3 3 Vol.3.3 0 3 JournalofHarbinEngineeringUniversity Mar.0 doi:0.3969/j.isn.006-7043.0.03.0 ARIMA GARCH,, 5000 :!""#$%&' *+&,$-.,/0 ' 3$,456$*+7&'89 $:;,/0 <=7*+&,.>?4@A$ ARI MA GARCHBCDE FG%&HIJKL$ B ARIMA GARCHB,MNO7&, BB PQ3IJRST<OUV 7&.!"#: ; ;ARIMA GARCHB;RST<O;:B $%&' :F830.9 :A :006 7043003 0389 06 Exchangepoweroptionpricingbasedonthe ARIMA GARCH stochasticmartingalegainproces ZHENGXiaoyang,ZHONGChongyu ColegeofScience,HarbinEngineeringUniversity,Harbin5000,China Abstract:Intraditionalpricingoffinancialasets,theimpactofdriftrateandvolatilityrateonasetpricesareof tennotpaidenoughatention.itisbetertoreflecttheimpactofthestockpricedriftrateandvolatilityontheaset price.throughthesolutionofstochasticdiferentialequationscontainingthedriftrateandvolatilityrate,thedrift rateandvolatilityalongwiththemartingalemethodwasaddedtotraditionalasetpricing,andmeasurechangewas usedtodetermineasetprices.asaresult,astochasticarima GARCHproceswasjointlysetupbasedonthe clasicalrandomprocesesofarimaandgarch,reflectingnon linearcharacteristicsofstockprice,therebyim provingitsaccuracyinexoticoptionpricing.frompastinformation,anexchangeoptionwithalinearpower type waspricedwithhigh precision. Keywords:driftrate;volatilityrate;ARIMA GARCHproces;exchangeoptionwithalinearpower type;martin galeproces ]O^_$`abc,O7& Y $.973, Y F.Black @A$ Black ScholesO7& S [], S4 E$ J3/, % &, Brown, % $ # ^_. $WX;,% $ $, # 0J $.4% OT ', $, $*+ *+ :009 0 0.,-./: CDWXCY* HEUF040. 0 : 957,Z,[ \,E mail:zhengxiaoyang@ hrbeu. edu.cn. 0:. $, $. `, Y $WX 7$. $ K B*+. 973,D.E.P.Box S auto regresivemovingaveragemode,ar MA, $ S. #, V ` $ P, ` =7, UV 0 '$ $! ",#, $% S, $ S& ' S auto regresive integrated moving average model,
390 3 ARIMA. 98,Engle N S autoregresiveconditionalheteroskedasticmodel, ARCH [ 5].986,L.Bolerslev4 ARCH S$ CDEU N S generalizedautoregresiveconditionalheteroskedastic model,garch, FG Y ^_E*+ $ " $*" [4]. GARCH S+, -. $Q!K, `//0 $ 7, 4 J$ EQ3 $!, Q3 $ 3 #0&4 [6 ]. $, GARCH S 5 Black ScholesO7& S' $ 6 7V 38$. 9:; 456$*+7&'89 $:;,4@A$ B ARIMA GARCHBCDE FG%&HI JKL$ B ARIMA ARCH,/0 <= 7*+&,, MNO7&. -.% &,Y : T ~ ={t=-t ~,-T ~ -,,} T={t t 0, T]}, t=0 ; ; t<0,0 T ~ ; B $ P ; t>0,0 T; $ *. <Ω,F,{F t } t 0,P B=>?, 'F t t 0 @A` σ-b PC, 7 F 0 ={0,Ω},<S t t [T ~,T] ; DE*+&,B, St S t, F $.S 0 ; DE F&,B,4 t 3 ds 0 =rdt,' r D E/. *+4 t $&,,*+ G y t = lns t /S t-,t T ~,H*+$ / e y t $IJ B. t [-T ~,0],μ t t $ : F t- μ t = lim E S t-s t- S t- Δt 0 Δt σ t t $ : σ t = lim E S t- Δt 0 S t-s t- Δt -μ t,t [-T~,0]. F t-. K $ & IJ, : y t =ln+r t =lns t /S t-. % &,$ ` $, L" *"., WX$C9M : {y t } ` N$ O N$, ` $.-. 