802
BACKTESTING VALUE AT RISK MODELS IN THE PRESENCE OF STRUCTURAL
BREAK ON THE ROMANIAN AND HUNGARIAN STOCK MARKETS
Zapodeanu Daniela, Kulcsa Edina, Cociuba Mihail Ioan
Uni e si y o O adea, Facul y o Economics, Depa men o Finance, O adea, Romania
[email protected]
kulcsa [email protected]
[email protected]
Abs ac : T ansac ions on inancial ma ke s a e associa ed wi h a iabili y, isk and
unce ain y, so quan i ica ion o isk has a g ea impo ance. Beside S anda d De ia ion
and Va iance, one o he mos in ol ed isk measu e me hods is Value-a -Risk (VaR). In
his s udy, we use daily e u n o he s ock index om Romania (BET) and Hunga y
(BUX) o he 01:2007 - 02:2013 pe iods in o de o es he in luence o s uc u al b eaks
on he VaR me ics. We ind ou ha he ARCH phenomenon is p esen , so we use he
GARCH amily models. The s uc u al b eaks in he se ies mean and a iance a e
iden i ied using he Zi o -And ews es and PELT algo i hm, he s uc u al b eak da es
a e cap u ed using dummy a iables in he GARCH models (s uc-GARCH), he selec ion
o models is done using he in o ma ional c i e ion [Akaike, Schwa z, Log-likelihood]. The
esul s o p esen esea ch show a g ea e ola ili y associa ed wi h a highe isk le el in
case o Romanian s ock index. The s ock ma ke indices e u n ollows a nega i ely
skewed and lep oku ic dis ibu ions o ms ei he in wo cases, so is unspeci ic a no mal
dis ibu ion. A e applying abo e men ioned es s we can conclude ha he e a e eigh
s uc u al b eaks in BET index e u ns a iance and he e a e i e b eakpoin s in case o
BUX. The b eakpoin s in mean show e y closely esul s in ime, o BET in Feb ua y
2009 and o BUX Ma ch 2009. Back es ing VaR models a e done by measu ing he
numbe o imes he loss is g ea e han he VaR o ecas . The i s s ep o uncondi ional
co e age es ing consis s in compa ing o ac ion o VaR iola ion o a pa icula isk
model. The independence es ing i is e y impo an ool in back- es ing, because i is
no he same ha he VaR iola ions a e di e en ia ed in ime o he e a e clus e ed in
some ce ain pe iod. By checking he independence es , we ha e he possibili y o
disco e and ejec he model wi h clus e ed hi sequence. Tes ing he in luence o
s uc u al b eaks on VaR we ind ha inco po a ing s uc u al b eaks in he GJR-GARCH
models gene a es lowe iola ions when compa ing wi h he plain GJR-GARCH o
RiskMe ics me hodology.
Keywo ds: s ock ma ke , s uc u al b eak, Value a Risk, GARCH
JEL classi ica ion: G17
1. In oduc ion
Nowadays, economic and inancial en i onmen mos dominan cha ac e is ics
a e ins abili y, a iabili y, isk and unce ain y, bu a well known economic p inciple says:
“no isk means no gains”. When we deal wi h isk, unce ain y and ola ili y, i s i is
essen ial o ea he main di e ence be ween hese h ee concep s. Acco ding o
Keynesian app oach, he e isn’ a signi ican di e ence be ween i s wo concep s, Knigh
conside s he con a y ha he e is a sha p dis inc ion be ween isk and unce ain y in his
amous wo k “Risk, Unce ain y and P o i ” (1921). He conside s ha he mos impo an
di e ence be ween isk and unce ain y consis s in he possibili y o quan i ying, so in case
o isk we could make measu emen s bu in he case o unce ain y we can’ . Wha abou
a iabili y? In some cases, he concep o a iabili y is conside ed o be he main
803
componen o isk besides unce ain y (Molak, 1997; Cullen–F ey, 1999). O he s conside
he essence o his ep esen he empo al and spa ial he e ogenei y o alues (Molak,
1997).
