Copy Of Stock Price Random Processes

Wiener Process Source: Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. P. 220

Wiener Process

Property 1 where ε is a random drawing from a standardized normal distribution, Φ (0,1)

Δz=ε  Δt

Property 2 The values of ∆ z for any two different short intervals of time ∆ t are independent.

Generalized Wiener Process

Δx=aΔtbε  Δt

The values of ∆ x has a normal distribution with - mean of a∆ T, - variance of b2∆ T Drift rate is a: average drift per unit of time Variance rate is b2: variance per unit of time

Numerical example (press F9 to generate new random numbers) Time (T) 1 year Drift rate (a) 4 per year Variance rate (b) 2 per year ∆T 0.0033

Wiener Process 7.00 6.00 5.00 4.00 3.00 2.00

Column N

1.00 0.00 -1.00 -2.00 0 1 2 Time

Column P

Column Q

Row 6

Column O

Column R

Column S

ntice-Hall. P. 220 Random Numbers Period

normal distribution, Φ (0,1)

s of time ∆ t are independent.

mn O

Column R

Column S

Time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Uniform 0 0 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.06 0.06 0.07 0.07 0.07 0.08 0.08 0.08 0.09 0.09 0.09 0.1 0.1 0.1 0.11 0.11 0.11 0.12 0.12 0.12 0.13 0.13 0.13 0.14 0.14 0.14 0.15 0.15 0.15 0.16

0.46 0.07 0.83 0.39 0.31 0.62 0.27 0.78 0.53 0.58 0.94 0.81 0.41 0.95 0.91 0.18 0.81 0.84 0.88 0.15 0.85 0.23 0.75 0.58 0.64 0.43 0.43 0.47 0.62 0.94 0.74 0.08 0.01 0.56 0.47 0.32 0.18 0.74 0.1 0.7 0.32 0.04 0.51 0.73 0.99 0.42 0.91

Normal 0.04 1.74 -2.02 1.51 0.39 -1.9 -1.38 0.75 -2.5 0.09 -0.85 -0.52 -0.56 0.21 1.57 0.33 -0.24 -0.95 -0.89 -0.4 -0.94 0.44 0.34 0.25 1.01 -1.06 -1.37 -0.66 2.02 -0.99 0.56 -0.02 -1.18 0.63 -0.18 0.75 0.72 0.03 0.11 0.74 -1.65 0.65 -1.48 -0.42 -0.46 -2.26 1.84

Z + ∆Z 0 0 0.1 -0.01 0.07 0.1 -0.01 -0.09 -0.05 -0.19 -0.19 -0.24 -0.27 -0.3 -0.29 -0.2 -0.18 -0.19 -0.25 -0.3 -0.32 -0.38 -0.35 -0.33 -0.32 -0.26 -0.32 -0.4 -0.44 -0.32 -0.38 -0.35 -0.35 -0.41 -0.38 -0.39 -0.35 -0.3 -0.3 -0.3 -0.25 -0.35 -0.31 -0.4 -0.42 -0.45 -0.58 -0.47

X + ∆X 0 0.02 0.23 0.01 0.2 0.26 0.05 -0.09 0.01 -0.27 -0.24 -0.33 -0.38 -0.43 -0.39 -0.19 -0.14 -0.16 -0.25 -0.34 -0.38 -0.47 -0.41 -0.35 -0.31 -0.18 -0.29 -0.44 -0.5 -0.25 -0.36 -0.28 -0.27 -0.39 -0.3 -0.31 -0.21 -0.11 -0.1 -0.07 0.03 -0.15 -0.06 -0.22 -0.25 -0.29 -0.54 -0.32

T0.5 (Stdev Z) 0 0.06 0.08 0.1 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.22 0.23 0.24 0.24 0.25 0.26 0.26 0.27 0.28 0.28 0.29 0.29 0.3 0.31 0.31 0.32 0.32 0.33 0.33 0.34 0.34 0.35 0.35 0.36 0.36 0.37 0.37 0.37 0.38 0.38 0.39 0.39 0.4

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101

0.16 0.16 0.17 0.17 0.17 0.18 0.18 0.18 0.19 0.19 0.19 0.2 0.2 0.2 0.21 0.21 0.21 0.22 0.22 0.22 0.23 0.23 0.23 0.24 0.24 0.24 0.25 0.25 0.25 0.26 0.26 0.26 0.27 0.27 0.27 0.28 0.28 0.28 0.29 0.29 0.29 0.3 0.3 0.3 0.31 0.31 0.31 0.32 0.32 0.32 0.33 0.33 0.33 0.34

