泊松分布在金融領域的應用

Random(Poissondistribution)inthefieldoffinancialapplications

【Abstract】mathematicalfinanceasasubject.Usingagreatdealofteachingtheoryandmethodstudyandsolvemajortheoriesinfinancialissues,practicalproblems,andsome,suchasthepricingoffinancialinnovation.Duetofinancialproblemsthecomplexityofthemathematicalknowledge,inadditiontothebaseofknowledge,thereareplentyoftheoriesandmethodsofmodernmathematics.Inthisarticleweintroducethevolumefluctuationsinstockpricemodel.ApplicationofPoissonprocesstheorydescribesthevolatilityofstockprices,andbasedonoptionpricingtheory,Europeancalloptionpricingformulaisderived.Inthecourseoffinancialinvestment,investorstypicallyshyawayfromrisks,andcontroltherisksinthefirstplace,sowefurtherriskaversioninthemarketofEuropeancalloptionspricerange.Inordertogiveinvestorsamorespecificreference.

【Keywords】stochasticprocessofcompoundPoissonprocesssharestradedoptionspricing

Alongwithrapideconomicdevelopment,avarietyoffinancialtoolscontinuetoproduce.Thecorrectvaluationoffinancialinstrumentsisanecessaryconditionforeffectivemanagementofrisk,weusedthepricesofsecuritiesdescribedingeometricBrownianmotionprocessiscontinuous.Withfairpricesandfinancialinstrumentsisthattheyarereasonableandthekey.Mathematicalfinanceis20centurieslaterdevelopedanewcrossdiscipline.Itisobservedwithauniquewaytomeetfinancialproblems,whichcombinemathematicaltoolsandfinancialproblems.Provideabasisforcreativeresearch,solvingfinancialproblemsandguidance.Throughmathematicsbuiltdie,andtheoryanalysis,andtheoryisderived,andnumericalcalculation,quantitativeanalysis,researchandanalysisfinancialtradingintheofvariousproblem,toprecisetodescriptionoutfinancialtradingprocessintheofsomebehaviorandmayofresults,whileresearchitscorrespondingofforecasttheory,reachedavoidedfinancialrisk,andachievedfinancialtradingreturnsmaximizeofpurpose,tomakesaboutfinancialtradingofdecisionmoresimpleandaccurate.

Becauseoffinancialphenomenastudiedinmathematicalfinancestronguncertainty,stochasticprocesstheoryasanimportantbranchofprobabilitytheory,andarewidelyusedinthefinancialresearch.Stochasticprocesstheoryinclude:theoryofprobabilityspaces.Poissonprocess,theupdatingprocess,discreteMarkovchainsandcontinuousparametersoftheMarkovchain,theBrowncampaign,martingalestheoryandstochasticintegration,stochasticdifferentialequations,andsoon.Inrecentdecades,theoryandapplicationsofstochasticprocesseshasbeendevelopingrapidly.Physics,automation,communicationsciences,economicsandManagementSciencesandmanyotherfieldsareactivefigureofthetheoryofstochasticprocesses.

ThisstochasticprocesstheoryofoptionpricingusingPoissonprocesstheorytothestudyofregularityofstockpricefluctuationinthestockmarket,considertheimpactoftransactionsonstockprices,stockpriceprocessmodelisconstructed.Andgivestheoptionofavoidingrisksintheinvestmentprocess.

AndthePoissonprocessconcepts

Definitions1.1randomprocess{Nt,T≥0}iscalledthecountingprocess,iftheIntimeintervals(0,t]occursinacertainevent(duetoapointonthetimelineofevents,sopeoplecalledtheevent)number.Therefore,acountingprocessmustmeet:

(1)NtTakenon-negativeintegervalues;

(2)Ifs<t,thenNs<Nt

(3)NtInR+=［0，∞）Therearecontinuousandpiecewisefetchconstants,

(4)Fors<t,Ns,t=NS-NtIsequaltothetime(s,t]thenumberofeventsoccurringin,

Saidthecountingprocess{Nt,T≥0}hasindependentincrements.Ifit'sinanyfinitenumberofdisjointeventsthatoccurinthetimeintervalofafewindependentofeachother,saidthecountingprocess{Nt,T≥0}withstationaryincrements,ifatanytimetheprobabilitydistributionofthenumberofeventsthatoccurredintheintervaldependsonlyonthelengthoftheinterval,andhasnothingtodowithitslocation.Thatforany0≤t1≤t2Ands≥0IncrementalNt1,t2AndNt1+s,t2+sHavethesameprobabilitydistribution.

Definitions1.2countingprocess{Nt,T≥0}iscalledintensity(orspeed)ThehomogeneousPoissonprocessifitmeetsthefollowingconditions:

(1)P(N0=0)=1，

(2)Hasindependentincrements.

(3)Forany0s<t,Ns,t=NS-NtWithparameter(t--s)ThePoissondistribution,which

Definitions1.3countprocess{,T≥0}iscalledthePoissonprocess,theargumentis，λ>0If

(1)N0=0

；

(2)Processeswithstationaryindependentincrements.

If

Youcanprove

Thatis,Ns+t-NtHasmeanm(t+s)m(t)ofthePoissondistribution.

Non-homogeneousPoissonprocessisimportantbecausenolongerrequiresastationaryincrements,allowingthepossibilityofeventsatcertaintimesthanothers.

Dangstrength(t)Territoriescanbenon-homogeneousPoissonprocessisregarded

③dSprobability

Typeintheαn>0,βn>0,λn>0Marketdepthcorrespondingtotherisk-neutralprobabilitycoefficientindexnsaid"neutral"(neutral).Becausetherisk-neutral,sotheexpectationsofstockyieldsequaltotheyieldsonbonds,namely:

We'vegotaEuropeanbuyingoptionpricingfor:

Four,riskavoidanceandriskcontrol

Onthestockmarket,investorsoftenAvoidRiskandriskcontrolasatoppriority,soconsiderhowtoavoidinvestmentrisksisofgreattheoreticalandpracticalsignificance.Assumetheexpectationsofstockreturnsthanyieldsonbonds,whichu-1αλEξ-（1-d）βλEξ>r

Probabilityisassumedbythemodelshowsthatastock'spricefluctuationsareasfollows:

Undertherisk-neutralprobability,accordingtothemodelassumptions,fluctuationsinstockpriceprobabilityis:

Five,modelthespecificpracticalproblemsofapplication

Select2009years6months22daysuntil2010years6months22daysofchinadotcom(stock)pricesastheresearchobject,theactualdatafromhttp://CN．finance.yahoo.corn。Theyearsharesofthestockpriceandtradingvolumedatastatisticsandanalysis,2010years6months22daysthestock'sopeningpriceof8.57,itsEuropeancalloptionexpirefor3months,2010years9months2