Trading Volume and Price Distortion: An Agent-Based Model with Heterogenous Knowledge of Fundamentals

Abstract

This paper investigates whether trading volume and price distortion can be explained by the investor’s bounded rationality. Assuming that agents are bounded by their information access and processing, what are the consequences on market dynamics? We expose the result of simulations in an ABM that considers the liquidity as an endogenous characteristic of the market and allows to design investors as bounded rational. In a call auction market, where two risky assets are exchanged, traders are defined as a mix between fundamentalist and trend-follower outlook. Each one differs as to behaviour, order-placement strategy, mood, knowledge, risk-aversion and investment horizon. We place agents in a context of evolving fundamental values and order placement strategy; they perceive the fundamental but they also have some heterogeneous belief perseverance; and they adapt their orders to the market depth so as to maximise their execution probability and their profit. By adding bounded rationality in their information processing, we show that (1) usual features as trend-follower outlook and heterogeneous investment horizon are important features to generate excess volatility of asset prices and market inefficiency; (2) the learning fundamental value stabilises the market price and the trading volume; (3) the order-placement strategy increases trading volume, but reduces market efficiency and stability; (4) the agent’s mood prevents illiquid market and weakly increases the market volatility as classical noise trader agents; (5) the impatience to sell of traders is always present in the market: the market sell orders are always more numerous than the market buy orders.

Appendix

1.1 Demand Function

Each trader has its own demand function and decides the amount of wealth she would like to invest in the risky assets. The residual wealth is invested in a bond with zero interest rate. To facilitate reading of the appendix, we drop the agent superscript i from all variables.

$$\begin{aligned} U(W,\alpha )=-\exp (-\alpha W) \end{aligned}$$

(16)

The demand function is defined as a constant absolute risk-aversion class, where $\alpha$ is the risk aversion coefficient of the trader, and $W_t$ the trader wealth at time t. The wealth is given by a cash amount $C_t$ and the quantity of the risky assets $z^j_t$ held times their spot prices $p^j_t$.

$$\begin{aligned} W_t=z^1_t \cdot p^1_t+z^2_t \cdot p^2_t+C_t \end{aligned}$$

(17)

The wealth at time $t+\tau$ (assuming the order is executed at a price $p^j$ before $t+\tau$) is given by:

$$\begin{aligned} W_{t+\tau }= & {} z_{t+\tau }^1\cdot p^1_{t+\tau }+z_{t+\tau }^2\cdot p^2_{t+\tau }+C_{t+\tau } \end{aligned}$$

(18)

$$\begin{aligned}= & {} W_t + s^1_t\cdot p^1_t\cdot r_{t+\tau }^1+s_{t}^2\cdot p^2_t\cdot r_{t+\tau }^2 \end{aligned}$$

(19)

where $r_{t+\tau }=p_{t+\tau }/p_t-1$ is the return from t to $t+\tau$. As a zero order approximation, the agent’s expectations on future returns are taken to be Gaussian where future return is assumed to be $r_{t+\tau }=p_{t+\tau }/p_t-1\simeq ln(p_{t+\tau }/p_t)$. Based on her knowledge, the trader tries to maximise at time t her forward utility ($U_{t+\tau }$):

$$\begin{aligned} \max _{W_{t+\tau }} \ \mathbb {E}_t[U(W_{t+\tau }, \alpha )] = \max _{W_{t+\tau }} \ \mathbb {E}_t[ -exp({-\alpha \cdot W_{t+\tau }})] \end{aligned}$$

(20)

Because the utility is exponential and the returns are assumed to be Gaussian, the utility may be expressed as:

$$\begin{aligned} \hat{U}(W_{t+\tau }, \alpha ) =- \exp \left( -\alpha \cdot \mathbb {E}_t[W_{t+\tau }] + \alpha ^2 \cdot \sigma ^2_t[W_{t+\tau }]/2 \right) \end{aligned}$$

(21)

where the risk of investment is assigned its variance. The expected wealth and the variance are:

$$\begin{aligned} \mathbb {E}_t[W_{t+\tau }]= & {} W_t + s_t^1\cdot p^1_t \mathbb {E}_t[r_1]+s^2_t.p^2_t \mathbb {E}_t[r_2] \end{aligned}$$