Risk canno be comple ely a oided o he inancial ma ke s pa icipan s, bu he e
a e many ways o managing and minimizing i . This pape aims o p esen he p incipal
isk ca ego ies which a e speci ic o inancial ma ke s p oduc s, how hey a ec he s ock
ma ke pa icipan ’s beha io and also he choosing o isk managemen al e na i es. The
main objec i e o his pape consis s in quan i ica ion o isk wi h VaR me hod o wo
neighbo ing coun ies main s ock index e u ns: BET o Romania and BUX o Hunga y,
and es ing he in luence o s uc u al b eaks in mean and a iance on he VaR me ics.
The emaining o he a icle is o ganized as ollows: in sec ion 2 we e iew he
li e a u e wi h he main s eps o isk managemen and he mos used isk quan i ica ion
me hods, concen a ing on Value-a -Risk me hod and on his ad an ages and limi s. In
Sec ion 3 we p esen he esea ch me hodology, consis ing in ma hema ical backg ound
o Value-a -Risk, he main cha ac e is ics o GARCH models and used es s o alida ion
o s uc u al b eak poin s. The da a analysis pa con ains he e olu ion o e u ns in he
analyzed pe iod and he main s a is ics o i , we also p esen ed in his pa he esul s o
s a iona y es and he da es o s uc u al b eaks in means and a iance o s udied s ock
ma ke index. The nex pa shows he esul s o s udy, exac ly he equa ion o mean. The
las pa o s udy show he main conclusions a e analyzing hese wo s ock ma ke index
isk.
2. Li e a u e e iew
Acco ding o Ho che (2005) he isk managemen is a e y b oad concep , which
includes mo e s eps. Fi s and he mos impo an s ep ep esen he iden i ica ion and
quan i ica ion o he in e nal and ex e nal isk ac o s, and he speci ic isk ca ego ies
which could a ec expec ed gains and e u ns; he second is anking o isks by p io i y
and possible losses; nex s ep de ine a isk ole ance le el, which can be suppo ed; he
las one and also he mos consis en s ep is de eloping he isk managemen s a egies,
which includes also isk minimizing me hods. In he isk managemen p ocess, we y o
concen a e in his pape o he i s s ep, on isk and isk ac o iden i ica ion p ocess and
especially o isk quan i ica ion me hods.
One o he mos used me hods in inancial ma ke s isk explo a ion is Value a
Risk (VaR), which was de eloped in he ’90 yea s by J.P. Mo gan. In his pe iod he
me hod was used wi h g ea success by cen al banks, and a e ha becomes mo e
popula among inancial ins i u ions (Chen, 2007). In ou days, his me hod i ’s also used
a company le el in inancial isk quan i ica ion like ma ke isk, c edi isk, liquidi y isk
e c. The VaR me hod is o en used o es ima e he le el o exchange a e isk, bu is also
sui able o po olio isk measu emen . Based on s a is ical p obabili y es ima ions, he
essence o VaR me hod consis in quan i ica ion o maximum po en ial loss, which esul
om ma ke ac o s a iabili y. The e o e, he VaR de e mine he le el o maximum
expec ed loss, o di e en ime pe iods om 1 day o 100 days, a speci ic con idence
le el 95%, 97, 5%, o 99%. The VaR is he only one me hod which cha ac e ized he le el
o po olio, in es men isk by a numbe . So, one o a g ea ad an age is ha
cha ac e ized isk wi h a numbe . Ano he big ad an age is ha could be well comple ed
by o he isk measu emen me hods, such as scena io analysis and s ess es ing and
sensi i i y analysis me hods. Manganelli & Engle, (2001) classi y he Value-a - isk models
in h ee ca ego ies: pa ame ic (RiskMe ics, Ga ch), nonpa ame ic (his o ical simula ion,
hyb id model), semi-pa ame ic (Ex eme Value Theo y, CAViaR, quasi-maximum
likelihood Ga ch). In p ac ice, he applica ion o VaR knows h ee me hods: i s based on
his o ical da a, second he me hod o a iance and co a iance o pa ame ic based
804
me hod, and he hi d based on Mon e Ca lo simula ion (Ho che , 2005). The ad an age
o he i s me hod is ha pe mi s quick and easy usage, bu he las wo me hods p o ide
much mo e accu a e esul s and ha e a wide ange o applica ions. The VaR calcula ion
based on his o ical da a assumes ha pas da a and e en s also cha ac e ized he u u e
e en s. The VaR es ima ion based on Mon e Ca lo simula ion is he mos lexible me hod,
which basically consis s in andom numbe gene a o , which is o en used in inancial
modeling. The success o his me hod is de e mined by he success o used alua ion
me hod, bu also depends on he used pa ame e s in he simula ion (Ray, 2010).
