1. Л.Ганцоож

3. S2
S1

5. Хугацааны цувааны тренд

7. Олон хүчин зүйлсийн регрессийн загвар

8. 1200
1000
800
600
300
400
200
200
100
0
0
-100
-200
50
100
150
Residual
200
250
Actual
300
350
400
Fitted

9. ADF Test Statistic
-21.77050
1% Critical Value*
-2.5703
5% Critical Value
-1.9402
10% Critical Value
-1.6160
*MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(DGOLD,2)
Method: Least Squares
Date: 05/29/10 Time: 00:59
Sample(adjusted): 1973:06 2010:03
Included observations: 9486 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
D(DGOLD(-1))
-2.380847
0.109361
-21.77050
0.0000
D(DGOLD(-1),2)
0.803371
0.079388
10.11962
0.0000
D(DGOLD(-2),2)
0.285714
0.045798
6.238609
0.0000
R-squared
0.940791
Mean dependent var
-0.047055
Adjusted R-squared
0.939610
S.D. dependent var
49.31727
S.E. of regression
25.16582
Akaike info criterion
9.295614
Sum squared resid
278026.7
Schwarz criterion
9.323383
Durbin-Watson stat
2.117979
Log likelihood
-2051.331

12. Хялбарчлах үүднээс:

13. 






Mandelbrot,1963
Fama , 1965
Barndorff-Nielsen, 1977 (GH, NIG)
Blaesild, 1983
Eberlein áîëîí Keller, 1995
Prause, 1999
Sorensen, 2003

14. 
(μ ) - áàéðëàë (measure location ),
(δ ) – øóãàìàí òýëýëò (scale),
(α ) – ñ¿¿ëèéí çóçààí (steepness of tail ),
(β ) – àññèìåòðèéí êîýôôèöèåíò (skewness of distribution ,and)
(λ ) - òàðõàëòûí àíãèéã (distribution class).

λ ∈R , μ ∈ R , δ ≥ 0, 0≤ β <α .





15. skewness,
Eberlein,
kurtosis, volatility
Keller /1998/

16. 
NIG тархалт нь өргөтгөсөн гиперболлог (GH) тархалтын
тухайн тохиолдол

GH тархалтыг 1977 онд Barndorff-Nielsen

Ялгаатай параметрчлэлийн хувьд тархалтын хэлбэрийн
тухайд 1999 онд Prause

18. 1990M11-2000M1
2000M1-2010M12

21. Parameters
Moments
alpha
Beta
Delta
Mu
Mean
Variance
Skewness
Kurtosis
1990-2000
110.6
-20.95
0.0196
0.0052
0
0.0187
-0.38913
1.6104
2000-2010
62.04
4.199
0.0294
-0.001
0.00099
0.00477
0.150513
1.67871
Parameters
Lamb
da
Alpha.bar.
19902000
-0.5
20002010
-0.5
Moments
Mu Location
Sigma
Dispersion
Gamma
Mean
Var
Skew
Kurt
2.1414
0.0052
0.0134
-0.003
0
0.0018
-0.37
2.31
1.82613
-0.001
0.0218
0.002
0
0.00047
0.15
2.53
Shape

22. Levy-èéí çàõ çýýëä ýêâèâàëåíò
îëîí òºðëèéí ÿëãààòàé ìàðòèíãàë õýìæýýñ
áàéäãààñ õºíãºëòòýé àêòèâûí ¿íèéí ïðîöåññ íü ìàðòèíãàë áîëäîã.
MGF îðøèí áàéäàã ãýæ ¿çüå

24. Put-call îïöèîíä òàâèãääàã òýãø õýìèéí íºõöëèéã
Òîðãóóëèéí ïàðàìåòð á¿õèé áîäëîãî:
Heston, 1993
Scott, 1997
Carr áîëîí Madan, 1999

25. CF àøèãëàí ïàðàìåòð¿¿äèéã ¿íýëýâ:
,
2011 îíû 3-ð ñàðûí ñ¿¿ëýýð äóóñãàâàð áîëîõ call áà put ôþ÷åðñèéí îïöèîíû
ãýðýý, ýðñäýëã¿é õ¿¿ 0.5
1449=1587exp(-0.163),
1439

26. Хугацаа
Бодит утга
Trend
Regress
ARMA
СТОХАСТИК
2010.11
1342,2
1092,92
1112,12
1271,87
1334,5
2010.12
1369,89
1102,64
1121,19
1345,67
1354,32
2011.1
1425,00
1110,41
1200,1
1279,56
1409,26
2011.2
1435,02
1119,99
1187,19
1269,24
1418,71
2011.3
1449,21
1123,91
1271,42
1301,52
1439,05
2011.4
1129,12
1162,44
1304,76
1454,37
2011.5
1131,15
1269,1
1147,2
1464,09