Black Scholes
I...1...1...3...4...6...6...19...28...28...29...63...63...68...70...72...74...78...82...92...92...93...95...98
-1...5-1...9-2...9-3...9-4...12-1...53-1...79-2...79-3...81-4...84-5...85-6...86-7...90-8...90 II
-1...15-2...16-3...18-4...18-5...19-1...33-2...35-3...36-4...38-5...39-6...40-7...41-8...42-9...44-10...45-11...47-12...47-13...52-14...54-15...56-16 5...57-17 4...58-18 4...58-19 4...59-20 3...60-21 3...60-22 3...61-23...61 III
-1...64-2...69-3...73-4 92.8.16...74-5 93.8.16...75-6...76-7...78-8...81-9...83-10...84-11...86-12...88-13...89 IV
1 2 1 1999 tow dimension tree model 1999 2 91 5 22 22 30 1
Least-Square Monte Carlo simulation 2
1. 2. 3. 4. 5. 3
-1 91 8 16 4
-1 5
1. 2. 6
1. 3 2. 3. 3 Tej 7
89 6 1. 2001-1 -2-3 8
( ) ( ) -1 ( ) -2-3 9
1 2 3 1 2 ( ) 1 2 (Call on Call) (Call on Convertible bonds) 10
max{ CB (r, T, X,S)-X CB 0} CB ( r,t, X, S) S X r CB X (100 ) (Two-Factor Model) 3 ( ) 11
2. 91 6 3 89 6-4 -4 12
91 12 4 = ( 1 + + ) ^ ( ) 4 13
1. 2. 1 3 91 13 6 5 2 5 92 3 11 14
-1-1 3. 1 6 10 6-2 6 92 3 11 15
-2 100 70 80 2 110 15 110 6 10 7 4. 7 92 3 11 16
1 2 3 4 1. -3 17
-3 TCRI /( ) 85062 1 100 100 2005/8/16 2.87 65018 4 100 100 2004/6/14 1.70 53262 7 100 100 2004/4/25 1.57 TCRI TEJ 2. 1-4 -4 20 40 105 17.71 31.88 16.18 109.27 13.84 28 12.61 110 13.52 27.69 12 18
2-5 -5 15 2002/12/26 2003/06/26 2003/12/26 2004/06/26 2002/06/26 98 100.69 103.39 106.09 Black Scholes 19
1. (Analytical Model) Black-Scholes Black and Sholes 1973 Black-Sholes Ingersoll 1977a Merton 1974 Contingent Claim Analysis CCA Black Sholes CCA CCA Black Scholes Ingersoll 1977a 1992 1996 Black Scholes 20
2. 1 Longstaff and Schwartz 2001 Least-Square Least-Square Monte Carlo simulation,lsm 2 Brennan and Schwartz 1977 21
Brennan and Schwartz 1980 McConnell and Schwartz 1986 LYON McConnell and Schwartz Peter Carayannopoulos 1996 Brennan and Schwartz 1980 Brennan and Schwartz 1980 Peter Carayannopoulos Cox,Ingersoll and Ross 1985 Hull and White 1990 1992 Merton 1974 1992 22
1994 1995 Cox,Ingersoll and Ross 1985 Hull and White 1990 1996 1998 Hull and White 1990 Hull and White 1994 Extended Vasicek Model 23
1999 Brennan and Schwartz 1977 King 1986 5 3 1997 Cox.Ross.Rubinstein 1997 Vasicek 1997 Hull and White 1998 24
1999 Cox.Ingersoll and Ross 1985 1999 Option Adjusted Spread OAS OAS OAS 1999 24.86 2000 Hull and White 25
3. 1994 1995 1. 2. 1994 3. 4. 5. 1994 2002 Least-Square Monte Carlo simulation 26
2002 Vasicek Longstuff Schwartz 2001 ( ) ( ) 27
5-10 1997 Brennan and Schwartz 1980 28
1. 38.4 2. 40 3. 35 4. 5 5. 6. 106.61 7. 2 8. 2 9. 29