(22)

$$\begin{aligned} \sigma ^2_t[W_{t+\tau }]= & {} s^1_t\cdot (p_t^1)^2\cdot \sigma ^2_t[r_1]+s^2_t\cdot (p^2_t)^2\cdot \sigma ^2_t[r_2]+2\cdot s^1_t\cdot p^1_t\cdot s^2_t\cdot p^2_t\cdot \sigma _t[r_1;r_2]\nonumber \\ \end{aligned}$$

(23)

Equation (21) could be rewritten as:

$$\begin{aligned} \hat{U}_{t+\tau }= & {} \hat{U}_t \times \exp \Big (-\alpha \big [ s_t^1\cdot p^1_t\cdot \mathbb {E}_t[r_1]+ s_t^2\cdot p_t^2\cdot \mathbb {E}_t[r_2] \big ] \nonumber \\&+\frac{\alpha ^2}{2} \big [ (s_t^1)^2(p_t^1)^2\sigma ^2_t[r_1] + (s_t^2)^2(p_t^2)^2\sigma ^2_t[r_2] + 2\cdot s^1_t \cdot p^1_t\cdot s^2_t\cdot p^2_t\cdot \sigma _t[r_1;r_2] \big ] \Big )\nonumber \\ \end{aligned}$$

(24)

Differentiating the expected utility function (24) with respect to $s_t^j$ gives:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d}{ds_t^1}\hat{U}_{t+\tau }= - \hat{U}_{t+\tau } \left[ \alpha \cdot p_t^1\cdot \ln \left( \frac{p_{t+\tau }^1}{p^1_t} \right) -\alpha ^2\cdot s_t^1(p^1_t)^2 \sigma ^2_t[r_1]-2\cdot s_t^2 \cdot p^1_t\cdot p^2_t\cdot \sigma _t[r_1;r_2]\right] \\ \frac{d}{ds_t^2}\hat{U}_{t+\tau }= - \hat{U}_{t+\tau } \left[ \alpha \cdot p_t^2\cdot \ln \left( \frac{p_{t+\tau }^2}{p^2_t} \right) -\alpha ^2\cdot s_t^2(p^2_t)^2 \sigma ^2_t[r_2]-2\cdot s_t^1\cdot p^1_t\cdot p^2_t\cdot \sigma _t[r_1;r_2] \right] \end{array}\right. } \end{aligned}$$

(25)

Setting the expression to zero we determine the optimal amount of stocks ($S_t^*=\pi (p)$) that the agent wishes to hold in her portfolio for a given price level p.

$$\begin{aligned} {\left\{ \begin{array}{ll} \pi ^{1}_t (\hat{p_1} ,\hat{p}_2)= \dfrac{\sigma ^2[r_1]. \ln \left( \dfrac{ \hat{p}_{2}}{p_2} \right) \alpha ^3 - 2.\sigma [r_1;r_2].\ln \left( \dfrac{ \hat{p}_{1}}{p_1} \right) \alpha }{p_2 \left( \sigma ^2[r_1].\sigma ^2[r_2].\alpha ^4-4.\sigma ^2[r_1;r_2] \right) } \\ \pi ^{2}_t (\hat{p_1} ,\hat{p}_2)= \dfrac{\sigma ^2[r_2]. \ln \left( \dfrac{ \hat{p}_{1}}{p_1} \right) \alpha ^3 - 2.\sigma [r_1;r_2].\ln \left( \dfrac{ \hat{p}_{2}}{p_2} \right) \alpha }{p_1 \left( \sigma ^2[r_1].\sigma ^2[r_2].\alpha ^4-4.\sigma ^2[r_1;r_2] \right) } \end{array}\right. } \end{aligned}$$

(26)

Notice that in Eq. (26), the parameters are time- and agent-dependent. If we assume the assets to be independent ($\sigma [r_1;r_2]=0$) and replace the variance notation $\sigma ^2[x]$ by $Var_x$, it corresponds to Eqs. (9) and (10).

Table 9 Comparison among perfect knowledge of the fundamental value, adaptive learning and adaptive learning with order-placement strategy in a market with fundamentalist and chartist outlook ($\sigma ^2_f=0.6/\sigma ^2_c=1$)

Table 10 Comparison among adaptive learning and adaptive learning with order-placement strategy in a market with fundamentalist and chartist outlook ($\sigma ^2_f=0.6/\sigma ^2_c=10$)