O O S T K S T K

max 0000, 69 S

400.00 4,000 397.00 600 600 3,400 396.0 80 780 3,0 3 398.0 40 360 3,640 4 397.0 0 580 3,40 5 396.70 80 660 3,340 8 395.40 60 90 3,080 9 393.30 40,340,660,340 0 393.60 60,80 4,060 39.80 360,640 3,700 39.70 80,460 3,880 5 387.00,40,600,740,60 6 387.00 0,600 4,000 7 388.0 0,380 4,0 8 388.70 0,60 4,340 9 39.00 460,800 4,800 39.30 60,540 5,060

47 4 46 3 4 49 4

= -

S * ( S F ) * * ( ) ( ) F + S F + S S * ( S S ) * ( ) S * F ( ) S S * NYMEX

ν = σ + h σ hρσ σ S F S F ν = hσ ρσ σ h F S F ν / h S h = ρ σ σ F (. ) 0. 03 0. 8 = 0. 64 0. 040, 000, 000 0. 64 = 5. 4, 000

A R m mm + 3.

Aλ Rn 3. 0 00λ. = 0. 5 mn R R n Aλ = A + m = + λ R R m m

R R = mln + ( 3. 3) m ( ) R m R = λ / m ( 3. 4) 0 0 4 λ. = 0. 0808 ( )

( ) F S r T = λ t 3.5 Sλ r( T t ) ( ) Sλ r T t Sλ r( T t ) Sλ r( T t ) r( T t ) Sλ F 0 05 0 5 F = 40λ.. = 40. 50

Ke r T t r( T t) f + Ke = S ( ) ( ) f = S Ke r T t 3.6 ( ) F = Se r T t 0 f = 930 950e. 5 0. 06 = 808. ( ) F ( S I ) e r T = t ( 3. 7) ( ) F ( S I ) e r T > t ( ) ( S I ) e r T t ( ) F ( S I ) e r T t ( ) F ( S I ) e r T < t r( T t ) ( S I ) e F

0. 0 0. 04 0. 06 I = 075. e + 0. 75e + 075. e = 6. 0. 08 0. 8333 F = ( 50 6. ) e = 54. ( ) f Ke r T t + = S I ( ) f = S I Ke r T t ( 38. ) ( ) F ( S I ) e r T = t 0 I = 60e. 09 0. 5 00 + 60e. =. 65 0. f = 900 65. 90e = 35. 05

e q( T t) f Ke r ( T t ) q( T t ) + = Se f = Se q T t Ke ( ) r ( T t) ( 39. ) ( F Se r q ) ( T t) = ( 30. ) 0 f = 5e. 04 0. 5 0 7e. 0. 5 = 8. 0 f = 5e. 06 0. 5 = 5. 76 f ( F K) e r ( = T t ) ( 3. ) f ( F K) e r ( > T t) ( f F K) e r ( < T t)

f ( F K) e r ( > T t) ( S F) ( K S ) ( F K) T + T = f ( F K) e r ( T t ) f ( F K) e r ( < T t) r( T t) ( F K) e f

F Se r q T t = ( ) ( ) ( 3. ) 0 F = 400e. 05 0. 5 = 405. 03 ( F Se r q ) ( T t > )

( F Se r q ) ( T t < ) ( F Se r q ) ( T t < ) ( F Se r q ) ( T t > ) ( )( T t) Se r q

= a + β as + βs β S F S N = β F,00,000.5 50, 000

S β β * F S β * β F Ke -r(t-t) -r (T-t) e f f Ke r ( T t ) Se r f ( + = T t ) r ( T t ) r ( T t ) f f = Se Ke ( 33. ) ( F Se r r f )( = T t ) ( 34. )

F = Se r T-t 3.5 rt-t F S + Ue 3.6 r +u T-t FSe 3.7

-0.07 Ue =.865 0.07 F = 450.865 e = 484.6 rt-t F S + Ue 3.8 F S U e r T-t r(t-t) F S + Ue 3.9 (S U)e r(t-t) F S + U e r(t-t) r(t-t) F S + Ue 3.0

(r+u)(t-t) FSe 3. F < Se r +u T-t yt-t rt-t Fe S + Ue Fe y T-t Se r+u T-t F Se r + u = y T t ( )( ) ( 3. ) r - r f ct-t F = Se 3.3 c-y T-t F = Se 3.4

- Fe -r T-t + S T S T r T t Fe + E( S ) e ( ) k( T t ) T r( T t) k( T t ) Fe + E( S ) e = 0 T ( r k)( T t) F = E( S ) e ( 35. ) T S T S T S T S T S T S T S T S T S T S T S T S T

S T S Ke r ( T t ) Se r( T t ) S I Ke r( T t) ( S I ) e r T t q T t Se Ke ( ) r( T t ) ( ) Se ( r q)( T t)

F i e δ e δ e 3δ e δi i (F F ) e δ i i δi ( n i) δ nδ (F F ) e e = (F F ) e i i n nδ (Fi Fi ) e [( n n ) + ( n n ) + + ( 0 )] nδ ( F F ) e F F F F F F e = n 0 i= i i nδ F n S T ( ST ) nδ F 0 e F 0 ( T ) n n F e + S F e = S e δ δ nδ 0 0 T F 0 S T e nδ G 0 G 0 e nδ S T e nδ F 0 G 0 S T e nδ F 0 = G 0

3.5 0 n- n F 0 F F F n- F n e e e 3 e n 0 / 0 (F -F 0 )e (F -F )e (F n -F n -)e n n / 0 (F -F 0 )e n (F -F )e n (F n -F n -)e n

0 05 00e. = $3. 7

00 0. 0 0. e e = $3. 37 00e 0. 05 r * T * T * T * r = r * * ) T rt * 4. T T T * r * r * ( * T r = r + r r) * T T r * r *

T * r * r + T r T. 5 4 ln + = 00. 97. 5 5. ln + = 0. 047 94. 9 ln + 0 = 0054. 90. 0

0. 047 05. 0. 054 5. R 4e + 4e + 04e = 96 e 5. R = 08596. ln ( 08596. ) R = = 0068. 5. 0047 0 5 0 054 0 0 068 5 6e.. 6e.. 6e. + +. + 06e R = 06. 00. 0. 5 0. 0505 0. 75 006.. 5 0. 0745. 75 5e + 5e + 5e + 5e = 8. 08

0. 08 + R 3 3 ( ) 5. 5 0. 07 + R/ e 3 05e. 75 R + = 878.

54 $5. 50 = $. 64 8 98 0 3

40 7 00 i + 40 = 59. 38 04. 04. i= 36 7 00 i + 36 = 6373. 04. 04. i= 0 04.

F ( S I ) e r ( = T t ) 4.

60 0 + 6 = 978. 8 0 334 0 6e.. = 5803. 0.7397 0. (.978-5.803)e = 5094. 5. 094 6 48 = 0. 4 83 0. 4 = 85. 887. 4000 T * T * T * r * V * V * = 00e r* T* V * e rt r T rt rt r T F = 00e * * e = 00e * * ( 4. 3)

F = e r T 00 ( * T ) r T * 35 0. 5 45 0 = 0. 75% 90 r * ln 0. 95 = 0. 7 0. 4000

r ln 0. 9695 = 056. 0. 466 r * r * * * r T r ( T T) r = T. 7 46. 56 90 = 80%. 56 360 ( 00 Y ) n 365 = 0. 088 98 90

0.08530-0.0840 0.0864 90 0 0864 0 466 00e.. = 97. 89

t i c i B = n i= c e i yt i ( 4. 4) n yti t icie i= D = ( 4. 5) B D = n i= t i cie B yt i t i t i t i n B yt i = citie ( 4. 6) y i= B = BD ( 4. 7) y B = BD ( 48. ) y B = D y B

5e -0. 0.5 = 4.709 BD y B + y D F

D S S = -SDS y 4.9 F = FD y 4.0 F * SDS N = 4. FD F 3, 300, 000 05. = 678. 973, 600 0. 5 y 4.