Kevin B.Connolly Pricing Convertible Bonds 6.4.4 u 2 S us S uds ds d 2 S C uu ( u S ) = max 2 k,0 C u C Cud = max( uds k,0) C d Cdd = max( d 2 S k,0) 1 u d p u = e σ t d = e σ t r t e d p = u d 2 u d p 30
3 max 0 4 roll back node max CB ( CV Bond ) uu = max uu, CB u CB Cud = max ( CV, ud Bond ) CB d Cdd = max( CV dd, Bond) 1 2 max 3 roll back node 31
max,,min 4 coc ( CV Bond ) uu = max uu, coc u call on CB coc ( CV Bond ) ud = max ud, coc d coc dd = max( CV dd, Bond) max,0 roll back node max 32
1. max max -1-1 Time t=0 t=1 t=2 t=3 t=4 t=5 CB Price 116.493 147.406 197.064 274.335 389.299 552.442 116.493 147.406 197.064 274.335 389.299 98.861 115.648 144.326 193.32 274.335 98.861 115.648 144.326 193.32 91.961 99.643 112.442 136.23 91.961 99.643 112.442 92.312 96.079 100 92.312 96.079 96.079 100 96.079 100-1 33
116.493 1 5 2 bond price par value/exp Risk dt Time t=0 t=1 t=2 t=3 t=4 t=5 Bond Price 81.873 85.214 88.692 92.312 96.079 100.000 1 38.4 2 40 3 5 4 max( 0) max -2 34
-2 Time t=0 t=1 t=2 t=3 t=4 t=5 Stock Option 32.441 59.059 104.447 178.256 291.279 452.442 32.441 59.059 104.447 178.256 291.279 12.559 25.293 49.841 95.3 174.335 12.559 25.293 49.841 95.3 2.94 6.79 15.685 36.23 2.94 6.79 15.685 0 0 0 0 0 0 0 0 0-2 -3 35
-3 t=0 t=1 t=2 t=3 t=4 t=5 2.180 3.133 3.925 3.767 1.941 0.000 1.089 1.663 2.174 1.941 0.000 0.330 0.541 0.678 0.000 0.000 0.000 0.000 0.000 0.000 0.000 36
1 38.4 2 3 5 4 max 0-4 37
-4 Time t=0 t=1 t=2 t=3 t=4 t=5 Stock Option 33.005 60.163 106.537 180.158 291.279 452.442 33.005 60.163 106.537 182.023 293.22 12.716 25.657 50.681 95.3 174.335 12.716 25.657 50.681 97.241 2.94 6.79 15.685 36.23 2.94 6.79 15.685 0 0 0 0 0 0 0 0 0 3 4-5 38
-5 t=0 t=1 t=2 t=3 t=4 t=5 1.616 2.029 1.834 0.000 0.000 0.000 0.931 1.299 1.333 0.000 0.000 0.330 0.541 0.678 0.000 0.000 0.000 0.000 0.000 0.000 0.000-5 39
1 2 3 5 4 max 0 max -6-6 Time t=0 t=1 t=2 t=3 t=4 t=5 Call on CB 34.393 61.668 107.267 180.158 291.279 452.442 34.62 62.192 108.372 182.023 293.22 13.459 26.53 51.052 95.3 174.335 13.647 26.956 52.015 97.241 3.174 7.084 15.685 36.23 3.269 7.331 16.363 0 0 0 0 0 0 0 0 0 40
-6-7 -7 t=0 t=1 t=2 t=3 t=4 t=5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-7 41
2. 106.61 max max -8-8 Time t=0 t=1 t=2 t=3 t=4 t=5 CB price 120.762 147.406 197.064 274.335 389.299 552.442 120.762 147.406 197.064 274.335 389.299 106.756 115.648 144.326 193.32 274.335 106.756 115.648 144.326 193.32 91.961 99.643 112.442 136.23 106.61 99.643 112.442 92.312 96.079 100 92.312 96.079 96.079 100 96.079-8 2 100 42
120.762 1 5 2 3 2 106.61 bond price put price/exp Risk dt Time t=0 t=1 t=2 t=3 t=4 t=5 Bond Price 98.413 102.430 106.610 92.312 96.079 100.000 1 2 3 5 4-9 43
