Implementation of the modified Monte Carlo simulation for evaluate the barrier option prices

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JournalofTaibahUniversityforSciencexxx(2015)xxx–xxx

ScienceDirect

Implementation

of

the

modified

Monte

Carlo

simulation

for

evaluate

the

barrier

option

prices

Kazem

Nouri

∗

,

Behzad

Abbasi

DepartmentofMathematics,FacultyofMathematics,StatisticsandComputerSciences,SemnanUniversity,P.O.Box35195-363,Semnan,Iran

Abstract

Inthispaper,weapplyanimprovedversionofMonteCarlomethodstopricingbarrieroptions.Thiskindofoptionsmaymatch withriskhedgingneedsmorecloselythanstandardoptions.Barrieroptionsbehavelikeaplainvanillaoptionwithoneexception. Azeropayoffmayoccurbeforeexpiry,iftheoptionceasestoexist;accordingly,barrieroptionsarecheaperthansimilarstandard vanillaoptions.WeapplyanewMonteCarlomethodtocomputethepricesofsingleanddoublebarrieroptionswrittenonstocks. Thebasicideaofthenewmethodistouseuniformlydistributedrandomnumbersandanexitprobabilityinordertoperformarobust estimationofthefirsttimethestockpricehitsthebarrier.Usinguniformlydistributedrandomnumbersdecreasestheestimationof firsthittingtimeerrorincomparisonwithstandardMonteCarloorsimilarmethods.Itisnumericallyshownthattheanswerofour methodisclosertotheexactvalueandthefirsthittingtimeerrorisreduced.

Keywords: Pricingbarrieroption;Doublebarrier;Uniformdistribution;Exitprobability;ModifiedMonteCarlomethod

1. Introduction

The payoff of a standard European vanilla option dependsontheunderlyingstockpriceattheexpirydate and the strike price. Hereupon, it is usually referred to as a path-independent option. Barrier options are path-dependentoptionsandcanalsohavecashrebates

∗_{Corresponding}_author._Tel.:₊₉₈₂₃₃₃₃₈_3204;

fax:+982333654082.

E-mailaddresses:[email protected](K.Nouri), [email protected](B.Abbasi).

PeerreviewunderresponsibilityofTaibahUniversity.

http://dx.doi.org/10.1016/j.jtusci.2015.02.010

associated withthem.The rebatecan benothingor it couldbesomefractionofthepremium.Rebatesare usu-allypaidimmediatelywhen anoptionisknocked out. However,paymentscanbedeferredtothematurityof the option.The most frequently used standard barrier optionsareknockinandknockoutoptions.Ifthe bar-rierlevel istouched atany timebefore maturity, then theoptioneithercomesintoexistenceorceasestoexit dependingonthetypeofabarrieroptioni.e.knockinor knockout.Theinstantpayoffiseitherthesameasfora vanillaoptionorzero,respectively.Thesebasicfeatures ofbarrieroptionsapplytobothcallandputoptions,for EuropeanandAmericantypeofoptions.Whenthe bar-rierisapproachedfrombelow,thebarrieroptioniscalled anup-option;otherwise,itiscalledadown-option.One canidentifyeighttypesofEuropeanbarrieroptions,such as down-and-out calls, up-and-out calls, down-and-in

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puts,up-and-inputs,etc.[1].Forexample,anup-and-in calloptionallocates the optionholder the payoffof a calloptioniftheunderlyingassetpricereachesahigher barrierlevelduringtheoption’slife,anditpaysoffzero unlesstheunderlyingassetpricereachesthatlevel.For anup-and-outcall,theoptionbecomesworthlessifthe asset price hitshigher barrier, andits payoff at expi-rationis a callotherwise. Following[2,3] inorder to analyzethenumericalresults,inthispaperwefocuson down-and-outcalloption,particularly.