The majo disad an age o VaR isk measu emen me hod is ha i couldn’ be
applied in he ex eme, shock conjunc u e, such as inancial c isis. The ab up and
signi ican luc ua ions o isk ac o s g ea ly de o m he e iciency VaR me hod. To
elimina e his p oblem, A zne e al. (1997, 1999) de eloped he Expec ed Sho all (ES)
concep , which cha ac e ized he condi ional expec ed loss which exceeds he alue o
loss ecei ed by using o VaR me hod (Yamai e al, 2002). The ES me hod is closely
ela ed o he VaR, because we could ob ain he Expec ed Sho all alue om VaR alue
by a aching p obabili y le els o expec ed loss. The g ea ad an age o ES me hod is ha
ake in o accoun he possibili y o ex eme si ua ions (Ke kho , 2003). A zne e al. (1999)
conside s Expec ed Sho all (ES) me hod a mo e cohe en isk measu e me hod han
Value a Risk (VaR). Cuoco, He, Issaenko (2001) in hei esea ch, conclude ha he
mul iple uses o VaR and ES me hods gene a e equally esul s.
2. Me hodology
The Value-a - isk is de ined by (McNeil e al, 2005) as a “... some con idence
le el he VaR o he po olio a he con idence le el is gi en by he smalles
numbe l such ha he p obabili y ha he loss L exceeds l is no la ge han “.
Ma hema ically we can w i e VaR as p obabili y:
(1)
Engle (1982) in his seminal pape p oposed he au o eg essi e condi ional
he e oskedas ici y models which iew he a iance as being dependen o he e o s, he
ARCH model was ex ended in he Gene alized Au o eg essi e Condi ional
He e oskedas ici y (GARCH) by Bolle sle [1986] which has he ollowing o m:
(2)
(3)
(4)
Because GARCH model ea s he shocks symme ically while on he inancial ma ke s
bad news gene a es mo e ola ili y han he good news. Glos en, Jaganna han and
Runkle [1993] p oposed Th eshold GARCH which ea s di e en ly he bad-good news
in luence on he asse s p ices. I is an asymme ic model in which he condi ional ola ili y
is:
(5)
whe e: d = 1 i e <0 o d = 0 i e > 0.
The de ec ion o b eakpoin s in ime se ies can be posed (Killick, Fea nhead, &
Eckley, 2012) as a hypo hesis es whe e H0 is he null hypo hesis whe e he e is no
changepoin (m=0) and he al e na i e hypo hesis H1 whe e we ha e a leas 1
changepoin (m>=1). Killick e al. (2012) de eloped he P uned Exac Linea Time (PELT)
me hod which es o changepoin s using he ollowing s a is ical c i e ia: penalized
likelihood, quasi-likelihood and CUSUM; he b eakpoin analysis is ca ied in he mean,
a iance and bo h mean/ a iance o he se ies. The PELT me hod is implemen ed as an
R packaged (changepoin package). The es s a is ics used in he PELT me hod
implemen a ion (Killick e al., 2012) has he ollowing null hypo hesis H0: no b eakpoin
and al e na i e hypo hesis H1: one b eakpoin τ1, wi h τ1 {1, 2,…, n-1}. By ejec ing he null
805
hypo hesis H0, a changepoin is de ec ed and i is es ima ed by maximizing he log-
likelihood.
