Heterogeneous-Agent Models in Asset Pricing: The Dynamic Programming Approach and Finite Difference Method

Hamilton Galindo Gil

doi:10.1515/bejte-2024-0065

Enjoy 40% off

academic books on De Gruyter Brill *

Article

Heterogeneous-Agent Models in Asset Pricing: The Dynamic Programming Approach and Finite Difference Method

Hamilton Galindo Gil

Published/Copyright: April 21, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal The B.E. Journal of Theoretical Economics Volume 25 Issue 1

Abstract

This paper provides a detailed guide to solving a model characterized by risk-aversion heterogeneity, utilizing the dynamic programming approach in conjunction with the finite difference method. Although this model is characterized by a system of three partial differential equations (PDEs) – two related to the agents’ value functions and one to the risky asset price – it is surprisingly unnecessary to solve the full 3-PDEs system. Solving the 2-PDEs system for the agents’ value functions is sufficient, as, in equilibrium, the risky asset price is a function of these values. This problem is further simplified since each agent’s PDE can be solved independently due to the properties of the value function under constant relative risk aversion (CRRA) preferences. Finally, we demonstrate that applying the finite difference method with the implicit approach and an upwind scheme is straightforward for this type of asset pricing model.

Keywords: heterogeneous agents; preferences; asset pricing; dynamic programming; finite difference; continuous time

JEL Classification: C02; G11; G12

Corresponding author: Hamilton Galindo Gil, Department of Finance and Economics, Monte Ahuja College of Business, Cleveland State University, 1860 E 18th St, Cleveland, OH 44114, USA; and Graduate School of Business, Universidad ESAN, Alonso de Molina 1652, Santiago de Surco, Lima, Peru, E-mail: h.galindogil@csuohio.edu

Appendix A: Lemma Proofs

A.1 Proof of Lemma 2.1

This lemma relies on the wealth dynamics results established by Merton (1971).

A.2 Proof of Lemma 2.2

This lemma is an adaptation of the HJB equation with constraints from the representative agent model of Cox, Ingersoll, and Ross (1985) to an economy with two heterogeneous agents.

A.3 Proof of Lemma 2.3

This lemma is an adaptation of the HJB equation from the representative agent model of Cox, Ingersoll, and Ross (1985) to an economy with two heterogeneous agents, where we have incorporated the properties of the value function under CRRA-type preferences.

A.4 Proof of Lemma 2.4

We know that S _t in equilibrium will depend on the state variable Y. We also know the SDE of Y _t. Both are given by

(58) S t = f ( t , Y t )

(59) d Y t = μ Y t d t + σ Y t d Z t

I then apply Itô’s lemma to expression (58) to find dS _t.

(60) d S t = ∂ S t ∂ t + μ Y t ∂ S t ∂ Y t + 1 2 ( σ Y t ) 2 ∂ 2 S t ∂ Y t 2 d t + σ Y t ∂ S t ∂ Y t d Z t .

I then compare the terms of the Eq. (60) with

(61) d S t = ( β t S t − Y t ) d t + ν t S t d Z t

Volatility term. Comparing the volatility terms of Eqs. (60) and (61), I have an expression of ν _t–the volatility of dS _t/S _t.

(62) σ Y t ∂ S t ∂ Y t = ν t S t → ν t = σ Y t S t ∂ S t ∂ Y t

Drift term. Comparing the drift terms of Eqs. (60) and (61), I have the following expression.

(63) ∂ S t ∂ t + μ Y t ∂ S t ∂ Y t + 1 2 ( σ Y t ) 2 ∂ 2 S t ∂ Y t 2 = β t S t − Y t ,

Using the definition of ψ _t from Martingale asset pricing theory, I can obtain β _t.

(64) ψ t = β t − r t ν t → β t = r t + ψ t ν t .

Then, the right side of Eq. (63) becomes

β t S t − Y t ≡ ( r t + ψ t ν t ) S t − Y t ,

and then

(65) ∂ S t ∂ t + μ Y t ∂ S t ∂ Y t + 1 2 ( σ Y t ) 2 ∂ 2 S t ∂ Y t 2 = r t S t + ψ t ν t S t − Y t ∂ S t ∂ t + μ Y t ∂ S t ∂ Y t + 1 2 ( σ Y t ) 2 ∂ 2 S t ∂ Y t 2 − r t S t + Y t = ψ t ν t S t

From the volatility term comparison, I know the expression of ν _t (see Eq. (62)).