-9 Time t=0 t=1 t=2 t=3 t=4 t=5 Call on CB 33.626 60.957 107.267 180.158 291.279 452.442 33.626 60.957 107.267 182.023 293.22 13.222 26.53 51.052 95.3 174.335 13.222 26.53 52.015 97.241 3.174 7.084 15.685 36.23 3.174 7.331 16.363 0 0 0 0 0 0 0-9 0 0-10 44
-10 t=0 t=1 t=2 t=3 t=4 t=5-11.277-15.981-16.813 0.000 0.000 0.000-8.896-17.492 0.000 0.000 0.000-3.174 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 max max 45
2 3 3 92.312 98.413 92.312 1 2 3 2 4-11 46
-11 Time t=0 t=1 t=2 Call on CB 21.839 44.105 90.454 22.348 44.976 3.913 9.038 4.326 0-11 -12-12 t=0 t=1 t=2 0.000 0.000 0.000 0.000 0.000 0.000 47
max max 48
compound option Boyle 1977 49
1 S t = S t 1 e 2 ( r 0.5σ ) t+ σε t S t 1 S t t 2 3 1 2 4 2 3 2 2 5 1 10 John C.Hull Options,Futures,AND OtherDerivatives Antithetic Method CB 1 1, 2 12 CB 2-1,- 2-12 CB 1 CB 2 CB + 1 CB 2 2 CB Var CB Var 1 C + 2 C 2 50
1 4 2 2 2 ( + + 2 ρ ) σ σ σ CB 1 CB 2 CB 1 CB 2 Var CB σ 2-10 N 10 N 100 122.578 3.901 1 120.151 0.714 500 121.991 3.069 3/4 1 119.519 0.623 6/7-17 51
-13 N=100 N=500 N=1000 N=5000 N=10000 122.578 119.617 119.908 120.386 120.151 3.901 2.827 1.071 0.893 0.714 121.991 118.687 119.190 119.722 119.591 3.069 2.728 1.059 0.828 0.623 119.591 2 1-1 52
-1 3.1 53
1-18 -14 T 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 CB 119.86 121.12 122.93 124.65 127.48 127.98 128.89 130.29 131.49 132.86 bond price 98.41 100.40 102.43 104.50 106.61 90.48 92.31 94.18 96.08 98.02 stock option 31.38 31.17 31.43 31.88 32.74 32.86 32.81 33.24 33.47 33.86 Bond +stock option 129.79 131.57 133.86 136.38 139.35 123.34 125.12 127.42 129.55 131.88 CB-bond -stock option -9.93-10.46-10.93-11.73-11.87 4.64 3.77 2.87 1.94 0.99 call on CB 31.96 31.69 31.96 32.09 33.31 32.86 32.81 33.24 33.47 33.86 Bond +call on CB 130.37 132.09 134.39 136.59 139.92 123.34 125.12 127.42 129.55 131.88 CB-bond -call on CB -10.51-10.98-11.46-11.93-12.43 4.64 3.77 2.87 1.94 0.99 call on CB 21.63 21.07 20.95 20.82 21.69 Bond +call on CB 120.04 121.47 123.38 125.32 128.30 CB-bond -call on CB -0.18-0.36-0.45-0.67-0.81 54
120.762 32.441 33.626 22.348 55
Least-Square Monte Carlo Simulation Longstaff and Schwartz 2001 Least-Square Monte Carlo Simulation Least-Square Monte Carlo Simulation M backward approach 8-19 -15 path t=0 t=1 t=2 t=3 t=4 t=5 1 38.4 25.2 21.3 29.1 30.0 29.4 2 38.4 48.9 51.8 43.8 54.0 52.8 3 38.4 64.8 63.9 22.7 16.8 10.3 4 38.4 38.7 57.8 78.0 90.3 42.6 5 38.4 30.2 34.2 35.6 49.2 16.5 6 38.4 20.1 22.7 27.2 32.1 43.4 7 38.4 47.8 41.8 51.1 46.1 39.6 8 38.4 25.7 29.5 29.7 45.6 6.3 56
