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A?>;L7CO 3@0=8G=K5 CA;>28O, =0;0305<K5 ?@028;0<8 2K?;0B ?> >?F8>=C, @5H0NB C@02=5=85 8 ?>;CG0NB D>@<C;C >F5=:8 AB>8<>AB8 >?F8>=>2. 06=59H8< <5B>48G5A:8< ?@85<>< ?@8 @5H5=88 B0:8E AB>E0AB8G5A:8E 48DD5@5=F80;L=KE C@02=5=89 O2;O5BAO 8A?>;L7>20=85 <5B>40 @8A: =59B@0;L=>9 >F5=:8 (risk-neutral valuation). 0==K9 ?@85< >A=>20= =0 D0:B5, GB> ?>;CG5==>5 48DD5@5=F80;L=>5 C@02=5=85 4;O AB>8<>AB8 >?F8>=0 A2>1>4=> >B ?0@0<5B@>2, >?8AK20NI8E A:;>==>ABL 8=25AB>@>2 : @8A:C. 5H5=85 B0:>3> C@02=5=8O B0:65 =5 7028A8B >B A:;>==>AB8 8=25AB>@>2 : @8A:C. >MB><C ?@8 @5H5=88 C@02=5=8O, =5 >3@0=8G820O >1I=>AB8 @5H5=8O, <>6=> A45;0BL ?@54?>;>65=85, GB> 2A5 8=25AB>@K =59B@0;L=K : @8A:C, 8 A>>B25BAB25==> 4>E>4=>ABL 2A5E D8=0=A>2KE 0:B82>2 >48=0:>20 8 @02=0 157@8A:>2>9 AB02:5. -B> ACI5AB25==> >1;53G05B ?>;CG5=85 @5H5=8O 48DD5@5=F80;L=>3> C@02=5=8O 2 O2=>< 2845. >45;L ;M:0-(>C;70 (Black-Scholes option pricing model)  <>45;L =07=0G5=8O F5=K >?F8>=0 :>;;, >A=>20==0O =0 0@18B@06=KE 0@3C<5=B0E. !>3;0A=> MB>9 <>45;8 ?@5<8O >?F8>=0 :>;; 52@>?59A:>3> AB8;O =0E>48BAO 2 ?@O<>9 7028A8<>AB8 >B F5=K 1078A=>3> 0:B820, 2>;0B8;L=>AB8, >AB02H53>AO A@>:0 4> 8AB5G5=8O :>=B@0:B0, 157@8A:>2>9 ?@>F5=B=>9 AB02:8 8 2 >1@0B=>9 7028A8<>AB8 >B F5=K 8A?>;=5=8O.  @57C;LB0B5 8<8 1K;0 ?>;CG5=0 D>@<C;0 >F5=:8 >?F8>=0 :>;; =0 0:F88. >A;5 MB>3> @0728B85 0=0;8B8G5A:8E <>45;59 ?@>8AE>48;> ?> 42C< =0?@02;5=8O<: >1>1I5=85 <>45;8 Black-Scholes. "0:85 <>45;8 AB@>8;8AL A 8A?>;L7>20=85< 1>;55 <O3:8E >3@0=8G5=89, G5< 2 <>45;8 Black-Scholes. @0A?@>AB@0=5=85 <>45;8 =0 >?F8>=K A 4@C38<8 107>2K<8 0:B820<8 8 =0 4@C385 284K >?F8>=>2. @82545< 40;55 >A=>2=K5 E0@0:B5@8AB8:8 ACI5AB2CNI8E <>45;59. 1>1I5==K5 <>45;8 >A;5 >?C1;8:>20=8O @01>BK Black-Scholes 2 B>< 65 1973 3>4C Robert Merton >F5=8; AB>8<>ABL 52@>?59A:>3> >?F8>=0 =0 0:F88, ?> :>B>@K< 2K?;0G820NBAO 482845=4K. @><5 MB>3> >= @0AA<>B@5; A;CG09 ?5@5<5==>9 157@8A:>2>9 ?@>F5=B=>9 AB02:8.  1975 3>4C Jonathan Ingersoll CA;>6=8; <>45;L, @0AA<>B@52 A;CG09, :>340 482845=4K >1;030NBAO =0;>3><. 4=> 87 ?