7 t- $ P F t-,y t $ 4,H μ r =Ey t F t-, S r =vary t F t- = E[y t -μ t F t- ],t [-T ~,0]. % &, $ ARIMAm,n-GARCH p,q:b,h ARIMA-GARCH: Δ d y t =μ t + m θ i Δ d y t + n y t =μ t +σ t ε t, h t =y t -μ t, φ i ε t +ε t, σ t = α i h t+ q β i σ t. ':d ; θ i,,,,m; φ i,,,,n;m 0,n 0;p 0,q 0;α i 0,,,,p;β i 0,,,,q. ε t F t- ~i.i.d. N0,,h t F t- ~N0,σ t, p α i+ q β i<. ARIMA GARCH!"#$ <Ω,F,{F t } t 0,P B=>?,7 ^_EP4 DE*+ S S,Q DE F S 0. 3R/ST $*+&,UVW $ stochasticdiferentialequations, SDE: ds i t=μ i -δ i S i tdt+σ i S i tdw i t. ',t [0,T],,. μ j =Δ d y ~ tj- m θ i Δ d y ~ ij- n φ i ε ij -ε ij F T~, σ j = ij+ q ij- F T~. ',t [-T ~,0],j=,. X,δ i R R/ST. Y3 ds 0 t=rdt, X t [0,T], D E / r R +. 4 PW,< Λ t = T ~ t=- e p t y t M P y t F t- ρ t, M P y t F t- +ρ t =e r M P y t F t- ρ t. ',M P y t F t- ρ t =E P [e ρ t y t F t- ]<, DE/
3, :ARIMA GARCH :BWRST<O7& 39 r R +. dq Es dp =Λ t. Escher <, 4 Q Es W: E QEs [y t F T μ t +ρ t σ t, var QEs [y t F T σ t. + ρ t = r-μ σ t - σ t U t E QEs [y t F T r- σ t. ' 3R/ST $*+&,U$ $ S i t=s i 0exp μ i -δ i - σ t t+σ iw 4 PW i. E P [S i t F T S i 0exp μ i -δ i - σ t t. " e -rt E P [S i t F T S i 0expμ i -δ i -r- σ t t. 4 Q Es W " R/ 7. e -rt e +δt E QEs S i t= e -rt e +δt E P [Λ t S i t F T~ ]=S i 0,,. 3 %& ARIMA GARCH' % + W i t, 4 P, dw dw =ρdt,+ W t W t B t B ds t=μ -δ S tdt+σ S tdb t, ds t=μ -δ S tdt+σ S tdb t+ σ -ρ S tdb t,t [0,T], μ j =Δ d y ~ tj- m θ i Δ d y ~ tj- n φ i ε ij -ε ij F T~, σ j = tj+ q tj F T~. XN μ j σ j ' t [-T ~,0],j=, X B t B t4ω,f,{f t } t 0,PE $,4 Z$^_[\]3^/ _ / P4 > P ~,# S i /S 0 :,.+ Girsanov7 B ~ T=B t+δ d y ~ - m θ i Δ d y ~ - n φ i ε t -ε t -r/σ t, B ~ t=b t+δ d y ~ - m n φ i ε t -ε t -r- ρ t j + q θ i Δ d y ~ - / Δ d y ~ - m φ i ε t -ε t -r θ i Δ d y ~ - n ds t=r-δ S tdt+ / -ρ t, S tdb ~ t, ds t=r-δ S tdt+ S tdb ~ t+ 4 P ~ 3 ds 0 =rdt,t [0,T]. -ρ S tdb ~ t. C ARIMA GARCH :B WQ3R/$IJRSST$T<O7&: ΦS T,S T=λ S T-λ S T +. 4 t $` arst<o ξ=r,,,λ,λ,δ,δ,σ,σ,4 b T>t3 PEt,S,S,ξ;T=R t,s,ξ:tnd - R t,s,ξ:tnd. ': d ={lnλ S /λ S +[ r-δ - r-δ - - /+ /+ v /]T-t}/v T-t, d ={lnλ S /λ S +[ r-δ - r-δ - - /+ - v /]T-t}/v T-t, /- v = +
39 3 R t,s,ξ:t=λ S exp{[ -rα R t,s,ξ:t=λ S exp{[ -rα ρ -, - - /]T-t}, - - N c. /]T-t}. 3 S Y S T &,$*+,7 N P ~ & P, Radon Nikod m dp /dp ~ =S T/E ~ 0[S T], S t=s 0exp{[ r-δ - ]t- ρ ρ # B ~ t}+ -ρ B ~ t} E ~ [S T F T S texp{[ r-δ - exp{- dp /dp ~ = + ]T-t}. T + B ~ T+ -ρ B ~ T} PEt,S,S,ξ:T=e -rt-t E ~ [ΦS T,S T]= e -rt-t E ~ t[λ S T-λ S T + F T e -rt-t E ~ t[λ S Tλ S T/λ S T- + F T e -rt-t E ~ t[λ S T F T ~] E ~ [λ S T/E ~ tλ S T] λ S T/λ S T- + F T e -rt-t E ~ t[λ S T F T ~] E ~ [λ S T/E ~ λ S T] λ S T/λ S T- + F T e -rt-t λ S texp{[ r-δ - + ]T-t} E ~ [λ S T/λ S T- + F T ~]. 4 P W3 Y =S t/s t= S 0/S 0exp{ r-δ - r-δ + /- - + ρb ~ t- ': - - -ρ B ~ ρt+ B ~ t}. B ~ t= B ~ t= t- ρt+b~ t, t+b ~ t. -ρ B ~ t B ~ t,+ Girsanov7 7 $ W W t={[ -