Since1967,barrieroptionshavebeentraded sporad-ically in the US markets and nowadays are the most popularclassofexoticoptions.Usually,tradersbuyor sellthistypeofoptionswhentheybelievethatthestock pricewouldeithergoupordown,butwouldnotexceed or become lower than acertain level. Barrier options aregenerallycheaperthanordinaryvanillaoptionsand it isone of the reasonsthat aninvestor prefers them. Theotherreasonisthatbarrieroptionsmaymatchwith riskhedgingneedsmorecloselythanstandardoptions, whichmake themparticularly attractiveto hedgers in thefinancialmarket.Thereforeitisreallyimportantto expandefficientandaccuratemethodstoevaluatebarrier optionpricesinfinancialderivativemarkets.

To evaluate barrier option prices, there are two major directions. The first approach is the solving Black–Scholes partial differential equation, see [4,5]. There is aclosed-form expression of the correspond-ingwellknownBlack–Scholesequation wheneverthe volatility of the underlying asset is constant, but this hypothesisisfar frombeingrealistic. Merton[5] pro-videdthefirstanalyticalformulaforadown-and-outcall optionwhichwasdevelopedforalleighttypesof barri-ersbyReinerandRubinstein[6];seealsoHaug[7],for ageneralization.However,sometimesitisverydifficult topricebarrieroptionsanalytically,andoneshouldrely onnumericalapproximationsasasecondapproach.For example,when the underlyingdynamics andthe con-tracts are complex or when the underlying asset has stochasticvolatility. Due totheir popularity ina mar-ket,morecomplicatedstructuresofbarrieroptionshave beenstudiedbysomeauthors.IkedaandKunitomo[8], derivedapricingformulafordoublebarrieroptionswith curvedboundariesasthesumofaninfiniteseries.Geman andYor[9],followedaprobabilisticapproachtoderive theLaplacetransformofthedoublebarrieroptionprice. Pasquali et al. [10], proposed a numerical technique througharecombiningmultinomialtreeforevaluation inastochasticvolatilitymodel.Evaluationimpliesthe useofnumericaltechniquesthatcanbedividedinthree maingroups:(a) finitedifference methods,(b) Monte Carlo(MC)simulation,and(c)multinomialtrees.Each

methodologyhasadvantagesaswellasdrawbacks, mak-ingitusefulforsomecontractsandalmostuseless for others[10].ForinstanceMCmethods,duetothetheir randomnature,areappropriateforstudyingtheobtained models of random phenomena such as occurrence of earthquakes,fault system,functionof heartandbrain, financialproblemsandetc.TheMCsimulationisvery popularandrobustnumericalmethod.Itisthough inher-entlystochasticandalsosimpletocodebutitisnoteasily extensible to multipleunderlyingassets. Onthe other hand,oneofthemaindrawbacksoftheMCmethodis aslowconvergence.Theorderofstatisticalerrorofthe MCmethodisO(1/√M)withMtimessimulations.In particular, for continuously monitored barrieroptions, thehittingtimeerrorisoforderO(1/√N)withNtimes steps,see[11],whiletheEuropeanvanillaoptionshave no time discretization error. There have been differ-entideasonhowtoreducethisfirsthittingtimeerror. Dzougoutovetal.in[12],usedadaptivemeshnearthe barriertoreducethiserror;alsoMetwallyandAtiyain

[13]usedaBrownianbridgeideaforjump-diffusion pro-cess.Inordertoefficientlyreducethishittingtimeerror nearthebarrierpriceateachfinitetimestep,inspiredby

[14,15],inthisstudytheuseofauniformlydistributed randomvariableandofaconditionalexitprobabilityhas beenapplied.Numericalresultsshowthatthemodified MonteCarlo(MMC)methodconvergesmuchfasterthan thestandardMCmethod.Thisideaofusingexit prob-ability for stopped diffusioniswellknown inphysics community,see[14,16].

