Basak-Cuoco and Gârleanu-Panageas
The full solution for Basak-Cuoco model can be found at basak_cuoco.ipynb. The solution for Gârleanu-Panageas model (with passive agent pricing) is at garleanu_panageas.ipynb.
The model has 3 agents and 2 shocks. Agent \(u\) is unconstrained; agent \(c\) is constrained; and agent \(p\) is passive. Endowment follows a GBM with two independent shocks. Here we assume that the passive agent is not participating tin pricing.
\[
\frac{dY_t}{Y_t} = \mu dt + \sigma^1 dW_t^1 + \sigma^2 dW_t^2
\]
Risky asset price \(P_t\) follows
\[
\frac{dP_t}{P_t} = \mu_{P} dt + \sigma_P^1 dW_t^1 + \sigma_P^2 dW_t^2
\]
Returns are given by
\[
\frac{dR_t}{R_t} = \frac{dP_t}{P_t} + \frac{Y_t}{P_t}dt = \left( \mu_P + \frac{Y_t}{P_t} \right)dt + \sigma_P^1 dW^1 + \sigma_P^2 dW^2
\]
Parameters
| Parameter | Definition | Value |
|---|---|---|
| \(n\) | Number of households | \(n=3\) |
| \(\gamma=(\gamma_u, \gamma_c, \gamma_p)\) | Risk aversion (unconstrained, constrained, passive) | \(\gamma_u=1,\gamma_c=1,\gamma_p=1\) |
| \(\rho\) | Discount rate | \(\rho=0.05\) |
| \(\bar{\alpha}\) | Leverage constrained ceiling | \(\bar{\alpha}=1000.95\) |
| \(\bar{\alpha_p}\) | Alpha passive agent | \(\bar{\alpha_p}=0.0\) |
| \(\theta\) | Alpha drift | \(\theta=0.0\) |
| \(\sigma_{\alpha_p}=(\sigma_{\alpha_p}^1, \sigma_{\alpha_p}^2)\) | Vol. of passive demand share | \(\sigma_{\alpha_p}^1=0, \sigma_{\alpha_p}^2=0\) |
| \(\nu\) | Correlation | \(\nu=0.5\) |
| \(\psi\) | IES | \(\psi=1/\gamma\) |
| \(\mu\) | Mean of the lognormal distribution of the asset return | \(\mu=0.0183\) |
| \(\sigma=(\sigma^1, \sigma^2)\) | Fundamental volatility | \(\sigma^1=0.0357, \sigma^2=0\) |
| \(\kappa\) | Death rate | \(\kappa=0\) |
| \(\omega=(\omega_u,\omega_c,\omega_p)\) | Mass of agents | \(\omega_u=0.25, \omega_c=0.25, \omega_p=0.5\) |
| \(\alpha_{p,min},\alpha_{p,max}\) | Max/Min alpha for passive agent | \(\alpha_{p,min}=\alpha_{p,max}=0\) |
For passive pricing, we change the following parameters,
| Parameter | Definition | Value |
|---|---|---|
| \(\gamma=(\gamma_u, \gamma_c, \gamma_p)\) | Risk aversion (unconstrained, constrained, passive) | \(\gamma_u=1,\gamma_c=1,\gamma_p=1\) |
| \(\psi\) | IES | \(\psi=1.5\) |
Variables
| Type | Definition |
|---|---|
| State Variables | \((x_u, x_c, \alpha_p)\), \(x_u,x_c\geq 0\), \(x_u+x_c\leq 1\), \(\alpha_p\in [\alpha_{p,min},\alpha_{p,max}]\) |
| Agents | \(\xi = (\xi_u, \xi_c, \xi_p)\) |