ν t = σ Y t S t ∂ S t ∂ Y t → ν t S t = σ Y t ∂ S t ∂ Y t

Putting ν _t S _t into the Eq. (65), I have

(66) ∂ S t ∂ t + μ Y t ∂ S t ∂ Y t + 1 2 ( σ Y t ) 2 ∂ 2 S t ∂ Y t 2 − r t S t + Y t = ψ t σ Y t ∂ S t ∂ Y t .

Ordering terms, this equation becomes

(67) ∂ S t ∂ t + Y t ( μ − ψ t σ ) ∂ S t ∂ Y t + 1 2 ( σ Y t ) 2 ∂ 2 S t ∂ Y t 2 − r t S t + Y t = 0 ,

which is a PDE of the risky asset price.

A.5 Proof of Lemma 2.5

This lemma is an adaptation of the results of Wang (1996) for any values of the relative risk aversion parameters (γ ₁ and γ ₂), in contrast to the specific values used by Wang (1996) (γ ₁ = 1 and γ ₂ = 0.5).

A.6 Proof of Lemma 2.6

Optimal consumption. The social planner solves the following static optimization problem P in period t:
(68) sup { c 1 t , c 2 t } E 0 e − ρ t λ u ( c 1 t ) + ( 1 − λ ) u ( c 2 t )
subject to
(69) c 1 t + c 2 t ≤ Y t

The Lagrange function associated with P is
(70) L = λ u ( c 1 t ) + ( 1 − λ ) u ( Y − c 2 t )

The FOC is given by
(71) c 1 t : ∂ L ∂ c 1 t = λ u c 1 t + ( 1 − λ ) u c 2 t ( − 1 ) = 0

Since u = c k t 1 − γ k / ( 1 − γ k ) , then
(72) λ c 1 t − γ 1 − ( 1 − λ ) c 2 t − γ 2 = 0 → c 1 t − γ 1 c 2 t − γ 2 = 1 − λ λ → c 1 t − γ 1 Y t − c 1 t − γ 2 = 1 − λ λ ≡ a
where a is a constant. Therefore,
(73) c 1 t = a − 1 / γ 1 ( Y t − c 1 t ) γ 2 / γ 1 .

The expression (73) is a nonlinear equation of c _1t which can be solved easily by the function fsolve in Matlab.
r _t and ψ _t. The marginal utility of the Representative Agent (RA) is the Stochastic Discount Factor (m _t). I first define the undiscounted utility function as
(74) U ( c 1 , t , c 2 , t , λ ) = λ u ( c 1 , t ) + ( 1 − λ ) u ( c 2 , t ) .

Considering the optimal consumption into the undiscounted utility function, we have
(75) U ( c 1 , t , c 2 , t , λ ) = U ( Y t , λ )

The discounted utility function is defined as continuous discount factor e^−ρt times the undiscounted utility function.
(76) U ( Y t , λ , t ) = e − ρ t U ( Y t , λ )

Then, m _t is the derivative of the discounted utility function U(Y _t, λ, t) with respect to the aggregate consumption Y _t as
(77) m t = ∂ U ( Y t , λ , t ) ∂ Y t ,
where
∂ U ( Y t , λ , t ) ∂ Y t = e − ρ t ∂ U ( Y t , λ ) ∂ Y t = e − ρ t ∂ ∂ Y t λ u ( c 1 , t ) + ( 1 − λ ) u ( c 2 , t ) .

Working on the last derivative, we have
∂ ∂ Y t λ u ( c 1 , t ) + ( 1 − λ ) u ( c 2 , t ) = λ u c 1 ∂ c 1 ∂ Y t + ( 1 − λ ) u c 2 ∂ c 2 ∂ Y t

= λ u c 1 ∂ c 1 ∂ Y t + ( 1 − λ ) u c 2 1 − ∂ c 1 ∂ Y t

= ∂ c 1 ∂ Y t λ u c 1 − ( 1 − λ ) u c 2 ︸ = 0 + ( 1 − λ ) u c 2

(78) = ( 1 − λ ) u c 2 , or equivalent

(79) = λ u c 1

Therefore,
(80) m t = ∂ U ( Y t , λ , t ) ∂ Y t = e − ρ t ( 1 − λ ) u c 2 ≡ e − ρ t λ u c 1

