微观经济学 数学基础 第9章 随机过程I

I1I7789101111919119129139149292192292393931932933949419429439595195295396961962-/——(randomness)11(Bernstein1990)I22()3/SamuelsonMertonBlackScholes——/9-1(Pareto)4(efficientmarket)(informationset)(RollS)(RobertsH)1967(FamaE1970)(weak-formmarketefficiency)()(technicalanalysis)((chartist)))(semistrong-form)(public)(basicanalysis)(strong-form)(private)29-134I3(beatthemarket)——(IvanBoesky)51986——(LevineD)()198613()A()91&(Dow&Jones)1998620015————9-1&bigchartscom911}{ω=ΩT()Tt∈),(tXωt),(tXωωt),0[∞],0[TR→×Ω],0[}{:),(TtXω5(MishkinF)(financialmarketsinstitutionsandmoney)1998I4}{=Ω21,tt),(1tX),(2tX],0[),(TttX∈],0[)(TttX∈1)ωt),(••X2)ωt),(•ωX(samplepath)63)t),(tX•4)ωt),(tXω9-2Ωω),(tXω0tT9-2),(tXωnTtXR→×Ω],0[}{:),(ω()()(discretetime)(stochasticserials)13651364()()1)ΩT76(realization)(trajectory)(simulation)7......}2,1,0:{==ktTk......}2,1,0:{==kkTI52)ΩT3)ΩT4)ΩT8(continuous-timefinance)9128],0[)(TttX∈911])([);(xtXPtx≤=D);(txD])([xtX≤t],0[)(TttX∈xω);(txD],0[)(TttX∈tx912xtxtx∂∂=);();(Dd],0[)(TttX∈1t2t913])(;)([),;,(22112121xtXxtXPttxx≤≤=D],0[)(TttX∈91421212122121),;,(),;,(xxttxxttxx∂∂∂=Dd915);(),;,(11211txttxDD=∞916∫+∞∞−∞=221111),;,();(dxttxtxddn810.1.2I6917])(;...;)(;)([),...,,;,...,,(22112121nnnnnxtXxtXxtXPtttxxx≤≤≤=D918nnnnnnnnxxxtttxxxtttxxx∂∂∂∂=...),...,,;,...,,(),...,,;,...,,(2121212121Ddnt],0[T],0[)(TttX∈8],0[)(TttX∈);(txd)(tX919∫+∞∞−==dxtxxtXEt);()]([)(dµ)(tXt()9t)(tµ],0[)(TttX∈9-3X(t)σµ+µσµ−t9-3],0[)(TttX∈911022)]()([)(ttXEtµσ−=)(tµ9-49-39-49-3I7X(t)σµ+µσµ−t9-4)(1tX)(2tX],0[)(TttX∈1t2t),;,(2121ttxxd9111212121212121),;,()]()([),(dxdxttxxxxtXtXEttre∫∫+∞∞−+∞∞−==d],0[)(TttX∈91122121212211221121),;,()]()][([)]()()][()({[),cov(dxdxttxxtxtxttXttXEtt∫∫∞+∞−∞+∞−−−=−−=dµµµµ9113)()(),(),cov(212121ttttrttµµ−=ttt==21tX9114)(),(),cov()(22tttretttµσ−==911)cos()(θ+=btatXba,θ]2,0[πθ≤=,020,2/1)(πθπθd021)cos()]cos([)(20∫=+=+=πθπθθµdbtabtaEt)](cos[221)cos()cos()]cos()[cos(),(2122021221221ttbadbtbtabtbtEattre−=++=++=∫πθπθθθθ)](cos[2)()(),(),cov(212212121ttbattttrett−=−=µµ9I82),(),()(222attttret=−=µσ9131)],0[)(TttX∈],0[)(TttY∈1t2tnt'1t'2t'mtmn+)]'(),...,'(),'();(),...,(),([2121mntYtYtYtXtXtX9115])'(,...,)'(,)'(;)(,...,)(,)([)',...,',';,...,,:,...,,;,...,,(2211221121212121,mmnnmmnnmnytYytYxtYxtXxtXxtXPtttyyytttxxx≤≤≤≤≤≤=D)(tX)(tYmn+9116mnmmnnmnmnmmnnmnyyyxxxtttyyytttxxxtttyyytttxxx∂∂∂∂∂∂∂=+......)',...,',';,...,,:,...,,;,...,,()',...,',';,...,,:,...,,;,...,,(212121212121,21212121,Dd9117)',...,',';,...,,(),...,,;,...,,()',...,',';,...,,:,...,,;,