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上海交通大学硕士学位论文随机波动率模型下使用期权对冲的风险最小化问题姓名：陆文杰申请学位级别：硕士专业：应用数学指导教师：林建忠20070101 ¯¯˜˙ .eƒ^ˇ Ø ”x zflK` †L ¥z 7K Nz!7KM# E ? ˇ K §‰|”x+nfi⁄ 7K § y 7Kn %SN ¥§ˇ ~^ ß)‹ §ˇ k(„ıC!⁄ $!V | ‘:, 7K¯ 2 H A^, 3Œ˘7Kn §|^ˇ ?1”x+n% 9 ] ‰d!‰|”x O !”x z ı ¡ SN§ˇd· an 5 r! (J flK '–”x (ValueatRisk) ”x § ˜ 3 E, d |˙ ¯¯˜˙ .e§ ƒ ¯ ”x zflK ' SN ˜⁄J8BXe 1.du–’uƒ^ˇ ?1”x zflK ˜ ·˜u~ ¯˜˙ .e§ '3 d ¯¯˜˙ d .e?1 dflK ˜§O d .eˇ] | d ' ) /“§–9”x ) /“§¿y† 3”xØ ⁄ ^ e VaR zflK§ – d/ˇL) =k C ' § 2.Øu E, ¯ ¯¯˜˙ .§ ,ˆ{ ı) /“ (J§ '3 « ¯¯˜˙ Fong-Vasicek|˙ .e§( ”x zflKø ¢SAiv ˇ ˘a‹˘ '^ §( ) Œ {J ¯¯˜˙ .e «ˇ ‰d ”x z {§¿ ‘† ø« {duc k) /“)§¿ ' ”x z |˙L§ [ˆ’§ˇd3O E,5 · 1 3.Øu ¯¯˜˙ Cox-Ingersoll-Ross d .§|^Monte-Carlo { Ø N ”x zflK?1 Œ [§ƒ^ '? { fl$/… ”x zflK )§¿ ~ ¯˜˙ .e „?1 ’ ’ c ¯¯˜˙,”x , {,Monte-Carlo {,Cox-Ingersoll-Ross .,Fong-Vasicek .–v–RiskMinimizingundertheStochasticVolatilityModelsAbstractUnderthebackgroundofglobalization,financialinnovationandtechnologyde-velopment,theproblemtomanagefinancialmarketriskshasbecomeoneofthehotissuesofthecontemporaryfinancialresearch.Amongvarioustechniques,itisawidelyusedpracticetohedgeportfoliobyusingoptions.However,suchaprobleminvolvesassetpricing,riskmeasuringandminimizingtheories,andisthusquitecomplextosolve.Thisdissertationaddressthequestionthatunderthesophisticatedstochasticvolatilitymodels,howtominimizeValue-at-Riskwithafixedhedgingcost.Bothstockandbondcasesarediscussed.Thekeyfindingscouldbesummarizedasbelow:1.Pastresearchesonriskminimizingmainlyfocusonconstantvolatilitymodels.Thisdissertation,however,studiesthisproblemunderthestochasticmodels.WeprovideananalyticalapproachtocalculatethepayoffdistributionofportfoliovalueattheterminationandtominimizeVaR.Itisdemonstratedthattheoptimalhedgingstrategycouldbefoundbysolvingasingle-variabledifferentialequation.2.Astothemorecomplexbondcase,althoughtheanalyticresultsisnearlyimpossibletoobtain,thisdissertationprovidesaanalytic-numericalapproachtopricevi ˇ ˘a‹˘ 'bondoptionsandtominimizeVaRundertheFong-Vasicekstochasticvolatilitymodel.Thisalgorithmprovestobepracticalintermsofcomputationalcomplexity.3.WealsorunanumericalsimulationtosolveaspecificCox-Ingersoll-Rossformedstochasticvolatilitystockpricemodel.ByusingMonte-Carlosimulation,theap-proacheddiscussedearliereffectivelyfindtheoptimalsolutiontominimizetherisk.Keywords Stochsticvolatility,Value-at-Risk,MartingaleMethods,Monte-Carlomethods,Cox-Ingersoll-RossModel,Fong-VasicekModel–vii– ˇ ˘˘ ' M5(† x›(† ⁄¥ ˘ '§· 3 e§ Æ?1 ˜ ⁄ ⁄J '¥fi†5† ^ SN § ' „? ƒ ‰8Nfi†uL‰ L ‹⁄JØ ' ˜ › z 8N§ fi3'¥–†( “I† ¿£ († {˘(Jd «œ˘ ' \¶ Fˇ c Fii ˇ ˘˘ ' ƒ^˙ ˘ ' )˘ k’ 3!