Then,
(81) m t = e − ρ t λ u c 1 ≡ f ( t , Y ; λ )

I then find the SDE of m _t by using Itô’s lemma because m _t depends on Y _t, which SDE is known.
d Y t = μ Y t d t + σ Y t d Z t

Using Ito’s lemma on expression (81), dm _t is given by
(82) d m t = f t + μ Y t f Y + 1 2 ( σ Y t ) 2 f Y Y d t + σ Y t f Y d Z t .
with
f t = − ρ m t

f Y = − γ 1 γ 2 m t a 1

f Y Y = − γ 1 γ 2 m t a 2 a 1 3

a 1 = γ 1 c 2 t + γ 2 c 1 t ≡ a 1 ( Y t )

a 2 = − c 1 t γ 2 2 ( 1 + γ 1 ) − Y t − c 1 t γ 1 2 ( 1 + γ 2 ) ≡ a 2 ( Y t )

I then compare it with the drift and diffusion terms of the SDE of m _t, which comes from the Martingale Asset Pricing theory.
(83) d m t = − r t m t d t − ψ t m t d Z t ,
where r _t is the risk-free interest rate and ψ _t is the price of risk. I start comparing the drift term and then the volatility term.

Drift term. Comparing the drifts terms of Eqs. (82) and (83), we an expression for r _t.
(84) − r t m t = m t − ρ + μ Y t f Y f + 1 2 ( σ Y t ) 2 f Y Y f .

As a result,
(85) r t = ρ + μ Y t γ 1 γ 2 a 1 + 1 2 ( σ Y t ) 2 γ 1 γ 2 a 2 a 1 3 .

This expression suggests r _t depends on Y and the RRA of agents

Volatility term. Comparing the volatility terms of Eqs. (82) and (83), I have an expression of the price of risk.
(86) − ψ t m t = σ Y t f Y ≡ σ Y t − γ 1 γ 2 m t a 1 → ψ t = ( σ Y t ) γ 1 γ 2 a 1

This clearly shows that ψ _t depends on Y and the RRA of agents.

A.7 Proof of Lemma 2.7

From the equilibrium in the financial markets, we have

(87) N 1 t ( 1 ) + N 2 t ( 1 ) = 1 ( risky asset market )

(88) N 1 t ( 2 ) + N 2 t ( 2 ) = 0 ( riskless asset market )

Multiplying the condition (87) and (88) by S _t and B _t, respectively. Then, multiplying and dividing by the agent’s wealth W _k.

(89) S t N 1 t ( 1 ) + S t N 2 t ( 1 ) = S → W 1 ω 1 ( 1 ) + W 2 ω 2 ( 1 ) = S t

(90) B t N 1 t ( 2 ) + B t N 2 t ( 2 ) = 0 → W 1 ω 1 ( 2 ) + W 2 ω 2 ( 2 ) = 0

Sum both equations,

(91) W 1 ω 1 ( 1 ) + W 2 ω 2 ( 1 ) + W 1 ω 1 ( 2 ) + W 2 ω 2 ( 2 ) = S W 1 ω 1 ( 1 ) + ω 1 ( 2 ) ︸ = 1 + W 2 ω 2 ( 1 ) + ω 2 ( 2 ) ︸ = 1 = S W 1 + W 2 = S

Then, from the FOC w.r.t. consumption (Eqs. (25) and (30)), we have the following.

(92) c k − γ k = A k W k − γ k → W 1 = c 1 A 1 1 / γ 1 & W 2 = c 2 A 2 1 / γ 2 ,

Introducing these expressions into Eq. (91), we have

(93) S t = c 1 t A 1 1 / γ 1 + c 2 t A 2 1 / γ 2

A.8 Proof of Lemma 2.8

Given the Eq. (93) and c _2t = Y _t − c _1t, we have

S t = c 1 t A 1 1 / γ 1 + ( Y t − c 1 t ) A 2 1 / γ 2

Taking derivative w.r.t. Y _t (without the subscript t), we have

S Y = c 1 Y A 1 1 / γ 1 + c 1 γ 1 A 1 1 / γ 1 − 1 A 1 , Y + ( 1 − c 1 Y ) A 2 1 / γ 2 + ( Y − c 1 ) γ 2 A 2 1 / γ 2 − 1 A 2 , Y ,

where,

c 1 Y ≡ ∂ c 1 ∂ Y = γ 2 c 1 a 1

This last expression comes from taking the derivative of Eq. (45) w.r.t. Y. Introducing c _1Y into S _y and grouping terms, we have the Eq. (51).