3, :ARIMA GARCH :BWRST<O7& 393 ρb ~ t- tb ~ t]}/ v, -ρ Y =S t/s t= S 0/S 0exp{ r-δ - r-δ + /+ - - + ρt+ { + - } W t}, Y =S t/s t= S 0/S 0exp{ r-δ - r-δ + /+ - - + ρt+ { + - ' W t~n0,t, 3 } W t}. E ~ t [λ S T/λ S T- + F T~ ]= ut-t,s t/s t- +, 0 t<t; { λ S T/λ S T- +, t=t + λ uτ,y= S 0/S 0{[α λ r-δ - - - -α r-δ τ+ + ρ τ+ v z]-} + π τ exp-z τ dz, + : λ /λ S 0/S 0{[ r-δ - - -α r-δ + / + ρ τ+ v z]>. z>z =/ v - [-lnλ /λ S 0/S 0- r-δ - - r-δ + + ρ τ. -
394 3 uτ,y= + - λ S 0/S 0exp{[α λ r-δ - - -α r-δ τ+ + ρ τ+ v z]-} + π τ exp-z τ dz- + - π τ exp-z τ dz. < z=-y τ+ v z z=-y τ E ~ t [λ S T/λ S T- + F T~ ]= ut-t,s t/s t= Nd S 0/λ S 0exp{[ r-δ - - -α r-δ + + R i t,s i,ξ:t,,, ρ T-t}-Nd. R i t,s i,ξ:t=e -rt-t S α i i texp{[α i r-δ i - α i σ i α 0+ p α ih ~ t j + q β iσ ~ t j - i - r-δ + i α 0+ p α ih ~ t j + q β iσ ~ t j T-t}= e -rt-t E ~ [S α i i T] F T ~]. 4 * + 9: B4*+7&'89 $ :;, FG%&HIJKL$ B ARIMA ARCHB,/0 <=7* +&,, MNO7&. *+ S t,s tt<, O. λ =λ = = =,; $T< O. λ =, R +, =0,λ R +,; RSaO. 3 λ =, =, =0,λ R +, V& λ ; $ Black Scholes S. 34 : []BLACKF,SCHOLESM.Thepricingofoptionsandcorpo rateliabilities[j].journalofpoliticaleconomy,973,8 :637 659. []BOXG,JENKINSG,REINSELG.Timeseriesanalysis: Forecastingandcontrol[M].3rded.NewJersey:Prentice Hal,003:36 39. [3]BOXG,PIERCED.Distributionofresiduanantocorela tionsinautore gresive integratedmovingaveragetimese riesmodels[j].journaloftheamericanstatisticalasocia tion,970,93:509 56. [4]BOLLERSLEY T. Generalizedautoregresiveconditional heteroskedasticity[j].journalofeconometrics,986,3 :307 37. [5]BARONEA,ENGLEG,MANCNIR.GARCH optionsin incompletemarkets[c]//nccr FinRisk,Zurich:Univer sityofzurich,004:55. [6]LLOYDB.Powerexchangeoptions[J].FinanceResearch Leters,005,8:97 06. [7]ALEXANDDRA B,REG K.GARCH optionpricing:a semiparametricapproach[j].insurance:mathematicsand Economics,008,43:69 84. [8]ESSERA.Generalvaluationprinciplesforarbitrarypayofs andapplicationtopoweroptionsunderstochasticvolatility models[j].financialmarketsandportfoliomanagement, 003,7:35 37. [9]ALISON E.A courseinfinancialcalculus[m].beijing: Post&TelecomPres,006:08. [0]GERBERU.Actuarialbridgestodynamichedgingandop tionpricing[j].insurance,mathematicsandeconomics, 996,8:83 8. []ALATONP,DJEHICHEB,STILLBERGERD.Onmodel ingandpricingweatherderivatives[j].appliedmathe maticalfinance,00,9: 0. []BATH E,SALTYTEJ.Thevolatilityoftemperatureand pricingofweatherderivatives[j].quantiativefinance, 007,7:553 56. [567:89:]