© 2019 Published by VŠB-TU Os a a. All igh s ese ed. ER-CEREI, Volume 22: 37–44 (2019).
ISSN 1212-3951 (P in ), 1805-9481 (Online) doi: 10.7327/ce ei.2019.06.01
Company alua ion unde in e ac ion in disc e e
ime ( eal game op ions model)
Dana DLUHOŠOVÁa, Zdeněk ZMEŠKALa
*
a Depa men o Finance, Facul y o Economics, VŠB – Technical Uni e si y o Os a a, Sokolská řída 33, 702
00 Os a a, Czech Republic
Abs ac
Company alua ion is a c ucial opic in inancial decision making. The ad anced alua ion me hod is he eal op ions
app oach ealised unde isk and lexibili y. I e lec s a s ochas ic ea u e o he unde lying asse and dynamic
manage ial decision making. Ano he aspec o alua ion, which is o en neglec ed, is in e ac ion, meaning he mu-
ual impac o o he companies on he calcula ed alue. Game heo y models his aspec . The pape ’s objec i e is o
desc ibe and apply company wo-phase eal game op ions alua ion in disc e e ime. A gene alised eal game op-
ions alua ion model based on he wo-phase me hod, disc e e ime, isk-neu al p obabili y, and swi ching cos is
o mula ed. The game ca ego isa ion is in oduced, especially ma ke s uc u e games, including equilib ium calcu-
la ions ollowing pu e and mixed s a egies, and he eal game op ions model is o mula ed. A company wo-phase
alua ion me hod in he Cou no p oduc ion duopoly ma ke s uc u e unde andom demand is de eloped, and an
illus a i e example is p esen ed. The pape con i ms he possibili y o modelling company wo-phase alue h ough
eal game op ions alua ion models. Neglec ing an in e ac ion unde a non-pe ec ma ke s uc u e can unde alue
a company, so his aspec is essen ial.
Keywo ds
Game heo y, eal game op ion, eal op ion, alua ion
JEL Classi ica ion: C7, D4, G13, G32, L1
*
[email p o ec ed] (co esponding au ho )
The esea ch was unded by he VSB-Technical Uni e si y o Os a a, SGS P ojec s SP2021/57 and SP2020/124.
Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019
38
Company alua ion unde in e ac ion in
disc e e ime ( eal game op ions model)
Dana DLUHOŠOVÁ, Zdeněk ZMEŠKAL
In oduc ion
Company alua ion is an impo an p oblem in inan-
cial decision making and managemen . The choice o
alua ion me hod depends on a ious aspec s. The
p ac ical alua ion app oach is a wo-phase discoun ed
cash low me hod. The complexi y o alua ion leads o
he equen applica ion o disc e e binomial models.
Due o he alua ion en i onmen , isk, lexibili y, and
in e ac ion a e subs an ial aspec s. Te m isk cha ac e -
ises andomness (s ochas ic p ocess), lexibili y ep e-
sen s a dynamic decision (op ion alua ion), and in e -
ac i i y means ha one company's decision is in lu-
enced by o he companies' decisions (game heo y).
The eal op ions me hod is applied unde isk and lex-
ibili y. Howe e , in e ac i i y is o en neglec ed, e en
hough i ep esen s a signi ican aspec o a company's
alua ion, supposing a non-pe ec ma ke .
Resea che s and p ac i ione s deal wi h game eal
op ions. Many aspec s conce ning goals, o mula ions,
me hodological concep ions, and applica ion possibili-
ies a e in es iga ed (see e.g. Aze edo and Paxson,
2014; Che alie -Roignan and T igeo gis, 2011; G en-
adie , 2000a, 2000b, 2002; Huisman, 2001; Huisman e
al., 2004; Smi and T igeo gis, 2004, 2017). The p ob-
lem can be o mula ed in disc e e o con inuous ime
using Bellman's dynamic p og amming p inciple. Va -
ious op imisa ion echniques can be applied, game
ypes in es iga ed, he p ojec NPV o alue o he
company compu ed, and p oblems o a ious sec o s
in es iga ed and analysed.
The pape ocuses on he eal game op ions alua-
ion o a company using he wo-phase me hod in dis-
c e e ime. This app oach has no ye been desc ibed in
dep h in he li e a u e. The pape 's objec i e is o de-
sc ibe and apply company wo-phase eal game op ions
alua ion in disc e e ime.
P ima ily, an in e ac ion ea u e is in es iga ed. The
wo-phase discoun ed cash low is applied, and he bi-
nomial model in disc e e ime is assumed. The i s sec-
ion is de o ed o he de elopmen o alua ion me h-
ods and a desc ip ion o he eal op ions alua ion p in-
ciples. The chosen aspec s o game heo y a e hen de-
sc ibed and analysed. Subsequen ly, he me hods o
game eal op ions a e ou lined, especially ma ke s uc-
u e games. The applica ion o he alua ion model wi h
andom demand and a duopoly ma ke is e i ied.
1. Valua ion me hods unde isk and lexibili y
( eal op ions)
The alue o company V can be s a ed h ough he dis-
coun ed cash low me hod as he p esen alue
o cash low
( )
1
1
V FCF R
−
=
=  +