0, 000, 000 680. = 79. 4 93, 0650. 9. 0 n B yti = ci ti e y i=

t i t i t i t i

A 9.95% Libor B B B A B A A B

B A Libor+% 0% Libor 9.95% B A Libor 9.9% Libor 0.0%

B A Libor 9.9% Libor+ Libor 0% 0.0%

00 0

5. -yr.tn 30bps.yr.TN 38bps 7.5 3 3.yr.TN 35bps 3.yr.TN 44bps 7.7 4 4-yr.TN 38bps 4-yr.TN 48bps 7.83 5 5-yr.TN 44bps 4-yr.TN 54bps 7.90 6 6-yr.TN 48bps 6-yr.TN 60bps 7.94 7 7-yr.TN 50bps 7.yr.TN 63bps 7.97 0 0-yr.TN 60bps 0-yr.TN 75bps 7.99

r i t i n ri ti B = Ke + Qe i= rt i i rt * r t B = Qe + k e V = B B

B = 4e + 4e + 04e 0. 5 0. 0. 75 0. 05 5. 0. 984 B = 5. e + 00e 0. 5 0. 0. 5 0. 05 ( k 05. i ) R Q e rt i i * ( k k ) e rt n rt ( 0 5 i ) * rt ( k k ) e + k. R Q e i= n ( 0 5 i ) * rt ( k k) e +. R Q k e rt i=

*.....,.... R R 3 r t rt = t - t r t r t = t - t 3 3 3 0. 75 005. 0. 5 0. 0 = = 0. 075 0. 5. 5 0. 0. 75 0. 05 = = 0. 75 0. 5 R 0.04 R 0.0 3 ( 4. 0 5. ) e + ( 4. 0 0. 5 004. 00) e 0.. 5 + ( 4. 0 0. 5 0. 0 00) e = 47 0. 0. 5 005. 0. 75

B A %Sterling Dollars8% Sterling% %Sterling Doiiars8% Dollars9.4%

B A %Sterling Dollars8% Sterling% %Sterling Doiiars8% Dollars8.4% B A %Sterling Dollars9.0% Sterling% %Sterling Doiiars8% Dollars9.4% V SB B F D =

0. 09 0. 09 0. 09 3 BD = 08. e + 08. e + 08. e = 964 BF = 60e + 60e + 60e 0. 04 0. 04 0. 04 3 = 3055 3055 964 = 55 0 t i r i t i F i t i t i (. Fi 4. ) e rt i i ( 0Fn 5) e rn tn

0. 05 0. 00909e = 0. 0096 0. 05 0. 00909e = 0. 000 0. 05 3 0. 00909e = 0. 006 0. 09 ( 60 00096. 08. ) e = 0. 0. 09 ( 60 000. 08. ) e = 06. 0. 09 3 ( 60 0006. 08. ) e = 03. 0. 09 3 ( 00 0. 006 0) e = 04( )

5 8 8

S T p Xe -r(t-t)

S - Xe -r(t-t) S - Xe -r(t-t) 0. = 0 8e = 37. Xe -r(t-t) c + Xe -r(t-t) > S c > S - Xe -r(t-t) c > max(s- Xe -r(t-t), 0) ( 7. )

S - Xe -r(t-t) -0. 0.5 5-50e = $3.9 Xe -r(t-t) S Xe -r(t-t) 0 S = 40e. 05 0. 5 37 =. 0 Xe -r(t-t) p + S > Xe -r(t-t)

p > Xe -r(t-t) S p > max(xe -r(t-t) S, 0) ( 7. ) Xe -r(t-t) S 0 0 5 40e.. 38 = $. 0 0 0 0833 50 40e.. = $0. 33 Xe -r(t-t) $0

Xe -r(t-t) S- X + Xe -r(t-t) c > S - Xe -r(t-t) C > S- Xe -r(t-t)

Xe -r(t-t) Xe -r(t-t) p > Xe -r(t-t) S

Xe -r(t-t) Xe -r(t-t) c + Xe -r(t-t) = p + S ( 7. 3) r T t c + Xe ( ) 0 = + e. 0 3 30. 5 = 3. 6 p + S =. 5+ 3 = 335.

r T t c + Xe ( ) 0 = + e. 0 3 30. 5 = 3. 5 p + S = + 3 = 3. 00 r T t P > c + Xe ( ) S r T t P > c + Xe ( ) S C P < S Xe r T t ( ) ( 7. 4)

r( T t) max( S, X ) + Xe X T Xe r( τ t) S X < C P < S Xe r T t ( ) ( 75. ) 0. 0. 467 50. + 0e 9 = $. 68 9 0 9 0 0 < C P < e. 0. 467

D + Xe r ( T τ ) c > S D Xe r ( T τ) ( 7. 6) D + Xe r ( T τ ) r T P > D + Xe ( τ ) S ( 7. 7) r T t c + D + Xe ( ) = p + S ( 7. 8) S D X < C P < S Xe r T t ( ) ( 7. 9)

C P S Xe r ( < T t ) S D Xe r ( T t)

max( S Xe r( T t), 0) r( T t) max( Xe S, 0) ( ) max( S D Xe r T t, 0) r( T t) max( Xe + D S, 0) r T t C + Xe ( ) = p + S r T t c + D + Xe ( ) = p + S

r T t p + S = c + Xe ( ) + D ( 8. ) r ( T t) Xe + D S c Xe r T t = ( ) + D p

r ( T t) Xe + D S T X S T X X S T X X X S T X S T X 0 S T X S T X 0 0 0

S T 30-30 S T 35 S T 3 S T 35 3

S T 30 + 30 S T 35 3 S T S T 35-3

S T X 0 X-S T X-S T S T > X S T -X 0 S T -X

S T X 0 X -S T X -S T X <S T <X 0 0 0 S T X S T -X 0 S T -X

S T X X X 3 8.3

IBM IBM

z = t ( 9. ) z = 0 z = t z t T N = t N z( T) z( 0) = t ( 9. ) i= i i i Y N Y Y Y

[ z(t) - z(0) ] [ z(t) - z(0) ] [ z(t) - z(0) ] = 0 = N t = T = T T t y x t 0 t 0 t 0 dz = dt dx dt = a

x = a t + b t x = a t x = b t x = b t x = at x = b T x = b T 900 05.

ds S = µdt S µ = S e t 0 σ σ t σ S t σ S µs σ S ds = µ Sdt + σsdz ds S = µ dt + σ dz ( 9. 6)

ds = 0. 5dt + 0. 30dz S S = 05. t + 0. 30 t S S = µ t + σ t ( 9. 7) S σ t σ t σ t S ~ ϕ ( µ t, σ t ) ( 9. 8) S ϕ( m, s) µ = 0. 4 σ = 0. 0

0. 0 S S ~ ϕ ( 0. 004, 0. 0 ) ( 9. 9 ) ϕ ϕ ϕ ϕ ϕ ϕ ϕ.4

σ t σ T σ T

p Su S -p Sb σ t u t σ t e d u = e, d = p = u u d 0. 30 0. 04 u = e = 068. d = = 0. 948 u 0. 0. 04 e 0. 948 p = = 0. 55. 068 0. 948 ud p t t t

T b T

dx = a (x, t) dt + b(x, t) dz (0.) G dg = x a + G t + G x b dt G + x bdz ( 0. ) G x a G G + + t x G b x

G dg = S S + G µ t + G σ S S G dt + S σsdz ( 04. ) F Se r ( = T t ) F S r T t F F = e, = 0, = rse S t ( ) r( T t ) r( T t ) r( T t) r( T t ) df = e µ S rse dt + e σsdz [ ] F = Se r(t-t) G G G =, = = 0 S S S S t dg = µ σ dt + σdz µ σ ( T t )

σ ( T t) S T S T ln S ln S T lns ln S ~ ( T t ) T ϕ µ σ, σ T t lns ln S ~ ( T t ), T t (. ) T ϕ µ σ σ 0 6 S T ϕ ln S ~ ln s ( T t ), T t (. ) T ϕ µ σ + σ 0 7 S T S T ( T t) S T 004. ln S T ~ ϕ ln 40+ 06. 05., 0. 05. ln S T ~ ϕ( 3. 759, 0. 4) 3.477 < lns T < 4.04 3.477 4.04 e < S T < e

3.36 < S < 56.88 T S T S T u( T t) E( S ) = Se ( 08. ) T S T S T u( T t ) σ ( T t) var( S ) = S e ( e ) ( 09. ) T S T S T 0. E( S ) = 0e = 4. 43 T 0. 4 0. 6 var( S ) = 400e ( e ) = 03. 54 T 0354. η( T t ) S = Se T ST η = ln ( 0. 0) T t S ST lnst lns = ln S ST ln ~ ϕ µ σ ( T t), σ T t ( 0. ) S

η ϕ µ σ ~, σ ( 0. ) T t µ σ σ T t 0. 04 0. 7 = 0. 5 0. = 05. 3 µ σ / 5 00.4 = 9.54

/ 5 ( 7940. ) = 04. µ σ / µ σ / S i S i µ i = ln S i ui S = S e, u i i i s = s = n u u i n n i= u i n i= ( u u) i n( n ) u i σ τ σ τ s * s s * = τ n i= ui

