A macroscopic portfolio model: from rational agents to bounded rationality

Abstract

We introduce a microscopic model of interacting financial agents, where each agent is characterized by two portfolios; money invested in bonds and money invested in stocks. Furthermore, each agent is faced with an optimization problem in order to determine the optimal asset allocation. Thus, we consider a differential game since all agents aim to invest optimal and we introduce the concept of Nash equilibrium solutions to ensure the existence of a solution. Especially, we denote an agent who solves this Nash equilibrium exactly a rational agent. As next step we use model predictive control to approximate the control problem. This enables us to derive a precise mathematical characterization of the degree of rationality of a financial agent. This is a novel concept in portfolio optimization and can be regarded as a general approach. In a second step we consider the case of a fully myopic agent, where we can solve the optimal investment decision of investors explicitly. We select the running cost to be the expected missed revenue of an agent which are determined by a combination of a fundamentalist and chartist strategy. Then we derive the mean field limit of the microscopic model in order to obtain a macroscopic portfolio model. The novelty in comparison to existent macroeconomic models in literature is that our model is derived from microeconomic dynamics. The resulting portfolio model is a three dimensional ODE system which enables us to derive analytical results. The conducted simulations reveal that the model shares many dynamical properties with existing models in literature. Thus, our model is able to replicate the most prominent features of financial markets, namely booms and crashes. In the case of random fundamental prices the model is even able to reproduce fat tails in logarithmic stock price return data. Mathematically, the model can be regarded as the moment model of the recently introduced mesoscopic kinetic portfolio model (Trimborn et al. in Portfolio optimization and model predictive con trol: a kinetic approach, arXiv:1711.03291, 2017).

Appendices

Appendix A

Derivation of myopic investment strategy In this setting the parameter p of the approximation scheme is set to one. The Lagrangian becomes:

$$\begin{aligned}&L_i(x_i,y_i, S, u_i, \lambda _{x_i}, \lambda _{y_i}, \lambda _{S})\\&\quad = \int \limits _{{\bar{t}}}^{{\bar{t}}+ \Delta t}\left( \frac{\nu \Delta t}{2} u_i^2(t) + \Psi _i(t)\right) dt \\&\qquad +\int \limits _{{\bar{t}}}^{{\bar{t}}+ \Delta t}{\dot{\lambda }}_{x_i}\ x_i+ \lambda _{x_i}\ \kappa \ ED_N \ x_i + \lambda _{x_i}\ \frac{D}{S} \ x_i +\lambda _{x_i}\ u_i\ dt -\lambda _{x_i}\ {\bar{x}}_i\\&\qquad +\int \limits _{{\bar{t}}}^{{\bar{t}}+ \Delta t}{\dot{\lambda }}_{y_i}\ y_i+ \lambda _{y_i}\ r\ y_i -\lambda _{y_i}\ u_i\ dt-\lambda _{y_i}\ {\bar{y}}_i,\\&\qquad +\int \limits _{{\bar{t}}}^{{\bar{t}}+ \Delta t}{\dot{\lambda }}_{S}\ S+ \lambda _{S}\ \kappa \ ED_N\ S \ dt-\lambda _{S}\ {\bar{S}}, \end{aligned}$$

Notice that the quantities $(x_j^*,y_j^*,u_j^*),\ j=1,...,i-1,i+1,...,N$ are assumed to be optimal because we consider Nash equilibrium solutions. Thus, $(x_j^*,y_j^*,u_j^*),\ j=1,...,i-1,i+1,...,N$ only enter as parameters in the ith Lagrangian $L_i$. The optimality conditions can be immediately derived from the Lagrangian and read:

