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AbstractInthispaper,westudythepriceofalongtermAsianoptionthepay-offofwhichisdeterminedbytheaveragepriceoftheunderlyingassetduringthelastfixednumberofdaysofitslife.Asonecanimagine,itcon-vergestothepriceofaplainvanillaoptionasthetimetomaturityincreases.WeexplicitlyobtainedtheasymptoticdifferencewhichwillbeusefulforcomputingthepriceofAsianoptioninpractice.KeywordsAsianoptionÆAsymptoticbehavior1IntroductionThepay-offoftheAsiancalloptionwewillstudyinthepresentpaperismax1NXN1i¼0STDiK;0!;ð1ÞwhereSisthepriceprocessoftheunderlyingasset,Tistheexerciseday,DisanintervaloftradingdaysandKisthestrikeprice.WefixNandDinthisY.HishidaFixedIncomeTradingDepartment,FixedIncomeGroup,MizuhoSecuritiesCo.,Ltd.,1-5-1,Otemach,Chiyoda-ku,Tokyo100-0004,Japane-mail:yuji.hishida@mizuho-sc.comK.Yasutomi(&)DepartmentofMathematicalSciences,RitsumeikanUniversity,1-1-1Nojihigashi,Kusatsu,Shiga525-8577,Japane-mail:yasutomi@se.ritsumei.ac.jp123Asia-PacificFinanMarkets(2005)12:289–306DOI10.1007/s10690-006-9027-4OntheasymptoticbehaviorofthepricesofAsianoptionsYujiHishidaÆKenjiYasutomiPublishedonline:5December2006SpringerScience+BusinessMediaB.V.2006paper.WeconsiderthecasewheretimetomaturityTissufficientlylargecomparedtoND.LetS¼St¼S0eZtt2½0;1ÞandZt¼aWtþbtwherea[0;b2R,S02ð0;1ÞandW¼fWtgt0isaBrownianmotionstartingat0.Then,ourmainresultisthefollowing.Theorem1ThereexistsanegativeconstantDsuchthatE1NXN1i¼0STDiK!þ#1NXN1i¼0E½ðSTDiKÞþDaTðT!1Þð2ÞwhereaT¼S0ffiffiffiTpexpb22a2T,xþ¼maxðx;0Þ,andthesymbol~heremeanstheasymptoticequivalence,i.e.,AðTÞBðTÞ,deflimT!1AðTÞBðTÞ¼1:Furthermore,wecanextendTheorem1toanarbitraryorder.Theorem2Fork2N,thereexistconstantsDðmÞðm¼0;1;...;kÞsuchthatE1NXN1i¼0STDiK!þ#1NXN1i¼0E½ðSTDiKÞþ¼aTXk1m¼0DðmÞTmþO1Tk!ðT!1Þ:ð3ÞFromtheaboveresults,wecanapproximatethepriceofanAsianoptionwhenthematuritytimeTissufficientlylarge.Inparticularfrom(2),wehaveE1NXN1i¼0STDiK!þ#1NXN1i¼0E½ðSTDiKÞþþDaT:ð4ÞThustheapproximatevalueofanAsianoptionepAisepAðTÞ¼erT1NXN1i¼0erðTDiÞpPðTDiÞþDaT!ð5ÞwherepPðsÞ¼erðsÞE½ðSsKÞþisthepriceofplainvanillasoftheexercisedays.ThetermstructurefpPðsÞgsofplainvanillasisusuallyquotedinthemarket,oreasilyobtainedviatheBlack–Scholesformula.Therefore,theaboveapproximation(5)withaT(givenas(6))givesusapricingschemeforAsianoptions.TogiveexactformulaofaT,weassumethattheunderlyingassertprocessfStgisthesolutionofSDE:dSt¼rStdWtþrStdtorjustSt¼S0expðrWtþðrr22ÞtÞ.Then290Y.Hishida,K.Yasutomi123aT¼S0ffiffiffiffiTpexp12r2rr222T!