* s n u = 00953., u = 0. 00333 i 0. 00333 00953. = 9 380 i 0. 03 τ = / 50 0.03 50 = 094. 094. = 003. 0

0. (S i / Si -) u i = ln(s i / Si -) 0 8 9 7 8 0 0 4 0 7 8 0 7 8 0 7 8 0 3 4 0 3 4 8 0 7 8 0 7 8 4 3 8 3 8 4 3 4

u i Si + D ui = ln S i Si ui = ln S i =0 = = =8 =0 8a = a - = 4.5

4. 5 f = 5 = 0. 5445. 0 ds = µ Sdt + σsdz 0.3

f df = S S + f f f S dt t + + µ σ S S σsdz ( 04. ) S = µ S t + σ S t ( 0. 5) f = f S S + f f f S t t + + µ σ S S σs z ( 06. ) t + f S + f S Π f = f + S S 0.7 t f = f + S S ( 08. ) = f f σ S t ( 09. ) t S

r = t f f σ t + f S t r f S = S S t f t rs f + f σ S S + S = rf ( 00. ) f = max( S X, 0) t = T f = max( X S, 0) t = T + f S f S ke r ( = T t ) f r T t f f = rke ( ) = = t, t, 0 S r( T t) rke + rs

p + 8( p) = 0 0. 055. + 0. 45 0 = $0. 55 0. 55 = 0. 5445 0.

S K T S T f e r ( T = t ) E ( S K ) 0. T E f e r ( T t ) r T t = E ( ( ) S ) Ke 0. T ( ) r( T t) E S = Se 0.3 T f S Ke r ( = T t ) 0.4 E [ max( S X, 0 ) T ] E [ T ] c e r ( T = t ) E max( S X, 0) 0.5

ln S T σ ln S ~ ln S r ( T t ), T t (. ) T ϕ + σ 0 6 r( T t ) c = SN( d) Xe N( d) 0.7 ln( S / N) + ( r + σ / )( T t) d = σ T t d ln( S / N ) + ( r σ / )( T t) = = d σ T t σ T t r T t p = Xe ( ) N( d ) SN( d ) (. ) 0 8 S Xe rt d d N d N d.0 g i0.6 ST 0.5

0.8 N d N d 0 Se rt max( Se rt X, 0) rt rt e max( Se X, 0) rt = max( S Xe, 0) rt 0.7 S Xe ln( S / N) + rt 0 σ 0 d d N d N d.0 0 7 c = S Xe rt rt S Xe ln( S / X ) + rt 0 σ 0 d d N d N d 0 0.7 0 σ 0 max( S Xe, 0) rt σ 0 max( Xe S, 0) rt N ( x)( a k + a k + a k 3 3 ) x 0 N( x) = N ( x) x 0 k = + γx γ = 0. 3367 a a a 3 = 0436836. = 00676. = 0937980. N ( x) = e x π /

3 4 N ( x)( a k + a k + a k + a k + a k 5 3 4 5 ) x 0 N( x) = N ( x) x 0 k = + γx γ0.03649 a 0.3938530 a 0.35656378 a.78477937 a.855978 a 5.334749 l 3 4 ln. 05 + 0. 0. 5 d = = 0. 7693 0. 0. 5 ln. 05 + 0. 08 0. 5 d = = 0. 678 0. 0. 5 r( T t ) 0. 05 Xe = 40e = 38. 049

γ V T MγX V + T M γ X N + Mγ V + T MγX N + Mγ V MγX γ + T X N + Mγ Nγ VT X N + Mγ N Nγ VT max X N + Mγ,0 N Nγ N + Mγ V = NS + MW V M S N = + N W Nγ / ( N + Mγ )

τ τ rτ c = SN( d ) Xe N( d ) r p = Xe N( d ) SN( d ) τ d d ln( S / N ) + rτ + ( σ / ) τ = σ τ ln( S / N ) + rτ ( σ / ) τ = σ τ = d σ τ

0 667 0 09 0 467 0 09 05... e 05. e. +. = 0947. ln 0. 9756 + 0. 35 0. 5 d = = 0. 07 0. 3 0. 5 ln 0. 9756 + 0. 045 0. 5 d = = 0. 004 0. 3 0. 5 N( d ) = 0. 5800, N( d ) = 0. 4959 0. 09 0. 5 39. 059 0. 5800 40e 0. 4959 = 3. 67 S/S-V V

t t t t 3 n t t t t 3 n DD D3 Dn t n t n S( t ) X n S( t ) D S( t ) D Xe n n n r( T t n ) r( T tn ) S( t ) D Xe S ( t ) X n n D ( X e ) ( 0. 9) n r T t ( n ) n t n D ( X e ) ( 030. ) n r T t ( n ) S( t n ) t n t n t n t n S( t ) X n t n S( tn ) D n t n t n S( t ) D Xe n n n r( tn tn ) r( tn tn ) S( t ) D Xe S ( t ) X n n n )

r( t t ) n n n D ( X e ) t n r ti ti D X ( e ( + ) ) 0.3 i t i D Xr( t t ) i i + i t n t n D = D = 0. 5, S = 40, X = 40, r = 0. 09 t t r( t t) 0. 09 0. 5 X ( e ) = 40( e ) = 0. 89 r( T t) 0. 09 0. 0833 X ( e ) = 40( e ) = 0. 30 0. 667 0. 09 0. 5e = 0. 496

r ti ti D ( X e ( + ) ) i r( t t D ( X e n ) ) n

G dg x ( 0A. ) dx x G dg 3 d G d G 3 = x + x + 3 x + Λ dx dx 6 dx G G x x G + y ( 0A. ) y G G x x G y y G G x x xdy x y G = + + + + y + Λ ( 0A. 3) y G dg = x dx + G y dy ( 0 A. 4) dx = a (x, t) dt + b (x, t) dz (0A.5) G G G G G G = x + t + x + x t + t + Λ ( 0A. 6) x t x x t t x = a (x, t) t + b (x, t) t

x = a t + b t ( 0A. 7) x x = b t + t ( 0A. 8) x ( ) [ E( )] E = ( ) E = t t t t b dt G dg = x dx + G t dt + G x b dt 0 A (. 9) G dg = x a + G G t + x b dt G + x bdz t D τ rτ rτ rτ C = ( S D e ) N( b ) + ( S D e ) M a, b ; Xe M a, b; τ rτ ( X D ) e N( b ) ( 0B. ) a a b b τ l = rτ [( S De ) X] + ( r + σ ) = a σ τ rτ [( S De ) S] + ( r + σ ) ln / / = σ τ = b σ τ = t t τ = T t ln / / σ τ τ τ τ τ

M a, b;ρ a b ρ S ( ) ( ) c S t = S + D X, c S, t S = S tt S = b = b = S D e rτ S, S(t ) S + D t D Ma, b; ρ, a 0, b 0, ρ 0, 4 ρ Ma, b; ρ= A ia jf ( Bi, B j) π i, j= [ ] (, ) = exp ( ) + ( ) + ρ( )( ) f x y a x a b y b x a y b a b a =, b = - - ( ρ ) ( ρ ) A 0.353030 B 0.337764 A 0.407 B = 0.64347 A = 0.33445 B.345378 3 3 A 0.00637433 B =. 66645 4 4 Ma, b; ρ= N( a) Ma, b; ρ Ma, b; ρ= N( b) M a, b; ρ Ma, b; ρ= N( a) + N( b) + M( a, b; ρ) Ma, b; ρ= Ma, 0; ρ + M( b,0; ρ) δ ( ρa b) sgn( a) ( ρb a) sgn( b) ρ =, ρ = a ρab + b a ρab + b ( ) ( ) 0 δ = sgn a sgn b ( x) = + x, sgn 4 x 0 t

S T q( T t) S e Se q( T t) S T Se q( T t) Se q( T t) Se q( T t) r T t ( ) ( ) ( ) ( ) q( T t) ( ) c = Se N d Xe N d r( T t) q( T t) p = Xe N d Se N d ( ) q T t Se S ln ln ( ) x = X q T t T (. ) (. ) d d d, d ( ) ( σ ) ln S / X + r - q + / ( T - t) = σ T - t ( ) ( σ ) ln S / X + r - q - / ( T - t) = = d ( T t) - σ - σ T - t

ln 03333. + 0. 07 0667. d = = 0. 5444 0. 0667. ln. 03333 + 0. 03 0. 667 d = = 0. 678 0. 0667. N( d ) = 0. 7069, N ( d ) = 0. 678 0. 03 0667. 0. 08 0. 667 c = 30 0. 7069e 300 0. 678e = 7. 8

r f r f r f ( ) ( ) f ( ) ( ) r ( T t) r( T t) f c = Se N d Xe N d p = Xe N d Se N d r( T t ) r ( T t) ln ( S / X ) + ( r r )( T t) f + σ / d = σ T t ( 3. ) ( 4. ) ln ( S / X ) + ( r r )( T t ) f + σ / d = = d σ T t σ T t r f X A X B X A X B ( F Se r r f )( = T t ) [ ( ) ( )] ( ) ( ) r( T t) c = e FN d XN d [ ] r( T t ) p = e XN d FN d ( 5. ) ( 6. )

ln ( F / X ) + ( σ / )( T t) d = σ T t ln ( F / X ) ( σ / )( T t ) d = = d σ T t σ T t r f