. In he case o he
wo-phase me hod
( ) ( )
1
1
1 2 1 1
T T
T
V V V FCF R V R
−−−
=
= + =  + +  +

, (1)
1V
is he alue o he i s phase,
2V
is he alue o
he second phase,
FCF
is he ee cash low,
R
is he
isk- ee a e, and
T
V
is he e minal (con inual) alue,
being a alue a he beginning o he second phase.
Thus,
FCF EAT DEP NWC INV= + − −
and, o ex-
ample, cons an pe pe ui y
1/
TT
V FCF R
+
=
o g owing
pe pe ui y
( )
1/
TT
V FCF R g
+
=−
. Then, he p esen
alue can be o mula ed as ollows:
( )
( )
( )
( )
( )
1
1
1
1
2
1
1
1
1
1
1 ......
TT
T
V R FCF
VR
R FCF
R FCF
−
−
−
−
−
−


 + +





=  +

 + +



+ + +

.
This implies ha he ecu en o mula o one s ep is
( )
1
11
V FCF V R −
+
= +  +
, (2)
and he alue equals he ee cash low plus he p esen
alue o he one-s ep u u e alue. The alue can be cal-
cula ed h ough he backwa d induc ion and dynamic
p og amming (Bellman's equa ion) me hod.
Valua ion unde isk means ha he unde lying as-
se ( ac o ) is a andom p ocess. In a disc e e binomial
model, he alue can be s a ed using isk-neu al alua-
ion and de i ed om he eplica ion s a egy.
The basic idea is o c ea e a po olio alue

om he unde lying ee cash low (asse ) FCF and
isk- ee asse B so a de i a i e alue
V
can be epli-
ca ed. Fo he po olio alue a ime , s a e
i
is
,, i i
a FCF B V   + =
,
D. Dluhošo á, Z. Zmeškal – Company alua ion unde in e ac ion in disc e e ime ( eal game op ions model)
39
he po olio alue a ime
1 +
in upwa d mo emen
1i+
is
( )
1, 1 1, 1 1, 1
1
i i i
a FCF B R V
+ + + + + +
   +  + =
,
he po olio alue a ime
1 +
in downwa d mo e-
men
1i−
is
( )
1, 1 1, 1 1, 1
1
i i i
a FCF B R V
− + − + − +
   +  + =
,
and he e symbol a is he unde lying asse quan i y.
Sol ing he h ee equa ions wi h a iables a, B, and
,i
V
, he alue o mula is he ollowing:
( ) ( )
1
, , 1, 1 1, 1
11
i i i i
V FCF R p V p V
−
+ + − +
= + +   + − 


