Chapter4BrownianMotion&Ilt6FormulaStochasticProcess1Thepricemovementofanunderlyingassetisastochasticprocess。TheFrenchmathematicianLouisBachelierwasthefirstonetodescribethestocksharepricemovement2SaBrownianmotioninhis1900doctoralthesis.5introductiontotheBrownianmotion*derivethecontinuousmodelofoptionpricing*dvingthedefihitionandreleyantpropertisBrawmianmotionderivestochasticcalculusbasedontheBrownianmotioninc_udingtheliointegral&toformulaAllofthedescriptionanddiscussionemphasizeclarityratherthanmathematicalrigor.Coin-tossingProblem目Definearandomvariable1五eadRi(酣)=,G=h2K)一1=10i目Itiseasytoshowthatithasthefollowingplopeltiss:E(R)=0,Var(R)=l,E(RR)=0,(i八刀&R&RGx丿areindependentRandomVariable盯Withtherandomvariable,definearandomvariable呓=VARandarandomsequence84.(k=0,LL):q人日弯=定砺=穴VER,g=l2L闯l闹RandomWalkConsideratimeperiod0,刀,whichcanbedividedintoNequalintervals.LetA=TVN,t_n=nA,(n=0,1,cdots,N),then0=白0;0片外CentralLimitTheoremlForanyrandomsequenceRdefinedabove,When大一yool火辅菩鸣X,whereihelahdomvaliahlsXN(01),与ie.therandomvariableXobeysthestandardnormaldistribution:E(X)=0,Var(X)-1.ApplicationofCentralLimitThem.盯Considerlimits(z)/更2S一亡0