$$\begin{aligned}&\partial _{\lambda _{x_i}} L_i(x_i,y_i, S, u_i, \lambda _{x_i}, \lambda _{y_i}, \lambda _{S}) = 0\\&\partial _{\lambda _{y_i}} L_i(x_i,y_i, S, u_i, \lambda _{x_i}, \lambda _{y_i}, \lambda _{S}) = 0\\&\partial _{\lambda _S} L_i(x_i,y_i, S, u_i, \lambda _{x_i}, \lambda _{y_i}, \lambda _{S}) = 0\\&\partial _{u_i} L_i(x_i,y_i, S, u_i, \lambda _{x_i}, \lambda _{y_i}, \lambda _{S}) = 0\\&\partial _{x_i} L_i(x_i,y_i, S, u_i, \lambda _{x_i}, \lambda _{y_i}, \lambda _{S}) = 0\\&\partial _{y_i} L_i(x_i,y_i, S, u_i, \lambda _{x_i}, \lambda _{y_i}, \lambda _{S}) = 0\\&\partial _{S} L_i(x_i,y_i, S, u_i, \lambda _{x_i}, \lambda _{y_i}, \lambda _{S}) = 0. \\ \end{aligned}$$

Thus, we get:

$$\begin{aligned}&\dot{x}(t) = \kappa \ ED_N(t)\ x_i(t) + \frac{D(t)}{S(t)}\ x_i+ u_i,\ x_i({\bar{t}}) ={\bar{x}}_i, \\&\dot{y}_i(t) = ry_i(t) - u_i(t),\ y_i({\bar{t}})={\bar{y}}_i, \\&{\dot{S}}(t) = \kappa \ ED_N(t)\ S(t),\ S({\bar{t}}) = {\bar{S}}, \\&\nu \ \Delta t\ u_i(t) = - \lambda _{x_i}(t)- \lambda _{x_i}(t)\ \frac{\kappa }{N}\ x_i(t) + \lambda _{y_i}(t)- \frac{\kappa }{N}\ S(t)\ \lambda _{S}(t),\\&{\dot{\lambda }}_{x_i}(t) = -\kappa \ ED_N(t)\ \lambda _{x_i}(t)- \frac{D(t)}{S(t)}\ \lambda _{x_i}(t)- \partial _{x_i} \Psi _i(t),\ \lambda _{x_i}({\bar{t}}+\Delta t)=0\\&{\dot{\lambda }}_{y_i}(t) = -r \lambda _{y_i}(t)- \partial _{y_i} \Psi _i(t),\ \lambda _{y_i}({\bar{t}}+\Delta t)=0\\&{\dot{\lambda }}_{S}(t) = \lambda _{x_i}(t)\frac{D(t)}{S^2(t)} x_i-\kappa \ ED_N(t)\ \lambda _{S}(t)- \partial _{S} \Psi _i(t),\ \lambda _{S}({\bar{t}}+\Delta t)=0. \end{aligned}$$

For an introduction to optimal control theory and the derivation of optimality conditions we refer to [29, 54]. We can rewrite the maximum condition

$$\begin{aligned} \begin{aligned}&\nu \ \Delta t\ u_i(t) = - \lambda _{x_i}(t)- \lambda _{x_i}(t)\ \frac{\kappa }{N}\ x_i(t) + \lambda _{y_i}(t)- \frac{\kappa }{N}\ S(t)\ \lambda _{S}(t)\\&\iff u_i(t)=\frac{1}{\nu \Delta t}\ \left( - \lambda _{x_i}(t)- \lambda _{x_i}(t)\ \frac{\kappa }{N}\ x_i(t) + \lambda _{y_i}(t)- \frac{\kappa }{N}\ S(t)\ \lambda _{S}(t) \right) \end{aligned} \end{aligned}$$

(9)

Thus, in order to obtain the optimal strategy $u_i^*$ we need to solve the adjoint equations. In fact we apply a backward Euler discretization on the time interval $[{\bar{t}}, {\bar{t}}+\Delta t]$ in order to compute the costates $\lambda _{x_i}({\bar{t}}),\lambda _{y_i}({\bar{t}}), \lambda _{S}({\bar{t}})$ explicitly.