0.81 0.26 0.79 0.96 0.1 0.02 0.71 0.68 0.65 0.14 0.11 0.12 0.76 0.06 0.86 0.84 0.07 0.42 0.31 0.39 0.6 0.05 0.49 0.3 0.37 0.53 0.81 0.1 0.52 0.23 0.01 0.33 0.49 0.8 0.29 0.59 0.82 1 0.27 0.47 0.13 0.38 0.59 0.89 0.44 0.45 0.73 0.5 0.87 0.05 0.89 0.46 0.1 0.38

0.61 -0.91 -1.03 -1.45 -0.37 0.93 0.55 1.7 0.43 1.04 -1.75 1.42 0.95 -0.84 0.34 -0.46 -0.47 -0.33 0.68 -1.33 -1.11 -0.63 0.35 1.29 -0.49 -0.31 1.91 0.39 0.53 2.14 0.3 -0.17 -0.97 0.74 0.01 0.06 0.23 -0.8 -0.05 -0.65 -1.52 0.05 -0.91 1.24 0.58 0.9 -0.79 -1.75 -0.9 1.83 -1.12 -0.47 -0.74 0.73

-0.44 -0.49 -0.55 -0.63 -0.65 -0.6 -0.57 -0.47 -0.44 -0.38 -0.48 -0.4 -0.35 -0.4 -0.38 -0.4 -0.43 -0.45 -0.41 -0.49 -0.55 -0.59 -0.57 -0.49 -0.52 -0.54 -0.43 -0.41 -0.38 -0.25 -0.23 -0.24 -0.3 -0.26 -0.26 -0.25 -0.24 -0.29 -0.29 -0.33 -0.41 -0.41 -0.46 -0.39 -0.36 -0.31 -0.35 -0.45 -0.51 -0.4 -0.46 -0.49 -0.53 -0.49

-0.23 -0.32 -0.43 -0.58 -0.61 -0.49 -0.41 -0.2 -0.14 -0.01 -0.2 -0.02 0.1 0.02 0.07 0.03 -0.01 -0.03 0.06 -0.08 -0.2 -0.26 -0.2 -0.04 -0.08 -0.1 0.13 0.19 0.26 0.52 0.57 0.56 0.47 0.56 0.58 0.6 0.64 0.56 0.57 0.51 0.34 0.36 0.27 0.43 0.51 0.63 0.55 0.36 0.27 0.49 0.38 0.34 0.27 0.36

0.4 0.4 0.41 0.41 0.42 0.42 0.42 0.43 0.43 0.44 0.44 0.44 0.45 0.45 0.45 0.46 0.46 0.47 0.47 0.47 0.48 0.48 0.48 0.49 0.49 0.49 0.5 0.5 0.5 0.51 0.51 0.51 0.52 0.52 0.52 0.53 0.53 0.53 0.54 0.54 0.54 0.54 0.55 0.55 0.55 0.56 0.56 0.56 0.57 0.57 0.57 0.57 0.58 0.58

102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155

0.34 0.34 0.35 0.35 0.35 0.36 0.36 0.36 0.37 0.37 0.37 0.38 0.38 0.38 0.39 0.39 0.39 0.4 0.4 0.4 0.41 0.41 0.41 0.42 0.42 0.42 0.43 0.43 0.43 0.44 0.44 0.44 0.45 0.45 0.45 0.46 0.46 0.46 0.47 0.47 0.47 0.48 0.48 0.48 0.49 0.49 0.49 0.5 0.5 0.5 0.51 0.51 0.51 0.52

0.76 0.47 0.91 0.57 0.57 0.43 0.8 0.58 0.76 0.28 0.38 0.05 0.87 0.19 0.04 0.14 0.66 0.17 0.52 0.26 0.07 0.96 0.7 0.8 0.46 0.57 0.85 0.36 0.03 0.94 0.74 0.79 0.41 0.65 0.36 0.99 0.08 0.15 0.57 0.84 0.43 0.94 0.89 0.3 0.14 0.93 0.44 0.8 0.11 0.97 0.06 0.17 0.93 0.76

