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U8V t JhG", .2}M2 hBK2 bi2T H;Q`Bi?K h?2 2[miBQM #Qp2 Bb i?2 bi`iBM; TQBMi 7Q` Mv /Bb+`2iBxiBQM b+?2K2X i iBK2 t- i?2 pHm2 Q7 St Bb FMQrM- M/ r2 rBb? iQ Q#iBM i?2 M2ti pHm2 St+dt X RX8 h?2 JBHbi2BM a+?2K2 h?2 b+?2K2 rQ`Fb 7Q` a.1b 7Q` r?B+? i?2 +Q2{+B2Mib µ(St ) M/ σ(St ) /2T2M/ QMHv QM S- M/ /Q MQi /2T2M/ QM t /B`2+iHvX >2M+2- r2 bbmK2 i?i i?2 biQ+F T`B+2 St Bb /`Bp2M #v i?2 a.1, dSt = µ(St )dt + σ(St )dWt UeV = µt dt + σt dWt y HH 2[miBQMb- bim/B2b- M/ `272`2M+2b 7Q` i?Bb b2+iBQM `2 +`2/Bi2/ iQ 6X _Qm?- EHQ2/2M  SHi2M- M/ :Hbb2`KMX +?2+F "B#HB;`QT?v j  A7 r2 Tmi BMiQ BMi2;`H 7Q`K St+dt = St + ! t+dt µs ds + t ! t+dt UdV σs dWs t q2 i?2M /Q 2tTMbBQMb- /2`Bp2 i  ;2M2`H bQHmiBQM 7i2` TTHvBM; AiQǶb G2KK  /Bz2`2MiBiBQM /2}M2/ b, √ 1 St+dt = St + µt dt + σt dtZ + σt′ σt dt(Z 2 − 1) U3V 2 RXe JBHbi2BM a+?2K2 7Q` i?2 "H+F@a+?QH2b JQ/2H 6B`bi- r2ǶHH /2}M2 h?2 "H+F@a+?QH2b biQ+F T`B+2 /vMKB+b mM/2` i?2 `BbF M2mi`H K2bm`2 b, UNV dSt = rSt dt + σSt dWt r2 ?p2 µ(St ) = rSt M/ σ(St ) = σSt X h?2 ;2M2`H bQHmiBQM iQ JBHbi2BM #Qp2 r?2M /Bz2`2M@ iBi2/c r2 ;2i i?2 7QHHQrBM; JBHbi2BM b+?2K2 /Dmbi2/ 7Q`  /`B7iH2bb J`iBM;H2 rBi? `BbF M2mi`H K2bm`2 rBi?Qmi i?2 µ r2 ;2i JBHbi2BMǶb "a1, √ 1 St+dt = St + rSt dt + σSt dtZ + σ 2 dt(Z 2 − 1) URyV 2 r?B+? //b i?2 +Q``2+iBQM i2`K 12 σ 2 dt(Z 2 − 1)- iQ i?2 1mH2` b+?2K2 BM U8VX AM i?2 "H+F@a+?QH2b KQ/2H 7Q` i?2 HQ;@biQ+F T`B+2- 7Q` i?2 1mH2` a+?2K2 mbBM; "a 6`K2rQ`Fb 1 dlnSt = (r σ 2 )dt + dWt URRV 2 - r2 ?p2 µSt = (r − 12 σ 2 ) M/ σSt = σ bQ i?i µ′t = σt′ = 0 h?2 JBHbi2BM b+?2K2 Bb i?2`27Q`2, √ 1 2 σ )dt + σ dtZ URkV 2 r?B+? Bb B/2MiB+H iQ i?2 1mH2` /Bb+`2iBxiBQM b+?2K2 7Q` i?2 "a biQ+F T`B+2 /vMKB+ rBi? `BbF M2mi`H K2bm`2, lnSt+dt = lnSt + (r − RXd 1mH2` J2i?Q/ q2 /Bb+mbb2/ i?2 JBHbi2BM /Bb+`2iBxiBQM Rk Ki+?2b i?2 1mH2` /Bb+`2iBxiBQM mM/2` i?2 "H+F@a+?QH2b `BbF M2mi`H K2bm`2 /2}M2/ #v, 9 √ G2iǶb bbmK2 i?i