The backward Euler discretization of the general ODE

$$\begin{aligned} {\dot{z}}(t)=G(t,z(t)),\quad z(t_0)=z_0,\quad t>0 \end{aligned}$$

is given by

$$\begin{aligned} z(t+\Delta t)= z(t)+\Delta t\ G(t+\Delta t, z(t+\Delta t)), \end{aligned}$$

for a positive small time step $\Delta t>0$. When we apply this numerical scheme on the adjoint equations, we obtain:

$$\begin{aligned}&\frac{\lambda _{x_i}({\bar{t}}+\Delta t)-\lambda _{x_i}({\bar{t}})}{\Delta t}= -\kappa \ ED_N({\bar{t}}+\Delta t)\ \lambda _{x_i}({\bar{t}}+\Delta t)- \frac{D({\bar{t}}+\Delta t)}{S({\bar{t}}+\Delta t)}\ \lambda _{x_i}({\bar{t}}+\Delta t)\\&\qquad - \partial _{x_i} \Psi _i({\bar{t}}+\Delta t),\\&\frac{\lambda _{y_i}({\bar{t}}+\Delta t)-\lambda _{y_i}({\bar{t}})}{\Delta t}= -r \lambda _{y_i}({\bar{t}}+\Delta t)- \partial _{y_i} \Psi _i({\bar{t}}+\Delta t),\\&\frac{\lambda _{S}({\bar{t}}+\Delta t)-\lambda _{S}({\bar{t}})}{\Delta t} = \lambda _{x_i}({\bar{t}}+\Delta t)\frac{D({\bar{t}}+\Delta t)}{S^2({\bar{t}}+\Delta t)} x_i-\kappa \ ED_N({\bar{t}}+\Delta t)\ \lambda _{S}({\bar{t}}+\Delta t)\\&\qquad - \partial _{S} \Psi _i({\bar{t}}+\Delta t). \end{aligned}$$

By assumption $\lambda _{x_i}({\bar{t}}+\Delta t)=\lambda _{y_i}({\bar{t}}+\Delta t)=\lambda _{S}({\bar{t}}+\Delta t)=0$ holds and thus the previously discretized adjoint equations reduce to

$$\begin{aligned}&\lambda _{x_i}({\bar{t}}) = \Delta t\ \partial _{x_i} \Psi _i({\bar{t}}+\Delta t)={\left\{ \begin{array}{ll} \Delta t\ |K|\ x_i,&{}\ K<0,\\ 0,&{}\ \text {sonst}, \end{array}\right. }\\&\lambda _{y_i}({\bar{t}}) = \Delta t\ \partial _{y_i} \Psi _i({\bar{t}}+\Delta t)={\left\{ \begin{array}{ll} \Delta t\ |K|\ y_i,&{}\ K>0,\\ 0,&{}\ \text {sonst}, \end{array}\right. }\\&\lambda _{S}({\bar{t}}) = \Delta t\ \partial _{S} \Psi _i({\bar{t}}+\Delta t)={\left\{ \begin{array}{ll} \partial _S\ |K|\ \frac{x_i^2}{2},&{}\ K<0,\\ 0,&{}\ K=0,\\ \partial _S\ |K|\ \frac{y_i^2}{2},&{}\ K>0. \end{array}\right. } \end{aligned}$$

Then we insert the costates in the maximum condition (9) in order to obtain the optimal investment strategy $u_N^*$.

$$\begin{aligned} u_N^*(x_i,y_i, S)= {\left\{ \begin{array}{ll} \frac{1}{\nu } ( {K}\ y_i - \frac{\kappa }{N}\ S\ (\partial _{S} {K})\ \frac{y_i^2}{2}),&{}\quad K>0,\\ 0,&{}\quad K=0,\\ \frac{1}{\nu } ( {K}\ x_i + {K}\ \frac{\kappa }{N}\ x_i^2 + \frac{\kappa }{N} S\ (\partial _{S} |{K}|)\ \frac{x_i^2}{2}),&{}\quad K<0. \end{array}\right. } \end{aligned}$$

Appendix B

Qualitative analysis The proof of Proposition 1 reads:

Proof

We show local Lipschitz continuity. Then existence and uniqueness directly follows by the Picard–Lindelöf theorem. We can rewrite the stock price equation into an explicit ODE system:

$$\begin{aligned} {\dot{S}} = {\left\{ \begin{array}{ll} \kappa \ (\chi \omega s^f+(1-\chi )\ D-(r+\chi \omega )\ S)\ \frac{Y}{1+(1-\chi )\kappa \ Y},\quad K(S)>0,\\ \kappa \ (\chi \omega s^f+(1-\chi )\ D-(r+\chi \omega )\ S)\ \frac{X}{1+(1-\chi )\kappa \ X},\quad K(S)<0. \end{array}\right. } \end{aligned}$$