We apply he ollowing uni - oo es : Augmen ed Dickey–Fulle es (ADF) and
Phillips–Pe on (PP) and he Zi o and And ews (1992) which ex ended he Dickey–Fulle
es by allowing o a b eak in in e cep , end and bo h (model C). The s uc u al b eak
will be in oduce in he GARCH model equa ions using a dummy a iable, also in o de o
elimina e au oco ela ion lags o dependen a iable will be in oduce in he main equa ion,
he model will be as ollows:
(6)
(7)
whe e Dmi ,..., Dhi a e dummy a iables which ake he alue 0 be o e he b eakpoin and
1 a e he b eakpoin un il he end o he pe iod.
Back es ing VaR models is done by measu ing he numbe o imes he loss is
g ea e han he VaR o ecas , he numbe o VaR iola ions can be de ine as:
(8)
Fo an imp o ed isk model i is necessa y o p edic he p obabili y o VaR iola ions,
no ed wi h p. The VaR iola ion p obabili y depends on he co e age a e o VaR, he hi
sequence om a co ec ly speci ied isk model looks like a sequence o andom osses o
coin (Ch is o e sen, 2012). The i s s ep o uncondi ional co e age es ing consis s in
compa ing o ac ion o VaR iola ion o a pa icula isk model. The independence
es ing i is e y impo an ool in back- es ing, because i is no he same ha he VaR
iola ions a e di e en ia ed in ime o he e a e clus e ed in some ce ain pe iod. By
checking he independence es , we ha e he possibili y o disco e and ejec he model
wi h clus e ed hi sequence. The i s s ep consis in assuming ha iola ions a e
dependen o e ime, which could be desc ibed be e by Ma ko ansi ion p obabili y
ma ix. The Ma ko p ope y e e s o he assump ion ha only oday’s ou come is
de e minan o omo ow ou come, he e olu ion om he pas doesn’ ma e , whe e π11
is he p obabili y o omo ow being a iola ion gi en oday is also a iola ion and π01 is
he p obabili y o omo ow being a iola ion gi en oday is no a iola ion. Fo checking
he independence π01 = π11, a likelihood a io es is used. A e using he independence
es , he nex s ep o co ec co e age is he condi ional co e age es , which checks ha
π01 = π11 = p. The es is compu ed by summing o uncondi ional co e age and
independence es .
3. Da a analysis
The analyzed se ies a e wo s ock exchange index: BET o Romania and BUX
o Hunga y, he analyzed pe iod is be ween 01:2007 - 03:2013, daily se ies; he da a a e
ob ained om www.b b. o and www.be .hu ; he econome ics so wa e used a e G e l
and R package s ucchange, in o de o ob ain e u ns om he daily se ies we apply he
ollowing ans o ma ion:
whe e i= BET, BUX.
806
Fig. 1 The e olu ion o _BET and _BUX
Table 1. Desc ip i e s a is ics
Se ies
Mean
SD
SK
KT
Q(12)
Q2(12)
JB
_BET
-0.0259
2.00
-1.19
17.40
48
290
13605
_BUX
-0.0212
1.88
-0.03
9.27
62
1088
2515
No es: SD, SK, KT, and JB deno e s anda d de ia ion, skewness, ku osis, and Ja que-Be a s a is ic,
espec i ely. The Ljung–Box s a is ics, Q and Q2 s a checks o se ial co ela ion o e u ns and squa ed e u ns
up o he 12 o de , he c i ical alue o he Q(12) espec i ely a e 26.21, a 1% signi icance le el. The c i ical
alue o he Ja que-Be a (JB) es is 5.991 a 5% signi icance le el.