Wt+dt − Wt M/ dtZ `2 B/2MiB+H BM /Bbi`B#miBQM- r?2`2 w Bb  biM/`/ MQ`KH p`B#H2X >2M+2 i?2 1mH2` /Bb+`2iBxiBQM Q7 bb2i Bb /2}M2/ #v i?2 7QHHQrBM; a.1 /q 4 `M/MURV b[`iU/iV √ 1 lnSt+dt = St + (r − σ 2 )dt + σ dtZ 2 √ 1 St+dt = St 2tT((r − σ 2 )dt + σ dtZ) 2 q?2`2 dt = ti − ti−1 bQ i?iX URjV h?2`27Q`2- r?BH2 i?2 JBHbi2BM b+?2K2 BKT`Qp2b i?2 /Bb+`2iBxiBQM Q7 St BM i?2 "H+F@a+?QH2b KQ/2H- Bi /Q2b MQi BKT`Qp2 i?2 /Bb+`2iBxiBQM Q7 HM St X y aKUM-iV 4 aKUM-i@RVY U`@[V aKUM-i@RV /i Y p aKUM-i@RV qXXXY yX8 pk aKUM-i@RV Uwk@RV /ic 9  RX3 R JBHbi2BM "a1 b+?2K2 JiH# .2}M2b i?2 iBK2 bi2T  /Bb+`2iBxiBQM b+?2K2 R k j 9 8 e d 3 N Ry RR Rk Rj R9 R8 Re Rd W JQMi2 *`HQ aBKmHiBQM b 2 i i B M ; b X L 4 8yyyyc W LmK#2` Q 7 J* b B K m H  i B Q M b iQ ` m M X h 4 Ryyc W LmK#2` Q 7 i n b i 2 T b X /i 4 Kifhc W hBK2 BM+`2K2Mi Q` /i U inbi2T V W A M B i B  H B x  i B Q M 7 Q ` i?2 i2 ` KB M  H b i Q + F T ` B + 2 K i` B+2b W 7 Q ` i?2 J B H b i 2 B M / B b + ` 2 i B x  i B Q M b+?2K2bX a 4 x 2 ` Q b UL-hV c a U , - R V 4 ay c W aBKmHi2 i?2 b i Q + F T ` B + 2 mM/2` i?2 J B H b i 2 B M b+?2K2bX 7 Q ` M4R,L 7 Q ` i 4k,h /q 4 `M/M U R V b [ ` i U /i V c aUM - i V 4 aUM - i @ R V Y U ` @ [ @ pkfkV aUM - i @ R V /i Y p aUM - i @ R V /qX X X Y y X 8 pk aUM - i @ R V /q k c 2M/ k AMi`Q/m+iBQM iQ "H+F@a+?QH2b "27Q`2 r2 #2;BM T`B+BM; Qm` bBM +HH QTiBQMb- r2 Kmbi Q#b2`p2 i?2 #bB+ BMimBiBQM #2?BM/ QTiBQMb +QMi`+ib M/ i?2B` T`B+BM; KQ/2HX hQ +HbbB7v r?i  }MM+BH QTiBQM Bb- r2 HQQF i i?2 bBKTH2bi 1m`QT2M +HH QTiBQMX Ai Bb  +QMi`+i rBi? i?2 7QHHQrBM; +QM/BiBQMb M/ /2b+`BTiBQM 7`QK W ilmott 2i HX URNN8V, i  T`2/2i2`KBM2/ iBK2 T BM i?2 7mim`2- Q` 1tTB`iBQM- i?2 QrM2` Q7 i?2 QTiBQM Kv Tm`+?b2 i?2 mM/2`HvBM; bb2i 7Q`  T`2/2i2`KBM2/ T`B+2- HbQ FMQrM b i?2 ai`BF2 T`B+2 KX h?2 Qi?2` T`iv Q7 i?2 QTiBQM Bb i?2 r`Bi2`c i?2v ?p2 M Q#HB;iBQM iQ b2HH i?2 bb2i iQ i?2 ?QH/2` B7 i?2v rMi iQ #mv BiX kXR "bB+ LQiiBQM * 4 *HH QTiBQM T`B+2 a 