(10)

Thus, we may denote the right hand side of the explicit ODE system by $F(z),\ z:=(X,Y,S)^T \in [0,\infty )\times [0,\infty )\times (0,\infty )$. Notice that the excess demand ED does no longer depend on ${\dot{S}}$, since we can insert the right hand side of the stock price equation (10). Local Lipschitz continuity is obvious except for the potential singularity in $S^*=\frac{\chi \ \omega \ s^f+(1-\chi )\ D}{r+\chi \ \omega }$, since $K(S^*)=0$ holds. Thus, we show Lipschitz continuity on $U=U_X\times U_Y\times U_S, \ U_X:= [X_0-\epsilon , X_0+\epsilon ],\ U_Y:= [Y_0-\epsilon , Y_0+\epsilon ],\ U_S:= [S^*-\epsilon , S^*+\epsilon ],\ \epsilon >0$ with $z_0:= (X_0, Y_0, S^*)$, where $X_0,Y_0\in [0,\infty )$ are arbitrary but fixed. First we discuss the excess demand ED:

$$\begin{aligned} |ED(X,Y,S)| \le&\kappa \ (r+\chi \ \omega )\ \max \left\{ \max \limits _{X \in U_X} |X|, \max \limits _{Y \in U_Y} |Y| \right\} \ \max \limits _{S\in U_S} \left| \frac{1}{S}\right| | S^*-S | \\&\left( 1+ \kappa \ (1-\chi )\ \max \left\{ \max \limits _{X \in U_X} \left| \frac{X}{1+(1-\chi )\kappa \ X} \right| , \max \limits _{Y \in U_Y} \left| \frac{Y}{1+(1-\chi )\kappa \ Y} \right| \right\} \right) \\&\le C_1\ |S^*-S| \end{aligned}$$

As next step we discuss each component of $F=(F_1,F_2,F_3)^T$ separately. For the stock price evolution we obtain:

$$\begin{aligned} |F_3(Z)-F_3(Z_0)|&\le \kappa \ (r+\chi \omega )\ \max \left\{ \max \limits _{X \in U_X} \left| \frac{X}{1+(1-\chi )\kappa \ X} \right| , \max \limits _{Y \in U_Y} \left| \frac{Y}{1+(1-\chi )\kappa \ Y} \right| \right\} \ |S^*-S|\\&\le C_2\ |S-S^*| \end{aligned}$$

For the portfolio dynamics we get:

$$\begin{aligned} |F_1(Z)-F_1(Z_0)| \le&D\ \max \limits _{S\in U_S} \left| \frac{1}{S} \right| \ |X-X_0|+ (1+\kappa \ \max \limits _{X\in U_X} |X|)\ C_1\ |S^*-S|\\ \le&C_3\ |X-X_0|+ C_4\ |S-S^*|,\\ |F_2(Z)-F_2(Z_0)| \le&r\ |Y-Y_0|+ \ C_1\ |S^*-S|. \end{aligned}$$

Hence, we conclude that

$$\begin{aligned} ||F(z)-F(z_0)||\le L\ ||z-z_0 ||,\quad z,z_0 \in U, \end{aligned}$$

holds on U with Lipschitz constant $L:= C (C_1+C_2+C_3+ C_4 +r)$, where the additionally constant C is due to the equivalence of norms. $\square$

The proof of Proposition 2 is given by:

Proof

We set $r=D=0$ and derive the explicit ODE system. Thus, for a continuous differentiable Lyapunov functional we can compute the Lie derivative. We define the Lyapunov functional as follows: $\psi : \mathbb {R}^3 \rightarrow \mathbb {R},\ (x,y,S)^T\mapsto -(s^f-S)\ (x+y-(s^f-s))$. We immediately obtain

$$\begin{aligned} \frac{d}{dt}\psi ((X(t),Y(t),S(t)))\le 0,\quad \text {with}\ \frac{d}{dt}\psi ((X(t),Y(t),S(t)))\Big |_{S=S_{\infty }}=0, \end{aligned}$$

and can conclude the asymptotic stability of $S_{\infty }$. $\square$

Proposition 3

In special cases, we can compute solutions of the stock price equation. We assume constant weights $\chi$ and assume that the utility function is described by the identity.