F Se a ( = T t ) ( 7. ) [ ( ) ( )] ( ) ( ) r( T t) c = e FN d XN d [ ] r( T t ) p = e XN d FN d ( 8. ) ( 9. ). q R -q

ln ( F / X ) + ( σ / )( T t) d = σ T t ln ( F / X ) ( σ / )( T t ) d = = d σ T t σ T t σ T t d = = 0. 076 σ T t d = = 0076. ( ) ( ) N d = 0. 47, N d = 0588. p = e 0. 09 0. 3333 ( 0 0588. 0 0. 47 ) =. Xe r( T t) ( FT X 0) + X = ( FT X) max, max, Fe r( T t ) ( T ) max ( T, 0) max ( T, ) F + F F + X F = F X ( ) ( ) c Xe r T t r T t + = p + Fe ( 0. )

-0.5 0. 0.56 + 8.50e 0. 5 8 00 0.. e = 04.

ds = µ Sdt + σsdz f df = S S + f f f S dt t + + µ σ S S σsdz + f S

f = f + S S ( A. ) = f t f σ S S t qs f S t f f W = S + qs f σ t ( A. ) t S S W = r t ( A. 3) f f + = + σ S qs f f t r f t S S S S t f ( r q) S f f + + σ S = rf ( A. 4) t S S F Se a ( = T t ) ds = µ Sdt + σsdz σ F σ F σ a T t F = S F = σse ( ) = σ S σ F = σ df = µ Fdt + σ Fdz ( B. ) f f f f df = F F dt F t F F Fdz B F + + + µ σ σ (. ) F

f + F = f ( B. 3) f F F W = f F F f F = µ F t + σf z F f f f f f = F F t F t F F F z F + + + µ σ σ f f W = σ F t ( B. 4) t F W = r t ( B. 5) f f σ F t = rf t t F f f + σ F = rf t F

dθ mdt sdz θ = + (. ) f f f, f df = µ dt + σdz f df f = µ dt + σ dz µ µ σ σ θ t f = µ f t + σ f z. f = µ f t + σ f z.3 σ f σ f = ( σf ) f ( σf) f (. 4) = σ f f σ f f

= ( µ σ ff µ σ ff ) t (. 5) = r t µ σ µ σ = rσ rσ µ r µ r = σ σ (. 6) µ r µ r = = λ σ σ df = µ fdt + σ fdz (. 7) µ r = λ σ (. 8) f θ µ r = λσ.9 dz dz -dz f

0. 0. 08 = 0. 0. f µ f mθ f θ f = + + s t θ θ σ θ f f = s θ θ ( λ ) θ f f f + m s + s = rf ( 0. ) t θ θ q = r m + λs r ( r m + λs) = m λs λs λs m r = λs m λs = r λs

m λs = 00. 05. 0. = 008. 0 08 05 800e.. = 768. 63 0. 03 0. 06 = 0. 5 0.

θ θ θ n n dθ θ i i = m dt + s dz (. ) i i i i, n, dz m s i i i θ A θ f i i n df = µ dt + σ idzi (. ) f i= dz θ i µ = λ σ n r i i i= σ i i.3 λ θ i i λ i σi λ i σ i θ i λ i σ i λ i σ i θ i λ i σ i θ i λ i σ i σ CAPM λ θ θ λ 0 i i i i m m λ s i i i i

θ i mi λ isi θ i θ i m m λ s i i i i m m λ s i i i i ( ) θ i n θ i m λ s i i i s i θ θ ρik i k f T ( T ) f = e r( T t) E f ( 4. ) E θ m λ s i i i i θ i λ rsr λ r s r f T r( T t) f = E e f ( 5. ) [ T ] r f T f T X A X B -r( T-t ) 00QAQ Be Q A X A Q B X B θ i i i. r

Q A ( S X ) ( r )( T t) A A + σa ln / / = N σa T t ( S X ) ( r )( T t ) B B + σb ln / / QB = N σb T t S S A B σ σ A B A B A B ( ) ( T ) f = e r T t E S K [ T ] ( t) ( ) r T f = e E S K E ( ) F = E S T.6

6. 70 ln = 0. 034 60. 60 6.95e -0. 56.96 r ( y u) = r y + u m λs m λs = r y + u y = r + u m + λs S(t) t Nikkei Q(t) t K qnikleei r r f F S( t) K ( ) S( t ) e ( rf q) T t

rf q S( t ) e ( r q)( T t) S( T) K Q T [ ] ( ) r ( T t f ) e E S ( T ) Q ( T ) KQ ( T ) [ ] { [ ( ) ( )] [ ( )]} e r ( T t ) f E S T Q T KE Q T E S T Q T F = [ ( ) ( )] E.7 [ Q ( T )] rf r E [ Q( T) ] Q( t) e ( r r)( T f = t) ( 8. ) E S( t) Q( t ) [ ] ( ) ( ) ( ) ( ) ds t = r q S t dt + σ S t dzs f ( ) ( ) ( ) ( ) S dq t = r r Q t dt + σ S t dzq ( 9. ) f Q σ σ S Q dz dz A ITO S Q S Q d S( t) Q( t) = r q + r r + ρσ σ S t Q t dt [ ] [ ] ( ) ( ) f f S Q + S( t ) Q( t)[ σ SdzS + σqdzq ] rf q r +ρσ SσQ ( ) E S( t) Q( t) S t Q t e r f q r + ρσs σq = T t. 0 [ ] ( ) ( ) Qt P.

F S( t) e r f q + S Q = T t ρσ σ ( ). r q r + ρσ σ f S Q q * * r q = r q + ρσ σ f S Q q * = r rf + q ρσ Sσ Q ( ) q * ( T t ) ( ) r ( T t ) ( ) S t e N d Xe N d d * ( ) ( ) ( ) Xe r ( T t ) N d S t e q ( T t ) N d d * ln ( S / X ) + ( r q + σ / )( T t ) = σ T t * ln ( S / X ) + ( r q + σ / )( T t ) = σ T t = d σ T t

x, x,λ x n x i a i b i dx = a dt + b dz ( A. ) i i i i dz i a b x i f f f f f = x t x x xi t ( A ) i + + i j Λ. x t x x x t i i i j i j x = a t + b t i i i i i i dz dz ρ ij i j lim t 0 lim t 0 x = b dt i i x x = b b ρ dt i j i j ij t 0 f df x dx f t dt f = x x b b dt i + + i jρij i i i j i j i j i i j df dx i f x a f = i + + i t i f x x b b dt f x b dz i jρ ij + i i i i j i j i A.3

θ i dθ = m θ dt + s θ dz B. i i i i i i dz m s θ θ m s n i i i i i i i ρikdz dz i,k n i k f j j n j r f r n j θ i df j = µ j f jdt + σ ij f jdzi ( B. ) f j f j µ j f j = + m θ + ρiks S θ θ t θ i i i i i i k i k i, k θ θ ( B. 3) f j σ θ ( ) ij f j = si i B. 4 θ i,µ f σ f j j ij j θ i = k f ( B ) j j. 5 j k j i f j k f j

k f ( B ) jσij j = 0. 6 j d = k jµ j f jdt j k f j j k µ f = r k f ( B. 7) j j j j j j j ( µ ) = (. ) k j f j j r 0 B 8 j j k j ( µ ) = λ σ (. 9) f r f B j j i ij j i µ r = λ σ ( B. 0) j i ij i ( ) λ i i n f θ ρ θ θ λ j f j f j f j + mi i iksi sk i k rf j i s θ t + = i i θ θ θ θ i i i, k θ f j f j f j + i ( mi λisi ) ρiksi skθiθ k rf t + = θ θ θ i i i, k i k θ i i n ( ) f t θ f f + i + = θ θ θ i i ( m λ s ) ρiks s θ θ rf ( B ) i i i i k i k. i, k i k i i k i j k j