4 *m``2Mi biQ+F T`B+2 E 4 ai`BF2 T`B+2 Q7 i?2 QTiBQM ` 4 `BbF@7`22 BMi2`2bi `i2 U MmK#2` #2ir22M y M/ RV σ 4 pQHiBHBiv Q7 i?2 biQ+Fb `2im`M U MmK#2` #2ir22M y M/ RV i 4 iBK2 iQ QTiBQM Kim`Biv UBM v2`bV L 4 MQ`KH +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM R 6#`B+2 .Qm;Hb _Qm? 8  kXk oMBHH QTiBQMb q?2M S > K i 2tTB`v- Bi rQmH/ KF2 }MM+BH b2Mb2 iQ 2t2`+Bb2 i?2 +HH QTiBQM- ?2M+2 ?M/BM; Qp2` i?2 KQmMi K- Q#iBMBM; M bb2i rQ`i? SX h?2 T`Q}i rQmH/ #2 S − KX A7 S < K i 2tTB`v-  HQbb Q7 K − S rQmH/ #2 K/2 M/ i?2 QTiBQM Bb rQ`i?H2bbX q2 +M r`Bi2  +HH QTiBQM b, C(S, T ) = max(S − K, 0)X kXj .Bz2`2Mi ivT2b Q7 QTiBQMb SHBM oMBHHb `2 ivTB+HHv European QTiBQMb r?B+? +M #2 2t2`+Bb2/ QMHv i 2tTB`iBQM M/ American QTiBQMb r?B+? +M #2 2t2`+Bb2/ i Mv iBK2 #27Q`2 2tTB`vX Si?@.2T2M/2Mi QTiBQMb `2 ivTB+HHv `272``2/ iQ b Ƕ1tQiB+bǶ M/ mMHBF2 oMBHHǶb i?2v `2 MQi Q7i2M i`/2/ QM 2t+?M;2b M/ Kmbi #2 K/2 Qp2`@i?2@+QmMi2`X AM Qm` bim/v- r2ǶHH #2 HQQFBM; i QM2 bT2+B}+ ivT2 Q7 Si?@.2T2M/2Mi QTiBQM +HH2/ M Asian QTiBQM i?iǶb Tv@Qz Bb +H+mHi2/ 7`QK i?2 p2`;2 T`B+2 Q7 i?2 bb2i Qp2`  T2`BQ/ Q7 iBK2X h?2`2 `2 Qi?2` Ti? /2T2M/2Mi QTiBQMb- bm+? b GQQF#+F QTiBQMb- M/ Qi?2` MQM@Ti?@/2T2M/2Mi 2tQiB+ QTiBQMb- #mi r2ǶHH T`BK`BHv #2 7Q+mbBM; QM T`B+BM; bBM QTiBQMb BM i?Bb 2tT2`BK2MiX kX9 "H+F@a+?QH2b 1[miBQM h?2 "H+F@a+?QH2b 2[miBQM b22Fb iQ /2b+`B#2 T`B+2 V Q7 M QTiBQM Qp2` iBK2X h?2 /2`BpiBQM Q7 i?Bb 2[miBQM Bb 2t+Hm/2/- r2ǶHH T`QpB/2  ;2M2`HBx2/ 2[miBQMX ∂* 1 2 2 ∂ 2 * ∂* + σ a + `a = `* UR9V ∂i 2 ∂*2 ∂a LQiB+2 i?i i?i 2[miBQM UR9V Bb  T`iBH /Bz2`2MiBH 2[miBQMX h?2 bQHmiBQM Q7 i?Bb 2[miBQM ;Bp2b mb i?2 "H+F@a+?QH2b 7Q`KmHX kX8 "H+F@a+?QH2b 7Q`KmH 7Q` 1m`QT2M QTiBQM T`B+2 h?2 "H+F@a+?QH2b 7Q`KmH HHQrb mb i?2 +H+mHi2 i?2 T`B+2 Q7 1m`QT2M +HH M/ Tmi QTiBQMbX *(a, i) = L(/1 )a − L(/2 )E2−rt " # $ # $% S σ2 HM +t r+ K 2 σ i " # $ # $% 1 S σ2 /2 = √ HM +t r− K 2 σ i ! x 1 2 1 N (x) = √ 2− 2 z dz 2π −∞ /1 = 1 √ e UR8V UReV URdV UR3V  j bBM PTiBQMb Pp2`pB2r q2ǶHH #2 T`B+BM; M 1tQiB+ Si?