, (3)
whe e he e m
p
is he isk-neu al (no ma ke ) p ob-
abili y such ha eplica ion is eached.
( )
, 1, 1
1, 1 1, 1
1i i
i i
R FCF FCF
pFCF FCF
−+
+ + − +
+  −
=−
.
I he ee cash low depends on ano he unde lying
asse
S
, which is a unc ion o
S
,
( )
FCF S=
, hen
he isk-neu al p obabili y is
( )
, 1, 1
1, 1 1, 1
1i i
i i
R S S
pSS
−+
+ + − +
+  −
=−
.
The gene alised eal op ions alua ion model is a
mul i-mode (mul i-swi ching model) allowing swi ch-
ing be ween mo e han wo modes. The ecu en equa-
ion is he ollowing:
( ) ( )
,,
,1
1, 1 1, 1
max 11
q
m q i
i qQ i i
C FCF
VR p V p V
−
+ + − +

++

=

+   + − 



. (4)
He e,
q
is a pa icula mode,
Q
is a mode se , and
m
is an ini ial mode.
,mq
C
is he swi ching cos be ween
modes, o which a nega i e alue means a cos and a
posi i e alue means e enue.
Fo mo e in o ma ion abou he eal op ions opic,
see, o example, T igeo gis (1998).
2. Game heo y appa a us
The impo an aspec o alua ion is in e ac ion. I
means ha a pa icula subjec ’s decision depends on
he decisions o o he in elligen subjec s and ice
e sa. This opic is he subjec o game heo y, which
could be conside ed o be a gene alised decision-mak-
ing heo y. Basic e e ences a e, o example, Dlouhý
and Fiala (2009), Maňas (1974), Pe e s (2015), and
Tadelis (2013).
Games a e ca ego ised acco ding o a ious c i e ia:
he numbe o playe s ( wo, mo e han wo), he numbe
o s a egies ( ini e – disc e e, in ini e – con inual), he
coope a ion ype (coope a i e, non-coope a i e), syn-
ch onisa ion (simul aneous, sequen ial), ime (s a ic,
dynamic), solu ion esul s a egies (pu e, mixed), in-
o ma ion ( ull, pa ), and he game o mula ion (s a e-
gic – no mal, ex ensi e).
The c ucial e m o game heo y is equilib ium, ha
is, playe s a egies sea ching o equilib ium. The
basic p inciple is he Nash equilib ium; in o he wo ds,
equilib ium playe s a egies ep esen he bes e-
sponses o pa icula playe s o o he playe s' s a egies.
Al e na i ely, i equilib ium exis s, i any playe s di e
om he equilib ium s a egy, hey a e damaged
(achie e less u ili y).
Usually, a pe ec ma ke is supposed in a alua ion;
all he pa icipan s a e p ice ake s. This does no o en
su icien ly e lec eali y, and he ma ke s uc u e
mus espond o i . I is necessa y o conside and pe -
o m a alua ion me hod in coincidence wi h a ma ke
s uc u e; o he wise, he alua ion canno be co ec
and habi ual asse s a e unde alued. When aluing
companies and p ojec s, games ha e o be ca ego ised
due o ma ke s uc u es: a pe ec ma ke (conside able
playe numbe s), oligopoly ( ini e playe numbe s), du-
opoly ( wo playe s), and monopoly (one playe ).
Fo he Cou no model, he c ucial a iable is he
p oduc ion quan i y and simul aneous decisions o
playe s. Ano he possibili y is a compe i ion by p ice,
as desc ibed by he Be and model. In he case o se-
quen ial decisions, he ma ke consis s o leade s and
ollowe s. The S ackelbe g model ep esen s his si ua-
ion.
Duopoly is he ma ke s uc u e o wo companies.
He e, he s a egy choice o he i s company in luence
he s a egy selec ion o he second company and ice
e sa, so mu ual in e ac ions a e espec ed. In he
Cou no p oduc ion duopoly, he goal is p o i maximi-
sa ion (
12
,zz
) and he s a egic decision abou p oduc-
ion quan i y (
12
,QQ
) conce ns gi en posi i e in e -
als. The in e se demand cu e p o ides he p ice.
Sales (
12
,TT
) and cos s (
12
,NN
) a e exp essed
h ough a linea unc ion;
12
,
a e uni a iable cos s.
The p ice is o mula ed as an in e se linea demand
cu e unc ion,
( )
12
P a b Q Q= −  +
. The p o i is
s a ed as ollows:
1 1 1 1 1 1
z T N P Q Q= − =  − 
, subs i u ing o a
p ice,
 