A.9 Proof of Lemma 2.9

Equation (54) is obtained by substituting expressions (52) and (53) into the HJB equation (36). The algebra is straightforward.

A.10 Proof of Lemma 2.10

Considering that θ _1t ∈ [0, 1] allows extreme cases in which only one of the agent lives in the economy, then

θ 1 t = Y t / S t − A 2 − 1 / γ 2 A 1 − 1 / γ 1 − A 2 − 1 / γ 2

(94) 0 ≤ θ 1 t ≤ 1 → A 2 − 1 / γ 2 ≤ Y t S t ≤ A 1 − 1 / γ 1 → A 2 − 1 / γ 2 ≤ A 1 − 1 / γ 1

Evaluating the relation (94) .
1. Agent 2. In the limit case when only agent 1 lives in the economy, we know that A 1 − 1 / γ 1 = N 1 . Considering this equivalence into (94), we have
  A 2 − 1 / γ 2 ≤ A 1 − 1 / γ 1 ≡ N 1 → A 2 − 1 / γ 2 ≤ N 1 → A 2 ≥ N 1 − γ 2
  
  This is the lower bound of A ₂.
2. Agent 1. In the limit case when only agent 2 lives in the economy, we know that A 2 − 1 / γ 2 = N 2 . Considering this equivalence into (94), we have
  N 2 ≡ A 2 − 1 / γ 2 ≤ A 1 − 1 / γ 1 → N 2 ≤ A 1 − 1 / γ 1 → A 1 ≤ N 2 − γ 1
  
  This is the upper bound of A ₁.
One-Agent economy. The consumption-wealth ratio of agent 1 (more risk averse) in a two-agent economy would be lower than the corresponding ratio of a one-agent economy populated by agent 1. This is not the case for agent 2 (the less risk-averse). His consumption-wealth ratio would be higher than the corresponding one in a one-agent economy populated by agent 2. This means the less risk-averse agent benefits from trading in the financial market to increase his consumption allocation.
A 1 − 1 / γ 1 ≤ N 1 → A 1 ≥ N 1 − γ 1

A 2 − 1 / γ 2 ≥ N 2 → A 2 ≤ N 2 − γ 2

A.11 Proof of Lemma 2.11

To obtain Eq. (57), substitute A _k,Y = A _k,YY = 0 in the HJB equation (54) and rearrange terms.

Appendix B: The Discretization and Solution Methods

B.1 Discretization Method

B.1.1 Space Discretization

We use a structured grid: an equispaced time t and endowment Y grid.

Time
(95) t = 0 , Δ t , 2 Δ t , … , t n , … , t N , t n = n Δ t n = 0,1,2 , … N t = t 0 , t 1 , … , t N

Then, we have N + 1 points.
Endowment
(96) Y = Y min , Δ Y , 2 Δ Y , … , Y i , … , Y I + 1 , Y i = i Δ Y i = 1,2 , … , I + 1 ,
where Y _I+1 = Y _max. Then, we have I + 1 points of Y.

B.1.2 The PDE Discretization

B.1.2.1 Time-Dependent Solution of the PDE

Before discretizing the PDE of A _k for k = {1, 2}, it is worth noting that we work with the time-dependent solution of these equations. In this case, we solve (and discretize) the PDE of A _k (Eq. (56)) without time subscript.

B.1.2.2 Finite Difference

Since the PDE of A _k (Eq. (56)) for agents k = {1, 2} contains the first and second derivatives of A _k, we define the forward and backward difference approximation of the first derivative of A _k and the central approximation for the second derivative as follows.