...,,(2121212121212121,mmmnnnmmnnmntttyyytttxxxtttyyytttxxxddd=2)],0[)(TttX∈],0[)(TttY∈9118∫∫∞∞−∞∞−==dxdytytxxytYtXEttYXXY),;,()]()([),(Re21,2121d),;,(21,tytxYXd)(tX)(tY9119)]}()()][()({[),(221121ttYttXEttCovYXXYµµ−−=)(tX)(tY)(1tXµ)(2tYµ)(tX)(tYtXtY1)1t2tnt'1t'2t'mt9120)',...,',';,...,,(),...,,;,...,,()',...,',';,...,,:,...,,;,...,,(2121212121212121,mmYnnXmmnnYXtttyyytttxxxtttyyytttxxxddd=),(),(),;,(2121,tytxtytxYXYXddd=I9)()(),(),()]()([),(Re21212121ttdytyydxtxxtYtXEttYXYXXYµµ===∫∫∞∞−∞∞−dd0)]()([)]()([)]}()()][()({[),(2211221121=−−=−−=ttYEttXEttYttXEttCovYXYXXYµµµµ2)1t2t00),(21=ttCovXYattXY=),(Re21())(tX)(tY3)1t2t00),(Re21=ttXY)(tX)(tY9141)(stable)(unstableprocess)tnnttt,...,,21t∆],0[)(TttX∈n...2,1),...,,;,...,,(),...,,;,...,,(21212121=∆+∆+∆+=nttttttxxxtttxxxnnnnnnDD)(tX2)3a)n∏==niiinnntxtxtxtx112211);(),;...;,;,(DDb)(Markovprocess)],0[)(TttX∈nTnttt...21),|,(),;...;,;,|,(11111111nnnnnnnnnntxtxtxtxtxtx++−−++=DD],0[)(TttX∈(Markovproperty)c)(independentincrementprocess)I10)(tX)()()(1−−=∆nnntXtXtXnttt...21)(),...(),(21tXtXtXn∆∆∆)(tX3)(Guassian))(tX(Poisson)1)],0[)(TttX∈()∞∆∑=TttX0||(boundedvariation)2)],0[)(TttX∈TtXEt∈∀∞,)(2(squareintegrable)929211979(Cox)(Ross)(Rubinstein)(optionpricing:asimplifiedapproach)(Binaryprocess)10(numericalmethod)(latticemethod)1189-510CRRChriss11Hull(1996)Willmott(2000)Rebonato(1996)I111/8)1(!)!(!)(=−−==−pnxppxxnnxXPxnxttSttS∆+ttS∆+p0,∆ata)1(p−ta∆tt∆+))(1()()(taSptaSpSEtttt∆−−+∆+=∆+22222)]([))(1()()]([)()(tttttttttttSSEtaptapSSESSESVar−−∆−−+∆=−−−=∆+∆+∆+∆+2/1=p0ta∆2ta∆921))(1()(taptapSSEt∆−−+∆=∆922])1()[1(])1([)(taaSptaSpSEtttt∆−−+∆+=∆+uta∆+)1(dta∆−)1(n923∑=−−∆+−−=nitiniinitntSduppiinnSE0)()1(!)!(!)(9-21212Chriss(1996)I129-64x-y0t(node)2t3SuuuuSuuuSuuSuS0SdSddSdddSdddd0t1t2t3t4t9-60t0S1SPuS)1(P−dS1)0SuSu0/SSuu=d0/SSdd=(recombining)udud=1(centeringcondition)2)1TuS50%(standardtree)3)TNNTt/=∆4)1T1SduSPPSSE)1()(1−+=13t∆teSSE∆=µ01)(µ()tdueSSPPS∆=−+µ0)1(1311-4I130uSSu=0dSSd=dudePt−−=∆µP5)(localvolatility)(annualrevenuerate)R01log1SStR∆=00log)1(log)(SStPSStPREdu∆−+∆=)log()1(1log)1(12uPPtSSPPtduloc−∆=−∆=σCRRteu∆=σud/1=dudePt−−=∆µ0833.0=∆t()2887.0=∆t%15=σ%10=µ/0443.1==∆teuσ9576.0/1==ud5853.0=−−=∆dudePtµ%7.14)log()1(12=−∆=uPPtlocσ9-714118.91113.87109.05109.05104.43104.4310010010095.7695.76S0S191.7091.70S287.8284.09S3S4T0T1T2T3T49-79221827(RobertBrown)1900(LouisBachelier)151905(Einstein)14ExcelBenninga(2000)152060(Cootner1964Savage1965Samuleson1969)I141918(NorbertWiener)()(Levy)16tXt1t2t12ttXX−12ttXX−921),0[)(∞∈ttWt∆W