@.2T2M/2Mi bBM QTiBQM BM Qm` KQ/2HX aQK2 2`Hv mb2b Q7 i?2b2 QTiBQMb r2`2 mb2/ 7Q` ?2/;BM; +`m/2 QBH 7mim`2b M/ ′ Asian′ `272`2M+2b BiǶb #B`i?TH+2 Q7 hQFvQX Ai Bb ′ exotic′ BM i?2 b2Mb2 i?i BiǶb Tv@Qz Bb  7mM+iBQM Q7 i?2 mM/2`HvBM; bb2i i KmHiBTH2 TQBMib i?`Qm;?Qmi i?2 +QMi`+iǶb HB72iBK2- `i?2` i?M Dmbi i?2 pHm2 i 2tTB`vX h?Bb bT2+B}+ 2tQiB+ mb2b i?2 mean Q7 i?2 mM/2`HvBM; bb2i- r2ǶHH +HH aiQ+F- St iQ T`B+2 At X hQ /Q i?Bb- r2ǶHH Q#b2`p2 St Ƕb T`B+2 bKTH2/ i TT`QT`Bi2 2[mH BMi2`pHb M/ mb2 b i?2 #bBb 7Q` Qm` +HH QTiBQMǶb Tv@QzX h?Bb Bb i?2 BMimBiBQM #2?BM/ i?2 i2`K ǴTi?@/2T2M/2MiǴ- #2+mb2 St Bb +QMiBMmQmbc #mi i?2 QTiBQM pHm2 i 2tTB`iBQM ;;`2;i2b /Bb+`2i2 bKTHBM;b Q7 St M/ +H+mHi2b  K2M T`B+2- i?Bb rBHH #2 Qm` bBM +HH bTQi pHm2 At i 2tTB`iBQM E r2ǶHH /2}M2 b AT X q2ǶHH #2 7Q+mb2/ QM i?2 irQ KBM ivT2b Q7 bBMbc i?2 arithmetic bBM +HH M/ i?2 geometric bBM +HHX q2 rQMǶi #Qi?2` /Bb+mbbBM; i?2Q`2iB+H +QMiBMmQmb T`B+BM; b+?2K2b bBM+2 r2Ƕ`2 7Q+mbBM; QM /Bb+`2i2 iBK2 KQ/2HBM;X h?2 KBM /Bz2`2M+2 #2ir22M i?2 arithmetic M/ geometric QTiBQM Bb ?Qr r2 +imHHv +H+mHi2 i?2 K2M Q7 i?2 mM/2`HvBM; pHm2bc Qm` K2i?Q/ rBHH z2+i Qm` 2tT2+i2/ /Bb+QmMi2/ TvQz i 2tTB`iBQMX q2ǶHH 7Q+mb QM  +HH QTiBQM QMHv BM i?Bb 2tKTH2X lMHBF2 T`B+BM;  pMBHH 1m`QT2M QTiBQM rBi?  JQMi2 *`HQ K2i?Q/- r?2`2 r2 QMHv M22/ iQ ;2M@ 2`i2 pHm2b Q7 +QMi`+ib i 2tTB`v- rBi? i?Bb KQ/2H r2 rBHH M22/ iQ ;2M2`i2 KmHiBTH2 bTQi Ti?bM/ i?2M i?Qb2 bTQi Ti?b Kmbi #2 bKTH2/ i +Q``2+i TQBMib iQ +Q``2+iHv T`B+2 Qm` Asian *HHX q2 rBHH biBHH #2 KQ/2HBM; Qm` bb2i T`B+2 Ti? pB  biM/`/ :2QK2i`B+ "`QrMBM JQiBQMc Gi2` QMr2 rBHH BKTH2K2Mi M/ /Bb+mbb Antithetical variance reduction K2i?Q/b BM Qm` KQ/2HX jXR `Bi?K2iB+ S`B+BM; AM Q`/2` iQ +H+mHi2 i?2 Arithmetic K2M A Q7 i?2 bTQi T`B+2 r2ǶHH BKTH2K2Mi i?2 7QHHQrBM; 7Q`KmH, A(0, T ) = N 1 & S(ti ) N t=1 URNV Pm` Q#D2+iBp2 Bb iQ /2i2`KBM2 i?2 pHm2 