Thispaperisorganizedasfollows:

InSection2,weintroducesomeconceptsofbarrier options andpresentpricing formula for down-and-out call option.In Section3,first, application of standard MCmethodandmodifiedMonteCarlomethodfor pri-cing down-and-out barrier options is proposed, while secondlywecompareourresultswithsomeother meth-ods. Then,wewill useMCandMMCsimulationsfor pricingdoubleknock-outcalloption.Also,algorithms are compared andwe propose a wayfor making bet-terourestimation.Finallyconclusionsofthisworkare summarizedinSection4.

2. Preliminariesandbasicconcepts

The evolution of the financial asset price can be written as a stochastic process {S_t}_t∈[0,T], defined on a suitable probability space (Ω,F,P). We consider the assumptionsofthe classicalBlack–Scholesoption pricing model. The price St of the underlying asset

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constant expectedrate of returnμ>0, andaconstant volatilityσ>0,i.e.,

dSt =μStdt+σStdWt, (1)

whereWt isastandardBrownianmotion process,see

[17–19].

Instead of solving Black–Scholes PDE, numerical approaches require the compute an expected value of thediscountedterminalpayoffunderarisk-neutral mea-sureQ,i.e.,μ=rin(1),wherer>0isaconstantrisk-less interestrate.ThebarrieroptionpriceV(s,t)atapresent timetcanbecomputedby

V(s,t)=EQ[Λ(Sτ,τ)|St =s], (2)

whereΛ(Sτ,τ)isadiscountedpayofffunctionandτis

thefirsttimewhichSthitstobarrier.Toapproximatethe

optionpricein(2),onemayapply eitherlattice meth-odsorMCmethods.Forexample,fordown-and-outcall barrieroptions,whichspotpricestartsabovethebarrier level (St>B) andhastomove downfor the optionto

becomenullandvoid,therandomvariableτ isdefined by

τ =inf{l≥t:Sl≤B},

whiletheoptionhaspayoff

Λ(Sτ,τ)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

e−r(T−t)max(ST −K,0), if Su>B,∀uT, i.e. τ =T,

e−r(τ−t)R, if τ<T, (3)

whereKisagivenexercisepriceatexpirationdateT,B

isabarrierprice,andRisaprescribedcashrebate. Indown-and-outoptions,barrierlevelsaresetbeneath theinitialunderlyingassetprice,andtheoptionbecomes worthlesswhentheunderlyingassetpricehitsthebarrier. Pricingformulafordown-and-outcalloptionis(see

[20]) Vdoc = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ S0(Φ(d1)−b(1−Φ(d8)))−Ke−rT(Φ(d2)−a(1−Φ(d7))), if K>B, S0(Φ(d3)−b(1−Φ(d6)))−Ke−rT(Φ(d4)−a(1−Φ(d5))), if K<B, (4)

Fig.1.Assetpricesamplepathwiththeloweranduppercurves. where S0 is the initial stock price, the function Φ(x) is the cumulative probability distributionfunction for astandardizednormaldistribution,and

a= B S0 _−1+(2r/σ2₎ , b= B S0 1+(2r/σ2) , d1=ln(S 0/K)+(r+1/2σ2)T σ√T , d3= ln(S0/B)+(r+1/2σ2)T σ√T , d5=ln(S0/B)−(r−1/2σ 2 )T σ√T , d7= ln(S0K/B2)−(r−1/2σ2)T σ√T , d2i=d2i−1−σ √ T, i=1,2,3,4.

Deﬁnition2.1. Wesayacurveisworthlesswithrespect tothestockpricemovementoveragiventimeintervalif withprobability1,thepathfollowedbyStwillnotbreach

thecurve(eitherfromaboveorfrombelow),overthat timeinterval.

We are interested in finding the minimum value

Sml(maximumvalueSmu)ofS0whichensuresthatthe stockpricewillnotbreachthelower(upper)curvebefore timeT[21].ConsiderEq.(1)withtwocontinuouscurves

Bl(t)andBu(t)overthetimeintervalI=[0,T],suchthat Bl=Bl(0)<S0<Bu(0)=Bu (see Fig. 1). Let P_tl be the

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probabilitythatSt>Bl(t),thenbyEq.(1)wehave P_tl =Φ(wl_t), with wl_t = (μ−(σ 2_/2))t₊ ln(S0/Bl(t)) σ√t .