ƒ^˘ ' 5‰§ ¿˘ 3¿ I[k’ ‰¯ x ' E f §#N ' / ˙ ˇ ˘ – ˘ ' ‰ 'SN?\k’Œ ¥?1u¢§ – ^K! ‰ £ E ˆª fi? ˘ ' §3c) •^ ˙ ˘ 'Æu £ 3– S /√0⁄˘ ' ' \¶: \¶ Fˇ c FFˇ c Fiii1 X 1.1flK J †L ¥z 7K Nz!7KM# E ? (7K § &EE ) ˇ K § ¥7K‰|u) ˜:5 ( 5Cz§5 * ! ˙†wJp§7K‰|3†L$1¥ ^ / ,§ ¥†L$1 J[z§ Jp§ ߣ †E \ 6u7K‰| d §C Ac5§7K‰| ¥y 5 ¯˜§c ?\20›V90c – ›. S 7K ¯“ u§›. 7K¯ § U37K‰| ˜ºA •{ ª ø ˇ ƒ 7K‰|”x⁄ ¥7K¯ 9i+ ’5 :§”x+n ˇd⁄ 7K § y 7Kn %SN‰|”x +n· 3O(E£ ”x ˜: §|^ «7K (X ˇ(futures)!ˇ (options)!Ø (swaps)) Ø7K”x5;!' ! L§ ¥§ˇ 7KM# §˜ku70c ¥ˇ3{I y§¿Øfl A^u‰|”x +n§ 7K¯ 2 H|^ˇ ?1”x+n ⁄– @ · p ”x+nE §·duˇ ‹ k(„ıC( – ”x+nI ˘‰1 d ºŒ§$ | ⁄ E, ”xØ )§⁄ $(ˇ d ˇ~ $u I ] d § ˇ~ I|G y7)§V |(• /|^ˇ –47K¯ 3y ‰|d , § 3d e ‹ ? |d ¯‹) ‘:, 3Œ˘7Kn §|^ˇ ?1”x+n%· n 5 r! (J ˜ Kø ¥ 9 –eA ¡ flK 1. ”x o ] | ( 1 ˇ ˘a‹˘ 'ƒ!¯! fi‰ | ) ‰dflK¶2.˜u] | ˇ ‰dflK¶3.‰|”x flK¶4.”x z‰”x+n Y ‘zflK{II[†L ˜⁄ 7K˘[Dong-HyunAhn[1999] 3JournalofFinanceuL ' ¥§3 .b d l~ ¯˜˙ A K$˜§ ƒ^”x (Value-at-Risk) ”x ø /e§ /£ ª4 flKƒ ·˜u ƒL§ ¯˜˙· fi ~Œø b e §3ø b e§ – /ˇA K$˜ †; Black-Scholes .… d ˇ d — 5 , ¯˜˙w ~Œ§¿ U ¢S‰|Œ Ø—/[ § X ’u ƒ¯˜ ¢y' L†¯˜˙ø ºŒˇ~ ·~Œˇ ‰|Œ ¥ „¯˜˙ v‰d Cz› / 0 / 0 A£volatilitysmileorvolatilityskew⁄y† ø :§ Blattberg-Gonedes(1974)§Rubinstein(1985)§Scott(1987) Ø „¯˜˙/ 0 / 0 A n )”·§¯˜˙ ·~Œ · ¯ §§Ø U m d k’§$ d ƒd – ¯ˇ ⁄ ˇd§3 ˜ƒ^ˇ ?1”x+n flK §ƒ^ ¯¯˜˙ .w, ˛ ‰|¢S „§ uyIS ˘ kø ¡ ; ˜1.2IS ˜yGØu ¯¯˜˙e ˇ ‰dflK§7KŒ˘.k2 &?§~ )ßg· )e A« 1.b ¯˜˙ ’u m …Œ§=σ≡σ(t)( Wilmottetal.(1995))Merton(1973)? ‰† ¯˜˙¯σ2T=1TRT0σ2udue§|˙ˇ ( §¥ ~¯˜˙*— σ≡σ(t) „2.b ¯˜˙ ’u m d …Œ§=σ≡σ(t,St)( Hull(2000))§=d dL§ dSt=μStdt+σ(t,St)StdZt ¥Zt· IOWienerL§3.b ¯˜˙· ¯ §= 3 d ¯ Zt , ¯ Wt§¯˜˙ l–2– ˇ ˘a‹˘ 'L§dσt=a(t,σt)dt+b(t,σt)dWt ¥Wt dL§¥ Zt· ’ WienerL§( Buff(2002),HullandWhite(1987),Heston(1993))4.b ¯˜˙ ßu ¯C x§=σ≡σ(x(t))§ ¥x(t)· ¯L§( ElliottandSwishchuk(2002)§GriegoandSwichchuk(2000)§Swishchuk(1995)§Swishchuk(2000)§Swishchuketal.(2000))5.b N ¯˜˙c (volatilitysurface)·l | ¯˜˙c ¥] 5 ø« (‰( ¢S ¯)¯˜˙ .˜uAvellanedaetal.(1995)J {§ ˜^ˆ@| nO ] | d (.§ ø (.·ˇL3 4 :σmin σmax?'OO ] | d (da { Buff(2002))6.b ¯˜˙· ¢ L§§=σ(t,St) ßuSt:=S(t+θ)§ ¥θ∈[−τ,0]( Kazmerchuk,SwishchukandWu(2002)), ¡§Øu‰|”x+