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3456789:; ;= (continuous-time¯nance) Merton 1990 1992 ! #$%&’($%&) %&*+,- ./ 01 %&23456789: ;=?. @ABCDEFG H I JK LMDN O*P Q3RS BrownTUVLMWDXY Z [Q\]!^_ Black{Scholes‘Vab-cQ3-‘ 8 I +lN )3= H s +u x@ ? x5.1defghijklmBrownnop(´n)n=0;1;2;:::; qLMrstu)v´0=0;´n+1=´n+n;n=1;2;:::;O*n,n=1;2;:::; wxyz {R|1/2D}~§1 LM U 01 LMrstu { 4 1.E[´n]=0;Var[´n]=n,n=0;1;2;:::;N 2. mnjk0, ´m¡´n ´j¡´kwxyz 3. m¡n=j¡k, ´m¡´n ´j¡´k 1D LM. LMrstu /) Brownno(Bt)t0 LM Y)v L t./ LM. )B0=0, u 4 1.( m ) st, Bt¡Bs u6s Buyz 2.( m ) st, Bt¡Bs Bt¡s¡B0 1 R|D 3.(jk ) (Bt) ! 4.(# ) $%t0,Bt &’E[Bt]=0VVar[Bt]=t () *D LM. 243Bachelier 1900 ( +c 4), BrownTU) Ein-stein 1905 + 01-./BrownTU 0 01O23 45 BrownTU 6 -7 895 :-;@ = 45?@ ABC D?@01 EF 4 BrownTU !+ GG =? HI J)0K!LM )N./(O)P G QR =S 0KFTUV^WXC YZ [\)] ^_ ,GG =? ‘IV^aI bcdef)T ghi Zj 8k5* E[Bt]=0VVar[Bt]=t K q/,p)0Var[Bt] t lH = 4m-?@ n2])o yz Vpq )zv=r 9s0 ) Var[B2s]=Var[B2s¡Bs+Bs¡B0]=Var[B2s¡Bs]+Var[Bs¡B0]=2Var[Bs¡B0]=2Var[Bs]:js5H=ri $%tMIk, Var[Bks]=kVar[Bs], [ uv w?r $% ‘I¹0, Var[B¹]=¹Var[B1]. x] )vr $%t0,Var[Bt]=tVar[B1]:,8k5* * H={ 45?@ 0mR| *fyz-‘s u{U)] | . BrownLM. +=} ~z( yz LM U x ‘I RS) C?@0 n2]) Ã(t;¸)=ÁBt(¸)=E[ei¸Bt]: Ã(t+s;¸)¡Ã(t;¸)=E[ei¸Bt+s]¡E[ei¸Bt]=E£ei¸Bt¡ei¸(Bt+s¡Bt)¡1¢¤=E[ei¸Bt]E[ei¸(Bt+s¡Bt)¡1](yz )=E[ei¸Bt]E[ei¸(Bs¡B0)¡1](pq )=Ã(t;¸)E[ei¸Bs¡1]:0E[ei¸Bs¡1]=¡12¸2Var[Bs]+o(s)=¡12¸2+o(s):][) 4O {s0, s!0, Iri uWDXY@Ã@t(t;¸)=¡12¸2Ã(t;¸):0 à t WDXY) $%¸2R Ã(0;¸)=E[1]=1:][)Ã(t;¸)=ÁBt(¸)=e¡12¸2t:244 ) *D LM. ‘I )’ J) D e¡12¸2t=Z1¡1ei¸xe¡x22tp2¼tdx;vrBt ) 0,X t *D N(0;t).01) BrownTU LMrs /I 82 8 BrownTU(Bt) o Y(Vst)t0=(Bt+s¡Bt)t0&’E[Vst]=0,E[Vst+¿Vst]=½0;¿s;R(¿);¿s;01R(¿) ¿ ‘I) O R(0)=Var[Bs]=s. LM Yft&’ E[ft]=¹{ E[ft+¿ft]=r(¿)+ t~ ) ( ) BrownTU o YI pq Y pq Y={!5J K# L .2 LM Y 0KLM Y $ - %( LM&’ I x ( LM&’ I ) gt=asin!t+bcos!t LM Y)O*!(*|) I)a;b I $+ , LM. ),aVb X -V./ 0 LM&’ 01 pqLM Y - z ’ ~z LM&’ x(2V’ 2 D) LM&’3D * 3D 01 }0) 0 pq YI45@ - 6$ 0KX7 8DN X7) H; -DN+c(’+9:;) 6$ 0 01{ )P= }+ctu(’+9:;tu) ? pq )’ p7aO pq/ (@ ABO* oCDE) BrownTU o Y *8 YF pq Y)v LM&’3D 01 + ( *| LM&’3D 01+ 1 pq Y GHI 01) HI J K~LM=S), G N GOQ PQRKSO(*|) Q Q7 BrownTU o YYTU GHI KI ] VWX J7Y )BrownTU ?I GHI LMrstu 1)BrownTUH={ Z RS[\ X [tMH] ¾-/I^ _‘ Y-j RS defg Z h@ 01 i =!5-‘ P=) LMrstu 1)BrownTU Z(LM Y))v st-J)Bt .s ¾-/IFs 5j$+I Bs.’ klmSJ) .s n- .t BrownTUvop#)O qrI .s BrownTU s01 BrownTU~/ @AcX)0I tuv .s~7p#n- .t cX wx 245 qBrownTUD=y3Oz Z ${| }) (aBt+b),01a;b I 4MH Z ~ }) vopXT 0 (B2t) Z)] B20=0,0E[B2t]=Var[Bt]=t6=0.M,)={ @)(B2t¡t) Z 0KZ KT JIT BrownTU{| JIT H y3Z @ )(expBt)I Z M,)={J@(exp(¾Bt¡(¾2=2)t)) Z)01¾ I 0KZ 04K${| Z @Z h@ ^ *)^ {|5,-+9cXSt ./&’ uLMWDXY dStSt=¹dt+¾dBt;01¹ (drift))¾ o (volatility) &’01 WDXY LM Y Brownno 0 XY WXI Q7 : )0 01={!5s ‘ O LM. @AcX = :;| )v+c ./ l+c ¹dt, X ¾2dt(] VWX J7)dBt=Bt+dt¡Bt X dt) HI J)01 8Markowitz X -. :;| 7 ) BrownTU ) = :;| D *D LM. X)s] XY ) ] 3dStdt=Stµ¹+¾dBtdt¶; D={J)+c oC|¹ i01 ¾ GHI 9M)01 K) )I ] t7 O ) ^!JBrownTUBt ! GG =? )),O?IdBtdt P t7 01-7 (St)D=T 5 (St)=(x0exp(¾Bt+(¹¡¾2=2)t)),O*x0=S0,- I #-J) Z 0 ¹=0,(St)I.3JIZ(exp(¾Bt¡(¾2=2)t)).[D={}@) $%t-J)LM. St() I *D +9cX 01 ,p $%,- ), %( # ^ b C$DNI& ri s] LMWDXY’(/(0= VWX5Y dt=1 ,ri )). Sn+1¡SnSn=¹+¾n+1;01fng LM Utu 0I J) %BrownTU &’ LMWDXY2)]@t *+,-m(.)/0 12(3456/07)89:;fgo12 0 ,p ] =*^? )@A246])V^ CY @A; B C;,p c @AcXtu)3LMrstu)OPD tMH Bachelier 0 0KJ7^ ‘V2@4X+;?- 2)])Z 2@DNJ@)+c Dtul pqtu),+c:;tuE pqtu)],E= )3LM Utu X) XYD=F dlogSt=¹dt+¾dBt: ’(/ logSn+1¡logSn=¹+¾n+1: tuv*+,-mG HI1289:;fgo121 04 ’(/ J)0 ] K* J@) LUs t7 J )v Us logµSn+1Sn¶¼Sn+1¡SnSn:0 Us -t 9jyjYM )log(1+y)¼y.][)9Sn+1YTUSn ) I=slog³Sn+1Sn´9 +9:; Us 2)]) I ])log³Sn+1Sn´lN CG‘) 2@DN )C ^ Is(O):;|01-7 T -PQ # LMWDXY dXt=Ktdt+HtdBt:01(Kt)V(Ht) 4 LM Y( ] +cXY*)Kt=St¹,Ht=St¾) (Xt). [-7 LM Y T PQ) Xt ‘If(Xt)‘R&’S1 LMWDXY ] XY*Ht=0,Kt t kl‘I) 0I T‘I2?I U )v df(Xt)=f0(Xt)dXt=f0(Xt)Ktdt:0 Ht , )}V [ JW]D )WD -7JX WD df(x)=f0(x)dx 2)Q7 9x o ¢x./YM )f(x) o ¢f(x)=f(x+¢x)¡f(x)= 5 f0(x)¢x+o(¢x),O*o(¢x) 5¢x XL~zM)vlim¢x!0o(¢x)=¢x=0:f0(x)¢x ¢f(x) ! 1 Lo MacKinlay 1988 Black{Scholes !#$ % &’( 247HY f mxG WDdf(x). dxI ¢x,Iri] WD xD t ‘Ix=x(t), D df(x(t))=f0(x(t))x0(t)dt.0 ] ¢f(x(t)) ¢t ! HY ={ ¢x ! HY V¢x ¢t ! HY-T3 M,) LMWDXY-J)}V [ n2])@ ) KtVHt+ t - ‘I }) ) VWX t7)Us ¢Xt¼Kt¢t+Ht¢Bt ZOtuv Kt¢t, q Htp¢t *D LM. LM. ¢X2t *DPQ¢t [ 01)W- WD 3z QR‘If=f(x), Taylor\] ¢f(x)=f(x+¢x)¡f(x)=f0(x)¢x+f00(x)2¢x2+o(¢x2); I=?r uIt^o(^_)‘a df(Xt)=f0(Xt)dXt+12f00(Xt)H