ItiswellknownthatΦisapositivedefiniteincreasing functionandconvergenceofitto1isveryfast.Denoteby

h=h(Φ)theaccuracyinagivenvalueofΦ.For simplic-ity,wemaywriteh(Φ)intheformh(Φ)=10−m,where

misthenumberofsignificantdigitstotherightofthe decimalpoint inΦ. Letν bethe smallest numberfor whichtheapproximationΦ(ν)=1holds,thatis,suchthat

Φ(ν)>1−h.Forinstanceforh=10−3,νis3.090231and

ν=4.753424ifh=10−6,thereforewithaccuracylevel considerationtodetermineh,theparameterνisuniquely determined.WehaveP_tl=1ifandonlyifwl_t≥ν,and solvingthisinequalityforS0showsthat

P_tl =1⇔S0≥Sl(t) where Sl(t)=Bl(t)e(νσ √ t−(μ−(σ2/2))t)_. ₍₅₎ Similarly,letPu

t betheprobabilitythatSt<Bu(t).Then

wehaveP_tu=(wu_t),where

wu_t =ln(Bu(t)/S0)−(μ−(σ

2_/2))t

σ√t ,

andusingagainthefactthatP_tu=1ifandonlyifwu_t ≥ ν,itfollowsthat P_tu =1⇔S0≤S_u(t) where Su(t)=Bu(t)e−(νσ √ t+(μ−(σ2_/2))t) . (6)

Foraboveresultsreflection,wehavethefollowing the-orems.

Theorem2.2. LetSml bethemaximumvalueofSl(t) andSmutheminimumvalueofSu(t)overthetimeinterval I=[0,T].Then

(a) SmlistheminimumvalueoftheinitialstockpriceS0

abovewhichthecurveBl(t)becomesworthless.

(b) Smuisthemaximumvalueoftheinitialstockprice S0belowwhichthecurveBu(t)becomesworthless. Deﬁnition2.3. Abarrieroptionofagiventypeiscalled atypicalbarrierofthat typeifnoneofitsbarrierscan beconsideredworthlessforthecorrespondingparameter set.

Theorem2.4. Consideradown-and-outbarrieroption withlowerbarrierBl(t),andagivenparameterset,

(a) IfS0≥Sml,thebarrierisworthlessandtheoptionis equivalenttoavanillaoptionwiththesame param-eterset.

(b) Theoptionisatypicaldown-and-outbarrieroption ifandonlyifS0<Sml.

Theproofsofabovetheoremsandmoredetailsaregiven in[21].

3. ApplicationofMMCalgorithm

Let us assumethat the evolutionof the underlying assetpricefollowsthegeometricBrownianmotion(1). FromIto’sformula,theanalyticsolutionsatisfies

St =S0e(r−(σ

2

/2))t+σWt_, ₀≤_t≤_T, ₍₇₎

wherer=μis the risk-lessrate of return, σ isa con-stantvolatilityandWt isastandardBrownianmotion.

ThebasicideaoftheMMCmethodistouseuniformly distributedrandomvariablesandanexitprobabilityto robustly estimate the first time that stock price hits the barrier. In order to generate stock paths by MC method,wediscretizethetimeinterval[0,T]intoN uni-form subinterval, each of length δt=T/N as timestep size, by the grid points ti=iδt, i=0, 1, 2, ...,N. Let Si =Sti,i=0,1,2,...,N;so,ateachofthegridpoints

t0,t1,t2,...,tN−1,Eq.(7)becomesinexplicitrecursive

form

Sn+1=Sne(r−(σ

2_/2))δt+σ√_δtz

n_, _n=_0,_1,_2,_._._.,_N−_1,

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Repeat-ing Mtimes MC simulationsof thestockprice,value of the discounted terminal payoff Λ(Sτ, τ) and

sub-sequently barrieroptionprice canbeapproximated as follows, V(s,t)=EQ[Λ(Sτ,τ)|St=s]= 1 M M j=1 Vj, (9)

whereVj,j=1,...,M,isdown-and-outcalloptionpayoff

ineachMCsimulation.ButinMMCsimulation,after the simulation of Sn, n=1, 2, ...,N, by relation (8),

