A multi-factor model of heterogeneous traders in a dynamic stock market

Figures & data

Figure 1. A day in the life of a typical trader.

Table 1. Summary of endogenous variables

List endogenous variables
Description	Admissibility conditions
Number of bonds	$B_{\mathit{it}}\in {\mathbf{R}}_{+}$
Stock price	$P_t$
Stock return	$r_t=\frac{P_t-P_{t-1}}{P_{t-1}}$
Number of stock shares	$S_{\mathit{it}}\in {\mathbf{R}}_{+}$
Realized portfolio weight on stock	$x_{\mathit{it}}=\frac{P_t{S}_{\mathit{it}}}{P_t{S}_{\mathit{it}}+Q_t{B}_{\mathit{it}}}\in [0,1]$
Temporary desired portfolio weight on stock	${\widehat{x}}_{\mathit{it}}\in [0,1]$
Final desired portfolio weight on stock	$x_{\mathit{it}}^{\ast }\in [0,1]$
Portfolio weights of neighborhood	${\mathbf{X}}_t^{Z_i}={\{x_{\mathit{jt}}\}}_{j\in Z_i}$
Wealth	$W_{\mathit{it}}$

Table 2. Summary of exogenous variables

Exogenous variables
Description	Admissibility conditions
Weights on neighborhood	${\mathbf{a}}_i^{Z_i}={\{a_{\mathit{ij}}\}}_{j\in Z_i}\text{s.t.}{\sum }_{j=1}^{N_i}{a}_{\mathit{ij}}=1$
Dividend	$d_t$
Moving average of dividend	${\overline{d}}_t^{h_i}=({\sum }_{k=t-h_i}^t{d}_k)/h_i$
Memory length	$h_i<\infty$
Total number of traders in the market	$N\in {\mathbf{R}}_{+}$
Number of traders in neighborhood	$N_i=|Z_i|$
Price of bonds	$Q_t=\frac{1}{r_f}$
Risk-free rate	$r_f\in [0,1]$
Tolerance level	$To{l}_i\in [0,1]$
Set of interacting traders	$Z_i$
Weight on network factor	${\mathit{\alpha }}_i\in [0,1]$
Weight on dividend factor	${\mathit{\beta }}_i\in [0,1]$
Aggression parameter of dividend factor	${\mathit{\gamma }}_i^f\in {\mathbf{R}}_{+}$
Aggression parameter of technical factor	${\mathit{\gamma }}_i^g\in {\mathbf{R}}_{+}$
Stochastic shock on the dividend	${\mathit{ϵ}}_t\sim N(0,1)$

Figure 2. The network structure of traders.

Note: This figure illustrates how 100 traders are linked. Note that each node represents a single trader.

Figure 3. Effects of changes in the agression parameters ${\mathit{\gamma }}^f$ and ${\mathit{\gamma }}^g$ on the curvatures of f(z) and g(z).

Figure 4. Bid/offer curve of a trader.

Note: The depicted curve is the graph of $\mathrm{\Delta }{S}_{\mathit{it}}^{\ast }(P)$ as a function of stock price P. Trader i switches his trading position at the point where the curve and the vertical line intersect.

Table 3. Six cases to be experimentally studied

Case number	Type description	${\mathit{\alpha }}_i$	${\mathit{\beta }}_i$	$h_i$
1	long-memory dividend trader	0	1	100
2	short-memory dividend trader	0	1	10
3	Technical trader	0	0	n.a.
4	Network trader	1	0	n.a.
5	long-memory hybrid trader	1/3	1/3	100
6	short-memory hybrid trader	1/3	1/3	10

Notes: The network and technical trader cases do not involve a data truncation issue. Hence, a varying degree of memory length ($h_i$) is not considered for these cases.

Table 4. Maintained parameter values

Parameters	Values
Number of traders (N)	100
Risk-free rate ($r_f$)	0
Aggression parameter for dividend factor (${\mathit{\gamma }}_i^f$)	5
Aggression parameter for technical factor (${\mathit{\gamma }}_i^g$)	5
Initial stock price ($P_0$)	166
Initial dividend ($d_0$)	10
Long-memory length ($h_i^{\mathit{long}}$)	100
Short-memory length ($h_i^{\mathit{short}}$)	10
Drift in the dividend process ($\overline{d}$)	0.0014
Volatility of the dividend process ($\mathit{\sigma }$)	0.0072
Lower bound of initial stock number ($S_L$)	0
Upper bound of initial stock number ($S_U$)	100
Lower bound of initial bond number ($B_L$)	0
Upper bound of initial bond number ($B_U$)	100
Lower bound of tolerance level ($To{l}_L$)	0
Upper bound of tolerance level ($To{l}_U$)	0.1
Neighborhood weights (${\mathbf{a}}_i$)	$(\frac{1}{N_i},\frac{1}{N_i},\cdots ,\frac{1}{N_i},\frac{1}{N_i})$

Notes: The neighborhood weights vary by trader since a trader’s number of links to other traders is heterogeneous. A trader i’s initial numbers of stocks and bonds will be maintained throughout all test cases.

