Speculative bubbles and crashes： Fundamentalists and positive‐feedback trading

Figures & data

Table 1. Initial settings

Parameter	Value
Fundamentalists’ eagerness toward profits (α^f)	1
Positive‐feedback traders’ eagerness toward profits (α^c)	1
Wealth effect (β)	0.3
Funding rate (r^f, r^c)	0.1
Adjustment scale to expected price (ν, μ)	0.5
Intensity to switch strategy (φ)	0.1
Noise traders’ reaction strength (γ)	3
Noise traders’ proportion in the market (ξ)	0.3
Market’s sensitivity to the demand of stock (θ)	0.001
Total number of traders in the market (n)	1,000
Initial positive‐feedback traders’ proportion in the market (κ0)	0.35
Initial fundamental value ($p_0^{\ast }$)	100
Initial stock price (p0)	100

Figure 1. Scenarios for bubbles, crashes, and normal times.

Figure 2. Scenarios for jump processes.

Table 2. Unit root test results

	Bubble	Crash	Normal—no deviation	Normal—deviation
Augmented Dickey‐Fuller test
p‐value	0.001	0.001	0.001	0.001
Test statistics	−30.559	−51.435	−107.790	−100.318
Critical value for the test	−2.866	−2.864	−2.861	−2.861
Phillips–Perron test
p‐value	0.001	0.001	0.001	0.001
Test statistics	−30.849	−51.436	−107.786	−100.692
Critical value for the test	−3.416	−3.414	−3.412	−3.412
Kwiatkowski–Phillips–Schmidt–Shin test
p‐value	0.100	0.100	0.100	0.010
Test statistics	0.093	0.030	0.052	0.577
Critical value for the test	0.146	0.146	0.146	0.146

Figure 3. Occurrence that fundamentalists dominate as α^f vary (Case 1).

Figure 4. Occurrence of bubbles (left) and crashes (right) as α^f vary (Case 1).

Figure 5. Occurrence of normal times‐no deviation as α^f vary (Case 1).

Figure 6. ARCH effects, traders’ dominance versus kurtosis (Case 1, t‐distributed).

Notes: The horizontal axis includes four situations: no ARCH effect with fundamentalists dominate (0), no ARCH effect with positive‐feedback traders dominate (0.7), the ARCH effect with fundamentalists dominate (1), and the ARCH effect with positive‐feedback traders dominate (1.7).

Figure 7. Average correlations of demands between fundamentalists and positive‐feedback traders (Case 1).

Figure 8. Shannon entropy H_f versus H_c (Case 1, t-distributed).

Figure 9. Average $\left|H_f-H_c\right|$ (Case 1).

Figure 10. Occurrence of crashes (left) and normal times-no deviation (right) as r^c vary (Case 1).

Figure 11. Occurrence that fundamentalists dominate (left) and bubbles (right) as r^c vary (Case 1).

Figure 12. Occurrence that fundamentalists dominate as ξ vary in Case 1 (left) and Case 2 (right).

Table

Parameter	Range
Fundamentalists’ eagerness towards profits (α^f)	$\left\{0.1,0.3,0.5,0.7,0.9,1,3,5,7,9,11,13,15\right\}$
Positive‐feedback traders’ funding rate (r^c)	$\left\{0.1,0.3,0.5,0.7,0.9\right\}$
Noise traders’ reaction strength (γ)	$\left\{1,3,5,7,9\right\}$

Figure 13. Fundamentalists’ eagerness toward profits over time (based on KS test).

Table 3. Correlations between estimated α^f and stock prices, returns, or absolute returns

	Prices		Returns		Absolute returns
	KS	AD	KS	AD	KS	AD
S&P 500	lead−	n	−	−	+	+
DAX	lag −	lag −	lead +/ lag −	lead +/ lag+then −	lead −	lead −/ lag +
FTSE 100	lead −/ lag +	lead −	−	−	lead +/ lag −	lead +
CAC 40	−	−	lead +	lead +	n	lead −/lag +
SMI	lead −/ lag +	−	−	−	lead +/lag + then −	lead +/ lag −
Nikkei 225	+	+	n	lag −	n	n
KOSPI	−	−	−	–	lead +	lead +
SSE	lead +	lead +	lead +/ lag −	lead +	lead +/ lag −	−
HSI	−	lag −	lead +/ lag −	lag + then −	lead −/ lag +	lag +

Notes: “+” represents positively correlated, “−” represents negatively correlated, and “n” indicates insignificant, “lead” means that α^f is a leading variable, and “lag” means that α^f is a lag variable.

Figure 14. Positive-feedback traders’ funding rate over time (based on KS test).

Figure 15. Positive-feedback traders’ funding rate versus Fed effective rate (S&P 500).

Figure 16. Noise traders’ reaction strength over time.

Figure 17. Occurrence that fundamentalists dominate as α^f vary (Case 1, 2, and 3; t-distributed).

Table 4. Statistics for rejecting the null hypothesis

	S&P 500	DAX	FTSE 100	CAC 40	SMI	Nikkei 225	KOSPI	SSE	HSI
KS test
Mean	353.34	362.18	361.67	340.38	361.69	331.81	370.98	387.81	364.67
Std	46.62	64.38	54.14	62.82	52.56	69.68	72.59	61.24	58.78
AD test
Mean	374.29	370.77	387.32	362.34	391.61	357.68	358.80	287.31	320.89
Std	97.49	112.18	77.72	106.04	62.21	102.40	131.65	147.29	128.44

Table 5. Correlations between first‐trial and second‐trial estimated parameters

	α^f	r^c	γ
KS test
Correlation	0.43	0.76	0.84
AD test
Correlation	0.23	0.81	0.84

Figure 18. Estimate the parameters several trials.

Figure 19. Likelihoods of rejecting the null hypothesis fixing two of the optimal estimated parameters.

Figure 20. Q–Q plot of historical returns versus simulated returns.