635
1 INTRODUCTION
Due to the financial crisis of the early 1970s seen at
the conventional money and capital markets, new
financialassetsaredecidedasarequirementinorder
toprotectandcontrolthefinancialrisks. Options as
oneoftheseassetsarefirstlyexercisedbytheChicago
Board Options Exchange in 1973. Developing
countries model financial markets and techniquesof
thedevelopedcountries,thenthederiva
tivemarkets
are expanded globally (Tural 2008). As a developed
financialtechnique,anoptionprovidesarighttobuy
or sell an asset on or after a pre‐defined time at a
particularstatedprice,whichdependsontheformof
theoption.
Infinance,anoptionalwaysprovidesarighttothe
owner (holder or buyer), but it is not obligatory for
the owner of the option. The owner only pa
ys the
option premium, but the seller is obligated to fulfill
therequirementsofthecondit
ions.Thefinancialasset
mightbeexercisedbytheholderduringthefavorable
conditions. The option contracts are commonly
exercised in underlying shares, stock market index,
and exchange and interest based tools (Akalın 2006,
Alpan1999,Yazir2011).
Black & Scholes (B&S) model has widely been
used by many scholars. Merton (1976) generates an
option pricing formula for the returns of the
underlyingshares(Merton1976).Geske(1979)studies
composite options. The study of Scott (1987) deals
with the pricing of European call options with
randomlychangingva
riances.TurnbulandWakeman
(1991) propose a faster algorithm of B&S. Demir
(2003)analysesthecall optionspricewhich thedata
derivedfromBorsaIstanbulbyusingfinit
edifference
method. In another study, Polat (2009) investigates
quantitative solution techniques particularly for
Americantypeofcalloptions.ThestudiesofEasleyet
al.(1998),Madanetal.(1998)aregiventodetermine
thevaluesofoptionsfortheint
erestedreaders.
Inthisstudy,alinearregressionmodelforthecall
option and put option is proposed to analyze the
changesbymanipulating the parameters of the B&S
model.Inorder tocomparethe results, weused the
sameexampleinthestudyofErolandDursun(2015)
inwhichtheycalculatecalloptionpremiumbyusing
Linear Regression Approach for the Financial Risks of
Shipping Industry
D.Yazir&B.Sahin
KaradenizTechnicalUniversity,Trabzon,Turkey
ABSTRACT:TheaimofthisstudyistoproposealinearregressionapproachforBlack&Scholesmodelthatis
usedforcalloptionandputoptiontoderivepricingandriskmanagement.Inthisstudy,theeffectsofshare
prices, exercise prices, volatility and interest rate on put options are observed in an int
eractive manner.
Financialrisksofmaritimetransportationarestudiedinordertoprovideafeasiblesolutionforthethreatsin
theglobalmaritimeeconomicsystem.Inconclusion,theunclearbehaviorsoftheBlack&Scholesmodelsare
revealedbythelinearregressionmodel.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 4
December 2017
DOI:10.12716/1001.11.04.09
636
B&Smodel. We observe thatthe proposed modelin
thestudyismoresuitablethantheirstudyofErol and
Dursun (2015). Effects of volatility and risk‐free
interest rate on options are obviously monitored by
thismodel.
2 PARTICULARSOFTHEOPTIONS
The options might be categorized into two
types as
call option and put option in terms of the intended
use. The holder of the call option expects profit
meaning that an increase of asset price. The call
optionconveystherighttotheholderafterthepriceis
increased under the specified conditions. If the
financial asset
decreases, the holder loses the
premiumpaid.Thesellerofthecallcontractsexpects
adecreasingtrend.Thesellersellsthecalloptionsby
obtainingthe premiumto minimize financialloss or
tomakeprofit.Similarly,theholderoftheputoption
expectsprofitondecreasingtrendofassetprice.The
put option conveys the right to the holder after the
priceisdecreasedunderthepredefinedconditions.If
thepriceoffinancialassetincreases,theholderloses
the premium paid. In order to take anadvantage of
theincrease,theoptionissoldforthereturnsofthe
premium.
The
relatedconceptsshouldbedefinedintermsof
profitabilityoftheoptions.
Market(spot)price is theprice ofthe underlying
share at that moment. Market price is a significant
indicatorbecauseitwillcontinuouslybecomparedto
the exercise price during the time period.Intrinsic
value represents the profit after
option exercised. In
otherwords,itistherelationshipbetweenthema rket
priceandexerciseprice.