• Fundamentalists alone ($\chi =1$): The stock price equation reads

  $$\begin{aligned} {\dot{S}} ={\left\{ \begin{array}{ll} \kappa \ (\omega \ s^f- (\omega +r)\ S)\ Y,\quad \frac{\omega s^f}{\omega +r}>S,\\ \kappa \ (\omega \ s^f-(\omega +r)\ S)\ X,\quad \frac{\omega s^f}{\omega +r}<S . \end{array}\right. } \end{aligned}$$

  This equation seems reasonable, so the investor shifts his capital into stocks if he expects a positive stock return, and vice versa. The solution is given by

  $$\begin{aligned} S(t)= {\left\{ \begin{array}{ll} (1-\exp \{- \kappa \ (\omega +r)\ \int \limits _0^t Y(\tau )\ d\tau )\})\ \frac{\omega \ s^f}{\omega +r} +S(0) \ \exp \{- \kappa \ (\omega +r)\ \int \limits _0^tY(\tau )\ d\tau \},\\ \quad \text {for}\ \frac{s^f}{\omega +r}>S,\\ (1-\exp \{- \kappa \ (\omega +r)\ \int \limits _0^t X(\tau )\ d\tau )\})\ \frac{\omega \ s^f}{\omega +r} +S(0) \ \exp \{- \kappa \ (\omega +r)\ \int \limits _0^tX(\tau )\ d\tau \},\\ \quad \text {for}\ \frac{s^f}{\omega +r}<S. \end{array}\right. } \end{aligned}$$

  Hence, the price is driven exponentially fast to the steady state $S_{\infty }=\frac{\omega \ s^f}{\omega +r}$.

• Chartists alone ($\chi = 0$): We get

  $$\begin{aligned} {\dot{S}} = {\left\{ \begin{array}{ll} \frac{\kappa \ D\ Y}{1-\kappa \ Y}- \frac{r\ \kappa Y}{1-\kappa Y}\ S,\quad \text {for}\ D>S,\\ \frac{\kappa \ D\ X}{1-\kappa \ X}- \frac{r\ \kappa X}{1-\kappa X}\ S,\quad \text {for}\ D<S. \end{array}\right. } \end{aligned}$$

  The solution is given by

  $$\begin{aligned} S(t)= {\left\{ \begin{array}{ll} \left( 1-\exp \left\{ -r \ \kappa \ \int \limits _0^t \frac{Y(\tau )}{1-\kappa Y(\tau )}\ d\tau \right\} \right) \ \frac{D}{r}+S(0)\ \exp \left\{ -r \ \kappa \ \int \limits _0^t \frac{Y(\tau )}{1-\kappa Y(\tau )}\ d\tau \right\} ,\\ \quad \text {for}\ \kappa \ D\ Y +D\ (1-\kappa \ Y)>S,\\ \left( 1+ \exp \left\{ -r \ \kappa \ \int \limits _0^t \frac{X(\tau )}{1-\kappa \ X(\tau )}\ d\tau \right\} \right) \ \frac{D}{r}+S(0)\ \exp \left\{ -r \ \kappa \ \int \limits _0^t \frac{X(\tau )}{1-\kappa \ X(\tau )}\ d\tau \right\} ,\\ \quad \text {for}\ \kappa \ D\ X+D\ (1-\kappa \ Y)<S. \end{array}\right. } \end{aligned}$$