backward approach max(, ) -20-16 5 path t=1 t=2 t=3 t=4 t=5 1 100.0 2 131.9 3 100.0 4 106.6 5 100.0 6 108.5 7 100.0 8 100.0 5 4 5 4 path2 path4 path5 path7 path8 5 4 Y=a+bX+cX 2 y 5 4 X time4 E[Y/X] -295.84 12.76X 0.092X 2-21 57
-17 4 path y x x^2 1 2 129.3 54.0 2917.6 3 4 104.5 90.3 8151.3 5 98.0 49.2 2418.8 6 7 98.0 46.1 2124.0 8 98.0 45.6 2076.8 4 4 4 4 4-22 4 max, -18 4 path conversion continue CB at time4 1 2 135.04 124.28 135.04 3 4 225.71 104.62 225.71 5 122.95 108.59 122.95 6 7 115.22 96.29 115.22 8 113.93 94.07 113.93 58
-22 path2 path4 path5 path7 path8 4 5 4 path1 path3 path6 CB 4 5 5 4 4-23 -19 4 path t=1 t=2 t=3 t=4 t=5 1 0 100 2 135.04 0 3 0 100 4 225.71 0 5 122.95 0 6 0 108.50 7 115.22 0 8 113.93 0 3 3 3 3 E[Y/X] 687.75-21.266X+0.196X 2 3 3 3-24 -25 59
-20 3 path y x x^2 1 2 132.36 43.80 1918.59 3 4 221.24 77.95 6076.85 5 6 7 112.94 51.10 2611.06 8-21 3 path conversion continue CB at time 1 2 109.50 132.36 132.36 3 4 194.89 221.24 221.24 5 6 7 127.75 112.94 127.75 8 3 path2 path4 path7 3-26 60
-22 3 path t=1 t=2 t=3 t=4 t=5 1 0 0 100 2 0 135.04 0 3 0 0 100 4 0 225.71 0 5 0 122.95 0 6 0 0 108.50 7 127.75 0 0 8 0 113.93 0-27 -23 path t=1 t=2 t=3 t=4 t=5 1 0 106.61 0 0 0 2 0 0 0 135.04 0 3 161.99 0 0 0 0 4 0 0 0 225.71 0 5 0 0 0 122.95 0 6 0 106.61 0 0 0 7 0 0 127.75 0 0 8 0 0 0 113.93 0-27 0 61
128.45 max 0 Least-Square Monte Carlo Simulation 3 Y=aX+bX 2 OLS a b 8 Least-Square Monte Carlo Simulation 8 Least-Square Monte Carlo Simulation Least-Square Monte Carlo Simulation Least-Square Monte Carlo Simulation 62
Tej -1 63
-1 91 8 16 96 8 15 100 0.0 NT 41.2 3 1 30 50 2 10% 94 8 16 NT 109.59 1. 64
8 2. 100 3. 10 8 65
1. 91 8 15 38.8 2. 41.2 3. 9 1999 91 8 2.11 4. 91 12 3.78 1.67 5. Cox Robinstein 1985 S j R i = j=1,2,,n S j 1 9 66
^ u ^ 2 = 1 n ln ( ) s R k n k = 1 1 σ = s n ^ ( ) = ln Rk u s n k 1 ^ 2 ^ σ 2 s = 1 1 n k 1 ^ n ^ ( ) = ln R k u s 2 89 9 19 89 9 19 91 8 15 40.89 128.66 100 67
10 91 8 16-2 100 10 68
-2 91 8 16 91 8 20 94 8 15 0~1 ( ) 100~103.10 6.0% 1~2 ( ) 103.1~106.30 3.0% 2~3 ( ) 106.3~109.59 100 9.5% 94 8 15 38.8 128.66 40.51 40.51 43.13 9.5% 100 128.66 69
10 100 13 13.5 9.5% 91 8 16 92 2 15 6.12 100 7.77 70
94 8 16 3-4 11 1. 11 71
2. 3. 4. 72
-3 128.66 100 43.13 9.5 3.78 / 97.84 3-4 / 97.43-3 43.13 43.13 43.13 9.5 73
-4-5 -4 92.8.16 159.95 99.75 62.21 38.79 24.19 15.09 9.41 (1) 388.24 242.13 151 132.8 111.34 103.58 101.88 (2) 101.61 101.61 101.61 101.61 101.61 101.61 101.61 (3) 379.19 200.59 92.42 34.53 9.74 1.97 0.27 (1)-(2)-(3) -92.56-60.07-43.03-3.34-0.01 0 0 (4) 379.83 201.83 93.49 35.96 14.34 6.58 4.88 (1)-(2)-(4) -93.2-61.31-44.1-4.77-4.61-4.61-4.61 74