@54?>;>65=89 2 <>45;8 Black-Scholes 70:;NG05BAO 2 B><, GB> 87<5=5=85 F5=K 8<55B =5?@5@K2=K9 E0@0:B5@ 8 >?8AK205BAO A ?><>ILN ?@>AB>3> 8=5@>2A:>3> ?@>F5AA0. 0 ?@0:B8:5 F5=K 8<5NB B5=45=F8N 87<5=OBLAO A:>@55 A:0G:0<8, G5< =5?@5@K2=>. >45;L, 345 F5=K <5=ONBAO =5 =5?@5@K2=>, 0 A:0G:0<8, 1K;0 @0AA<>B@5=0 Cox 8 Ross 2 1976 3>4C. -B0 845O 70B5< ?@825;0 : ?>O2;5=8N 18=><80;L=>9 <>45;8 Cox-Ross-Rubinstein. Merton @0728; 845N > A:0G:>>1@07=>< ?>2545=85 F5=, ?@54?>;>682, GB> @0728B85 F5=K 5ABL :><18=0F8O A:0G:>2 F5=K 8 48DDC78>==>3> 42865=8O.  53> <>45;8 70405BAO G0AB>B0 ?>O2;5=8O A:0G:>2 F5=K, 45;05BAO ?@54?>;>65=85, GB> 25;8G8=0 A:0G:>2 8<55B ;>3=>@<0;L=>5 @0A?@545;5=85. @>2545==K5 ?@0:B8G5A:85 8AA;54>20=8O @0A?@545;5=8O F5= ?>:070;8, GB> ;>3=>@<0;L=>5 @0A?@545;5=85 =5:>@@5:B=> >?8AK205B  E2>ABK @50;L=>3> @0A?@545;5=8O. >MB><C F5;K9 :;0AA <>45;59 @0AA<0B@8205B A;CG09, :>340 F5=0 0:B820 A;54C5B 4@C3><C ?@>F5AAC, >B;8G=><C >B ;>3=>@<0;L=>3>. Jarrow 8 Rudd ?>AB@>8;8 <>45;L, CG8BK20NICN @0AE>645=85 <564C @50;L=K< @0A?@545;5=85< F5= 8 ;>3=>@<0;L=K<. I5 1>;55 A8;L=K< 4>?CI5=85< 2 <>45;8 Black-Scholes, =5 ?>4B25@640NI8<AO =0 ?@0:B8:5, O2;O5BAO ?@54?>;>65=85 > ?>AB>O==>9 2>;0B8;L=>AB8 F5=K 0:F88. !5@8O <>45;59, ?>O282H0OAO 2 1987 3>4C, 87CG05B A;CG09, :>340 2>;0B8;L=>ABL =5 ?>AB>O==0, 0 >?8AK205BAO =5:>B>@K< AB>E0AB8G5A:8< ?@>F5AA><. >45;8 =0 4@C385 >?F8>=K  1976 3>4C Fischer Black @0AH8@8; 8AE>4=CN <>45;L, @0AA<>B@52 A;CG09, :>340 107>2K< 0:B82>< O2;O5BAO DLNG5@A=K9 :>=B@0:B.  1983 3>4C Garman 8 Kohlhagen, Orlin 8 Grabbe ?>;CG8;8 @5H5=85 4;O 52@>?59A:>3> >?F8>=0 =0 :C@A 20;NB. &5;K9 @O4 <>45;59 ?>A2OI5= >F5=:5 0<5@8:0=A:8E >?F8>=>2. #G8BK20O, GB> @5H5=85 4;O 0<5@8:0=A:8E >?F8>=>2 2 0=0;8B8G5A:>< 2845 ?>;CG8BL 7=0G8B5;L=> A;>6=55, 0 G0AB> 8 =52>7<>6=>, 1>;LH8=AB2> <>45;59 >F5=:8 0<5@8:0=A:8E >?F8>=>2 8A?>;L7CNB G8A;5==K5 <5B>4K. @C3>5 =0?@02;5=85 8AA;54>20=89 A2O70=> A >F5=:>9 M:7>B8G5A:8E >?F8>=>2.  1979 3>4C Barry Goldman, Sosin, Mary Ann Gatto @0AA<>B@5;8 >F5=:C AB>8<>AB8 path dependent 52@>?59A:8E >?F8>=>2. Robert Geske ?