ ¯ D − ( κ − κ )( D − ¯ D )( λ + κ )( λ + κ ) + Em ( D ) if ¯ D < D ≤ ˜ DF u ( D ) if D > ˜ D .
Note that ¯ D may be negative and if so, we need to consider only 2 cases for B ( D ).One may be curious about the exact value of B ( D ), as the size of the confluent hypergeo-metric functions are not apparent. We give a visualization of the bubble size for some sampleparameters in our conclusion. We can check that the bubble is most apparent at 0 and quicklyconverges to 0 after ˜ D . θ > θ It is intuitive that bubbles exist when κ > κ and θ > θ . Indeed, we can easily prove itsexistence by proving the following lemma, which was originally mentioned in the closing remarksof Section 2 in Chen and Kohn [2011]. Lemma 4.7. If ¯ D > , the minimal equilibrium price is strictly larger than the intrinsic price.Proof. We prove the bubble existence by showing that P ( D ) > I ( D ). Here, P is the pricedefined in (3.1) during the proof of Theorem 3.1. P ( D ) = max i =1 , sup τ ≥ E Q i (cid:20)(cid:90) τ e − λt D t dt + e − λτ I ( D τ ) | D = D (cid:21) ≥ max i =1 , E Q i (cid:20)(cid:90) e − λt D t dt + e − λ I ( D ) | D = D (cid:21) > max i =1 , E Q i (cid:20)(cid:90) ∞ e − λt D t dt | D = D (cid:21) = I ( D )The final inequality holds because both { D ≤ ¯ D } and { D > ¯ D } have positive probability, and E Q i [ I ( D ) | D = D ] > E Q i (cid:20) θ i λ + D − θ i λ + κ i | D = D (cid:21) = E Q i (cid:20)(cid:90) ∞ e − λt D t dt | D = D (cid:21) n our case, ¯ D = κ κ ( θ − θ )+ λ ( κ θ − κ θ ) λ ( κ − κ ) > E . E and F must be positive forΦ to be an equilibrium price, since Φ( D ) must be at least I ( D ). While the positiveness of F trivially follows from the assumption θ > θ , E is not necessarily positive (see the formula inthe Appendix). In fact, for extreme cases when θ (cid:29) θ and b (cid:29) x , E becomes negative.In this section, we first calculate the bubble size under the assumption that E ≥
0. Next, weprovide a more general result that provides a lower bound of the bubble size.First, we assume that E ≥
0. Random simulations for the parameter values indicate thatthis assumption is valid for most of the cases. Under the assumption that E ≥
0, Lemma 4.4and 4.5 holds and Φ is the minimal equilibrium price by Theorem 4.6. In this case, ¯
D > ˜ D andthe formula of the bubble size B ( D ) is different from Section 4.2.1. To be more precise, theformula when ˜ D < D ≤ ¯ D changes and the bubble size is B ( D ) = Em ( D ) if D ≤ ˜ D ( κ − κ )( D − ¯ D )( λ + κ )( λ + κ ) + F u ( D ) if ˜ D < D ≤ ¯ DF u ( D ) if D > ¯ D Next, we consider the case when E may be negative. If so, Φ( D ) < I ( D ) for D ≤ ˜ D andΦ( D ) is not an equilibrium price. However, Φ( D ) is still the unique C solution that has lineargrowth at infinity and P ∗ is a viscosity supersolution of (4.3). Therefore, we can use the proofof Corollary 4.6 to show that P ∗ ( D ) ≥ Φ( D ). Hence, B ( D ) = P ∗ ( D ) − I ( D ) ≥ Φ( D ) − I ( D ) = F u ( D )for D > ˜ D . We can also calculate the lower bound when D ≤ ˜ D by finding another positive,increasing C solution of (4.1). However, the choice of this solution heavily depends on theinitial parameters and we omit the details. Throughout the previous section, we proved that bubbles exist if and only if κ > κ and κ θ > κ θ . This condition means that bubbles do not exist if one group uniformly dominatesthe drift term of the dividend stream dynamics. We also explicitly calculated the bubble size formost of the cases. Similar to the Chen–Kohn paper, an existing bubble is permanent as long asour investors have consistent heterogeneous beliefs. The bubble size is expressed via confluenthypergeometric functions, and we provide some numerical examples for its visualization.First, we present an example of the intrinsic value and the minimal equilibrium price when θ < θ and κ θ ≥ κ θ in Figure 1. Here, we set the parameter values as κ = 0 . , κ = 0 . , θ = 0 . , θ = 0 . , λ = 0 . , σ = 0 . . (5.1)This initial values are chosen similar to those of Chen and Kohn [2011] and the mean levelbeliefs are set to realistic values of 0 .
015 and 0 .
02. For this initial value, ¯ D = − . < D = 0 and rapidlydisappears. In particular, the bubble is nearly indistinguishable after D > ˜ D = 0 . R ( D ) := P ∗ ( D ) /I ( D ) − R ( D ) is 2.71% when D = 0 and linearly decrease to 0.35% as D increase to ˜ D = 0 .