-5 93.8.16 411.25 202.55 99.76 49.13 24.19 11.91 5.87 (1) 998.18 491.61 302.65 153.85 110.55 105.64 105.53 (2) 105.52 105.52 105.52 105.52 105.52 105.52 105.52 (3) 1072.77 507.63 199.13 50.79 5.03 0.12 0.01 (1)-(2)-(3) -180.11-121.54-2 -2.46 0 0 0 (4) 1072.60 507.14 199.41 50.63 7.30 2.39 2.28 (1)-(2)-(4) -179.94-121.05-2.28-2.27-2.27-2.27-2.27-4 -5 140.97 75
1-6 -6 76
t 0 0.5 1 1.5 2 2.5 (1) 127.13 128.12 131.03 132.41 135.29 136.87 (2) 97.84 99.71 101.61 103.55 105.52 107.54 (3) 33.69 33.07 34.33 31.03 32.19 29.05 (2)+(3) 131.53 132.78 135.94 134.58 137.71 136.59 (1)-(2)-(3) -4.4-4.66-4.91-2.17-2.42 0.28 t=0 127.13 33.69-6 77
91 9 28 31.8 91 11 15 29.1-7 -1-2 -7 910821 910916 911016 911115 911216 38.1 37.6 30.2 29.2 28 41.2 41.2 31.8 29.1 29.1 CB 102.95 105.05 103 103.8 104.1 CB 126.46 125.87 127.19 128.48 125.76 0.2284 0.1982 0.2349 0.2378 0.2081 920116 920217 920317 27.5 26.9 24.5 28.642 29.1 29.1 29.1 32.463 CB 106.1 105.8 105 104.496 CB 125.63 125.04 121.6 125.75 0.1841 0.1819 0.1581 0.2039 78
-1-2 CB CB 20.39 CB 91/9/15 91/12/16 79
3-7 -2 91 11 16-8 12-3 12 91 5 22 37 80
-8 910821 910916 911016 911115 911216 (1) 102.95 105.05 103 103.8 104.1 (2) 97.86 98.15 98.46 98.77 99.08 40.3 39.1 43.09 46.32 42.3 (3) (1)-(2)-(3) -35.21-32.2-38.55-41.29-37.28 920116 920217 920317 (1) 106.1 105.8 105 104.496 (2) 99.39 99.71 100.02 98.93 41.08 39.08 32.74 40.50 (3) (1)-(2)-(3) -35.1-33.51-28.11-34.95-3 81
-3 implied credit spread 40.89-9 82
40.8 23.71-9 91.5.16 91.2.16 90.11.16 90.8.16 91.8.15 91.5.15 91.2.15 90.11.15 23.71 24.03 39.10 36.27 100 11 1.67 10 90 100 5.89 100 5.89 23.71 10 5.89-10 -4-5 83
-10 910821 910916 911016 911115 911216 38.1 37.6 30.20 29.20 28.00 41.20 41.20 31.80 29.10 29.10 CB 102.95 105.05 103.00 103.80 104.10 CB 99.69 99.77 103.99 105.79 104.36-0.032-0.050 0.010 0.019 0.002 920116 920217 920317 27.50 26.90 24.50 28.642 29.10 29.10 29.10 32.46 CB 106.10 105.80 105.00 104.48 CB 103.87 103.63 102.04 102.89-0.021-0.021-0.028-0.015-4 84
-5-10 -5 0.2039-0.015-5 -11-6 85
-11 910821 910916 911016 911115 911216 (1) 102.95 105.05 103 103.8 104.1 (2) 86.21 86.8 87.4 87.95 88.54 9.45 8.81 18.8 22.37 19.1 (3) (1)-(2)-(3) 7.29 9.44-3.2-6.52-3.54 920116 920217 920317 (1) 106.1 105.8 105 104.475 (2) 89.13 89.72 90.32 88.259 (3) 17.64 15.85 10.76 15.348 (1)-(2)-(3) -0.67 0.23 3.92 0.869-6 86
8-6 1 13-12 -12 0.2039 0.168 13 87