Q = max S X, 0 3. ( ) ( ) 40,000

t X X t X

= c S (f(s f S

= N( d ) d N d ( ) = N d ( ) d 8 3 8

3. Delta =63,400 Delta ((, ( $000 $000 $000 ) ) ) 0 49 0.5 5,00,557.8,557.8.5 48 8 0.458 (6,400) (308.0),5.3. 47 3 8 3 50 4 4 5 3 4 5 53 8 0.400 (5,800) (74.8),979.7.9 0.596 9,600 984.9,966.5.9 0.693 9,700 50.0 3,47.3 3.3 0.744 8,00 430.3 3,904.9 3.8 6 53 0.77 (300) (5.9) 3,89.8 3.7 7 5 7 8 0.706 6,500 (337.) 3,559.3 3.4 8 5 3 8 0.674 (3,00) 64.4 3,398.4 3.3 9 53 0.787,300 598.9 4,000.5 3.8 0 49 7 8 0.550 (3,700) (,8.0),8.3.7 48 49 7 8 3 50 3 8 4 5 8 5 5 7 8 6 5 7 8 7 54 7 8 8 54 5 8 9 55 7 8 0 57 4 0.43 (3,700) (664.4),60.6. 0.54,900 643.4.806..7 0.59 4,900 46.8 3,055.6.9 0.768 7,700 9.6 3.98. 3.8 0.759 (900) (46.7) 3,983.3 3.8 0.865 0,600 560.5 4,50.6 4.3 0.978,300 60. 5,7.0 4.9 0.990,00 65.6 5,97.5 5.0.000.000 55.9 5,58.3 5..000 0 0.0 5,63.4

3. Delta =56,600 Delta (, ( $000 $000 $000 0 49 0.5 5,00,557.8,557.8.5 49 3 4 0.568 4,600 8.9,789..7 5 0.705 3,700 7.4 3,504. 3.4 3 50 0.579 (,600) (630.0),877.6.8 4 48 3 8 0.459 (,000) (580.5),99.8. 5 48 4 6 48 3 4 7 49 5 8 8 48 4 9 48 4 0 5 8 5 49 7 8 3 49 7 8 4 5 8 5 5 7 8 6 5 7 8 7 54 7 8 8 54 5 8 9 55 7 8 0 57 4 0.443 (,600) (77.),4.8. 0.475 3,00 56.0,383.0.3 0.540 6,500 3.6,707.8.6 0.40 (,000) (579.0),3.4.0 0.40 (,000) (48.),085..0 0.658 4,800,67.9 3,355. 3. 0.69 3,400 75. 3,533.5 3.4 0.54 (5,000) (748.),788.7.7 0.538 (400) (0.0),77.5.7 0.400 (3,800) (67.7),0.4.0 0.36 (6,400) (779.0),34.4.3 0.6 (,500) 0.0,445.7.4 0.06 (9,900) (90.4) 56.7 0.5 0.83,00 58.3,09.5. 0.007 (7,600) (80.6) 90.0 0.3 0.000 (700) (33.7) 56.6

= e ( ) N( d) q T t = e [ ] ( ) N( d ) q T t = rf e ( T t) N( d) r f d = e r f [ ] ( T t) N( d ) = r T e ( t) N( d )

d = e [ ] ( t) N( d ) r T r f ( ) [ N( d ) ] e r f T t d d 0.087 N d 0.55 * T H t Delta A H t Delta F F Se r T / = t ( ) Se r( T / t) e r T e r T ( / t) ( / t) r T t H = e ( / ) H F A r q T t H = e ( )( / ) H F A ( 3. ) r rf T t H = e ( )( / ) H F A

* T - t = 0.75 e r r f T * ( )( t ) = 08. e r * (T -t) ω i i n = ω i i= i ( 0. 08 0. 04) 0. 5 4, 900e = 4, 605

( ) SN d σ r( T t) Θ = rxe N( d ) T t d d N ( x) = e x / π ( ) SN d σ r( T t) Θ = + rxe N( d ) T t ( ) q T t SN d σ e ( ) q ( T t) r ( T t) Θ = + qsn( d) e rxe N( d) T t d d ( ) q T t SN d σ e ( ) q ( T t) r ( T t) Θ = qsn( d) e + rxe N( d ) T t r f ( ) q T t SN d σ e ( ) q( T t) r( T t) qsn( d) e + rxe N( d ) = 85. T t rxe r( T t) (=(/(t.

t = Θ t + Γ S ( 3. 3) 38 (( 3.8 Deta

Γ T ω T ω T Γ T + Γ Γ / Γ T Γ / Γ T 3000 =, 000 5. d ( ) N d Γ = Sσ T t ( ) q( T t) N d e Γ = Sσ T t d

( ) q( T t) N d e = 000857. Sσ T t f rs f f + + σ S = rf t S S f f f Θ =, =, Γ = t S S Θ + rs + σ S Γ = rf ( 3. 4) Θ + σ S Γ = rf (((/(( H Vega LambdaKappa Sigma

Λ T Λ / Λ T ω, ω 5, 000 + 05. ω+ 08. ω = 0 8, 000+. 0ω+. ω = 0 ω = 400, ω = 6, 000 Λ = S T tn ( d ) d d Λ = S T ( ) tn d e q ( T t ) d r f

( ) S T tn d e q ( T t ) = 66. 44 r( T t) ( ) ( ) rho = X T t e N d ( ) r ( T t ) ( ) rho = X T t e N d d d ( ) r X T t e ( T t ) N( d ) = 457. rho ( T t ) e r f ( T t) = SN( d ) rf T t ( ) ( ) ( ) rho = T t e SN d rho((/(r H

q( T t) = e N( d) ( 35. ) [ ] ln ( S / X) + ( r q + σ / )( T t ) d = σ T t

e [ N( d )] q( T t ) [ ( ) ] = ( ) * * * [ ] e q( T t) e ( r q)( T t) N d e q( T T) e r( T t ) N d T * K K * * q( T T) r( T t) [ ( ) ] e e N d K K q( T t) e N( d) = 03. [ ] T * * T0. 5T t = 0. 75, k = 00, 000, k = 500, [ ( ) ] q( T K e * T) r( T t) e * N d = 6. 6 K

beta beat 3.

Π Π Π Π Π Π = + + + S t + + ( ) S t S t Λ 3A. S t S t S t Π = Θ t + Γ S

Π Π Π Π Π Π = S + σ + t + + σ + S σ t S S Λ σ

f T rt f = E f e 4. [ T ] ( ) E r rt f = e E ft 4. ( ) ( ) r E ( ft ) λσ λ

m θ = m θ t + sθ t ( 4. 3) r r θ i ( i n) s i θ i m i θ i ρik θ i θ k θ i θ = m θ t + s θ t ( 4. 4) i i i i i i θ i θ i i i k ρik i, k n i ( i n) θ i 0. 4.3

= R i 6 ( 4. 5) i= R i i ) x x = x = ρx + x ρ ρ ρ ij x ( i i n ) i ( i n) = k = i a x i ik k k= i ( j i) k i a ik = k a a ik jk j = ρij x 3 ω M f f f f + f f =

f f ω M f * A f * B f A * * f f f f ( ) A = A B + B 4. 6 f B m s i i f * f * f q f f *

t t Se r t r Se t = psu + p Sd 4. 7 r e t = pu + p d ( ) ( ) ( ) ( 48. ) t [ ] ( ) ( ) S σ t E Q E Q [ ] ( ) ( ) S σ t = ps u + p S d S pu + p d [ ] ( ) ( ) ( ) σ t pu p d pu p d = + + 4. 9 p,u d u = d r r

t a d p = u d u = e d e σ t = σ t ( 4. 0) ( 4. ) ( 4. ) a = e r t ( 43. ) t Su Sd t Su Sud Sd i t i + j i Su d j j = 0, Λ, i Su d = Su max( X - S,0 maxs X 0 S T T T t t T t T t t t T pu d t0 a b S(t t St) 0.80.9