(97) A k , Y i ≈ A k , i + 1 − A k , i Δ Y ≡ A k , Y i , F : Forward difference approximation

(98) A k , Y i ≈ A k , i − A k , i − 1 Δ Y ≡ A k , Y i , B : Backward difference approximation

(99) A k , Y Y i ≈ A k , i + 1 − 2 A k , i + A k , i − 1 ( Δ Y ) 2 ≡ A k , Y Y i , c : Central approximation

The discretized PDEs system is given by

ρ A k i 1 − γ k + A k , t i 1 − γ k = γ k A k i 1 − 1 / γ k 1 − γ k + ψ i 2 2 γ k + r i A k i

(100) + σ Y i 2 2 γ k A k , Y i A k i + μ Y i 1 − γ k + σ Y i ψ i γ k A k , Y i + ( σ Y i ) 2 2 ( 1 − γ k ) A k , Y Y i

where A _k,t = ∂A _k/∂t. This discretized PDE is accompanied by

(101) ψ i = ( σ Y i ) γ 1 γ 2 a 1 i

(102) r i = ρ + ( μ Y i ) γ 1 γ 2 a 1 i + ( σ Y i ) 2 2 γ 1 γ 2 a 2 i a 1 i 3

where A k , Y i is either the forward or the backward difference approximation when the state variable Y takes the value of Y _i for i = 1, …, I + 1. For A k , Y Y i , we use the central approximation.

B.1.2.3 Upwind Scheme

It is clear that for the second derivative of A _k, the central approximation is appropriate. However, which approximation should we use for the first derivative – the forward or backward? A criterion is provided by the Upwind scheme. We first define the coefficient of A k , Y i .

(103) a ̃ k i = σ Y i 2 2 γ k A k , Y i A k i + μ Y i 1 − γ k + σ Y i ψ i γ k

The rule. The Upwind scheme suggests the following rule:
1. Case 1. Use forward difference approximation when the coefficient is positive.
  (104) a ̃ k i > 0 → A k , Y i ≈ A k , Y i , F , a ̃ k i = a ̃ k i , F
2. Case 2. Use backward difference approximation when the coefficient is negative.
  (105) a ̃ k i < 0 → A k , Y i ≈ A k , Y i , B , a ̃ k i = a ̃ k i , B
3. Case 3. When the coefficient is zero, we can use either forward or backward. We choose forward for simplicity.
  (106) a ̃ k i = 0 → A k , Y i ≈ A k , Y i , F
Rule’s implications.
1. PDE of A _k. Because a ̃ k i depends on A _k,Y, it could also be forward or backward. We then incorporate this fact into the coefficient a ̃ k i , which is given by
  (107) a ̃ k i , F = σ Y i 2 2 γ k A k , Y i , F A k i + μ Y i 1 − γ k + σ Y i ψ i γ k
  
  (108) a ̃ k i , B = σ Y i 2 2 γ k A k , Y i , B A k i + μ Y i 1 − γ k + σ Y i ψ i γ k ,
The approximation of A _k,Y . The three cases stated in the rule of the Upwind scheme for the first derivative of A _k with respect to Y can be expressed in a single expression, as follows:
(109) A k , Y i = A k , Y i , F 1 a ̃ k i , F ≥ 0 + A k , Y i , B 1 a ̃ k i , B < 0
where 1 _{⋅} denotes an indicator function. Equation (109) represents the Upwind scheme, indicating when to use the forward or backward difference approximation for the first derivative of A _k with respect to Y. It also specifies the appropriate approximation to use when a ̃ k i is zero.

Therefore, the discretized PDE of A _k is given by

(110) 1 1 − γ k A k , i n + 1 − A k , i n Δ t + ρ A k i 1 − γ k = γ k A k i 1 − 1 / γ k 1 − γ k + ψ i 2 2 γ k + r i A k i + a ̃ k i , F A k , Y i , F + a ̃ k i , B A k , Y i , B + ( σ Y i ) 2 2 ( 1 − γ k ) A k , Y Y i

with Finite Difference approximation (Eqs. (97)–(99)) and the Upwind scheme (Eq. (109)).

B.2 Solution Method

Up to this point, I have the discretized PDE of A _k using the Finite Difference method and the Upwind scheme. Two additional steps are needed. The first step is the solution method, which could be either the explicit or implicit method. We use the implicit method for its outstanding convergence properties (Candler 2001; Achdou, Han, and Lasry 2022). The second step is to express the system of (I + 1) equations from (110) in a matrix form, where I + 1 is the number of grid points.