-12 910821 910916 911016 911115 911216 38.1 37.6 30.2 29.2 28 41.2 41.2 31.8 29.1 29.1 0.021 0.021 0.021 0.0185 0.0185 CB 102.95 105.05 103 103.8 104.1 CB 120.61 120.31 127.19 128.48 125.76 CB 118.13 117.84 124.57 126.15 123.48 0.15 0.12 0.21 0.22 0.19 920116 920217 920317 27.5 26.9 24.5 28.64 29.1 29.1 29.1 32.463 0.0185 0.0155 0.0155 0.019 CB 106.1 105.8 105 104.475 CB 125.63 125.04 121.6 124.328 CB 123.35 123.13 119.74 122.048 0.16 0.16 0.14 0.168 2 88
0.2866-13 -7-8 -13 910821 910916 911016 911115 911216 38.1 37.6 30.2 29.2 28 41.2 41.2 31.8 29.1 29.1 CB 102.95 105.05 103 103.8 104.1 CB 920116 920217 920317 27.5 26.9 24.5 28.642 29.1 29.1 29.1 32.463 CB 106.1 105.8 105 104.496 CB 0.001 0.0001-0.018 0.005 89
-7-8 90
-8 91
Least-Square Monte Carlo simulation 1. 1 2 1 2 2. 1 2 1 2 4. 92
5. 6. 93
ECB 94
95
Brenna,M.J.,and E.S.Schwartz, 1977 Convertible bonds : Valuation and optimal strategies for call and conversion, Journal of Finance,vol.32,351-367. Brenna,M.J.,and E.S.Schwartz, 1980 Analyzing convertible bonds, Journal of Finance and Quantitative Analysis, Vol.15, 907-929. Cox,J.C.,S.A.Ross and M.Rubinstein, 1985 Option Pricing A Simplified of Interest Rates. Econometrica,Vol.53,No.2,March,385-407. Hull,J., and A. White, 1990a Valuation Derivatives Securities Using the Explicit Finite Difference Method Journal of Financial and Quantitative Analysis,May,87-100. Ingersoll, J.E., 1977An Contingent Claims Valuation of Convertible Securities, Journal of Financial Economics, Vol.4, 289-322. John C.Hull, Options, Futures, and Other Derivatives,Fourth Wdition,1999 96
Kevin B.Connolly, Pricing Convertible Bonds,1998 King, R., 1986 Convertible Bond Valution An Empirical Test, Journal of Financial Research, Vol.9, No.1, 53-69. Longstaff, F. A.,and E.S. Schwartz (2001): Valuing American Options by Simulation: A Simple Least-Square Approach, The Review of Financial Studies, Vol. 14(1),113-147. Longstaff, F. A.,and E.S. Schwartz, (1995) A Simple Approach to Valuing Risky Fixed and Floating Rate Debt, Journal of Finance, Vol. 50(3), 789-819. McConnell,J.J., and E.S.Schwartz, 1986 LYON Taming,Journal of Finance 41,561-577. Merton, R. C., 1974 On the pricing of corporate debt The risk structure of interest rates, Journal of Finance29,449-470. Peter Carayannopoulos, 1996Valuing Convertible Bonds Under the Assumption of Stochastic Interest Rate An Empirical Investigation, Quarterly Journal of Business and Economics, Vol.35, No.3, 17-31. R.Geske, 1979 The Valuation of Compound Option, Journal of Financial Economics7,63-81. 97
91 8 16 60 10 5 91 8 16 96 8 15 0% ( 117.63% 3.3%) 3 10 100% 98
41.2 ( ) 1 ( ) 1. 2. 1 99
2 3 4 1 1. + 2. + 1 2 3 100
( ) 1. : (1) ( ) (2) 80% 2. 1 1. 80% (1) 82.96% (2) 77.29% 101
3. 2 (1)(2) ( ) 3 40 30 50 ( ) 30 3 40 10% ( 109.59% 3.1 ) 102