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lawofBrownianbridgethatgivesthefollowingformula [15,22], Pn+1=P max t∈[tn,tn+1] St ≥B|Sn=s1,Sn+1=s2 =exp −2(B−s1)(B−s2) σ2_s2 1δt , n=0,1,...,N−1. (10)

Furthermoretoapproximatehittingevent,wegenerate a standard uniformly distributedrandom variable Un, n=1, 2, ..., N, and compare it with the exit proba-bilityPn,n=1,2,...,N,obtainedbyrelation(10).In

down-and-outoptionsamomentbeforeoptionbecame worthless,thatmeansbeforehittingtobarrier,Sn→B−

andwithEq.(10)Pn→1.IntheotherwordPnwillget

itsmaximumvalue.Thusintheuniformandequitable state,locatedPnininterval(0.5,1)andUnin(0,0.5),

i.e.Un<Pnfollowshittingtobarrier,andifPnbelongs

to interval (0, 0.5)and Un∈(0.5, 1), i.e. Pn<Un, we

accept that the continuous pathSt does not hitto the

barrierandpricingprocessresume.Formoredetailssee Refs.[11,23,24].Also,toestimateVj,j=1,...,M,we

replacerelation(3)whent=0with

Vj =

e−rT max(ST −K,0), if Pn<Un, n=1,2,...,N(St >B,∀ 0tT),

0, o.w. (11)

3.1. Down-and-outcalloption

Withoutlossofgeneralityletusassumethatthecash rebateiszero,i.e.,R=0.Asanexample[3],letus con-siderpricingadown-and-outcalloptionbyapplyingthe abovealgorithms(MCandMMC),andcomparethe dif-ferencebetweentheexactandtheapproximatedvalues. Forthispurpose,accordingtoexampleof[3],weassume thattheparametersetareunderlyingstockpriceS0=100, theriskfreerater=0.1,timetomaturityT=0.2,lower barrierB=85,strikepriceK=100,volatilityσ=0.3and numberofsimulationsM=100,000.Theexactsolution withtheseparametersis Vexact=6.3076.Weapply the

MCandMMCsimulationalgorithmswithNasthe num-ber oftimestepsandcompare the differencebetween the exact andthe approximated values. By observing results in Table 1, we see that in MC simulation the errorsareinevitableanditconverges veryslowly,and theMMCmethodperformsmoreefficientlyforpricing barrieroptions.

In Table 2, we see the prices of above down-and-out call option, calculated with binomial, trinomial, standardMCandMMC(withM=1000)techniques[3].

Table1

ComparisonoftheexactandtheMC,MMCapproximatedvaluesfor down-and-outcalloption.

M N Standard MC MMC Error MC Error MMC 100,000 50 6.0267 6.3064 0.2809 0.0012 100,000 100 6.8212 6.3068 0.5136 0.0008 100,000 200 6.3094 6.3076 0.0018 0.0000 Table2

Thepricesofthedown-and-outcalloptioncalculatedwithdifferent methods.

N Binomial Trinomial Standard

MC MMC 1000 6.3111 6.3152 6.3467 6.2996 2000 6.3109 6.3119 6.3257 6.3105 3000 6.3098 6.3079 6.2940 6.3065 5000 6.3084 6.3093 6.3094 6.3075

Also,Fig.2showstheerrorscomparisonbetweenthese methods.