B.2.1 The Solution Method

Using the implicit method, the discretized equations (110) are now evaluated in n + 1.

B.2.1.1 The Discretized Equation of A _k with the Implicit Method

The discretized equations (110) becomes

1 1 − γ k A k , i n + 1 − A k , i n Δ t + ρ A k , i n + 1 1 − γ k = γ k A k , i n 1 − 1 / γ k 1 − γ k + ψ i 2 2 γ k + r i A k , i n + 1

(111) + a ̃ k i , F n A k , Y i , F n + 1 + a ̃ k i , B n A k , Y i , B n + 1 + ( σ Y i ) 2 2 ( 1 − γ k ) A k , Y Y i n + 1

The implementation of the implicit method deserves some comments.

The variables in the system (111) are the terms of A k i n + 1 for i = 1 + I. Then, to make this system linear, the coefficients that depend on A k i should be evaluated in n. This allows the coefficients to be known. This justifies the following comments.
The method should be called “semi-implicit method” because the term
γ k A k , i n 1 − 1 / γ k 1 − γ k
should be
γ k A k , i n + 1 1 − 1 / γ k 1 − γ k .

However, we evaluate it in n because this allows us to have linear equations in A k , i n + 1 , simplifying the solution of the system. This strategy is based on Achdou, Han, and Lasry (2022).
The same approach is used in the coefficient of A k , Y i , F n + 1 and A k , Y i , B n + 1 . These coefficients are evaluated at n as follows.
(112) a ̃ k i , F n = σ Y i 2 2 γ k A k , Y i , F n A k , i n + μ Y i 1 − γ k + σ Y i ψ i γ k

(113) a ̃ k i , B n = σ Y i 2 2 γ k A k , Y i , B n A k , i n + μ Y i 1 − γ k + σ Y i ψ i γ k .

This implies that these coefficients are known in the iteration n + 1. Thus, it also facilitates having a linear system.

B.2.2 Algebraic System with Boundaries

We then introduce the definitions of forward, backward, and central difference approximations into Eq. (111) and evaluate this equation at each point of the grid. As a result, we obtain a linear system in the elements of A k n + 1 . Importantly, the first equation, evaluated at i = 1, is replaced by the lower boundary condition of A _k. Similarly, the last equation (for i = I + 1) is replaced by the upper boundary condition of A _k.

0 = N 1 − γ k − A k , 1 n + 1 , i = 1

A k , 2 n + 1 − A k , 2 n ( 1 − γ k ) Δ t + ρ A k , 2 n + 1 1 − γ k = γ k A k , 2 n 1 − 1 / γ k 1 − γ k + A k , 1 n + 1 X k 2 + A k , 2 n + 1 H k 2 + A k , 3 n + 1 Z k 2 , i = 2

A k , 3 n + 1 − A k , 3 n ( 1 − γ k ) Δ t + ρ A k , 3 n + 1 1 − γ k = γ k A k , 3 n 1 − 1 / γ k 1 − γ k + A k , 2 n + 1 X k 3 + A k , 3 n + 1 H k 3 + A k , 4 n + 1 Z k 3 , i = 3

⋮ = ⋮

A k , I n + 1 − A k , I n ( 1 − γ k ) Δ t + ρ A k , I n + 1 1 − γ k = γ k A k , I n 1 − 1 / γ k 1 − γ k + A k , I n + 1 X k I + A k , I n + 1 H k I + A k , I n + 1 Z k I , i = I

0 = N 2 − γ k − A k , I + 1 n + 1 , i = I + 1

where X _ki, H _ki, and Z _ki are known coefficients since they depend on Y _i. Then, we express this system in matrix form as

(114) P k A k n + 1 − P k A k n + Q k A k n + 1 = Y ̃ k + M k A k n + 1 , k ∈ { 1,2 }

where P _k and Q _k are diagonal matrices, and M _k is a coefficient matrix that collects the coefficients X _ki, H _ki, and Z _ki for i = 1, …, I + 1. The Online Appendix provides details about these matrices and the solution technique. Finally, I obtain A k n + 1 by solving the system (114) as follows:

(115) P k + Q k − M k ︸ = B n A k n + 1 = Y ̃ k + P k A k n ︸ = b n B n A k n + 1 = b n ,

where B ⁿ is an (I + 1) × (I + 1) matrix and b ⁿ is an (I + 1) × 1 vector. Both are filled with known coefficients. To find the optimal A k n + 1 , it is common to use a loop in which the difference between A k n + 1 and A k n is less than a convergence criterion. When this criterion is met, A k n + 1 represents the optimal value function for agent k.