@54;>68; <>45;L >F5=:8 AB>8<>AB8 compound >?F8>=>2. F5=:0 exchange - >?F8>=>2 ?@82545=0 2 @01>B5 William Margrabe. Rene Stulz >F5=8; >48= 87 B8?>2 multi factor >?F8>=>2. '8A;5==K5 <>45;8 '8A;5==K5 <>45;8 >A=>20=K =0 B@5E ?>4E>40E. 5@2K9 ?>4E>4 8725AB5= :0: 18=><80;L=K9, 2B>@>9 ?>4E>4 >A=>20= =0 8A?>;L7>20=88 <5B>40 :>=5G=KE @07=>AB59, B@5B89  =0 8A?>;L7>20=88 <5B>40 >=B5-0@;> . 8=><80;L=K5 <>45;8 -B>B ?>4E>4 1K; ?@54;>65= William Sharpe 2 1978 3>4C 8 @07@01>B0= J. Cox, S. Ross 8 M. Rubinstein. 3> 845O 70:;NG05BAO 2 B><, GB> 2@5<O 4> ?>30H5=8O >?F8>=0 @0718205BAO =0 8=B5@20;K.  :064>< 8=B5@20;5, :0: ?@54?>;0305BAO, F5=0 0:B820 A;54C5B 18=><80;L=><C ?@>F5AAC: F5=0 8;8 2>7@0AB05B =0 7040==CN 25;8G8=C, 8;8 ?0405B =0 7040==CN 25;8G8=C. "0:8< >1@07><, ?>;CG0NB <=>65AB2> 2>7<>6=KE 7=0G5=89 F5=K 0:B820 2 <><5=B 8A?>;=5=8O >?F8>=0, 4;O :064>3> 7=0G5=8O F5=K F5=0 >?F8>=0 8725AB=0. >A;5 MB>3>, 8A?>;L7CO 2>7<>6=>ABL ?>AB@>5=8O 157@8A:>2>3> ?>@BD5;O 87 0:B820 8 >?F8>=>2, ?@>872>4OB A25@B:C ?>;CG5==>3> 45@520 F5=. @58<CI5AB2> B0:>3> ?>4E>40 A>AB>8B 2 2>7<>6=>AB8 53> 8A?>;L7>20=8O 4;O ;N1KE 284>2 >?F8>=>2. 081>;LH55 @0A?@>AB@0=5=85 18=><80;L=K5 <>45;8 ?>;CG8;8 ?@8 >F5=:5 >?F8>=>2 =0 ?@>F5=B=K5 8=AB@C<5=BK. 5@20O <>45;L 1K;0 ?@54;>65=0 Richard Rendleman 8 Brit Bartter.  1986 3>4C Thomas Ho 8 Sang-Bin Lee @0AA<>B@5;8 <>45;L 87<5=5=8O AB@C:BC@K ?@>F5=B=KE AB02>:, => 2>;0B8;L=>ABL 2A5E ?@>F5=B=KE AB02>: 1K;0 >48=0:>20.  1990 Fischer Black, Derman 8 Toy CAB@0=8;8 MB> >3@0=8G5=85. 5B>4 :>=5G=KE @07=>AB59 -B>B ?>4E>4 70:;NG05BAO 2 =0E>645=85 G8A;5==>3> @5H5=8O 48DD5@5=F80;L=>3> C@02=5=8O 4;O AB>8<>AB8 >?F8>=0. 5@20O @01>B0 1K;0 ?@54;>65=0 Eduardo Schwartz 8 @0AH8@5=0 Courtadon 2 1982. Brennan 8 Schwartz ?>:070;8, GB> MB>B ?>4E>4 <>6=> @0AA<0B@820BL :0: B@8=><80;L=>5 45@52> F5=. John Hull 8 Alan White <>48D8F8@>20;8 8E <>45;L, 4>:0702, GB> <5B>4>;>38O B@8=><80;L=>3> 45@520 F5= 8 48DD5@5=F80;L=>5 C@02=5=85 ?@82>4OB : >4=><C @5H5=8N. 5B>4 >=B5-0@;> 0==K9 ?>4E>4 ?