Table 1 p esen s he desc ip i e s a is ics o he e u ns on daily se ies o closing
p ices o wo s ock exchange indices: BET o Romania and BUX o Hunga y. Fo he
analyzed pe iod, 01:2007 - 03:2013 bo h e u ns a e nega i e, he lowes alue is
obse ed in case o Hunga ian s ock index, -0.0212; he s anda d de ia ion, e eals a
highe ola ili y in he case o Romanian s ock index, BET. The s ock ma ke indices
dis ibu ions a e nega i ely skewed, he ku osis alue is highe han he no mal
dis ibu ion ku osis alue, which is 3, so we could see o he bo h analyzed se ies, ha
he se ies ha e lep oku ic dis ibu ions. The Ja que-Be a es indica es ha he no mali y
o dis ibu ion o _BET and _BUX is ejec ed, also he Q-s a is ics indica es se ial
co ela ion o e u ns which will be emo ed using lag e ms and he se ial co ela ion o
squa ed e u ns Q2 sugges s he exis ence o he ARCH e ec . Tes ing o he ARCH
e ec is done using he LM es , he LM alue o he _BET, _BUX a e 148.19,
espec i ely 381.40, which compa ed wi h he c i ical alues shows he p esence o
ARCH(1) e ec s, so he se ies will be modeled using he GARCH amily models.
Table 2. Uni Roo /s a iona i y es
Se ies
ADF
PP
KPSS
_BET
-9.0041
-37.419
0.27484
_BUX
-16.197
-36.523
0.088335
MacKinnon’s 1% c i ical alue is -3.46 o he ADF and PP es s, he c i ical alue o he KPSS es is 0.739 a
1% signi icance le el, * deno e signi icance a 1% le els.

807
Table 2 p esen he esul o he uni oo es , whe e ADF and PP es has he null
hypo hesis ha he se ies is in eg a ed o o de 1, while KPSS null hypo hesis is ha he
se ies is s a iona y, and Zi o -And ews es allows o a b eak in in e cep , end o bo h.
We ind ha o _BET and _BUX we ejec he uni - oo hypo hesis based on he ADF
es , PP es . We ind ha he e u ns don’ ha e a uni - oo , he KPSS accep he
s a iona i y o e u ns o bo h coun ies. Based on he esul s om he ADF, PP and KPSS
es s we apply he Zi o -And ews es and b eakpoin analysis in o de o cap u e any
b eak in mean, a iance o bo h.
Table 3. B eakpoin da es
Se ies
Mean b eakpoin
Zi o -And ews
Va iance b eakpoin
_BET
2009-02-13
2009-02-24***
(-11.01)
2007-12-19, 2008-09-11, 2008-
10
-27, 2009-08-19, 2010-04-
30,
2010
-07-01, 2011-07-
29, 2011-
10-28
_BUX
2009-03-11
2009-03-11***
( -12.51)
2008-09-12, 2008-11-21, 2010-
06
-03, 2011-07-28, 2012-01-
19
Tes s a is ics in pa en hesis, he c i ical alue o Zi o -And ews es is -5.34 a 1% signi icance le el.
Table 3. and Figu e 2. p esen s he es ima ed b eakpoin s, and in o de o cap u e he
changes in mean and a iance dummy a iables will be in oduce in he GARCH model
equa ions. We ind o he _BET se ies he e is wo b eaks in he mean equa ion bo h o
hem in Feb ua y 2009 and eigh b eaks in he a iance: 2007-12-19, 2008-09-1, 2008-
10-27, 2009-08-19, 2010-04-30, 2010-07-01, 2011-07-29, 2011-10-28; o he _BUX
se ies he e is one b eak in he mean equa ion in Ma ch 2009 and i e b eaks in he
a iance: 2008-09-12, 2008-11-21, 2010-06-03, 2011-07-28, 2012-01-19.