3.2. Doublebarrieroptions

A double barrier option is a combination of two dependentknock-inorknock-outoptions.Itisobviously

1000 1300 1600 2000 2300 2600 3,000 3600 4300 5000 0 0.005 0.01 0.015 0.02 0.025 0.03 Number of Steps Errors V−Binomial V−Trinomial V−MMC V−MC

Fig.2.Errorscomparisonofthebinomial,trinomial,MCandMMC techniquesforpricingthedown-and-outcalloption.

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cheaper than its equivalent single barrier and subse-quently vanilla counterparty. This is because it has doubleriskofbeingknockedout,orofnotbeingknocked in.Oneofbarriersissetabovethepriceofthe underly-ingasset,andoneotherissetbelowit.Theunderlying assetmustonlycrossoneofthebarriers,eitherthelower boundary Lt or the upper boundary Ut to be

deacti-vated(activated)forknock-out(knock-in)options. Thepriceof adoubleknock-in callisequal tothe priceofaportfolioconsistingofalongstandardcalland ashortdoubleknock-outcall,withidenticalstrikesand timetoexpiration. Similarly,adoubleknock-inputis equaltoalongstandardputandashortdouble knock-output.Doublebarrieroptionscanbepricedusingthe IkedaandKunitomoformula[8].

3.2.1. Doubleknock-outcalloptionprice

Thepayoff atexpiry for agivenknock-out double barriercalloptionbyknownparameterset,is

Λ= ⎧ ⎪ ⎨ ⎪ ⎩ max(ST −K,0), if Lt <St<Ut before T, 0, else,

andanalyticalpricingformulaasfollows[7,25].

Cdko(ST,T) =STe(b−r)T +∞ n=−∞ U_Tn Ln T μ1n LT ST μ2n [N(d1n)−N(d2n)] − Ln+1_T U_TnST μ3n [N(d3n)−N(d4n)] −Ke−rT +∞ n=−∞ UTn Ln_T μ1n−2 LT ST μ2n ×[N(d1n−σ √ T)−N(d2n−σ√T)] − Ln+1_T U_TnST μ3n−2 [N(d3n−σ √ T)−N(d4n−σ√T)] , where d1n= ln(STU 2n T /KL2nT )+(b+(σ2/2))T σ√T , d2n= ln(STUT2n/FL2nT )+(b+(σ2/2))T σ√T , d3n= ln(L 2n+2 T /KSTUT2n)+(b+(σ2/2))T σ√T , d4n= ln(L2n+2_T /FSTUT2n)+(b+(σ2/2))T σ√T , μ1n=2[b−δ2−n(δ1−δ2 )] σ2 +1, μ2n=2n (δ1−δ2) σ2 , μ3n=2[b−δ 2+n(δ1−δ2)] σ2 +1, F =UTeδ1T. Table3

AbsoluteerrorsoftheMMCalgorithmtoapproximatethedouble barrieroptionpricebyusinguniformdistributionU(0,1).

M N MMCwithU(0,1) Error

100,000 50 3.9896 0.0108

100,000 100 3.9899 0.0105

100,000 200 3.9901 0.0103

100,000 400 3.9902 0.0102

100,000 800 3.9906 0.0098

Alsob=μ−randδ1,δ2determinethecurvatureLtand Ut,respectively;sotheyarezerocorrespondingtoflat

boundaries.

Theorem3.1. Consideradoublebarrieroptionwith lowerbarrierBl(t)andupperbarrierBu(t),andaﬁxed parameterset.

(a) IfS0<Sml,Smu,theoptiondegeneratesintoa down-and-outbarrieroption.

(b) IfSml,Smu<S0,theoptiondegeneratesintoan

up-and-outbarrieroption.

(c) The optionisequivalenttothevanillaoptionwith thesameparametersetifandonlyifSml≤S0≤Smu.

(d) Theoptionisatypicaldoublebarrieroptionifand onlyifSmu<S0<Sml.

Forproofsee[21].