Theintrinsicvalueofthecalloption=max(market
price‐exerciseprice,0)
The intrinsic value of the put option = max
(exerciseprice‐marketprice,0)
Optionpriceisthesumof
intrinsicvalueandtime
value.Timevalueisobtainedbysubtractingthespot
price from option premium. Since the In the money
optionsindicatesprofitandoutofthemoneyoptions
shows the financial loss, at the money options
representneitherprofitnorloss(Hull2006,Wilmotet
al.1995,Yazir
2011).
3 FINANCIALRISKSFORTHESHIPPING
INDUSTRY
FinancialrisksaredenotedintheirstudyofEroland
Dursun (2015) in detail. Table 1 shows the possible
risksinshippingindustryandthederivativeproducts
correspondingtheserisksinordertopreventthem.
Table1. Financial risks of maritime transportation and
derivative products (Gilleshammer and Hansen 2010, Erol
andDursun2015)
_______________________________________________
RisksFuture Forward Swap Option
_______________________________________________
FreightRateRisk
BulkCarrier++
Tanker++
Assetpricerisk
NewshipPrice
Secondhand+
Demolition+
Exchangeraterisk + + ++
Interestraterisk+ + ++
Bunkerrisk+ ++
_______________________________________________
4 METHODOLOGY
4.1 Pricingoffuturescontract
AlizadehandNomikos (2009)describetransportcost
model as transport cost until contract time, forward
priceandspotprice.Theformulaisgivenbelow:
rt
00
FSe
0
F
:forwardprice
0
S
:spotprice
r :riskfreeinterestrrate
T :time(year)
Forwardpriceiscomparedwiththespotpriceat
expiration date. At this point since one has some
profitsothergetsinloss(Alizadeh&Nomikos2009).
4.2 Black&Scholes(B&S) Model
The price (premium) of the call option is obtained
fromthefollowingB&Sformula
(Wilmotetal.1995):
rT t
12
CS,t SNd Ee Nd
where,
S:marketpriceoftheshare.
rT t
Ee
:discountedvalueoftheusingpriceofthe
optionbeforetheexpirydate.
j
Nd : probability derived from the standard
normaldistribution,j=1,2.
CS,t :valueofthecalloption.
B&Sformula,
rT t
12
CS,t SNd Ee Nd
2
1
S1
ln r σ T t
E2
d
σTt
21
ddσTt
so,
637
2
2
S1
ln( ) r σ T t
E2
d
σTt
ParametersofthecalloptionusedinB&Sformula
are given below, which depend on S value of the
share;
12
Nd ,Nd
: Cumulative normal probability
distributionvaluesfor(
12
d,d )
1
Nd :Therateoftheshareintheportfolio
2
Nd :Theusageprobabilityoftheshareoption
S:Currentpriceoftheshare
E:Exercisepriceorstrikingpriceoftheoption
r:Therateofriskfreeinterest
σ :Annualstandarddeviationforthereturnrates of
theshare
T‐t:Thetimeremainingtheexpirydate oftheoption
rT t
e
:Rateofdiscount
ln :Logarithmsymbol
4.3
j
Nd
,coefficientsoftheB&Smodel
The average of the values at the standard
distributioncurveiszero,standarddeviationis1,and
standardnormaldistribution is expressed as
N0,1
.
Thevaluesof
12
d,d
arethedeviationvaluesfromthe
averagevaluesofthestandardnormaldistribution.
The value of cumulative standard normal
distribution with the probability of
zd
on the
standard normal distribution curve is found by
addingoftheprobabilityvaluesuntilthegivenvalue
2
z
2
1
fz;0,1 e
2π
ofstandardnormalvariable.
Figure1. Representation of
1
Nd probability value on
standardnormaldistributioncurve
The value in the Figure 1,
1
d 1.0318
means that
thevalueof
1
d deviatesfromtheaveragezerovalue
as much as 1.0318.
N 1, 0318
is the cumulative
probability of standard normal distribution, which
means the cumulative addition of the probability
valuesattheshadedarea.Inthenormaldistribution
curve,the value of the area from theleft to average
zerobecomes
0 . 5 .
Thevalueof
2
d0.9516
issimilarforthestandard
normal distribution.
N0.9516
represents the
deviation from the average of standard normal
distributionasmuchas0.9516.
Since
1
d and
2
d valuesarepositive,probability
of the values are calculated as given below where
j
Nd
(j=1,2),aretheprobabilityvaluesof
j
d .
j
Nd
= (The value for the area under the left
side of the normal distribution curve=0.5 ) + (
j
d
value,theprobabilityvalueofthedeviationfromthe
averagezero=tablevalueofthenormaldistribution)
Since
1
d and
2
d values are negative, the
probabilityofthesevaluesarecalculatedas
j
Nd
=1‐
j
Nd
,(j=1,2).