808
Fig. 2. Va iance b eakpoin s in BET, BUX
4. Resul s
The GARCH model will inco po a e he non-no mali y o BET and BUX e u n, he
Ja que-Be a es indica es ha he se ies ollow a non-no mal dis ibu ion, by using he
no mal and s uden dis ibu ion o he e o s, also h eshold models will be es ed due o
he asymme ies o inancial se ies, he _BET and _BUX se ies a e skewed o he le
(nega i ely skewed); in o de o elimina e he se ial co ela ion au o eg essi e lags will be
used in he mean equa ion.
Table 4. GARCH models
_BET
_BUX
cons an
no
no
AR lags
1**, 13**
2**, 4**
Mean b eakpoin
(1)
no
no
Zi o -And ews
no
no
Va iance
b eakpoin
2007-12-19**, 2008-09-11*, 2009-
08-19**, 2010-04-30*, 2010-
07-
01**, 2011
-07-29**, 2011-10-
28**
2008-09-12**
2010-06-03**
GARCH (p,q)
(1,1)***
GJR
(1,1)***
GJR
Q(6)
3.80
7.01
Q(12)
11.21
10.52
Q2(6)
6.05
2.77
Q2(12)
14.00
5.25
Whe e p is he numbe o lagged h e ms and q he numbe o e2 e ms. *, **, *** deno e signi icance a 10%,
5%, and 1% le els.
809
The GARCH models a e selec ed based on hei in o ma ion c i e ion, no
p esen ed he e, and he s abili y o he models. Based on he esul s o GARCH models
es ima ion, we can obse e in Table 4., wi h bold, he signi ican b eakpoin s, we ind no
b eaks in he in e cep and we ind 7 s uc u al b eaks in he a iance e m in he case o
Romanian s ock ma ke and 2 s uc u al b eaks in he a iance e m o Hunga ian s ock
ma ke .
Back es ing he Value a Risk is done on he RiskMe ics model, he anilla
GJR(1,1) and he a iance b eak GJR model, he GARCH in mean model is no
ep esen a i e o he analyzed se ies.
Table 5. Numbe o hi s
BET
RiskMe
ics 1%
RiskMe
ics 5%
GJR(1,1
)
1%
GJR(1,1
)
5%
GJR(1,1
) 1%
wi h
b eaks
GJR(1,1)
5%
wi h
b eaks
hi s
2
14
2
9
1
6
consecu i e
hi s
0
2
0
0
0
0
β
0.13%
0.92%
0.13%
0.59%
0.07%
0.40%
BUX
RiskMe
ics 1%
RiskMe
ics 5%
GJR(1,1
)
1%
GJR(1,1
)
5%
GJR(1,1
) 1%
wi h
b eaks
GJR(1,1)
5%
wi h
b eaks
hi s
1
5
0
3
0
1
consecu i e
hi s
0
0
0
0
0
0
β
0.07%
0.33%
0.00%
0.20%
0.00%
0.07%
5. Conclusion
We can obse e om he analyzed imes se ies ha nei he BET o BUX indices
don’ ollow a no mal dis ibu ion, in bo h cases we can see nega i ely skewed and
lep oku ic dis ibu ions o daily e u ns. Analyzing s anda d de ia ion we can conclude
ha a g ea e isk and ola ili y is speci ic o Romanian s ock ma ke index e u n.
Because he no mali y and LB es s e eal ha he ARCH e ec is p esen in case o BET
and BUX we applied GARCH models. A e we applied he b eakpoin s analysis we ind
ou ha s uc u al b eakpoin s a e p esen in bo h cases. Fo BET index we obse e eigh
s uc u al b eaks in a iance, while in he case o BUX only i e b eakpoin s a e p esen .