σ t σ t u = e = 4., d = e = 0. 8909 r t a = e =. 0084, a d p = u d = 0. 5076 p = 0. 494 i t Su j d i j 3 50.4 0.8909 = $39.69 max X S 0 T 0. 0 0. 0833 ( 05076. 0 + 0. 494 545. ) e =. 66 0. 0 0. 0833 ( 05076. 545. + 0. 494 4. 64) e = 9. 90 0. 0 0. 0833 ( 05076. 6. 37 + 0. 494 4. 64) e = 0. 35 j i- fij i t Su d j ( 0 i N 0 ji ) f ij max ( X S T, ) 0 j N j [ ] f = max X Su d, 0 j = 0Λ,, N Nj

i t (i, j) i t i, j + i t (i, j) i t i, j (- p) [ i, j ( ) + + i+, j ] fij e r = t pf + p f 0 in,0 ji f i, j [ i+ j+ i+ j ]} { j i j r t ( ) fij = max X Su d, e pf, + p f, i t i t = f S S f t, Su, f Sd, f, S = Su Sd f = f - f, t 0 f f0 = Su Sd 0 Gamma Γ, t S = ( Su S) / ( 3 ), Delta ( f f ) / ( Su S) S = (SSd ) / ( ) Delta ( f f ) / ( S Sd ) S h h0.5su Sd Γ = [( f f ) / ( Su S) ] ( f f 0 ) / ( S Sd ) h 0 [ ] ( ) 44. t t t f f 00 Sd S Su

f f 0 = ( 4. 5) Su Sd f f 00 Θ = t t f 4 f 00 Θ = ( 4. 6) 4 t σ t f * f Vega = σ f* f t t 0. 63 0. 35 = 0. 4 6. 99 39. 69 [( 0. 63 376. ) / ( 6. 99 50. 00) ] [( 376. 0. 35) / ( 50. 00 39. 69) ] 65.. 66 4. 48 = 5. 5 0. 333 t

r q ( ) ( r q) t Se = psu + p Sd e ( r q ) t = pu + p d ( ) a = e ( r q) t ( 47. ) t a = σ t u = e = 4., d = = 08909. u a d p = = 0. 473, p = 0587. u d r f t q = r f

a = e ( 0. 08 00. ) 0. 5 = 09950. u = eσ t = 068., d = = 0. 948 u a d p = = 0. 4433, p = 05567. u d i t j i Su d j j = 0, Λ, i i t S( δ ) u j d i j j = 0, Λ, i δ i i t i t S( δi ) u d j i j τk t k t ik i k l i t j i Su d j D j = 0,, Λ, i i k

j i ( j j i j ) ( ) Su d D u Su d D d j = 0 i - i i l k m t m k k m l k t τ k S * S * ( x) = S( x) x > τ r( τ x) S * ( x) = S( x) De x τ σ * S * σ * σ * σ σ * p u d S * i t i tτ j i j r i t S * ( t ) u d De ( τ + ) j = 0, Λ, i i tτ j i j S * ( t ) u d j = 0, Λ, i i t i + S * S * 0. 97 0.. 06e =. 00 S * S * S * S *

m a = e t m m m t 4. 48 + 4. 08 4. 3 = 4. 5

f * f * f q t 0. 0. a = e =. 0 0. 0 0. u = e =. 003 d = = 0. 9968 u 0. 0. 9968 p = =. 39 003. 0. 9968 p = 39. ( S Fe r q )( = T t ) ( 48. ) r f t = 0. 5 a = 0. 04 0. 5 u = e =. 00 d = = 0. 980 u a d p = = 0. 4950 u d p = 0. 5050

( 0. 06 0. 0) 0. 79e = 07590. f rs f σ ( ) t + f S S + S = rf 49. t = T / N 0, t, t, Λ, T S max S = S max / M 0, S, S, Λ, Smax i, j i t j S f ij i, j i, j, f / S f f i, j+ f ij = ( 4. 0) S S f f ij f i, j = ( 4. ) S S

f f i, j+ fi, j = ( 4. ) S S f / S i t ( i + ) t f f = t, f i+ j ij t ( 4. 3) ( i, j) f / S ( i, j +) f, + f i j ij S i, j f / S f S f =, f f f, S S i j+ ij ij i j f f i, j+ + f i, j f ij ( ) = 4. 4 S S S = j t f f rj S f f i +,, j ij i j +, i j f i, j+ + f i, j f ij + + σ j S = rf ij t S S j =,, Λ, M, i = 0, Λ, N a j f i, j + bj f ij + c j f i, j+ = f i+, j ( 4. 5) a j = rj t σ j t b = + σ j t + r t j c j = rj t σ j t [ X S T ] max, 0 S T f Nj = max[ X j S, 0] j = 0, Λ, M ( 4. 6) S

f X i N ( ) i0 = = 0,, Λ, 4. 7 f i N ( ) im = 0 = 0,, Λ, 4. 8 S = 0, S = Smaxt = T T t i = N a j f N, j + bj f N, j + c j f N, j+ = f Nj ( 4. 9) j, Λ,M f = X f N, 0 N, M ( 4. 30) = 0 ( 4. 3) M M f N,, f N,, Λ f N, M f N, j f N, j < X j S, T t f N, j X j S T- t f, f, f, Λ, f, 0 0 03 0 M- 4. 07 + 4. 08 39. = $4. 4

Stock Price (dollars) 4. 4. Time to Maqturity(Months) 5 4 4 3 3 0 00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 95 0.0 0.0 0.0 0.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 90 0.05 0.04 0.03 0.0 0.0 0.0 0.00 0.00 0.00 0.00 0.00 85 0.09 0.07 0.05 0.03 0.0 0.0 0.0 0.00 0.00 0.00 0.00 80 0.6 0. 0.09 0.07 0.04 0.03 0.0 0.0 0.00 0.00 0.00 75 0.7 0. 0.7 0.3 0.09 0.06 0.03 0.0 0.0 0.00 0.00 70 0.47 0.39 0.3 0.5 0.8 0.3 0.08 0.04 0.0 0.00 0.00 65 0.8 0.7 0.60 0.49 0.38 0.8 0.9 0. 0.05 0.0 0.00 60.4.7. 0.95 0.78 0.6 0.45 0.30 0.6 0.05 0.00 55.43.4.05.83.6.36.09 0.8 0.5 0. 0.00 50 4.07 3.88 3.67 3.45 3.9.9.57.7.66 0.99 0.00 45 6.58 6.44 6.9 6.3 5.96 5.77 5.57 5.36 5.7 5.0 5.00 40 0.5 0.0 0.05 0.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 35 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 0 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 5 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 0 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 5 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 0 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 S t f i+, j f i, j i, j ( ) f / S f / S i +, j f f i+, j+ f i+, j = S S f f + f f = S S i+, j+ i+, j i+, j ( )

* * * fij = a j fi+, j + bj fi+, j + c j fi+, j+ ( 4. 3) * a j = rj t + σ j t + r t * bj = ( σ j t) + r t * c j = rj t + σ j t + r t f,, f f, ( i + ) t f, 4.3 i+ j i t ( f, ) i + t f,, f,, f i+ j i+ j i+, j+ i j i j ij i j+ * * * a, b, c j j j t + σ j t t j S ( j ) S σ j t t j S fj t + σ j t: t j S ( j +) S rj S t = rs t XlnS 4.9

t t σ j S t = σ S t i + t i t i t i + t rj t + σ j t σ j t rj t + σ j t σ j t Stock Price (dollars) 4. 4. Time to Maqturity(Months) 5 4 4 3 3 0 00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 95 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 90-0. 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 85 0.8-0.05 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 80-0.3 0.0 0.00 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 75 0.46 0.06 0.0 0.04 0.06 0.00 0.00 0.00 0.00 0.00 0.00 70 0.3 0.46 0.3 0.5 0.0 0.09 0.00 0.00 0.00 0.00 0.00 65 0.9 0.68 0.63 0.44 0.37 0. 0.4 0.00 0.00 0.00 0.00 60.48.37.7.0 0.8 0.65 0.4 0.7 0.00 0.00 0.00 55.59.39..99.77.50.4 0.90 0.59 0.00 0.00 50 4.6 4.08 3.89 3.68 3.44 3.8.87.53.07.56 0.00 45 6.76 6.6 6.47 6.3 6.5 5.96 5.75 5.50 5.4 5.00 5.00 40 0.8 0.0 0.3 0.06 0.0 0.00 0.00 0.00 0.00 0.00 0.00 35 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 0 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 5 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 0 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 5 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 0 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00

fij

v ( r q) S v σ t + v S S + S = rv τ = T t rτ h( τ) = e r a = σ ( r q) β = σ v = h τ g S, h ( ) ( ) g S S g a ( ) S S h g h a g β + = 0 h g S S g a ( ) S S h g A + β = 0 4. g / h r S c( S) + A S S * C( S) = S * S X S S * S *

q( T t) S * S * X = c( S *) + { e N[ d ( S *) ]} r r S p( S) + A S > S ** p( S) = S ** X S S S ** S ** q( T t) S ** X S ** = P( S * *) { e N[ d ( S **) ]} r 4a r = ( β ) ( β ) + h 4a r = ( β ) + ( β ) + h S q T t A { e N[ d ( S ) = ]} ** ( ) ** r S q T t A { e N[ d ( S ) = ]} * ( ) * r d ( S) = ln ( S / X ) + ( r q + σ / )( T t ) σ T t

4 4 64 4 = 065%. 64 00. 00 98. 00 065. = 0. 9375

Black Scholes

0.5 0 max(r - 0.,0) 0.5l0,000,0000.0 $5,000 max(r - 0.,0) R X τ, τ, Λ nτ ( k + ) ( ) ( ) k X 0 5 τl max R R,.