@8=F8?80;L=> >B;8G05BAO >B >AB0;L=KE. A5 >?8A0==K5 2KH5 <5B>4K >F5=:8 >?F8>=>2 A2>48;8AL : =5>1E>48<>AB8 @5H0BL AB>E0AB8G5A:>5 48DD5@5=F80;L=>5 C@02=5=85. >4E>4 >=B5-0@;> 8<8B0F88 70:;NG05BAO 2 =5?>A@54AB25==>9 ?@O<>9 >F5=:5 >68405<>9 AB>8<>AB8 >?F8>=0 ?>A@54AB2>< A8<C;OF88 2>7<>6=KE 7=0G5=89 F5=K 0:B820 2 <><5=B 8A?>;=5=8O >?F8>=0. @0240 MB>B ?>4E>4 B0:65 B@51C5B ?@54?>;>65=8O > 2845 AB>E0AB8G5A:>3> ?@>F5AA0, :>B>@><C A;54C5B F5=0 107>2>3> 0:B820. 5@20O <>45;L ?>O28;0AL 2 1977 3>4C 2 @01>B5 Phelim Boyle. !53>4=O MB>B <5B>4 >F5=:8 >?F8>=>2 ?>;CG8; =081>;LH55 @0A?@>AB@0=5=85 8 ?@>4>;605B 48=0<8G=> @072820BLAO. "0:, Jim Tilley 2 1993 3>4C 2?5@2K5 ?@54;>68; <5B>4 >F5=:8 >?F8>=>2 0<5@8:0=A:>3> B8?0. >A;5 MB>3> ?>A;54>20;8 <>45;8 Grant, Vora 8 Weeks (1994), Barraquand 8 Martineau (1995), Broadie 8 Glasserman (1995) . >45;8 0=0;8B8G5A:>9 0??@>:A8<0F88 -B8 <>45;8 >A=>20=K =0 A>G5B0=88 0=0;8B8G5A:>3> 8 G8A;5==>3> ?>4E>4>2. 1KG=> >=8 ?@8<5=ONBAO 4;O >F5=:8 0<5@8:0=A:8E 284>2 >?F8>=>2. >4E>4 70:;NG05BAO 2 ?>;CG5=88 0=0;8B8G5A:>9 >F5=:8 AB>8<>AB8 >?F8>=0 2 <><5=B 8A?>;=5=8O, 0 70B5< A ?><>ILN G8A;5==KE <5B>4>2 >F5=8205BAO 4>102:0 : AB>8<>AB8 >?F8>=0 87-70 2>7<>6=>AB8 @0==53> 8A?>;=5=8O. Lionel Macmillan ?@54;>68; >F5=:C >?F8>=>2, 8A?>;L7CO ?>4E>4 :204@0B8G=>9 0??@>:A8<0F88, MB0 845O ?>;CG8;0 >B@065=85 2 @01>B5 Giovanni Barone-Adesu 8 Robert Whaley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lack-Scholes $8H5@ ;M: (Fischer Black) 87 '8:03A:>3> C=825@A8B5B0 8 09@>= (>C;7 (Myron Scholes) 87 M.I.T @07@01>B0;8 <>45;L F5=>>1@07>20=8O ?@5<88 >?F8>=0 :>;;. ?5@2K5 >=0 1K;0 >?8A0=0 2 1973 3>4C (May  June 1973 Journal of Political Economy). Black-Scholes AG8B05BAO ?@545;L=>9 D>@<>9 18=><80;L=>9 <>45;8.  18=><80;L=>9 <>45;8 =C6=> 7040BL G8A;> B8:>2, >?@545;ONI55 42865=85 225@E 8;8 2=87, ?@5645 G5< 1C45B 70D8:A8@>20=> 2>7<>6=>5 7=0G5=85 F5=K. "5:CI0O F5=0 =0 ?5@2>< H035 <>65B ?>9B8 2 2-E =0?@02;5=8OE. 0 2B>@>< H035 2 4-E =0?@02;5=8OE.  18=><80;L=>9 <>45;8 4;O @0AG5B0 A?@0254;82>9 F5=K >?F8>=0 2K 4>;6=K 70@0=55 >?@545;8BL, A:>;L:> 2A53> ?5@8>4>2 8A?>;L7>20BL. Black-Scholes AG8B05BAO ?