References

Achdou, Yves, Jiequn Han, and Jean-Michel Lasry. 2022. “Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach.” The Review of Economic Studies: 42.Search in Google Scholar

Atmaz, Adem, and Suleyman Basak. 2018. “Belief Dispersion in the Stock Market.” The Journal of Finance 73 (3): 1225–79. https://doi.org/10.1111/jofi.12618.Search in Google Scholar

Basak, Suleyman. 2000. “A Model of Dynamic Equilibrium Asset Pricing with Heterogeneous Beliefs and Extraneous Risk.” Journal of Economic Dynamics and Control 24 (1): 63–95. https://doi.org/10.1016/s0165-1889(98)00064-5.Search in Google Scholar

Basak, Suleyman, and Domenico Cuoco. 1998. “An Equilibrium Model with Restricted Stock Market Participation.” Review of Financial Studies 11 (2): 309–41. https://doi.org/10.1093/rfs/11.2.309.Search in Google Scholar

Candler, Graham V. 2001. “Finite-Difference Methods for Continuous-Time Dynamic Programming.” In Computational Methods for the Study of Dynamic Economies, edited by R. Marimon, and A. Scott, 172–94. Oxford University Press.10.1093/0199248273.003.0008Search in Google Scholar

Cass, David, and Joseph E. Stiglitz. 1970. “The Structure of Investor Preferences and Asset Returns, and Separability in Portfolio Allocation: A Contribution to the Pure Theory of Mutual Funds.” Journal of Economic Theory 2 (2): 122–60. https://doi.org/10.1016/0022-0531(70)90002-5.Search in Google Scholar

Chabakauri, Georgy. 2013. “Dynamic Equilibrium with Two Stocks, Heterogeneous Investors, and Portfolio Constraints.” Review of Financial Studies 26 (12): 3104–41. https://doi.org/10.1093/rfs/hht030.Search in Google Scholar

Chan, Yeung Lewis, and Leonid Kogan. 2002. “Catching up with the Joneses: Heterogeneous Preferences and the Dynamics of Asset Prices.” Journal of Political Economy: 31.10.3386/w8607Search in Google Scholar

Constantinides, George M., and Anisha Ghosh. 2017. “Asset Pricing with Countercyclical Household Consumption Risk.” The Journal of Finance 72 (1): 415–60. https://doi.org/10.1111/jofi.12471.Search in Google Scholar

Cox, John C., Jonathan E. Ingersoll, and Stephen A. Ross. 1985. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407. https://doi.org/10.2307/1911242.Search in Google Scholar

Dumas, Bernard. 1989. “Two-Person Dynamic Equilibrium in the Capital Market.” Review of Financial Studies 2 (2): 157–88. https://doi.org/10.1093/rfs/2.2.157.Search in Google Scholar

Ferziger, Joel H., and Milovan Peric. 2002. Computational Methods for Fluid Dynamics, 3rd ed. Berlin, Heidelberg: Springer-Verlag.Search in Google Scholar

Galindo Gil, Hamilton. 2025. “How to Build and Solve Continuous-Time Heterogeneous Agents Models in Asset Pricing? The Martingale Approach and the Finite Difference Method.” Journal of Mathematical Economics 116: 103078. https://doi.org/10.1016/j.jmateco.2024.103078.Search in Google Scholar

Gârleanu, Nicolae, and Stavros Panageas. 2015. “Young, Old, Conservative, and Bold: The Implications of Heterogeneity and Finite Lives for Asset Pricing.” Journal of Political Economy 123 (3): 670–85. https://doi.org/10.1086/680996.Search in Google Scholar

Gârleanu, Nicolae, and Stavros Panageas. 2023. “Heterogeneity and Asset Prices: An Intergenerational Approach.” Journal of Political Economy 131 (4): 839–76. https://doi.org/10.1086/722224.Search in Google Scholar

Hakansson, Nils H. 1970. “Optimal Investment and Consumption Strategies Under Risk for a Class of Utility Functions.” Econometrica 38 (5): 587–607. https://doi.org/10.2307/1912196.Search in Google Scholar