Q7 i?2 bBM QTiBQM "H+F@a+?QH2b K2i?Q/ Q7 bBM PTiBQMb q2 rBHH #2;BM #v /2}MBM; i?2 "H+F a+?QH2b KQ/2H r?2`2 i?2 /vMKB+b Q7 i?2 mM/2`HvBM; bb2i `2, dS = S(µdt + σdWt ) jXk bBM SvQz h?2 bBM QTiBQM r2 +QMbB/2` ?b i?2 /Bb+QmMi2/ TvQz Pˆ = 2tT(ƐrT ), Kt(0, SƐK) q?2`2 S=T −1 ! T S(t)dt; 0 d f = ma;  h?2 bBKTH2bi TT`QtBKiBQM Q7 S- +Bi2/ #v 7QHHQrBM;, S = T −1 nT −1 & 0 1 ˆ h(Sn + Sˆn+1 ); 2 r?2`2 nT = T /h Bb i?2 MmK#2` Q7 iBK2bi2TbX h?Bb +Q``2bTQM/b iQ  HBM2` TT`QtBKiBQM iQ S(t) #mi BKT`Qp2/ ++m`+v +M #2 +?B2p2/ #v TT`QtBKiBM; i?2 #2?pBQ` rBi?BM  iBK2bi2T b bBKTH2 "`QrMBM KQiBQM- rBi? +QMbiMi /`B7i M/ pQHiBHBiv- +QM/BiBQMH QM i?2 +QKTmi2/ pHm2b Sˆn X hFBM; bn iQ #2 i?2 +QMbiMi pQHiBHBiv rBi?BM i?2 iBK2 BMi2`pH [tn , tn+1 ]- biM/`/ "`QrMBM "`B/;2 `2bmHib Ub22 b2+iBQM jXR BM (:Hy9)V ;Bp2 `Bi?K2iB+ SvQz 7 m M + i B Q M ` 2 i 4 SvQ77n`Bi?K2iB+bBM Ua - E- *V Ka 4 K2MUa - R V c W K2M rBi? ` 2 b T 2 + i iQ QM2 Ti? B 7 *44R ` 2 i 4 KtUKa@E- y V c 2Hb2 ` 2 i 4 KtUE@Ka- y V c R k j 9 8 e jXj :2QK2i`B+ S`B+BM; AM i?2 bK2 7b?BQM- 7Q` Geometric K2M A Q7 i?2 bTQi r2ǶHH mb2, ' ( N 1 & A(0, T ) = 2tT HQ;(S(ti )) N i=1 UkyV h?2 ;2QK2i`B+ Bb ;QBM; iQ #2 bBKBH` iQ i?2 `Bi?K2iB+ T`B+2- #mi Bb 2bB2` iQ }M/ QM+2 r2 FMQr i?2 `Bi?K2iB+ p2`;2X B7 S(t)  ;2QK2i`B+ "`QrMBM KQiBQM- i?2M AˆT Bb i?2 bmK Q7 HQ;MQ`KHHv /Bb@ i`B#mi2/ `M/QK p`B#H2bX AM +QMi`bi iQ i?2 `Bi?K2iB+ p2`;2- i?2 /Bbi`B#miBQM Q7 i?2 ;2QK2i`B+ p2`;2 +M 2bBHv #2 Q#iBM2/X h?2 ;2QK2i`B+ K2M Q7 n pHm2b HQ;X h?2 2tTQM2Mi Bb  r2B;?i2/ p2`;2 Q7 i?2 BM/2T2M/2Mi MQ`KH BM+`2K2Mib Q7 i?2 T`Q+2bb M/ i?2`27Q`2 MQ`KHHv /Bbi`B#mi2/X :2QK2i`B+ SvQz R k j 9 8 e d 3 N Ry RR 7 m M + i B Q M ` 2 i 4 :2QK2i`B+bBM Ua - E- *V W a 4 LaBK t Li Ki`Bt Q 7 bBKmHi2/ T ` B + 2 b W E 4 ai`BF2 T`B+2 W * 4 R @= *  H H c * 4 y @= Smi Ka 4 T`Q/ Ua U , - k ,2M/ V X URf b B x 2 Ua - k V V - k V c B 7 *44R ` 2 i 4 KtUKa@E- y V c 2Hb2 ` 2 i 4 KtUE@Ka- y V c 2M/ ` 2 i 4 K2MU ` 2 i V c 3