@545;L=>9 D>@<>9 18=><80;L=>9 <>45;8, B0: :0: 4>?CA:05B 15A:>=5G=>5 G8A;> ?5@8>4>2 (2 B5>@88), B> 5ABL Black-Scholes ?>4@07C<5205B, GB> MB0 =51>;LH0O 4803@0<<0 1C45B @0AH8@OBLAO 4> 15A:>=5G=>AB8. A;8 2K >?@545;8B5 A?@0254;82CN F5=C >?F8>=0 ?> Black-Scholes, B> ?>;CG8B5 B>B 65 >B25B, GB> 8 2 A;CG05 A 18=><80;L=>9 <>45;LN, 5A;8 G8A;> ?5@8>4>2, 8A?>;L7C5<KE 2 18=><80;L=>9 <>45;8, 1C45B AB@5<8BLAO : 15A:>=5G=>AB8. (">B D0:B, GB> Black-Scholes O2;O5BAO ?@545;L=>9 D>@<>9 18=><80;L=>9 <>45;8, ?>4@07C<5205B, GB> 18=><80;L=0O <>45;L ?>O28;0AL ?5@2>9, => =0 A0<>< 45;5 A=0G0;0 ?>O28;0AL 8<5==> <>45;L Black-Scholes). !?@0254;820O AB>8<>ABL D>=4>2>3> :>;;->?F8>=0 ?> Black-Scholes @0AAG8BK205BAO A;54CNI8< >1@07><: , 345 !  F5=0 call >?F8>=0 (the call option price); N( )  A>2<5AB=0O DC=:F8O =>@<0;L=>3> @0A?@545;5=8O (the cumulative normal distribution function);  2>;0B8;L=>ABL F5=K 0:B820 (the volatility expressed in annual terms); X  F5=0 AB@09: >?F8>=0 (the exercise price); S  F5=0 0:B820 (stock price);   F5=0 AB@09: =0 45=L @0AG5B>2 (the present value of the exercise price)  EMBED Equation.3   MB>9 <>45;8 4>;6=> 2K?>;=OBLAO A;54CNI55 @025=AB2> (F5=0 >?F8>=0 call 4>;6=0 1KBL @02=0 8;8 ?@52KH0BL F5=C 0:B820 (stock price) <8=CA F5=0 AB@09: =0 45=L @0AG5B>2 (the present value of the exercise price)):  07@01>BG8:8 MB>9 <>45;8 8AE>48;8 87 B@5E CB25@645=89. 5A<>B@O =0 =54>AB0B:8 MB8E CB25@645=89, ?@54;>65==0O <>45;L 2A5-B0:8 4>2>;L=> B>G=0, 8 F5=K >?F8>=>2 1C4CB AB@5<8BLAO : 7=0G5=8O<, ?>;CG5==K< 87 <>45;8. 5@2>5 87 MB8E CB25@645=89 A>AB>8B 2 B><, GB> >?F8>= =5 <>65B 1KBL 8A?>;=5= 4> 8AB5G5=8O A@>:0. -B> ?@82>48B : =54>>F5=:5 >?F8>=>2 0<5@8:0=A:>3> B8?0, :>B>@K5 <>3CB 8A?>;=OBLAO 4> 8AB5G5=8O A@>:0. B>@>5 CB25@645=85 ?@54?>;0305B, GB> <K 7=05< 1C4CICN 2>;0B8;L=>ABL 107>2>3> 8=AB@C<5=B0, 8 >=0 1C45B >AB020BLAO ?>AB>O==>9 2 B5G5=85 A@>:0 459AB28O >?F8>=0. 0 A0<>< 45;5 MB> =5 B0: (B.5. 2>;0B8;L=>ABL 87<5=8BAO). @><5 B>3>, @0A?@545;5=85 87<5=5=89 2>;0B8;L=>AB8 ;>30@8D<8G5A:8 =>@<0;L=>, 8 MBC ?@>1;5<C <>45;8 =5 CG8BK20NB. I5 >4=> 4>?CI5=85 <>45;8 A>AB>8B 2 B><, GB> 157@8A:>20O ?