Rk kτ F k kτ( k + ) τ R X R X F k F k kτ( k + ) τ ( k + ) τ kτ τl max ( R ) ( ) k RX, 0 5. + τf k ( F k ) τl / + τ kτ Fk = Rk kτ( k + ) τ F k R k F k σ F τl τ [ ( ) ( )] ( ) τf e rk F N d R N d k X 5. 3 + k d d ( ) ln Fk / RX + σ Fkτ / = σ kτ F ( ) ln Fk / RX σ Fkτ / = = d σ σ kτ F kτ ( k + ) τ kτ [ ( ) ( )] ( ) k X 53. r( k+ ) τle F N d R N d Fk0. 07, 0. 5, L0,000, R = 0. 08, r = 0. 065, σ = X 00., k τ = 0. τl 0. 5 0, 000 = =, 457 + τ + 0. 5 0. 07 d d F k ln 0. 875 + 0. 0 = = 0. 5677 0. 0 = d 0. 0 = 0. 7677

0. 065 457e [ 007. N( 05677. ) 008. N( 0. 7677) ] = 59. σ F F k F k σ F F k σ F ( k ) τ k τ τl + τr R R k ( k X, 0) L( + RX τ) max L, 0 ( 5. 4) + R τ k L ( + RX τ) + τr k

k τ L l R X τ kτ kτ k τ L l R X τ R( T t) ( ) ( ) ( ) 55. c = BN d e XN d ( ) ( ) (. ) 56 R T t p = e ( ) XN d BN d d d ln ( B / X ) + ( R + σ / )( T t ) = σ T t ln ( B / X ) + ( R σ / )( T t ) = = d σ T t σ T t vt,t 0

50 0. 5 0. 09 e 50e 0. 75 + 0. 095 = 95. 45 B = 960-95.45 = 864.55,X = 000,R = 0., σ = 0.09,T- t = 0.8333 000 + 50 0. 6667 = 008. 33 B = 864.55,R = 0., σ = 0.09,T- t = 0.8333

R T t c = e ( ) FN( d ) XN( d ) 57. [ ] ( ) [ ( ) ( )] (. ) 58 R T t p = e ( ) XN d FN d ln ( F / X ) + σ ( T t ) / d = σ T t ln ( F / X ) σ ( T t) / d = = d σ T t σ T t F ( B I ) e R ( = T t ) c + Xe r T t = p + Fe ( ) r( T t) ( 59. )

Y X B T Y T BT X = D( YT YX ) X ( ) B X = DX Y Y T X T [ DX( YX YT ) 0] max, [ DX( YT YX ) 0] max,

dr = m( r) dt + s( r) dz ( 50. ) r ( T t ) E e f 5. [ ] ( ) T r r t T r E P t, T t ( ) P( t, T ) E [ e r ( T = t) ] ( 5. ) ( R t, T) t (, )( ) (, ) = ( 53. ) P t T e R t T T t R( t, T) = ln P( t, T ) ( 54. ) T t R( t, T) ln E[ e r T t ] ( ) T t = ( ) 55. ( m r) s ( r )

m( r) s ( r ) m( r ) = Mr, s( r) = Sr u = e d S = e S t t a d p = u d a = e M t t =

j i j rij = ru d, P ij t + i t r ij [ ( ) ] ( ) i+, j+ i+, j 56. rij t P = e pp + p p + c ij f ij t + i t r ij i 4 [ P j ] f 4 j = max 4 000, 0 rij t [ ij 000 ( i+ j+ ( ) i+ j )] f = max p, e pf, + p p, ij

dr = a(b - r)dt + σ dz ( 57. ) σdz P t, T = A t, T e B( t, T ) r 58. A( t, T ) = exp ( ) ( ) ( ) a( T t) e B( t, T) = ( 59. ) a ( B( t, T) T + t )( a b σ / ) σ B( t, T) a 4a ( 50. ) a = 0 B( t,t) = T - t,a(r,t) = exp[ σ ( T t ) 3 / 6] (, ) ( ) (, ) ( σ ) ( 5. ) P t s N h XP t T N h P

h σ P( t, s) σ P = ln + σ P P( t, T ) X P = v( t, T) B( T, s) a T t ( e ) ( ) ( σ ), = v t T a (, ) ( + σ ) (, ) ( ) ( 5. ) XP t T N h P t s N h ( ) ( ) a = 0, v t, T = σ T t, σ = σ s T T t P P i n s T i r * T r X r = r * s $ T i i r * n max 0, ci P( T, si ) X i= r r * r r * s i c i X i n i= [ 0 ( i ) i ] c max, P r, T, s X i B 3 35 r B 3 4 r B 3 4 5 r B 3 5 r 5A 3, 35. e (,. ) 5A 3, 4 e (, ) 5A 3, 4. 5 e (,. ) + + + 05A 3, 5 e (, ) ( ) ( ) ( ) ( ) A( t, T) B t,t

5 0. 9988e + 5 0. 995e + 5 0. 9895e + 05 0. 989e 0. 4877r 0. 956r. 399r. 87r r * r * r 0 r + 0 k r r t r = σ 3 t i t r t r j 0 (i ) t r + 0 k r (i ) t ( r + k ) r, r + k r r + ( k + ) r 0 0 0 r / + r / r / + 3 r /

dr = 0.(0.5 - r)dt + 0.0dz r 0 r = 0.0 3 0.50.0087 a( b r) t = 0. 00375 0.0 0.5.005 0 r / + r / p, p p u m d pd + p m + pu = 0. 0087 pu 0. 0087 pd = 0. 00375 0. 0087 p + ( 0. 0087) p = 0. 005 + 0. 00375 u d p = 0.043 p = 0.483p = 0.474 d m u A B C D E F G H I r% 5.00 4.3 5.00 5.87 3.6 4.3 5.00 5.87 6.74 p u 0.474 0.5 0.474 0.48 0.039 0.5 0.474 0.48 0.385 p m 0.483 0.439 0.483 0.55 0.453 0.439 0.483 0.55 0.56 p d 0.043 0.040 0.043 0.047 0.508 0.040 0.043 0.047 0.054 a( b r) t = 0. 0046 r / 3 r / pu, p m pd pd + pm + pu =

0. 074 p + 0. 0087 p = 0. 0046 u m p p u + m = + 0. 074 0. 0087 0. 005 0. 0046 p = 0. 039, p = 0. 453, p = 0. 508 u m d fg f H f I [. I +. H +. G ] e 0. 0587 0. 5 048 f 055 f 0 047 f [. L +. K +. J ] e 0. 036 0. 5 0039 f 0 453 f 0508 f dr = a( b r) dt + σ rdz r P t T A t T e B (,, t, T ) = r ( ) ( ) γ( T t) ( e ) B( t, T) = γ( T t) γ + a e + γ (, ) A t T = ( )( ) ( a+ γ )( T t)/ γe + + γ γ( T t ) ( γ a)( e ) ab / σ γ = a + σ x = r

( x ) a b dx = σ dt + σdz x 4 x r( x / ) 4abσ f(t T T T T dp t, T = r t P t, T dt + v t, T P t, T dz t 5. 3 ( ) ( ) ( ) ( ) ( ) ( ) ( )

v( t. t ) = 0 f t, T, T ( ) (,, ) f t T T (,, ) df t T T = [ P( t T )] P( t T ) ln, ln, T T [ ] ( ) (, ) v t T d ln [ P( t, T )] r( t ) = (, ) v t T d ln [ P( t, T )] r( t ) = ( ) v( t T) ( T T) dt + v t T dz t (, ) ( ) dt + v t T dz t ( ) v( t T ) v t, T, v t, T, = dt + T T (, ) ( ) 54. dz( t ) ( 55. ) T = T T T T l f t T T F t T dz( t) v t T dt T [ v( t, T) ] = v( t, T) v ( t, T) T T df t, T = v t, T v t, T dt v t, T dz t ( ) ( ) ( ) ( ) ( ) T T df t, T = v t, T v t, T dt + v t, T dz t 5. 6 ( ) ( ) ( ) ( ) ( ) ( ) T T T (, ) (, ) (, ) v t T v t t = v t τ dτ t T 5.85.9 d