1 1 2 1 1 1
2
1 1 1 1 2
()
()
z a b Q Q Q Q
a Q b Q b Q Q
= −  +  −  =
−  −  −  
.
Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019
40
The maximal p o i is an ex emal alue as ollows:
11 1 2
1
( ) 2 0
za b Q b Q
Q
= − −  −  =

.
I implies
12
1
()
2
a b Q
Qb
− − 
=
; analogically,
21
2
()
2
a b Q
Qb
− − 
=
. Bo h unc ions a e so-called e-
sponse unc ions s a ing equilib ium p oduc ion in
coun e pa y p oduc ion. By mu ual subs i u ion, he
equilib ium p oduc ion is
12
1
2
3
a
Qb
−+
=
and
21
2
2
3
a
Qb
−+
=
. The o al p oduc ion is he ollow-
ing:
12
12
2 ( )
3
a
Q Q Q b
−+
= + =
. Subs i u ing he o al
p oduc ion in o he equilib ium p ice equa ion, he
p ice is
12
()
3
a
P++
=
. The las s ep is o s a e he
p o i wi h knowledge o p oduc ion,
( )
2
12
1
2
9
a
zb
−+
=
,
( )
2
21
2
2
9
a
zb
−+
=
. The a io
a
Qb

has o be posi i e o gi e he p oblem an eco-
nomic a ionale.
Simila ly, equa ions o o he ma ke s uc u es (ol-
igopoly, monopoly, and pe ec ma ke ) can be ob-
ained. The esul s a e p esen ed in Table 3–1. The uni
a iable cos and many companies a e supposed o he
pe ec ma ke .
Table 3–1 P oduc ion, p ice, and p o i due o he ma ke s uc u e
Ma ke
s uc u e
P oduc ion o he company i
i
Q
To al p oduc ion
Q
P ice
P
P o i o he company i
i
z
Monopoly
2
i
a
b
−
2
i
a
b
−
2
i
a +
( )
2
4
i
a
b
−
Cou no duo-
poly
2
3
ij
a
b
−+
( )
2
3
ij
a
b
−+
12
()
3
a ++
( )
2
2
9
ij
a
b
−+
Cou no oli-
gopoly
( )
1
1
ii
a n n
n
nb
−
−  + −
+
1
n a
nb
−


+
1
a n
n
+
+
( )
2
1
1
1
ii
a n n
bn
−
 −  + − 

+

Pe ec ma -
ke
1
1
a
nb
−
+
1
n a
nb
−


+
1
a n
n
+
+
2
1
1
a
bn
−


+

Legend:
1n
ij
ji
−
−

=
,
n
j
j
=
A bima ix game signi ies a wo-playe game wi h dis-
c e e s a egies. The ma ix o each playe p esen s pay-
o s o all combina ions o s a egies. Commonly, he
game is a non-ze o-sum one. The p ima y objec i e o
he playe s' choice o s a egy is o ind an equilib ium
s a egy. The solu ion can in ol e pu e s a egies o
mixed s a egies. Fi s ly, e e y playe selec s only one
s a egy. The me hods o bes esponses and i e a i e
elimina ion non-domina ed s a egies can be applied.
Secondly, he playe s selec s a egies wi h p obabili y.
The op imisa ion mixed p oblem is used.
The playe sea ches o he bes esponses (maximal
alue) o coun e -playe s a egies by applying he bes
esponse me hod. The bes esponses a e shaded in Fig-
u e 3–1. Fo playe A, i playe B selec s s a egy B1,
he bes eac ion is A1; i he playe selec s B2, he bes
esponse is A1. The eac ion o playe B is he ollow-
ing: i playe A chooses A1, B selec s B2; o choice
A1, playe B selec s B1. The equilib ium poin p esen s
he combina ion o A1 and B2 wi h a payo o 500 o
A and 700 o B.
Payo ma ix
Playe A
Playe B
S a egy B1
S a egy A1
Playe A
S a egy A2
600
300
S a egy B2
500
400
Payo ma ix
Playe B
Playe B
S a egy B1
S a egy A1
Playe A
S a egy A2
500
300
S a egy B2
700
200
Figu e 3–1 Bes esponse equilib ia in a bima ix game
D. Dluhošo á, Z. Zmeškal – Company alua ion unde in e ac ion in disc e e ime ( eal game op ions model)
41
In pu e s a egies, excep o one unique equilib-
ium, he e can be mo e o no solu ions. In he case o
mo e solu ions, he mos s able solu ion is sough , and
he concep o he so-called minimisa ion o shaken
hands is used. The common solu ion is in mixed s a e-
gies, encompassing pu e s a egies as a subse . The o -
mula ion o he non-linea op imisa ion p oblem is e-
po ed, o example, by Dlouhý and Fiala (2009),
Maňas (1974), and Pe e s (2015) as he ollowing:
P oblem I (bima ix game)
, , ,
max
ij
M N M N
i ij j i ij j
p q w i j i j
p a q pb q w+ − −
 