In this paper, we investigate which shares are
linear in case of change in call options based on
increasing share prices for the changing risk free
interest rates and volatility values (Wilmot et al.
1995).Yazir(2011)indicatesthatifsharepriceisatthe
neighborhood of exercise price,
Cr,σ behaves as
linear.Inthisstudy,weproposetorevealtheunclear
behaviors
Cr,σ of the B&S models. By the linear
regressionmodel,
L
Cαrβσc
theeffectsof the
α,β,c coefficients,related interest
rateandvolatilityoncalloptionvalueareanalyzed.
The value of call option based on B&S model is
calculatedas,
rT t
B& 1 2
C SNd Ee Nd
S
However, due to it is too hard to calculate semi‐
infinite integrals, regression method based on
changes in r and
σ
values are used. After finding
the acceptable
α,β and c values,
L
C linear
equationistransformedwhenthesharepricesareat
theneighborhoodvaluesfortheexerciseprices.Here,
Cr,σ
represents the price of call option, r is the
risk free interest rate,
σ
is volatility and c is a
constant(Yazir2011).
4.4 Linearregression approachfor
,Cr
Yazir(2011)finds
L
Cαrβσc
as aconvenient
approach. Therefore, let find the
α,β ve
c
values
by the least squares method, which minimizes the
followingequation(Yazir2011):
N
2
ii ii
i1
E(αrβσcCr,σ)
fromtheequationsof
EEE
0, 0, 0
αβ c
N
ii iii
i1
E
2αrβσcCr,σr0,
α
638
N
ii iii
i1
E
2(αrβσcCr,σ)σ 0,
β
N
ii ii
i1
E
2(αrβσ cCr,σ)0
c
are obtained. After conducting the required
modifications
NN
ii i
i1 i1
cαr βσ C
NN
2
i i ii ii
i1 i1
cr αr βσ r C r
NN
2
iii i ii
i1 i1
cσ αr σ βσ C σ
arefound. Here,inordertoselectthemostacceptable
α,β,c coefficients, thefollowing systemis obtained
byminimizingsumoferrorsquares.
ii i
2
ii ii ii
2
iii i ii
Nrσc C
rr rσα Cr
σrσσβ Cσ
5 APPLICATION
ErolandDursun(2015)calculatecalloptionpremium
byusingB&Smodel.Inthisstudy,weobtainthedata
of their study, and compute again to check, and
compare our results of linear regression model. It is
importantto say here that this approach is not only
suitable
for this particular case study but also it is
convenientforallotherpricingcallandputoptions.
5.1 Case1
The relevant data are given as S=F0=778.65 USD,
T=14,r=%0.25,K=E=778.19USD,
σ
=%10.78.
The GUI of proposed algorithm is simulated as
follows:
In their study, call option premium is found as
9.6268USD.Inthisstudy,wefindtheresultas9.6252
withthe9.5555e‐10errorrate(Figure2).
Figure2.Linearregressionmodelinterface
As it seen, the error rate is negligibly low.
Additionally, effects of risk‐free interest rate and
volatilityon option premium are explicitly analyzed
in our model. For instance, rho and vega are the
coefficients of risk free interest rate and volatility,
respectively. Figure 3a and Figure 3b show that the
valuesofvegaandrhocoefficientswiththevaluesof
18.5830and57.3341respectively.
Figure3.Coefficientsoflinearoptions (a‐Rhocoefficient,b
‐Vegacoefficient)
ThesimulatedresultsareprovidedintheFigure4.
Figure4. Comparison of B&S model and linear regression
model
5.2 Case2
If risk free interest rate is taken as 0.0029 with all
otherparametersareassumedconstant,rhoandvega
values are changed to 18.5908 and 57.3197
respectively.Calloptionpriceisfound9.6343.Asitis
seen,whenriskfreeinterestrateincreases,thevalue
639
of rho increases, but vega decreases. It means that,
volatility and call option premium are inversely
proportional, and risk free interest rate and call
optionpremiumaredirectlyproportional.
5.3 Case3
If volatility is taken as 0.1080 with all other
parameters are assumed constant, rho and vega
values are
changed to 18.5758 and 57.3459
respectively.Calloptionpriceisfound9.6383.Asitis
seen,whenvolatilityincreases,bothvaluesofrhoand
vegadecreases.
5.4 Case4
If share price is taken as 780 USD with all other
parametersareassumedconstant,calloptionpriceis
found9.6269.
Asitiseasilyseeninthismodel,when
share price increases, call option premium increases
too.