| Endogenous Variables | \(\alpha_u\), \(\alpha_c\) |
Equations
\[
\begin{align*}
x_p &= 1 - x_u - x_c\\
\alpha_{all} &= (\alpha_u, \alpha_c, \alpha_p)\\
y &= x_u \xi_u + x_c \xi_c + x_p \xi_p\\
\sigma_R &= \sigma - \sigma_y\\
\sigma_{\alpha_p} &= \begin{bmatrix}
\sigma_{\alpha_p}^1 \nu & \sigma_{\alpha_p}^1 \sqrt{1-\nu^2}\\
\sigma_{\alpha_p}^1 \nu & \sigma_{\alpha_p}^1 \sqrt{1-\nu^2}
\end{bmatrix}\\
A &= \frac{\partial y}{\partial x_u} x_u (\alpha_u-1) + \frac{\partial y}{\partial x_c} x_c (\alpha_c-1)\\
\sigma_y &= \frac{A\sigma + \frac{\partial y}{\partial \alpha_p} \sigma_{\alpha_p}}{y+A}\\
\sigma_R &= \sigma - \sigma_y\\
\sigma_x &= \begin{bmatrix}
x_u (\alpha_u-1)\sigma_R & x_c(\alpha_c-1)\sigma_R & \sigma_{\alpha_p}
\end{bmatrix}\\
\sigma_\xi &= \frac{1}{\xi} \nabla \xi \sigma_x\\
Var(\sigma) &= \frac{1-\frac{1}{\gamma}}{1-\psi} \frac{\sigma_\xi \cdot \sigma_R}{\|\sigma_R\|^2}\\
q &= \gamma_u (\alpha_u + Var(\sigma)_u)\\
\pi &= q \|\sigma_R\|^2\\
\eta &= q \|\sigma_R\|\\
\mu_x &= \begin{bmatrix}
x_u(y-\xi_u+(1-\alpha_u)(1-q)\|\sigma_R\|^2) + \kappa (\omega_u-x_u)\\
x_c(y-\xi_c+(1-\alpha_u)(1-q)\|\sigma_R\|^2) + \kappa (\omega_c-x_c)\\
\theta(\bar{\alpha_p}-\alpha_p)\\
\end{bmatrix}^T\\
\Sigma &= \sigma_x \sigma_x^T\\
\mu_\xi &= \frac{1}{\xi} \nabla_x \xi \mu_x + \frac{1}{2} \frac{1}{\xi} \text{Tr}(D^2\xi \Sigma)\\
\mu_y &= \frac{1}{y} \nabla_x y \mu_x + \frac{1}{2} \frac{1}{y} \text{Tr}(D^2y \Sigma)\\
\mu_P &= \mu - \mu_y + \sigma_y \cdot(\sigma_y -\sigma)\\
r &= y + \mu_P - \pi\\
\end{align*}
\]
\[
HJB = \rho\psi + (1-\psi) \left(r+\eta \alpha_{all}\|\sigma_R\| - \frac{1}{2} \gamma(\alpha_{all}\|\sigma_R\|)^2 \right) + \mu_\xi + (1-\gamma) \sigma_\xi \sigma_R \alpha_{all} + \frac{1}{2} \frac{\psi-\gamma}{1-\psi} \sigma_\xi\cdot \sigma_\xi - \xi
\]
Loss Functions
HJB residual (scaled to impose stronger penalty):
\[HJB/\rho = 0\]
Market clearning:
\[x_u\alpha_u+x_c\alpha_c+x_p\alpha_p=1\]
First order conditions:
\[\alpha_c = \min \left(\frac{q}{\gamma_c} - Var(\sigma)_c, \frac{\bar{\alpha_p}}{\|\sigma_R\|}\right)\]
If passive agent is pricing:
\[\alpha_p = \frac{q}{\gamma_p} - Var(\sigma)_p\]
Pricing:
\[\pi = \frac{(1+x_uVar(\sigma)_u + x_c Var(\sigma)_c)\|\sigma_R\|^2}{\frac{x_u}{\gamma_u} + \frac{x_c}{\gamma_c}}\]
If passive agent is pricing:
\[\pi = \frac{(1+x_uVar(\sigma)_u + x_c Var(\sigma)_c + x_p Var(\sigma)_p)\|\sigma_R\|^2}{\frac{x_u}{\gamma_u} + \frac{x_c}{\gamma_c} + \frac{x_p}{\gamma_p}}\]