,
N
ij j
j
a q i  

,
M
ij i
i
b p w j  

He e,
i
p
and
j
q
a e he s a egy p obabili ies o playe
A and, espec i ely, playe B,
ij
a
and
ij
b
a e he payo s
o playe A and, espec i ely, playe B,
and
w
a e
a iables,
i
and
j
a e he indexes o he s a egies o
playe A and, espec i ely, playe B, and
M
and
N
a e he numbe o s a egies o playe A and, espec-
i ely, playe B.
P oblem I can be modi ied in o a mo e sui able
compu a ion o mula ion. A e di iding he equa ions
by
and
w
and subs i u ing
/
ii
x p =
and
/
ii
x p w=
, P oblem II is as ollows:
P oblem II (modi ied bima ix game)
,,,
max 1 1
M N M N
i ij j i ij j
p q a b i i i i
x a y xb y+ − −
 
1,
N
ij j
j
a y i  

1,
M
ij i
i
b x j  

3. Real game op ions alua ion model
The payo unc ion depends on he unde lying asse
andom p ocess and independen ly on in e ac ions du -
ing he alua ion in he eal op ions me hod. In his sec-
ion, he app oach is gene alised. I is supposed ha a
payo unc ion is de e mined, excep o he unde ly-
ing andom asse mo emen , by mu ual in e ac ions, so
i depends on o he playe s’ decision (choice o s a e-
gies). The alua ion o he game eal op ions is simila
o he eal op ions alua ion, excep he payo unc ion
is only gi en by game heo y wi h in e ac ions. The
opic in disc e e ime is wo ked ou o example by
Che alie -Roignan (2011) and Smi and T igeo gis
(2004, 2017).
Valua ion conside ing he ac ions o o he compa-
nies is a gene alised app oach including isk, lexibil-
i y, and in e ac ion. Game heo y ins umen s se e o
model in e ac i i y. The c ucial e m o game heo y is
equilib ium. The e m mode is subs i u ed by he e m
s a egy in compa ing he eal op ions model. The gen-
e alised mul i-mode eal op ions wi h in e ac ion model
is modi ied o a gene alised mul i-s a egy model as ol-
lows:
( ) ( )
,
,,
,1
1, 1 1, 1
11
kk
k
ww
n w i
i
i i
C FCF
VR p V p V
−
−
+ + − +

++

=
+   + − 




, (5)
whe e, o he k h playe ,
k
k
wW
(scala o ec o ) is
he equilib ium s a egy and
k
k
wW
−
−
( ec o o ma-
ix) is he s a egy se , o o he playe s,
k
w−
( ec o
o ma ix) is he equilib ium s a egy and
k
W−
(ma ix)
is he s a egy se , and
,k
nw
C
(scala o ec o ) is he
swi ching cos be ween s a egies.
The binomial model wi h isk-neu al alua ion, he
wo-phase me hod, and he game payo unc ion is p e-
sen ed.
Valua ion p ocedu e o he eal game op ions model
(i) De e mina ion o he unde lying asse ( ac-
o ) andom p ocess
An app oach based on an expe 's es ima ion
o andom p ocess calib a ion (e.g., B owns,
CIR, Ho-Lee).
(ii) Equilib ium game payo de e mina ion
Equilib ium payo s a e calcula ed due o he
game model.
(iii) S a ing e minal (con inuum) alue
Te minal alue calcula ion (e.g. pe pe ui y
and g owing pe pe ui y) due o s a es
,iT
V
.
(i ) De e mina ion alue o pa icula s a es
Backwa d induc ion p ocedu e due o (5):
( ) ( )
,,
,1
1, 1 1, 1
11
k
k
w
n w i
i
i i
C FCF
VR p V p V
−
+ + − +