ð6Þwhereristhevolatilityoftheunderlyingassertandristheinterestrate.UnfortunatelythedirectnumericalcalculationofDisquitedifficultsincetheexactformulaofDinvolveshigherdimensionalmultipleintegrals1.ButwecaneasilyquoteDifweknowthemarketpriceoftheAsianoptionpAðeTÞ¼ereTE1NXN1i¼0SeTDiK!þ#ofanexercisedayeTandthatofofplainvanillaspPðeTDiÞ¼erðeTDiÞE½ðSeTDiKÞþfortheexercisedayseT;eTD;...;eTðN1ÞD.Indeed,wecanquoteDbyD1aeTereTpAðeTÞ1NXN1i¼0erðeTDiÞpPðeTDiÞ!¼:eD:ð7ÞThenwecanapproximatethepriceofAsianoptionforanyexercisedayTbyepAðTÞ:¼erT1NXN1i¼0erðTDiÞpPðTDiÞþeDaT!:ð8ÞRemark1.ThepricesoftheAsianoptionssuchas(1)areobtainedviatheMonteCarlomethodinAkahorietal.(2004)whenNissmallenough.SeealsoAkahorietal.(2005)fordetailedarguments.2ProofofTheoremsInthissectionweproveTheorems1and2.Letussett:¼TðN1ÞDsinceitisbetterforthemathematicaldescriptiontoregardthefirstdayoftakingaverageasTðN1ÞD.BeforeprovingTheorem1,weintroducetwolemmaswhichwillbeprovedinthenextsection.1Simplesubstitutionof(10),(23),(13),(16),(20),(17),(21),(18),(22),and(19)givestheformula.And(16),(17),(18),and(19)involvehigherdimensionalmultipleintegral.TheconstantDdependsonN,D,aandb,i.e.,thenumberoftradingdaysthatwetaketheaverageduring,theintervaloftradingdays,thevolatilityandtheinterestrate.OntheasymptoticbehaviorofthepricesofAsianoptions291123Lemma1Foreveryx;y2R,ðxþyÞþ¼xþþyþminðxþ;yÞminðx;yþÞwherezþ:¼maxðz;0Þandz:¼maxðz;0Þforz2R.Lemma2Foranyn2N;thereexistsapositiveconstantCnsuchthatCnatEminXn1i¼0StþDinK!þ;ðStþDnKÞ()#þEminXn1i¼0StþDinK!;ðStþDnKÞþ()#ðt!1Þð9ÞwhereatisthesameasdefinedinTheorem1.ProofofTheorem1Letx¼PN2i¼0StþDiðN1ÞKandy¼StþDðN1ÞKinLemma1andn=N–1inLemma2.Then,bytheline-arityofexpectationEðÞ,wehaveCN1atEXN1i¼0StþDiNK!þ#EXN2i¼0StþDiðN1ÞK!þ#EStþDðN1ÞK þhi:Byrepeatingthisprocedureforn¼N2;...;1andaddingthemall,weobtainthatXN1i¼1CiatEXN1i¼0StþDiNK!þ#XN1i¼0EStþDiKðÞþ :Now,itholdsthatE1NXN1i¼0StþDiK!þ#1NXN1i¼0E½ðStþDiKÞþ1NatXN1i¼1Ciðt!1Þ:Moreover,thedefinitionofaTandt¼TðN1ÞDgivethataTeb22a2ðN1ÞDat:ThereforebysettingD:¼1Neb22a2ðN1ÞDXN1i¼1Cið10Þ292Y.Hishida,K.Yasutomi123weobtainthatE1NXN1i¼0STDiK!þ#1NXN1i¼0E½ðSTDiKÞþDaTðT!1Þ:WehavethuscompletedtheproofofTheorem1.Lemma2isthekeypartoftheproofofTheorem1.Then,wecanextendTheorem1toTheorem2byextendingLemma2toLemma3asfollows;Lemma3Forn;k2N;thereexistconstantsCðmÞnðm¼0;...;k1ÞsuchthatatXk1m¼0CðmÞntmþOðtkÞ!¼EminXn1i¼0StþDinK!þ;ðStþDnKÞ()#þEminXn1i¼0StþDinK!;ðStþDnKÞþ()#ðt!1Þ:WewillproveLemma3alsointhenextsection.Proofof