01. After D for the initialvalues in (5.1) Figure 2: The relative size of the bubble for the parameters in (5.1) then, R ( D ) converges to 0. We see that even though the bubble mathematically exists, its sizeis very small and is most apparent at low initial dividend rates.Next, we consider the case when θ > θ . Here, we set the parameter values as κ = 0 . , κ = 0 . , θ = 0 . , θ = 0 . , λ = 0 . , σ = 0 . . (5.2)For these values E = 1 . × − > P ∗ = Φ. We can see in Figure 3 that the markettransactions smooth the price, in contrast to the non-differentiable intrinsic value. Additionally,the bubble size when D at 0 is almost 0 and increases until ¯ D = 0 .
26. This result sharplycontrasts the results in (5.1) and we see that the sign of θ − θ significantly contributes to thebubble size. Even though it is not apparent in Figure 3, the bubble size converges to 0 as D goes to infinity. We also provide the relative bubble size R ( D ) in Figure 4. R ( D ) is close to0 when D ≤ ˜ D = 0 .
06 and increases to almost 18% as D increase to ¯ D = 0 .
26. We can notethat the price bubble is most apparent around ˜ D and disappears as D → ∞ . The comparisonof bubbles from the initial values (5.2) and (5.1) provides some interesting results. Even thoughthe minimal equilibrium price is defined by the same function Φ, the bubble size is heavily D for the initialvalues in (5.2) dependant to the parameter values. When θ < θ , the bubble is significant when D < ˜ D .However, when θ > θ , the bubble size is nearly 0 when D < ˜ D , and is most apparent when˜ D < D < ¯ D .Finally, we end our paper by confirming that our results are analogous with the trivial cases.These results are under the assumption that E ≥ E and F in the appendix.(i) The bubble disappears as κ − κ → θ − θ → κ − κ increases, with the assumption that θ < θ or E > θ = θ . (This result is displayed in Figures 5 and 6 and issimilar to the analysis in Chen and Kohn [2011].)(iv) The bubble gets larger as σ increase or λ decrease.Figures 5 and 6 are based on the initial parameters κ = 0 . , κ = 0 . , θ = θ = 0 . , λ = 0 . , σ = 0 . . (5.3)Note that ¯ D = ˜ D when θ = θ , and hence the formula of the bubble size changes only at thisvalue. Here, the bubble is apparent for all values of D and can be regarded as a compromisebetween the case when θ < θ and θ < θ . A Appendix: Explicit formula of
E , F
The explicit value of the constants
E, F from section 4 can be seen in the following formula.We recall that
E, F are constants depending on the model parameters κ , κ , θ , θ , λ, and σ . E = 1 A (cid:20) u ( κ − κ ) − U κ κ ( θ − θ ) σ (cid:21) , (A.1) F = 1 A (cid:20) M κ θ − θ θ + m ( κ − κ ) (cid:21) (A.2) D for the parametersin (5.3) Here, m i := M ( a i , b i , x i ) , u i := U ( a i , b i , x i ) , M i := M ( a i + 1 , b i + 1 , x i ) , U i := U ( a i + 1 , b i + 1 , x i )where a i , b i are the constants defined in Lemma 4.4 and x i = κ i ˜ Dσ . A is a constant defined as A := 2 m U σ + M u κ θ . Since m i , u i , M i , U i are positive, A is clearly positive. From form (A.1) and (A.2), we cantrivially see that E and F are both positive when θ = θ .Now, we prove that E and F are positive under the assumptions of Lemma 4.4 as well. Weare assuming that κ > κ and θ ≤ θ and the formula of E in (A.1) trivially shows that E >
F >
F >
0. By the formula of F in (A.2), it is suffices to show that M ( a , b , x ) M ( a + 1 , b + 1 , x ) ≥ κ ( θ − θ )( κ − κ ) θ . Note that the ratio of confluent hypergeometric functions of the first type can be expressed bycontinued fractions. We use the formula M ( a , b , x ) M ( a + 1 , b + 1 , x ) = b − x b + 1 b ∞ K m =1 mx b + m − x from Cuyt et al. [2008]. Since b − x = κ ( θ − ˜ D ) σ ≥
0, K ∞ m =1 mx b + m − x is positive. Meanwhile, b − x b = θ − ˜ Dθ = ( κ − κ ) θ − ( κ θ − κ θ )( κ − κ ) θ = κ ( θ − θ )( κ − κ ) θ and the proof is complete.As stated in Section 4.2, E may be negative in certain extreme cases. We end this appendixby stating equivalent conditions for E ≥
0. By the formula of E in (A.1), this is equivalent to( κ − κ ) U ( a , b , x ) ≥ κ κ ( θ − θ ) σ U ( a + 1 , b + 1 , x ) . (A.3)Since the left hand side of (A.3) is nonnegative, it is enough to prove (A.3) for θ > θ . Byrewriting (A.3), it is equivalent to U ( a + 1 , b + 1 , x ) U ( a , b , x ) ≤ ( κ − κ ) σ κ κ ( θ − θ ) = 1 x − b . We can take a = 1 , b = 4 . , x = 1000 to see that this is not always true. References
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