probability density functions and first-exit times

《probability density functions and first-exit times》由会员上传分享，免费在线阅读，更多相关内容在工程资料-天天文库。

1、CHAPTER10probabilitydensityfunctionsandfirst-exittimesInthisChapter...•thetransitionprobabilitydensityfunction•howtoderivetheforwardandbackwardequationsforthetransitionprobabilitydensityfunction•howtousethetransitionprobabilitydensityfunctiontosolveavarietyofproblems•first-exittimesandtheirr

2、elevancetoAmericanoptions10.1INTRODUCTIONModernfinancetheory,especiallyderivativestheory,isbasedontherandommovementoffinancialquantities.Inthemain,thebuildingblockistheWienerprocessandNormaldis-tributions.Ihaveshownhowtoderivedeterministicequationsforthevaluesofoptionsinthisrandomworld,butIh

3、avesaidlittleaboutthewaythatthefuturemayactuallyevolve,whichdirectionastockisexpectedtomove,orwhattheprobabilityisoftheoptionexpiringinthemoney.Thismayseemperverse,butthemajorityofderivativetheoryusesideasofhedgingandnoarbitragesoastoavoiddealingwiththeissueofrandomness;uncertaintyisbad.Ne

4、vertheless,itisimportanttoacknowledgetheunderlyingrandomness,tostudyit,todeterminepropertiesaboutpossiblefutureoutcomes,ifoneistohaveathoroughunderstandingoffinancialmarkets.10.2THETRANSITIONPROBABILITYDENSITYFUNCTIONTheresultsofthischapterwillbeusefulforequities,currencies,interestratesora

5、nythingthatevolvesaccordingtoastochasticdifferentialequation.Forthatreason,Iwilldescribethetheoriesintermsofthegeneralstochasticdifferentialequationdy=A(y,t)dt+B(y,t)dX(10.1)170PartOnemathematicalandfinancialfoundationsforthevariabley.InourlognormalequityworldwewouldhaveA=µyandB=σy,andthenw

6、ewouldwriteSinplaceofy.Toanalyzetheprobabilisticpropertiesoftherandomwalk,Iwillintroducethetransitionprobabilitydensityfunctionp(y,t;y
,t
)definedby
bProb(a

7、tstartedoutwithvalueyattimet.’Thinkofyandtasbeingcurrentvalueswithy
andt
beingfuturevalues.Thetransitionprobabilitydensityfunctioncanbeusedtoanswerthequestion,‘Whatistheprobabilityofthevariableybeinginacertainrangeattimet
giventhatitstartedoutwithvalueyattimet