Appendix to Empirical Results, Model Group 2
A.1 Summary Statistics
The summary statistics are in Tables A.1 and A.2, the histograms in Figs. A.1, A.2, A.3, A.4, A.5, A.6, A.7, A.8, A.9, A.10, A.11, A.12 and A.13.
Table A.1.Summary Statistics models H, HXJ, HVJ, HVJG, HXVJ and HXVJG
For each trading day between 01 October 1999 and 30 September 2002, all parameters for the relevant models were independently estimated. Futures and forward contracts lead to implicit estimates min-imising the RMSE (root mean squared error) for all contracts. In this summary statistics table, mean, median and standard deviation of each parameter are reported, as well as of the RMSE. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
α β √v κ θ σ ρ
Model H
Mean -4,24 0,01 2,56 52,83 8,65 0,60 0,59
Median -1,01 0,01 1,90 13,50 1,86 0,44 0,95
Std. Dev. 11,57 0,02 2,76 70,01 25,81 0,71 0,65
Model HXJ
Mean -2,73 0,01 2,42 84,96 5,85 0,47 0,23
Median -0,78 0,01 2,27 39,42 1,06 0,24 0,57
Std. Dev. 9,04 0,02 1,55 84,67 25,37 0,82 0,79
Model HVJ
Mean -4,97 0,01 2,66 52,24 10,14 0,63 0,45
Median -0,96 0,01 2,20 11,81 1,66 0,29 0,71
Std. Dev. 12,19 0,02 1,82 70,47 37,30 1,34 0,69
Model HVJG
Mean -3,86 0,01 2,30 47,14 6,46 0,40 0,60
Median -0,65 0,003 1,89 10,56 1,09 0,30 0,80
Std. Dev. 8,01 0,01 1,56 68,83 13,71 0,37 0,55
Model HXVJ
Mean -3,37 0,002 2,10 37,15 6,28 0,39 0,47
Median -1,30 0,001 1,80 9,61 1,57 0,28 0,64
Std. Dev. 5,56 0,01 1,44 59,32 14,14 0,50 0,60
Model HXVJG
Mean -3,45 0,005 2,16 45,59 8,09 0,25 0,20
Median -1,54 0,003 1,81 11,08 1,68 0,16 0,28
Std. Dev. 6,42 0,02 1,99 66,78 34,37 0,53 0,59
Table A.2.Summary Statistics models H, HXJ, HVJ, HVJG, HXVJ and HXVJG – Continued
λ µJ σJ η γ ρJ RMSE
Model H
Mean 15,31
Median 14,48
Std. Dev. 6,20
Model HXJ
Mean 3,51 0,12 0,29 16,42
Median 0,62 0,13 0,18 15,02
Std. Dev. 15,12 0,30 0,43 7,65
Model HVJ
Mean 12,15 0,28 14,65
Median 2,98 0,02 14,06
Std. Dev. 40,98 1,43 4,05
Model HVJG
Mean 7,54 0,08 18,04 14,20
Median 2,64 0,02 2,20 13,82
Std. Dev. 28,15 0,29 294,07 2,86
Model HXVJ
Mean 3,66 0,14 0,29 0,12 0,98 14,20
Median 1,00 0,13 0,20 0,03 0,86 13,86
Std. Dev. 15,77 0,22 0,39 0,96 1,37 2,79
Model HXVJG
Mean 6,24 0,07 0,24 0,06 12,69 0,67 14,45
Median 3,34 0,09 0,20 0,01 1,73 0,80 13,97
Std. Dev. 18,71 0,29 0,20 0,67 103,28 1,49 3,60
0 5 10 15 20 25 30
The RMSEs for all estimations are plotted in a histogram for each model. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
-5 -4 -3 -2 -1 0 1 2
The parameter values ofαfor all estimations are plotted in a histogram for each model. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
-0.04 -0.02 0.00 0.02 0.04
-0.04 -0.02 0.00 0.02 0.04
0 100 200 300
HXJ
-0.04 -0.02 0.00 0.02 0.04
0
-0.04 -0.02 0.00 0.02 0.04
0
-0.04 -0.02 0.00 0.02 0.04
0
-0.04 -0.02 0.00 0.02 0.04
0
The parameter values ofβfor all estimations are plotted in a histogram for each model. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
0 1 2 3 4 5 6 7 8 9 10 The parameter values of√
vfor all estimations are plotted in a histogram for each model. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
0 50 100 150 200 250
The parameter values ofκfor all estimations are plotted in a histogram for each model. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
0.0 0.5 1.0 1.5 2.0
The parameter values ofθfor all estimations are plotted in a histogram for each model. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
0.0 0.1 0.2 0.3 0.4 0.5
The parameter values ofσfor all estimations are plotted in a histogram for each model. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
-1.0 -0.5 0.0 0.5 1.0
The parameter values ofρfor all estimations are plotted in a histogram for each model. The models are:
• H: a Heston-like model,
• HXJ: the same model with a normal-distributed jump in the first state variable,
• HVJ: with an exponential-distributed jump in the second state variable,
• HVJG: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0
20
40 HXVJG
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0 50 100 150
HXVJ
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0 20
40 HVJG
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0 20
40 HVJ
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0 50 100 150
200 HXJ
λ
Figure A.9.Histogramsλ
The parameter values ofλfor all estimations are plotted in a histogram for each model. The models are:
• HXJ: a Heston-like model with a normal-distributed jump in the first state variable,
• HVJ: the same Heston-like model with an exponential-distributed jump in the second state variable,
• HVJGS: with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
-0.4 -0.2 0.0 0.2 0.4
Figure A.10.HistogramsµJ andσJ
The parameter values ofµJ andσJ for all estimations are plotted in a histogram for each parameter and for each model. The models are:
• HXJ: a Heston-like model with a normal-distributed jump in the first state variable,
• HXVJ: the same Heston-like model with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
0.00 0.05 0.10 0.15 0.20 0
100 200
HVJ
0.00 0.05 0.10 0.15 0.20
0 100 200
HVJG
0.00 0.05 0.10 0.15 0.20
0 100 200
HXVJ
0.00 0.05 0.10 0.15 0.20
0 100 200 300
HXVJG
η
Figure A.11.Histogramsη
The parameter values ofηfor all estimations are plotted in a histogram for each model. The models are:
• HVJ: a Heston-like model with an exponential-distributed jump in the second state variable,
• HVJG: the same Heston-like model with a Γ-distributed jump in the second state variable,
• HXVJ: with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
0 5 10 15 20 25 30 0
50 100 150 200
HXVJG
0 5 10 15 20 25 30
0 50 100 150
HVJG
γ
Figure A.12.Histogramsγ
The parameter values ofγfor all estimations are plotted in a histogram for each model. The models are:
• HVJG: a Heston-like model with a Γ-distributed jump in the second state variable,
• HXVJG: the same Heston-like model with a normal-distributed jump in the first state variable and a Γ-distributed jump in the second state variable, both jumps also occurring simultaneously.
-3 -2 -1 0 1 2 3 0
50 100
HXVJG
ρJ
-3 -2 -1 0 1 2 3
0
50 HXVJ
Figure A.13.HistogramsρJ
The parameter values ofρJ for all estimations are plotted in a histogram for each model. The models are:
• HXVJ: a Heston-like model with a normal-distributed jump in the first state variable and an exponential-distributed jump in the second state variable, both jumps occurring simultaneously,
• HXVJG: the same model, but with a Γ-distributed jump in the second state variable.