• Chartists and fundamentalists with a constant weight $\chi \in (0,1)$: The corresponding stock price equation reads

  $$\begin{aligned} {\dot{S}} = {\left\{ \begin{array}{ll} \kappa \ (\chi \omega s^f+(1-\chi )\ D-(r+\chi \omega )\ S)\ \frac{Y}{1+(1-\chi )\kappa \ Y},\quad K(S)>0,\\ \kappa \ (\chi \omega s^f+(1-\chi )\ D-(r+\chi \omega )\ S)\ \frac{X}{1+(1-\chi )\kappa \ X},\quad K(S)<0. \end{array}\right. } \end{aligned}$$

  The solution is given by

  $$\begin{aligned} S(t)= {\left\{ \begin{array}{ll} (1-\exp \{- \kappa \ (\chi \omega +r)\ \int \limits _0^t \frac{Y(\tau )}{1+\kappa (1-\chi ) Y(\tau )}\ d\tau )\})\ \frac{\chi \omega \ s^f+(1-\chi )\ D}{\chi \omega +r} \\ \quad +S(0) \ \exp \{- \kappa \ (\chi \omega +r)\ \int \limits _0^t\frac{Y(\tau )}{1+\kappa (1-\chi ) Y(\tau )}\ d\tau \},\quad \text {for}\ K(S)>0,\\ (1-\exp \{- \kappa \ (\chi \omega +r)\ \int \limits _0^t \frac{X(\tau )}{1+\kappa (1-\chi ) X(\tau )}\ d\tau )\})\ \frac{\chi \omega \ s^f+(1-\chi )\ D}{\chi \omega +r} \\ \quad +S(0) \ \exp \{- \kappa \ (\chi \omega +r)\ \int \limits _0^t\frac{X(\tau )}{1+\kappa (1-\chi ) X(\tau )}\ d\tau \},\quad \text {for}\ K(S)<0. \end{array}\right. } \end{aligned}$$

Proposition 4

For the wealth evolution, we consider the stock and bond portfolio separately.

• In the stock portfolio, the wealth evolution is given by

  $$\begin{aligned} {\dot{X}} = {\left\{ \begin{array}{ll} (\kappa \ K(S)\ Y+ \frac{D}{S})\ X + K(S)\ Y,&{}\quad \text {for}\ K(S)>0,\\ (\kappa \ K(S)\ X +\frac{D}{S})\ X+ K(S)\ X,&{}\quad \text {for}\ K(S)<0. \end{array}\right. } \end{aligned}$$

  The solution is given by

  $$\begin{aligned} X(t) = {\left\{ \begin{array}{ll} X(0)\exp \left\{ \int \limits _0^t\kappa \ K(S) Y\ + \frac{D}{S}\ d\tau \right\} +\left( 1 - \exp \left\{ -\int \limits _0^t\kappa \ K(S) Y\ d\tau \right\} \right) \frac{1}{\kappa },\\ \quad \text {for}\ K(S)>0,\\ \frac{X(0)\ \exp \left\{ \int \limits _0^t K(s)+\frac{D}{S}\ d\tau \right\} }{ 1+\kappa \int \limits _0^t K(S)\ \exp \left\{ \int \limits _0^{\zeta } K(S)+\frac{D}{S}\ d\zeta \right\} \ d\tau },\quad \text {for}\ K(S)<0. \end{array}\right. } \end{aligned}$$

• The bond portfolio is given by

  $$\begin{aligned} {\dot{Y}} = {\left\{ \begin{array}{ll} r \ Y - K(S)\ Y,\quad &{}\text {for}\ K(S)>0,\\ r\ Y - K(S)\ X,\quad &{}\text {for}\ K(S)<0, \end{array}\right. } \end{aligned}$$

  with the solution

  $$\begin{aligned} Y(t) = {\left\{ \begin{array}{ll} Y(0)\ \exp \left\{ \int \limits _0^t (r-K(S))\ d\tau \right\} ,&{}\quad \text {for}\ K(S)>0,\\ \exp \left\{ r\ t \right\} \left( Y(0)- \int \limits _0^t K(S)\ X \exp \left\{ -r \ \tau \right\} \ d\tau \right) ,&{}\quad \text {for}\ K(S)<0. \end{array}\right. } \end{aligned}$$

Appendix C

Parameters of simulation If not indicated differently the parameters are set to:

Fig. 10

Standard parameter setting