++

=

+   + − 



( ) De e mina ion alue o an op ion
The alue o he eal game op ion a he begin-
ning
0
V
.

Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019
42
4. Company alua ion in a duopoly ma ke s uc-
u e unde andom demand (simpli ied example)
The objec i e is o s a e he alue o he companies
(company A and company B) ope a ing in a p oduc ion
duopoly. We simplis ically assume ha dep ecia ion
equals he in es men , o he ixed cos s a e no consid-
e ed, he ne wo king capi al change is ze o, and he
ansac ion cos and axes a e neglec ed. The e o e, he
ee cash low is iden ical o he p o i . A wo-phase
discoun ed cash low alua ion me hod is applied, and
a non-coope a i e Cou no p oduc ion duopoly de e -
mines he equilib ium p o i . An in e se linea demand
cu e gi es he p oduc ion p ice. Random de elopmen
is gi en by pa ame e
a
o he demand cu e, which
obeys a geome ic B ownian p ocess. Fo he calcula-
ion, a binomial model is used.
Wi h inpu pa ame e alue
a
= 10, he companies’
uni a iable cos s a e
A
= 2, and
B
= 3. Hence, com-
pany A is mo e e ec i e han company B, he cos o
capi al o bo h companies
AB
R
= 20%, and he isk-
ee a e
R=
10%. The de elopmen o pa ame e
a
is appa en om he binomial model (Figu e 5–1), and
he inpu pa ame e s, including he calcula ed up-index
and down-index, along wi h he isk-neu al p obabili-
ies, a e shown in Table 5–1.
210
ime
s a e
15,63
10,00
6,40
12,50
8,00
10,00
2
1
0
-1
-2
u
uu
ud = du
d
dd
Figu e 5–1 De elopmen o pa ame e
a
(binomial model)
Table 5–1 Inpu and calcula ed pa ame e s
I em
Pa ame e
Value
Pa ame e
Value
P ice
a
10
b
0,5
Uni
a iable
cos
A
2
B
3
Ra es
R
0,10
AB
R
0,2
Indices
U
1,25
D
0,8
P obabili-
ies
p
0,67
1p−
0,33
Valua ions applying he wo-phase me hod, bino-
mial model, eplica ion app oach, Ame ican op ions,
and wo payo ma ices a e implemen ed, and a pu e
s a egy is supposed. The ecu en alua ion equa ion
o mula is as ollows:
( ) ( )
,
,
,
1
1, 1 1, 1
11
AB
i
ww
i
A
AA
i i
z
VR p V p V
−
+ + − +