The Zi o -And ews and PELT me hod shows ha he mean b eakpoin s a e e y close o
he wo ma ke s, Feb ua y 2009 o BET index e u ns and Ma ch 2009 o BUX index
810
e u ns, which could be ela ed wi h he consequences o economic and inancial c isis
which debu ed in Cen al and Eas e n Eu ope egion in Oc obe 2008.
Tes ing he in luence o s uc u al b eaks on VaR we ind ha inco po a ing
s uc u al b eaks in he GJR-GARCH models gene a es lowe iola ions when compa ing
wi h he plain GJR-GARCH o RiskMe ics me hodology.
Re e ences
■ A zne , P., Delbaen, F., Ebe , J.-M., Hea h, D. (1997) Thinking cohe en ly. RISK 10(11).
■ A zne , P., Delbaen, F., Ebe , J.-M., Hea h, D. (1999) Cohe en measu es o isk.
Ma h. Fin. 9(3), 203–228.
■ Bolle sle , T. (1986). Gene alized au o eg essi e condi ional he e oskedas ici y. Jou nal
o econome ics, 31(3), 307–327. Else ie .
■ Chen, F. - M.Sc. Ying (2007): Adap i e Risk Managemen , Disse a ion, 27 p.
■ Ch is o e sen, Pe e F. Elemen s o inancial isk managemen . Academic P ess, 2012.
■ Cullen, A.C., F ey C.H. (1999): P obabilis ic Techniques in Exposu e Assessmen : A
Handbook o Dealing Va iabili y and Unce ain y in Models and Inpu s, Plenum P ess,
New Yo k, 335 p
■ Cuoco, D., H. He, and S. Issaenko (2001), Op imal Dynamic T ading S a egies wi h Risk
Limi s, Manusc ip , Yale Uni e si y.
■ Engle, R. F. (1982). Au o eg essi e Condi ional He e oscedas ici y wi h Es ima es o he
Va iance o Uni ed Kingdom In la ion. Econome ica, 50(4), 987. doi:10.2307/1912773
■ Glos en, Law ence R., Ra i Jaganna han, and Da id E. Runkle. "On he ela ion be ween
he expec ed alue and he ola ili y o he nominal excess e u n on s ocks." The jou nal
o inance 48.5 (1993): 1779-1801.
■ Ho che , K. A. (2005): Essen ials o inancial isk managemen , John Wiley & Sons, Inc.,
Hoboken, New Je sey., 217 p.
■ Ke kho , F., Lambe us, J. (2003): Model Risk Analysis o Risk Managemen and Op ion
P icing, dok o i é ekezés, Tilbu g
■ Killick, Rebecca, Paul Fea nhead, and I. A. Eckley. "Op imal de ec ion o changepoin s
wi h a linea compu a ional cos ." Jou nal o he Ame ican S a is ical Associa ion 107.500
(2012): 1590-1598.
■ Knigh , F. H. (1921): Risk, Unce ain y and P o i . Rep in London School o Economics,
1933
■ Manganelli, S., & Engle, R. F. (2001). Value a isk models in inance. ECB Wo king Pape
No. 75.
■ McNeil, A.; F ey, R.; Emb ech s, P. (2005). Quan i a i e Risk Managemen : Concep s
Techniques and Tools. P ince on Uni e si y P ess
■ Molak, V. (1997): Fundamen als o isk analysis and isk managemen . Lewis Publishe s
(CRC P ess, Inc.), New Yo k, 457 p.
■ Ray, C. (2010): Ex eme isk managemen . Re olu iona y app oaches o e alua ing and
measu ing isk, McG aw-Hill Companies, 37 p.
■ Yamai, Y. – Yoshiba, T. (2002): Compa a i e analyses o expec ed sho all and alue-a -
isk unde ma ke s ess, IMES Discussion Pape No 2002-E-2, Bank o Japan
■ Zi o E., DWK And ews; Fu he e idence on he g ea c ash, he oil-p ice shock, and he
uni - oo hypo hesis. Jou nal o Business and Economic S a is ics, 10 (1992), pp. 251–
270.