Basak Cuoco Problem

The full solution can be found at basak_cuoco.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

\[ \frac{dY_t}{Y_t} = \mu dt + \sigma^1 dW_t^1 + \sigma^2 dW_t^2 \]

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	\(\psi=1\)
\(\mu\)	Mean of the lognormal distribution of the asset return	\(\mu=0.22\)
\(\sigma=(\sigma^1, \sigma^2)\)	Fundamental volatility	\(\sigma^1=0.035, \sigma^2=0\)
\(\kappa\)	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	\(\alpha_{p,min}=\alpha_{p,max}=0\)

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	\(\pi\), \(r\) (risk-free rate), \(\mu_y\) (drift of endowment), \(\sigma_y= (\sigma_y^1, \sigma_y^2)\) (volatility of endowment), \(\alpha_c\) (alpha for constrained)

Equations

\[ \begin{align*} x_p &= 1 - x_u - x_c\\ y &= x_u \xi_u + x_c \xi_c + x_p \xi_p\\ \alpha_u &= \frac{1 - x_c \alpha_c - x_p \alpha_p}{x_u}\\ \alpha_{state} & = (\alpha_u, \alpha_c)\\ \alpha_{all} &= (\alpha_u, \alpha_c, \alpha_p)\\ \sigma_R &= \sigma - \sigma_y\\ \mu_p &= -\mu_{y,model} - \sigma_y\cdot \sigma_R\\ \mu_P &= \mu_p + \mu + \sigma_R \cdot \sigma_R - \sigma_R \cdot \sigma\\ \eta & = \frac{\pi}{\|\sigma_R\|}\\ \sigma_{state} &= (\alpha_{state} - 1) \sigma_R^T\\ \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}\\ \sigma_x & = \begin{bmatrix} \sigma_{state}\\ \frac{\sigma_{\alpha_p}}{\alpha_p} \end{bmatrix}\\ \mu_{\alpha_p} &= \frac{\theta (\bar{\alpha_p} - \alpha_p)}{\alpha_p}\\ \sigma_{\xi} &= \frac{\partial \xi}{\partial x} \sigma_x \begin{bmatrix} x_u\\ x_c\\ \alpha_p \end{bmatrix}\\ \hat{x} &= (x_u, x_c, x_p)\\ \mu_{x,state} &= r + \eta \alpha_{all} \|\sigma_R\| - \xi - \mu_P + (1 - \alpha_{all}) \sigma_R\cdot \sigma_R + \kappa \frac{\omega-\hat{x}}{\hat{x}}\\ \mu_{x} &= (\mu_{x,u}, \mu_{x,c}, \mu_{\alpha_p})\\ \mu_{\xi_i,1} &= \frac{1}{\xi_i} \sum_j \frac{\partial\xi_i}{\partial x_j} \mu_{x,j}x_j\\ \mu_{y_i,1} &= \frac{1}{y_i} \sum_j \frac{\partial y_i}{\partial x_j} \mu_{x,j}x_j\\ \Sigma_x &= \sigma_x x\\ a &= \Sigma_x \Sigma_x^T\\ \mu_{\xi_i, 2} &= \frac{1}{2} \text{Tr}\left(\frac{D^2\xi_i}{\xi_i} a\right)\\ \mu_{y_i, 2} &= \frac{1}{2} \text{Tr}\left(\frac{D^2y_i}{y_i} a\right)\\ \mu_{\xi_i} &= \mu_{\xi_i,1}+\mu_{\xi_i,2}\\ \mu_{y_i} &= \mu_{y_i,1} + \mu_{y_i,2}\\ Var(\sigma) &= \frac{1-\gamma}{1-\psi} \frac{(\sigma_\xi^T \sigma_R)^T}{\sigma_R\cdot \sigma_R}\\ HJB &= \rho\psi + (1-\psi) ( r + \eta \alpha_{all} \|\sigma_R\| - \frac{\gamma^T}{2} (\alpha_{all} \|\sigma_R\|)^2) + \mu_{\xi}\\ &\quad + (1-\gamma^T) \sigma_\xi^T \sigma_R \alpha_{all} + \frac{\psi-\gamma^T}{1-\psi} \frac{1}{2} \sigma_{\xi}\cdot \sigma_{\xi} - \xi\\ \Sigma_s &= \begin{bmatrix} \sigma^1, \sigma^2\\ \sigma^1, \sigma^2\\ \sigma_{\alpha_p}^1\nu, \sigma_{\alpha_p}^1\sqrt{1-\nu^2}\\ \end{bmatrix} \end{align*} \]

When \(\psi=1\):

\[ \begin{align*} Var(\sigma) &= 0\\ HJB &= \rho + \mu_{\xi} - \xi + (1-\gamma^T) \sigma_\xi^T \sigma_R \alpha_{all} \end{align*} \]

Loss Functions

\[ \begin{align*} HJB/\rho &= 0\\ \alpha_u &= \frac{\pi}{\gamma_u \|\sigma_R\|^2} - Var(\sigma)_u\\ \alpha_c &= \min \left\{\frac{\pi}{\gamma_c \|\sigma_R\|^2} - Var(\sigma)_c, \frac{\bar{\alpha}}{\|\sigma_R\|}\right\}\\ \frac{\mu_P + y - r - \pi}{\rho} &= 0\\ \frac{\mu_y - \mu_{y,model}}{y} &= 0\\ \sigma_y &= \frac{\nabla y \cdot ((\alpha - 1, 1)\odot \sigma )}{y + \nabla y \cdot (\alpha - 1)} \end{align*} \]