@>F5=B=0O AB02:0 >AB05BAO ?>AB>O==>9 2 B5G5=85 2@5<5=8 459AB28O >?F8>=0. -B> B0:65 =5 >1O70B5;L=>. >;55 B>3>, :@0B:>A@>G=K5 AB02:8 ;>30@8D<8G5A:8 =>@<0;L=> @0A?@545;5=K. "> >1AB>OB5;LAB2>, GB>, G5< 2KH5 :@0B:>A@>G=K5 AB02:8, B5< 2KH5 1C4CB F5=K >?F8>=>2, 8 CB25@645=85 >B=>A8B5;L=> =587<5==>AB8 :@0B:>A@>G=KE AB02>: <>65B ?@825AB8 : 5I5 1>;LH59 =54>>F5=:5 >?F8>=0 ?> >B=>H5=8N : >68405<>9 F5=5 (53> ?@028;L=><C 0@8D<5B8G5A:><C <0B5<0B8G5A:><C >6840=8N). I5 >4=> CB25@645=85 (2>7<>6=> =081>;55 206=>5), :>B>@>5 <>65B ?@825AB8 : =54>>F5=:5 AB>8<>AB8 >?F8>=0, @0AAG8B0==>9 A ?><>ILN <>45;8, ?> >B=>H5=8N : 459AB28B5;L=> >68405<>9 AB>8<>AB8, A>AB>8B 2 B><, GB> ;>30@8D<K 87<5=5=89 F5=K @0A?@545;ONBAO =>@<0;L=>. A;8 1K >?F8>=K E0@0:B5@87>20;8AL =5 G8A;>< 4=59 4> 40BK 8AB5G5=8O A@>:0, 0 G8A;>< B8:>2 225@E 8;8 2=87 4> 8AB5G5=8O, 0 F5=0 70 >48= @07 <>3;0 1K 87<5=OBLAO B>;L:> =0 1 B8:, 8 >= 1K; 1K AB0B8AB8G5A:8 =57028A8< >B ?@54K4CI53> B8:0, B> <K <>3;8 1K 4>?CAB8BL ACI5AB2>20=85 =>@<0;L=>3> @0A?@545;5=8O.  =0H5< A;CG05 ;>30@8D<K 87<5=5=89 F5=K =5 8<5NB B0:8E E0@0:B5@8AB8:. 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DC=:F88 EMBED Equation.3 <>6=> 70D8:A8@>20BL M;5<5=B0@=>5 A>1KB85 EMBED Equation.3 8 ?>;CG8BL DC=:F8N EMBED Equation.3. -B>  C65 =5A;CG09=0O DC=:F8O >B EMBED Equation.3; >=0 =07K205BAO @50;870F859 A;CG09=>9 DC=:F88 (5I5 2K1>@>G=>9 DC=:F859; 4;O A;CG09=KE ?@>F5AA>2 B0:65 B@05:B>@859). 0>1>@>B, 70D8:A8@>202 EMBED Equation.3, <K ?>;CG8< A;CG09=CN 25;8G8=C. 0;55, 5A;8 EMBED Equation.3 ?@8 ;N1>< EMBED Equation.3  A;CG09=0O 25;8G8=0, B> EMBED Equation.3 ?@8 ;N1KE EMBED Equation.3  A;CG09=K9 25:B>@, ?@8=8<0NI89 7=0G5=8O 2 EMBED Equation.3. 3> @0A?@545;5=85 EMBED Equation.3 (:@0B:> EMBED Equation.3): EMBED Equation.3 (7.1) -B8 @0A?@545;5=8O ?@8 2A52>7<>6=KE EMBED Equation.3 =07K20NBAO :>=5G=><5@=K<8 @0A?@545;5=8O<8 A;CG09=>9 DC=:F88. 8=5@>2A:89 ?@>F5AA. 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(7.2) #A;>28O 0  II 2 >?@545;5=88 8=5@>2A:>3> ?@>F5AA0 <>6=> 70<5=8BL CA;>28O<8 0 8 (6.2). =>3><5@=K9 8=5@>2A:89 ?@>F5AA.  !;CG09=K9 ?@>F5AA EMBED Equation.3 A> 7=0G5=8O<8 2 EMBED Equation.3, C4>2;5B2>@ONI89 CA;>28O< I 8 III ?. 2 8 CA;>28N II . !;CG09=K9 25:B>@ EMBED Equation.3, EMBED Equation.3, 8<55B =>@<0;L=>5 @0A?@545;5=85 A <0B5<0B8G5A:8< >6840=85< 0 8 <0B@8F59 :>20@80F89 EMBED Equation.3 (E  548=8G=0O <0B@8F0 ?>@O4:0 r)."0:>9 ?@>F5AA <>6=> ?@54AB028BL A515 :0: r =57028A8<KE 4@C3 >B 4@C30 8=5@>2A:8E ?@>F5AA>2. !B>E0AB8G5A:85 48DD5@5=F80;L=K5 C@02=5=8O =0;8B8G5A:>5 @5H5=85 >B@51=>ABL 2 @5H5=88 AB>E0AB8G5A:8E 48DD5@5=F80;L=KE C@02=5=89 (!#) 2>7=8:05B =5 B>;L:> 2 7040G0E D8=0=A>2>9 <0B5<0B8:8, => 8 2> <=>65AB25 4@C38E, A0<KE @07=>>1@07=KE 7040G0E. 0?@8<5@, 2 7040G0E D8;LB@0F88, 7040G0E >?8A0=8O 18>;>38G5A:8E ?>?C;OF89 8 4@. 0G8=0O A A5@548=K XX 25:0, 2 <0B5<0B8:5 =0G8=05B 1C@=> @072820BLAO <0B5<0B8G5A:89 0??0@0B >?8A0=8O 8 @5H5=8O AB>E0AB8G5A:8E 48DD5@5=F80;L=KE C@02=5=89. >;LH8=AB2> !# 2AB@5G0NI8EAO =0 ?@0:B8:5 2 >1I5< 2845 <>6=> ?@54AB028BL :0: EMBED Equation.3 (8.1) 345 EMBED Equation.3 25:B>@ DC=:F88, EMBED Equation.3 - <0B@8G=0O DC=:F8O, EMBED Equation.3 AB>E0AB8G5A:89 ?@>F5AA. 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B>@>9 B8? 8=B53@0;0 1>;55 D878G5=, => <5=55 8A?>;L7C5BAO ?@8 2KG8A;5=8OE, B.:. 4;O =53> <0B5<0B8G5A:0O B5E=8:0 ?>;CG05BAO EC65. /2;O5BAO 206=K< B>B D0:B, GB> <564C MB8<8 42C<O 8=B53@0;0<8 ACI5AB2C5B >?@545;Q==>5 A>>B=>H5=85; 8 ?@8 @5H5=88 ?@0:B8G5A:>9 7040G8 >B !# 2 A<KA;5 !B@0B>=>28G0 ?5@5E>4OB : M:2820;5=B=><C !# 2 A<KA;5 B>, 0 70B5< @5H0NB. @8<5@. >45;L ;M:0 (>C;70 8<55B 284  EMBED Equation.3 , (8.6) 345  EMBED Equation.3   F5=0 0:F88,  EMBED Equation.3   A:>@>ABL @>AB0,  EMBED Equation.3   2>;0B8;L=>ABL,  EMBED Equation.3   AB0=40@B=K9 8=5@>2A:89 ?@>F5AA. A;8 C@02=5=85 (8.6) @0AA<0B@820BL 2 A<KA;5 B>, B> @5H5=85< 1C45B A;CG09=K9 ?@>F5AA 2840:  EMBED Equation.3 . (8.7)  5A;8 C@02=5=85 (8.6) @0AA<0B@820BL 2 A<KA;5 !B@0B>=>28G0, B> @5H5=85< 1C45B A;CG09=K9 ?@>F5AA 2840:  EMBED Equation.3  (8.8) 0<5G0=85:  40==>< A;CG05 ?@>F5AA (8.7) 1;865 : @50;L=>AB8. 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