InthiscaseforimplementationoftheMMCmethod, we compute two exit probabilitiesP_nL andP_nU, n=1, 2, ...,N,for downandupbarriers.Alsowe letLn= Ltn,Un=Utn, n=0,1,...,N,andcheckthe

follow-ingcriteria

Ln<Sn, Sn<Un, for n=0,1,...,N; and P_nL<U_nL, P_nU <U_nU for n=1,2,...,N,

otherwise,theoptionwillbeobviouslyworthless. Inordertodemonstratetheeffectivenessoftheabove proposedalgorithm,wepresentanexample.Weusethe parameters which the present asset price is S0=100, theexercisepriceK=100,thebarrierpricesLt=L=70;

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Table4

AbsoluteerrorsoftheMMC algorithmtoapproximatethedouble barrieroptionpricebyusinguniformdistributionU(0.5,1).

M N MMCwith U(0.5,1) Error 100,000 50 3.9916 0.0088 100,000 100 3.9942 0.0062 100,000 200 3.9963 0.0041 100,000 400 3.9972 0.0032 100,000 800 3.9993 0.0011 Table5

Comparisonofdifferentmethodstoevaluationdoublebarrieroption.

Fourier Cox MMCwith

U(0,1)

MMCwith

U(0.5,1)

0.923 0.937 0.5465 0.6163

Ut=U=130,therisklessinterestrater=0.1,the

expira-tiondateis6months,thevolatilityσ=0.25andnaturally

δ1=δ2=0. Double barriercall option canbe computed byusinganalyticalformula.Thefairoptionpricewith the above parametersis Vexact=4.0004. Weapply the

MMCsimulationalgorithmandcomparethedifference betweentheexactandapproximatedvalues.By observ-ingresultsinTable3,weseethatourmethodconverges slowly.

Indoublebarrieroptionsthehittingeventisrareto occurandwecheckhittingtobarrierbycomparisonof

PnandUn.SoifselectionofUnbeofthesecondhalfof

interval(0,1),thatmeans(0.5,1),probabilityofPn<Un

isgreaterthanofwhenselectUnisthroughofinterval

(0,1).Inadditionouraimishittinglesstothebarrier,so it’smoreeligibletoselectrandomnumbersofU(0.5,1). Logicallyfornumerousbarrierschoosingrandom num-bersofU(0,0.5)ismoreeligible.Thenumericalresults obtainedfromthecombinedMMCmethodwiththeidea ofchoosingarandomnumberofU(0.5,1),aregivenin

Table4.Thistableindicatesthathybridapproachisbetter andproducessmallerapproximationerror.

Finally,inordertocompare theefficiencyofMMC algorithmforpricingdoublebarrieroptionswithother methods,letusconsideranothercontractwithriskfree rate0.05,spotvalueS=100,strikeprice100,volatility 0.1,Lt=90,Ut=110,timestepsN=100andthe

expira-tiondate1year[2].Thefairoptionpricewiththeabove parametersisVexact=0.6564.Comparisonofthe

approx-imatedvalueswithMMCsimulation(M=10,000)and FouriermethodandCoxapproach[2],forpricingabove doublebarrieroptionaregiveninTable5.

4. Conclusion

In this paper, we have considered standard Monte Carlo methods and its modified version to price bar-rieranddoublebarrieroptions.Implementationofanew MCmethodhasbeenproposedandimprovedinorder tocorrectlycomputethefirsthittingtimeofthebarrier pricebytheunderlyingasset.Wecomparedtheaccuracy ofthestandardMonteCarloandmodifiedMonteCarlo algorithms.Theapproximateerror ofthe newmethod converges much faster thanthe standardMC method. Ourfutureworkwillbedevotedtoextendthisideafor othertypesofoptionsandalsotheoreticallytostudythe rateofconvergenceoftheapproximateerrors.

Acknowledgements

Theauthorsaregratefultotherefereesfortheir care-fulreading,insightfulcommentsandhelpfulsuggestions whichhaveledtoimprovementofthepaper.

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