5.5 Case5
Iftimeistakenas30dayswithallotherparameters
are assumed constant, rho and vega values are
changed to 37.2006 and 86.3758 respectively. Call
option price is found 9.6269.
It is seen that if time
days to expiry date increases, volatility, risk free
interestrateandcalloptionpremiumincrease.
InconventionalB&Smodel,itishardlypossibleto
observe these results because of hard calculations
such as semi infinitive integrals. Linear regression
model explicitly gives these inferences easier
and
faster.
6 CONCLUSIONS
The major aim of this study is to provide an
alternativemodelforB&Smodelinwhichtheeffects
ofinterestrateandvolatilityareanalyzedeasilyand
rapidly. In our approach, it is observed that since
there exists a positive relationship between price of
the call
options and price of underlying share, risk‐
free interest rate, volatility and time, there is a
negativerelationshipbetweenthepriceofcalloptions
and exercise price. In this study, linear regression
model is obtained for option price when the share
priceisattheneighborhoodofexerciseprice.
Coefficients
of the volatility in the linear
regressionmodelincreaseasmuchastheincreasein
share prices. However, if the share price becomes
equaltoexerciseprice,theincreaseatthecoefficients
of volatility stops. Then, the coefficients of the
volatility decrease when share price soon after
exceeds the exercise price.
Coefficients of risk‐free
interest continuously increase with the increasing
shareprice.
Linear regression models are more advantageous
than the B&S models for calculation of put and call
options.Linearregressionmodelistimeefficient,and
the effects of changing parameters (i.e. risk free
interest rate and volatility) on option
premium can
easilybeanalyzed.Thecoefficientsof
c,α, β inthe
linear regression model, and the spot price of
underlyingsharecanalsobecalculatedinashorttime
easily.ProposedlinearregressionmodelandB&Sare
implemented by using a MATLAB GUI application,
and overlapping figures are generated for the
comparison.Theerrorratesarefoundthen.
For
the further studies, B&S model might be
analyzed for the time, dependent volatility and
interest rate. Free boundary conditions of the
problems that require applications before the expiry
date should be applied in a not defined boundary.
Thismethod can be applied for Americanoption by
thepropernumericalapproaches.
ACKNOWLEDGEMENT
The authors would like to thank Prof. Dr. Erhan
Coşkun, (Karadeniz Technical University, Science
Faculty, Math Department, Trabzon, Turkey) for his
kind help for constructing the model during the
Dr.Yazir’smaster’seducation.
APPENDIX
MATLAB SOURCE CODE OF LINEAR
REGRESSIONMODEL
clear,clc
close all
S=778.58;
T=14/365;
E=773.19;
r=[0.0023 0.0025 0.0025 0.0425 0.002 0.001
0.0027 0.0525 0.0035 0.0025];
s=[0.2078 0.1076 0.1047 0.1078 0.1059
0.30111 0.1078 0.1066 0.10781 0.1078];
d1=(log(S/E)+(r+0.5*s.^2)*T)./(s*sqrt(T));
d2=d1-s*sqrt(T);
Nd1=normcdf(d1);
Nd2=normcdf(d2);
C=S.*Nd1-E*exp(-r*T).*Nd2;
A=[length(C) sum(r) sum(s);sum(r) sum(r.^2)
sum(r.*s);sum(s) sum(r.*s) sum(s.^2)];
B=[sum(C);sum(C.*r);sum(C.*s)];
C,alpha,beta=inv(A)*B
r=0.0025;
s=0.1078;
linear_regression=xx(2)*r+xx(3)*s+xx(1)
S=778.58;
T=14/365;
E=773.19;
r=0.0025;
s=0.1078;
d11=(log(S/E)+(r+0.5*s^2)*T)/(s*sqrt(T));
d22=d11-s*sqrt(T);
Nd11=normcdf(d11);
Nd22=normcdf(d22);
C=S*Nd11-E*exp(-r*T)*Nd22;
Black_Scholes=C
x=0.0024:0.0001:0.0028;
y=0.1076:0.0001:0.1080;
[r s]=meshgrid(x,y);
d1=(log(S/E)+(r+0.5*s.^2)*T)./(s*sqrt(T));
d2=d1-s*sqrt(T);
Nd1=normcdf(d1);
Nd2=normcdf(d2);
640
C=S.*Nd1-E*exp(-r*T).*Nd2;
ff=xx(2)*r+xx(3)*s+xx(1);
Error=sum((sum((xx(2)*r+xx(3)*s+xx(1)-
C).^2)).^2)
mesh(x,y,ff)
hold on
mesh(x,y,C)
hold off
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