泊松分布在金融领域的应用.docx

泊松分布在金融领域的应用.docx

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泊松分布在金融领域的应用

泊松分布在金融领域的应用

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{

T≥0}iscalledthecountingprocess,iftheIntimeintervals（0,t]occursinacertainevent（duetoapointonthetimelineofevents,sopeoplecalledtheevent）number.Therefore,acountingprocessmustmeet:

（1）

Takenon-negativeintegervalues;

（2）Ifs

<

（3）

In

Therearecontinuousandpiecewisefetchconstants,

（4）Fors

Isequaltothetime（s,t]thenumberofeventsoccurringin,

Saidthecountingprocess{

T≥0}hasindependentincrements.Ifit'sinanyfinitenumberofdisjointeventsthatoccurinthetimeintervalofafewindependentofeachother,saidthecountingprocess{

T≥0}withstationaryincrements,ifatanytimetheprobabilitydistributionofthenumberofeventsthatoccurredintheintervaldependsonlyonthelengthoftheinterval,andhasnothingtodowithitslocation.Thatforany

Ands

0Incremental

And

Havethesameprobabilitydistribution.

Definitions1.2countingprocess{

T≥0}iscalledintensity（orspeed）ThehomogeneousPoissonprocessifitmeetsthefollowingconditions:

（1）P（

）=1，

（2）Hasindependentincrements.

（3）Forany0s

Withparameter（t--s）ThePoissondistribution,which

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

>0If

（1）

 ；

（2）Processeswithstationaryindependentincrements.

If

Youcanprove

Thatis,

Hasmeanm（t+s）m（t）ofthePoissondistribution.

Non-homogeneousPoissonprocessisimportantbecausenolongerrequiresastationaryincrements,allowingthepossibilityofeventsatcertaintimesthanothers.

Dangstrength（t）Territoriescanbenon-homogeneousPoissonprocessisregardedasahomogeneousPoissonrandomsampling.Establishedspecificallytomeet（t）≤And,forallt≥0andconsideredastrengthforPoissonprocess.Setuptheprocessattimetwithprobability（t）/Count,wascountofeventsistheprocessofwithintensityfunction（t）Non-homogeneousPoissonprocess.

Second,basedoncomplexPoissonprocessmodelofstockprices

1.modelconstruction

Assumptionsinthestockmarket,ayinandthestrengthofeachtransactionisasequenceofindependentidenticallydistributedrandomvariables.WeuseItradeintensity,thenforanyi>O,Havethesamedistribution.SetthestocktradesIsaparameterfor（

>o）Poissonprocess,itstradingvolume

forthecompoundPoissonprocess.

Webelievethatthetradingvolumeinthestockmarketwillhaveanimpactonstockprices,establishedthefollowingmodeltosimulatethevolatility.SettingthetimeparameterissettoT=[o,

）。

Stockpriceprocessiss（t）ands（0）=S0timestockprices.Thefollowingdefinition（t）²（Sufficientlysmall）changesinstockpricewithinatimeinterval:

①uSprobabilities

IIsprobability

③dSprobability

U,d（u>l>d>0）isaconstant,o（（

t）2）meet

.Thuswegetthestockpriceprocess,inaverysmalltime（

t）2,pricechangehasthreeStates,eachStateisassociatedwiththetradingvolumewasexpected,stockpricesriseorfallbyparametersofeach

，

（

>0，

>O）lation（alsoreferredtoasmarketdepthparameter）.CompoundPoissonprocess,theexpectedvaluecanbeobtained

andthemodel

①uSprobabilities

IIsprobability

③dSprobability

Table1volumeimpactontheshareprice

Volumecomparisoninordertomakeitclearthatthemodelofstockpricelevelsandmarketparameters

，

Extentoftheimpactonthestockprice,wegivespecificexamplestoillustratethis.AssumingstockstrengthObeytheinterval[a,,b]oftheuniformdistributionon（unit:

shares）,intervals

0.1,theparameter

=0.08,

=O.02。

SharessuptouStheprobabilityprise,felltodSprobabilitypdown,punchangedstockpricesdidnotchangetheprobability.Afterprobabilitiescalculatedfromtable1itcanbeseenthatwhenthestocktransactionsnumberparameters=1Shi,tradingintensityincreasedprobabilityofincreasespushedupshareprices,whichledtorisingshareprices.

Similarly,whenstocktradingintensityisconstant.Theincreaseinnumberoftransactionscanalsocausestockpricestochange,forexample:

inthisexample,becausethe

Andsothenumberoftransactionsincreasedprobabilityofrisingstockprices.Thusitcanbeseen,dealstrengthDependsonthedistributionofthenumberofvolumesandmarketparameters，Thesizeoftheriseandfallinstockprices.Onthestockmarketofwhichthepracticalproblems,wecanobtainawealthofhistoricaldata,byreasonableanalysis,processingthesedataandapplyeffectivestatisticalanalysis,wecanpending,andapproximatevolumeofdistributionandthecorrespondingparametersofthemarket.

Three,modelanalysis

Inthemodelassumptionsabove,weconsidertheEuropeancalloptionpricingproblem.Setthereisabondandastockmarket,bondsarerisk-freeasset,therisk-freeinterestrateisr,stocksareriskierassets,thepriceprocessasdescribedinthemodel.SetassociatedwiththestocksofEuropeancalloptionsexpireast,thestrikepriceforthek.Int=0times,bondpricesforb,then（t）²timebondvalue（principalandinterest）

.Assumingriskneutralprobability,volatilityinstockpricesareasfollows:

①uSprobabilities

IIsprobability

③dSprobability

Typeinthe

Marketdepthcorrespondingtotherisk-neutralprobabilitycoefficientindexnsaid"neutral"（neutral）.Becausetherisk-neutral,sotheexpectationsofstockyieldsequaltotheyieldsonbonds,namely:

We'vegotaEuropeanbuyingoptionpricingfor:

Four,riskavoidanceandriskcontrol

Onthestockmarket,investorsoftenAvoidRiskandriskcontrolasatoppriority,soconsiderhowtoavoidinvestmentrisksisofgreattheoreticalandpracticalsignificance.Assumetheexpectationsofstockreturnsthanyieldsonbonds,which

Probabilityisassumedbythemodelshowsthatastock'spricefluctuationsareasfollows:

Undertherisk-neutralprobability,accordingtothemodelassumptions,fluctuationsinstockpriceprobabilityis:

Five,modelthespecificpracticalproblemsofapplication

Select2009years6months22daysuntil2010years6months22daysofchinadotcom（stock）pricesastheresearchobject,theactualdatafromhttp:

//CN．finance.yahoo.corn。

Theyearsharesofthestockpriceandtradingvolumedatastatisticsandanalysis,2010years6months22daysthestock'sopeningpriceof8.57,itsEuropeancalloptionexpirefor3months,2010years9months22daysexpire,dothepricesfor7.5,thecurrentannualinterestrateof0.04,E