Isaenko, Sergei. 2008. “The Term Structure of Interest Rates in a Pure Exchange Economy where Investors Have Heterogeneous Recursive Preferences.” The Quarterly Review of Economics and Finance: 25.10.1016/j.qref.2007.02.003Search in Google Scholar

Longstaff, Francis A., and Jiang Wang. 2012. “Asset Pricing and the Credit Market.” Review of Financial Studies 25 (11): 3169–215. https://doi.org/10.1093/rfs/hhs086.Search in Google Scholar

Merton, Robert C. 1971. “Optimum Consumption and Portfolio Rules in a Continuous-Time Model.” Journal of Economic Theory 3 (4): 373–413. https://doi.org/10.1016/0022-0531(71)90038-x.Search in Google Scholar

Merton, C., and F. Samuelson. 1992. “Labor Supply Flexibility and Portfolio Choice in a Life Cycle Model.” Journal of Economic Dynamics and Control: 23.Search in Google Scholar

Panageas, Stavros. 2020. “The Implications of Heterogeneity and Inequality for Asset Pricing.” Foundations and Trends® in Finance 12 (3): 199–275. https://doi.org/10.1561/0500000057.Search in Google Scholar

Radner, Roy. 1972. “Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets.” Econometrica 40 (2): 289–303. https://doi.org/10.2307/1909407.Search in Google Scholar

Roche, Hervé. 2011. “Asset Prices in an Exchange Economy when Agents Have Heterogeneous Homothetic Recursive Preferences and No Risk Free Bond is Available.” Journal of Economic Dynamics and Control 35: 17.10.1016/j.jedc.2010.08.005Search in Google Scholar

Schneider, Andrés. 2022. “Risk-Sharing and the Term Structure of Interest Rates.” The Journal of Finance 77 (4): 2331–74. https://doi.org/10.1111/jofi.13139.Search in Google Scholar

Wang, Jiang. 1996. “The Term Structure of Interest Rates in a Pure Exchange Economy with Heterogeneous Investors.” Journal of Financial Economics 41: 75–110. https://doi.org/10.1016/0304-405x(95)00854-8.Search in Google Scholar

Weinbaum, David. 2009. “Investor Heterogeneity, Asset Pricing and Volatility Dynamics.” Journal of Economic Dynamics and Control: 19.10.1016/j.jedc.2008.12.002Search in Google Scholar

Weinbaum, David. 2010. “Preference Heterogeneity and Asset Prices: An Exact Solution.” Journal of Banking & Finance: 9.10.1016/j.jbankfin.2010.02.009Search in Google Scholar

Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/bejte-2024-0065).

Received: 2024-05-27

Accepted: 2025-03-18

Published Online: 2025-04-21

You are currently not able to access this content.

Supplementary Material Details

Articles in the same Issue

https://doi.org/10.1515/bejte-2024-0065

Keywords for this article

heterogeneous agents; preferences; asset pricing; dynamic programming; finite difference; continuous time

Heterogeneous-Agent Models in Asset Pricing: The Dynamic Programming Approach and Finite Difference Method

Article

Abstract

Appendix A: Lemma Proofs

A.1 Proof of Lemma 2.1

A.2 Proof of Lemma 2.2

A.3 Proof of Lemma 2.3

A.4 Proof of Lemma 2.4

A.5 Proof of Lemma 2.5

A.6 Proof of Lemma 2.6

A.7 Proof of Lemma 2.7

A.8 Proof of Lemma 2.8

A.9 Proof of Lemma 2.9

A.10 Proof of Lemma 2.10

A.11 Proof of Lemma 2.11

Appendix B: The Discretization and Solution Methods

B.1 Discretization Method

B.1.1 Space Discretization

B.1.2 The PDE Discretization

B.1.2.1 Time-Dependent Solution of the PDE

B.1.2.2 Finite Difference

B.1.2.3 Upwind Scheme

B.2 Solution Method

B.2.1 The Solution Method

B.2.1.1 The Discretized Equation of A k with the Implicit Method

B.2.2 Algebraic System with Boundaries

References

Supplementary Material

Supplementary Material

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

B.2.1.1 The Discretized Equation of A _k with the Implicit Method