Figure 5. Stock market dynamics: Long-memory dividend trader (Case 1).

The horizontal x-axis denotes trading periods.

Figure 6. Stock market dynamics: Short-memory dividend trader (Case 2).

The horizontal x-axis denotes trading periods.

Table 5. Summary of stock return statistical properties: Dividend traders

	1st moment	2nd moment	Skewness	Excessive Kurtosis
Long-memory	0.0015	0.0127	0.5885	5.7744
	(0.0001)	(0.0006)	(0.1863)	(0.779)
Short-memory	0.0015	0.0099	1.4434	8.5657
	(0.0001)	(0.0003)	(0.2097)	(1.4881)

Notes: Values in parentheses denote standard deviation from 1,000 runs under the calibrated dividend process. Excessive kurtosis is defined as the fourth moment of a distribution less the fourth moment of a standard normal distribution.

Table 6. ARCH effect tests for stock returns: Dividend traders

	$l=1$	$l=2$	$l=3$	$l=4$	$l=5$	$l=6$
Long-memory	606.3	310	214.6	163.1	130.6	108.8
	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)
Short-memory	79.8	40.91	27.3	20.75	16.72	14.1
	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.535e-16)	(6.753e-16)

Note: Estimates are F-statistics of LM tests proposed by Engle (1982). Values in parentheses denote p-values of F-statistics of LM tests. Notice that l denotes a number of lags in each test.

Figure 7. Stock market dynamics: Technical trader (Case 3).

The horizontal x-axis denotes trading periods. The disconnected portions of plots denote the periods in which no trades occur.

Figure 8. Trading profile: Technical trader (Case 3).

The horizontal x-axis denotes trading periods.

Figure 9. Stock market dynamics: Network trader (Case 4).

The horizontal x-axis denotes trading periods.

Figure 10. Stock holding comparison between network traders.

Figure 11. Stock market dynamics: long-memory hybrid trader (Case 5).

The horizontal x-axis denotes trading periods.

Table 7. Summary of stock return statistical properties: Hybrid traders ARCH effect tests for stock returns: Hybrid traders

	1st moment	2nd moment	Skewness	Excessive Kurtosis
Long-memory	0.0014	0.0062	4.9082	31.5835
	(0.0001)	(0.0004)	(0.47)01	(6.3364)
Short-memory	0.0017	0.0064	4.7670	28.8932
	(0.0001)	(0.0004)	(0.3735)	(5.3781)

Notes: Values in parentheses denote standard deviation from 1,000 runs under calibrated dividend process. Excessive kurtosis is defined as the fourth moment of a distribution less the fourth moment of a standard normal distribution.

Table 8. ARCH effect tests for stock returns: Hybrid traders

	$l=1$	$l=2$	$l=3$	$l=4$	$l=5$	$l=6$
Long-memory	3,991	3,224	2,358	1,785	1,427	1,189
	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)
Short-memory	2,575	1,943	1,434	1,102	883.3	736.1
	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)	(2.2e-16)

Notes: Estimates are F-statistics of LM tests proposed by Engle (1982). Values in parentheses denote p-values of F-statistics of LM tests. Notice that l denotes a number of lags in each test.

Figure 12. Stock market dynamics: Short-memory hybrid trader (Case 6).

The horizontal x-axis denotes trading periods.

Figure 13. Recovery phase dynamics: Long-memory sases.

Figure 14. Recovery phase dynamics: Short-memory cases.

Figure 15. Average wealth by trading type (AvgMt).

Figures are based on 100 simulation runs. The benchmark denotes the average wealth in which stock shares are equally distributed to all traders and no trades occur. Note that real wealth levels are normalized by a constant number for easier illustrations.

Table 9. Growth rate of average wealth

	Mean (%)	Standard deviation (%)
Long-memory Diviend Trader	0.15	0.07
Short-memory Diviend Trader	0.14	0.07
Technical Trader	0.13	0.12
Network Trader	0.13	0.09
Long-memory Hybrid Trader	0.14	0.07
Short-memory Hybrid Trader	0.13	0.07

Note: These estimates are based on 100 simulation runs. In each run, the growth rate of average wealth is based on $Avg{M}_r^t)$, and the initial 2,000 observations are discarded to eliminate the effects of initial conditions; i.e. Mean = $\left({\sum }_{r=1}^{N_r}{\sum }_{t=2001}^T\frac{\mathrm{\Delta }Avg{M}_t^r}{Avg{M}_{t-1}^r}\right)/(N_r+3000)$ where $N_r$ = 100 and T = 5000.

Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987–1007.

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