T (, ) (, ) v t T = v t τ dτ t T T ( ) ( ) ( ) m t, T = s t, T s t, τ dτ ( 5. 7) t t (, ) = ( 0, ) + ( τ, ) F t t F t df T t t r t = F 0, t + v τ, t vt τ, t dτ + vt τ, t dz τ 58. ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 { [ τ τ τ ] τ} 0 t + v ( t) dz( ) dt [ v t ] dz( t ) tt τ τ + t τ τ t t ( ) = t ( 0, ) + (, ) tt (, ) + t (, ) dr t F t dt v t v t v t d dt { } ( ) 0 = 0 0,, ( 5. 9) ( ) v τ, t = t t τ t τt v( τ t) r

dr = θ( t ) dt + σdz θ t = F 0, t + σ t ( ) ( ) t P t T A t T e r (,, T t ) = ( ) ( )

P ln P( 0, t) ln A( t, T) = ln ( T t ) σ t( T t ) P t ( 0, T) ( 0, t) σ σ( ) P = s T T t dr = θ( t ) ar dt + σdz 530. ( ) ( ) P t T A t T e B t,, =, T r ( ) ( ) ( ) a( T t) e B( t, T) = a P( 0, T) ln P( 0, t) ln A( t, T) = ln B( t, T ) P( 0, t) t at at at 3 σ ( e e ) ( e ) 4a

σ [ e a( T t ) ] a σ ( ) [ e a( T t ) ] a T t σe a( T r) r + 0 j r r 0 i t r = σ 3 t t i t r r0 j r i i j R i i t rjr 0 + j r, i j r µ i j n t n 0 R i i t i t ( i + ) t n t i n R i n t n t θ( n t) t σ t t θ( n t ) ( n ) R( n ) Q( n j) e r t ar t j + j = + + + + ( 5 3) ln,. j Q i j i j Q i j

rj* t (, ) = (, *) ( *, ) ( 53. ) Q i j Q i j q j j e j* q j* j i j* i j j* θ( n t) n t, µ n j µ, = θ( ) ( + ) (. ) n j n t a r0 j r 5 33 n t, µ n j θ 0 Q 0 0 r0 0. θ( 0) 0.00 Q Q 0 d lnr = θ( t ) aln r dt + σdz 534. [ ] ( )

c A ce r T t max S- X - A maxs- X- A- A (B (B 0 30

t t t S t cs / S, c t t e E c S r ( t t ) S [ ] E ( ) E S = Se r t t q E ( r q)( t t) S, Se q( t t) ce ο T X X T qt e rt rt Se M( a, b ; T / T ) X M( a, b ; T / T ) e X N( a )

a b ln( S / S*) + ( r q + σ / ) T = ; σ T ln( S / X ) + ( r q + σ / ) T = ; σ T a = a σ T b = b σ T S* T X T S * S * rt qt rt e X M( a, b ; T / T ) Se M( a, b ; T / T ) + e X N( a ) rt qt rt e X M( a, b ; T / T ) Se M( a, b ; T / T ) e X N( a ) qt Se M( a, b ; T / T ) X M( a, b ; T / T ) + e X N( a ) e rt rt t maxc, p S t l t r( t t ) q( t t ) q( t t ) ( r q)( t t ) max( c, p) = max( c, c + Xe S e ) = c + e max( 0, Xe S ) t q( t t ) ( r q)( t t) e Xe t

qt λ rt λ Xe ( H / S) N( y) Xe ( H / S) N( y σ T) rt λ qt λ Xe ( H / S) N( y + σ T) Se ( H / S) N( y) r σ λ = rf + σ [ H SX ] ln / ( ) y σ T / + λσ T N d

-r(t-t) Qe N d ( ) SN d S S S T max 0 ST S ST -S max 0 S - ST S - S T Se qt N a Se qt r q N a S e rt σ N a σ r q e Y N a ( ) ( ) min ( ) ( 3 ) ( ) ( ) S min a a = a σ T a Y 3 ln( S / Smin) + ( r q + σ / ) T = σ T ln( S / Smin) + ( r + q + σ / ) T = σ T ( r q σ / )ln( S / Smin) = σ S max e rt N b r q e Y N b σ ( ) ( ) Se qt r q N ( b ) Se qt ( ) ( ) N ( b ) + σ 3 S max b b = b σ T b Y 3 ln( Smax/ S) + ( r + q + σ / ) T = σ T ln( Smax/ S) + ( r q σ / ) T = σ ( r q σ / ) ln( Smax/ S) = σ

max, max, ( 0 S ave X ) ( 0 X S ave ) S ave max, max, ( 0 S S ave ) ( 0 S ave S) S ave r - q - σ / 6 / σ / 3 σ / 3 σ σ r ( r q ) = ( r + q + ) 6 6 M e ( r q) T = ( r q) T

M [ σ ] ( r q) + T ( r q) T e e = + ( r q + σ )( r q + σ ) T ( r q) T ( r q) + σ r q + σ SM S M σ (r-q )T e q A [ ( r qa) + σa] A = M ; e = M ln M ln M q A = r ; σa = ( r q A ) T T S S Sl S σl σ Sl S ρ, S S q q l e q ( T t ) e q ( T t ) S N( d ) S N( d ) d log( S / S) + ( q q + σ / )( T t) = σ T t d = d σ T t σ = σ + σ ρσ σ σ S S S S S q q S S S q q T A

min S S S maxs S 0 max S S S + maxs - S 0

0. 0. 08333 ( 0 05076. + 545. 0. 494) e =. 66 0. 0 0. 08333 ( 0 05076. + 57. 0. 409) e = 565.

n 3 / 6 Cea + Cag Ceg C ea C ag C eg C ag C C ea eg S S l ds = rs dt + σ S dz ds = rs dt + σ S dz dz dz ρ d ln S = ( r σ / ) dt + σ dz d ln S = ( r σ / ) dt + σ dz x = σ lns + σ lns x = σ lns + σ lns

[ ] [ ] dx = σ ( r σ / ) + σ ( r σ / ) dt + σ σ ( + ρ) dz dx = σ ( r σ / ) σ ( r σ / ) dt + σ σ ( ρ) dz dz dz A B t x h p i i i - p i h i p i x x t x x p p x h x h l ( p ) ( ) ( )( p ) p x h x h l - p p x h x h l - p x h x h l S S x x S S x + x = exp σ x x = exp σ A B h i

σ σ db = υ Bdt + σb dzb b µ B σ B dz B

d c = SN(d ) - BXN(d ) p = BXN(-d ) -SN(-d ) n( S / X) nb = ( σ / )( T t) = σ T t d = d σ T t t T ( T t ) = ( + B ) dt ( ) B 7. σ σ σ ρσσ ρ B( t) e R ( T t ) = σ (0 B(( 7.

p S c Xe r ( + = + T t ) 7. 7. 7.(a) 7.(b) 7.(c) 7.(d) Black-Scholes Black-Scholes Black-Scholes Black-Scholes

c( V ) g( V ) dv V σ V T * T * T * T * T * σ v r( T* t) S = VN ( d) Ae N( d) ( 7. )

d n( V / A) + ( r + σ v / )( T * t) = σv T * t d = d σv T * t α = α( + β) S = S A + SB S A S B S A S A σs a

λ τ Su s -λ τ Se -w t z dq dz dq

τ τ rτ rτ c = VM a, b; Ae M a, b ; Xe N( a ) τ τ

a b ln( V / V*) + ( r + σv) τ = σv τ ln( V / A) + ( r + σv) τ = σv τ a = a σv b = b σv τ τ = T t = T * t τ τ r( T t) c = asn( d ) ( X bs) e N( d ) d [ ] ln as / ( X bs) + ( r σr / )( T t) = σ T t d = d σ T t a = a( + β) b = ( a) e R r( T t) R c = (S- Xe -r(t-t) [ ] )N(y ) + (S - Xe -r(t-t) ) N( y ) υ n( y ) n( y ) υ = σ e r = S Xe y υ = S Xe y υ n( y) = e π r ( T t ) r( T t) r ( T t) y /

c S x y Xe x y r( T t) = Ψ( ; ) Ψ(, ) u β β Ψ( α; β) = e i! i= a i ( r + ω)( T t) u y = u ln( X / S) + ω( T t) ln u n e c = λ ' τ ( λ' τ) fn n= 0 n! δ σ + n τ nγ r λk + τ

B f = f * B* ( f f e y = y *)( T * t ) 0 05 3e. =. 9

max(, ) B B 0

max( B B S, 0) D F B D BF max S, 0 B F T * ( yτ yτ )( τ t) f * f = υ( τ) e a( t, τ) dτ t ) υ( τ) = E e max(, ) r ( τ t ) * [ f τ 0 ] 8.

> 4% > 8%

8. <yr nil nil.0% >yr 0.5% nil 5.0% f * f