+

=

+   + − 



, (6)
( ) ( )
,
,
,
1
1, 1 1, 1
11
BA
i
ww
i
B
BB
i i
z
VR p V p V
−
+ + − +

+

=

+   + − 



, (7)
whe e
,i
A
V
and
,i
B
V
a e alues,
,
A
i
z
and
,
B
i
z
a e p o i ,
A
A
wW
and
B
B
wW
(scala ) a e he equilib ium
s a egy,
A
W
and
B
W
a e ec o s o s a egies,
p
is
he isk-neu al p obabili y, and
R
is he isk- ee a e.
Duopoly companies' equilib ium p o i o conc e e
nodes, shown in Table 5–1, a e
( )
2
2
9
AB
A
a
zb
−+
=
and
( )
2
2
9
BA
B
a
zb
−+
=
. The backwa d induc ion p ocedu e,
wi h he i s alue a a e minal ime using pe pe ui y,
is calcula ed as
AB
z
VR
=
, hen he node alues a e cal-
cula ed using (6) and (7). Table 5–1 p esen s he esul s
o nodes, including p ice
()
3
AB
a
P++
=
and p o-
duc ion
2
3
AB
A
a
Qb
−+
=
,
2
3
BA
B
a
Qb
−+
=
. The
de elopmen equilib ium p o i and company alue a e
shown in Figu e 5–1.
D. Dluhošo á, Z. Zmeškal – Company alua ion unde in e ac ion in disc e e ime ( eal game op ions model)
43
Table 5–2 Calcula ed alues, p ocess, p oduc ion, and p o i o companies A and B
S a e
beginning
u
d
uu
ud = du
dd
company
A
B
A
B
A
B
A
B
A
B
A
B
coe .
a
10,00
12,50
8,00
15,63
10,00
6,40
p ice
P
5,00
5,83
4,33
6,88
5,00
3,80
p oduc ion
Q
1,50
1,00
1,92
1,42
1,17
0,67
2,44
1,94
1,50
1,00
0,90
0,40
p o i
z
18,00
8,00
29,39
16,06
10,89
3,56
47,53
30,03
18,00
8,00
6,48
1,28
237,66
90,00
32,40
200,70
75,25
162,44
210
2
1
0
-1
-2
ime
s a e
Company A
47,53
18,00
6,48
29,39
10,89
18,00
2
1
0
-1
-2
P o i
Value
150,16
40,00
6,40
119,18
29,74
89,24
210
Company B
30,03
8,00
1,28
16,06
3,56
8,00
u
uu
ud = du
d
dd
u
uu
ud = du
d
dd
Figu e 5–2 Equilib ium p o i and alue de elopmen o company A and company B
I is e iden ha company A’s alue is 162,44 m. u.,
company B’s alue is 89.24 m. u., and he impac o he
companies' e ec i eness is appa en . The compu ed
alues e lec he p oduc ion duopoly condi ions and
he economic le el o he pa icula companies. Ac-
co ding o s a es and ime, he p oblem allows he anal-
ysis o he alua ion ci cums ances and equilib ium pa-
ame e s; see Table 5–1, which p esen s he p ices
P
,
companies’ p oduc ion
A
Q
and
B
Q
, and p o i
A
z
and
A
z
. I is easy o show ha company alues in a duopoly
ma ke could be compu ed compa ably unde an oli-
gopoly ma ke s uc u e by applying he o mulas o
Table 3–1.
5. Conclusion
The eal op ions app oach could be conside ed as a
company alua ion concep ha e lec s unce ain y
and lexibili y. An essen ial elemen o he alua ion
en i onmen is in e ac ion. This phenomenon, embod-
ying he mu ual ela ionships among companies, is
deal wi h using game heo y. Howe e , he in oduced
aspec is o en neglec ed e en i i subs an ially in lu-
ences he company alue unde speci ic non-pe ec
ma ke s uc u es. Hence, he eal game op ions me hod
encompasses his phenomenon.
The me hodology o he game eal op ions alua ion
model, based on a wo-phase me hod in disc e e ime,
was de eloped and o mula ed and an illus a i e ex-
ample was p esen ed in he pape . The compu a ion
p ocedu e o eal game op ions was desc ibed. Games
Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019
44
wi h non-pe ec ma ke s uc u es we e o mula ed,
speci ically duopoly and oligopoly Cou no p oduc ion
games. The duopoly ma ke s uc u e was implemen ed
and calcula ed in he illus a i e example.
