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Figure 1: 1987 Crash, Realized Vs. Predicted (GARCH) Volatility
12
350
300
10
250
S&P 500
200
6
Volatility
8
150
4
100
2
50
0
10
/0
2
10
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5
10
/0
8
10
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1
10
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4
10
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7
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0
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3
10
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6
10
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9
11
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1
11
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4
11
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7
11
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0
11
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3
11
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6
11
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11
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2
11
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5
11
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8
12
/0
1
12
/0
4
12
/0
7
12
/1
0
0
Date
S&P 500
< Doo
Realized Volatility
Predicted Volatility
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1.2
1
Price
0.8
0.6
0.4
0.2
75
72
69
66
63
60
57
54
51
48
45
42
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30
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6
3
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0
Days After Crash/Frenzy
Crash price
Frenzy Price
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Figure 3: Prices and Rational Agent Stock Ownership in a Crash
1.2
1.2
1
1
0.8
0.8
0.6
0.4
Rational Share
Crash Price
0.6
0.2
0.4
0
0.2
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72
69
66
63
60
57
54
51
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45
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30
27
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21
18
15
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12
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6
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Days After Crash/Frenzy
Crash price
Rational Share
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Figure 4: Current Consumption of the Old in a Crash
1.2
1.8
1.4
0.8
Price
1.2
0.6
1
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0.8
0.2
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0
Days After Crash
Price
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Figure 5: Equilibrium Crash Sizes by Proportion of Rational Agents
for Four Different Degress of Risk Aversion
Equilibrium Crash Size (As Proportion of Initial Price)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.049
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0.045
0.043
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0.011
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0.005
0.003
0.001
0
Proportion of Rational Agents
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CARA=0.11348
CARA=0.05674
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6}hi Figure 6: Prices and Rational Agent Stock Ownership in a Crash
when Naive Traders Believe that Expected Returns are Increasing in Volatility
1.2
0.6
0.5
1
0.4
0.8
0.6
0.2
Rational Share
Crash Price
0.3
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Crash price
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