I was ound ha wo-phase eal game op ions al-
ua ion in disc e e ime is a sui able alua ion app oach
o companies e lec ing non-pe ec ma ke s uc u es.
Re e ences
AZEVEDO A., PAXSON, D. (2014). De eloping Real
Op ion Game Models. Eu opean Jou nal o Ope a io-
nal Resea ch 237 (3): 909–920.
h ps://doi.o g/10.1016/j.ejo .2014.02.002
DLOUHÝ, M., FIALA, P. (2009). Ú od do eo ie he .
2. up a ené ydání, P aha: Oeconomica.
DLUHOŠOVÁ, D. (2004). Přís upy k analýze inanční
ýkonnos i i em a od ě í na bázi me ody EVA – Eco-
nomic Value Added. Finance a ú ě – Czech Jou nal
o Economics and Finance 11–12, 54:541–559.
DLUHOŠOVÁ, D. e al. (2014). Financial Ma-
nagemen and Decision-Making o a Company. VSB-
Technical Uni e si y o Os a a, Facul y o Econo-
mics.
CHEVALIER-ROIGNANT, B., TRIGEORGIS, L.
(2011). Compe i i e S a egy: Op ions and Games.
USA: Massachuse s Ins i u e o Technology.
h ps://doi.o g/10.7551/mp ess/9780262015998.001.0
001
GRENADIER, S. R. (2000a). Op ion Exe cise Games:
he In e sec ion o Real Op ions and Game. Theo .
Jou nal o Applied Co po a e Finance. 13: 99–107.
h ps://doi.o g/10.1111/j.1745-6622.2000. b00057.x
GRENADIER, S. R. (2002). Op ion Exe cise Games:
An Applica ion o he Equilib ium In es men S a e-
gies o Fi ms. Re iew o Financial S udies. 15: 691–
721. h ps://doi.o g/10.1093/ s/15.3.691
GRENADIER, S. R. (2000b). Game Choices: The In-
e sec ion o Real Op ions and Game Theo y. Risk Bo-
oks.
h ps://doi.o g/10.1111/j.1745-6622.2000. b00057.x
HUISMAN, K. (2001). Technology In es men : A
Game Theo e ic Real Op ions App oach. Sp inge
h ps://doi.o g/10.1007/978-1-4757-3423-2
HUISMAN, K. J. M., KORT, P. M., PAWLINA, G.,
THIJSSEN, J. J. J. (2004). S a egic in es men unde
unce ain y: Me ging eal op ions wi h game heo y.
Zei sch i ü Be iebswi scha . 3: 97–123.
h ps://doi.o g/10.1007/978-3-663-12338-5_4
GUTHRIE, G. (2009). Real Op ions in Theo y and
P ac ice. Ox o d Uni e si y P ess.
MAŇAS, M. (1974). Teo ie he a op imální ozhodo-
ání. P aha: SNTL.
PETERS, H. (2015). Game Theo y: A Mul i-Le eled
App oach. Sp inge . h ps://doi.o g/10.1007/978-4-
431-55339-7, h ps://doi.o g/10.1007/978-4-431-
55336-6
SMIT, H. T. J., TRIGEORGIS, L. (2004). S a egic in-
es men : eal op ions and games. P ince on: P ince on
Uni e si y P ess.
h ps://doi.o g/10.1515/9781400829392
SMIT, H. T. J., TRIGEORGIS, L. (2017). S a egic
NPV: Real Op ions and S a egic Games unde Di e-
en In o ma ion S uc u e. S a egic Managemen
Jou nal, 38: 2555–2578.
h ps://doi.o g/10.1002/smj.2665
TADELIS, S. (2013). Game Theo y-In oduc ion. P in-
ce on Uni e si y P ess.
TRIGEORGIS, L. (1998). Real Op ions – Manage ial
Flexibili y and S a egy in Resou ce Alloca ion. Ha-
a d Uni e si y.
ZMEŠKAL, Z. (2008). Applica ion o he Ame ican
Real Flexible Swi ch Op ions Me hodology A Gene a-
lized App oach. Finance a ú ě -Czech Jou nal o Eco-
nomics and Finance. 5–6, 58:261-275.
ZMEŠKAL, Z. (2010). Gene alised so binomial
Ame ican eal op ion p icing model ( uzzy-s ochas ic
app oach). Eu opean Jou nal o Ope a ional Resea ch.
207(2):1096–1103.
h ps://doi.o g/10.1016/j.ejo .2010.05.045
ZMEŠKAL, Z. (2013). Game heo y and eal op ions
decision-making hyb id models unde andom demand.
In: 31 h In e na ional Con e ence on Ma hema ical
Me hods in Economics. P ague: 1057–1062.