-0.77 0.1 -1.05 -0.87 -0.87 1.03 -0.93 0.87 -0.58 -0.41 0.2 0.79 1.11 -0.95 2.97 -0.4 -0.18 -1.54 1.11 0.28 1.7 0.22 -0.19 -0.95 0.32 0.28 -1.09 0.7 1.46 -0.34 0.07 -1.05 1.34 0.45 -0.41 -1.27 0.06 0.04 1.34 0.85 1.07 -0.03 0.23 0.6 0.41 0.22 -1.47 -1.36 0.39 1.53 0.52 0.27 0.06 -1.16

-0.54 -0.53 -0.59 -0.64 -0.69 -0.63 -0.69 -0.64 -0.67 -0.69 -0.68 -0.63 -0.57 -0.63 -0.45 -0.48 -0.49 -0.58 -0.51 -0.5 -0.4 -0.38 -0.4 -0.45 -0.43 -0.42 -0.48 -0.44 -0.35 -0.37 -0.37 -0.43 -0.35 -0.33 -0.35 -0.42 -0.42 -0.42 -0.34 -0.29 -0.23 -0.23 -0.22 -0.18 -0.16 -0.15 -0.23 -0.31 -0.29 -0.2 -0.17 -0.15 -0.15 -0.22

0.29 0.31 0.2 0.12 0.03 0.16 0.07 0.18 0.13 0.1 0.13 0.24 0.38 0.28 0.64 0.61 0.6 0.43 0.58 0.62 0.83 0.87 0.86 0.77 0.82 0.86 0.75 0.84 1.03 1 1.02 0.91 1.08 1.15 1.11 0.98 1 1.02 1.18 1.3 1.43 1.44 1.48 1.57 1.63 1.67 1.51 1.37 1.42 1.61 1.69 1.73 1.75 1.63

0.58 0.59 0.59 0.59 0.59 0.6 0.6 0.6 0.61 0.61 0.61 0.61 0.62 0.62 0.62 0.62 0.63 0.63 0.63 0.64 0.64 0.64 0.64 0.65 0.65 0.65 0.65 0.66 0.66 0.66 0.66 0.67 0.67 0.67 0.67 0.68 0.68 0.68 0.68 0.69 0.69 0.69 0.69 0.7 0.7 0.7 0.7 0.7 0.71 0.71 0.71 0.71 0.72 0.72

156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209

0.52 0.52 0.53 0.53 0.53 0.54 0.54 0.54 0.55 0.55 0.55 0.56 0.56 0.56 0.57 0.57 0.57 0.58 0.58 0.58 0.59 0.59 0.59 0.6 0.6 0.6 0.61 0.61 0.61 0.62 0.62 0.62 0.63 0.63 0.63 0.64 0.64 0.64 0.65 0.65 0.65 0.66 0.66 0.66 0.67 0.67 0.67 0.68 0.68 0.68 0.69 0.69 0.69 0.7

0.97 0.39 0.33 0.82 0.75 0.36 0.76 0.48 0.15 0.18 0.13 0.5 0.16 0.21 0.65 0.73 0.05 0.09 0.67 0.94 0.39 0.81 0.87 0.84 0.61 0.98 0.8 0.67 0.15 0.73 0.43 0.12 0.12 0.76 0.94 0.86 0.12 0.7 0.35 0.26 0.88 0.47 0.76 0.04 0.68 0.42 0.77 0.73 0.5 0.44 0.67 0.9 0.26 0.54

0.76 -1.03 0.62 1.38 1.34 0.41 -0.57 -0.16 0.87 -0.86 -1.23 -1.04 -0.54 0.34 0.42 -0.82 -0.16 0.06 0.51 -1.97 -0.68 -0.38 0.28 -0.47 -0.15 -0.64 -0.65 -1.09 1.12 0.77 -0.64 0.37 1.49 2.47 0.16 1 0.4 1.02 -0.59 -0.09 -1.75 -0.29 0.28 -0.42 -2.15 -0.6 0.1 -0.12 0.83 -0.72 -0.06 -1.7 0.22 -1.36

-0.17 -0.23 -0.2 -0.12 -0.04 -0.02 -0.05 -0.06 -0.01 -0.06 -0.13 -0.19 -0.22 -0.2 -0.18 -0.22 -0.23 -0.23 -0.2 -0.32 -0.35 -0.38 -0.36 -0.39 -0.4 -0.43 -0.47 -0.53 -0.47 -0.42 -0.46 -0.44 -0.35 -0.21 -0.2 -0.15 -0.12 -0.06 -0.1 -0.1 -0.2 -0.22 -0.2 -0.23 -0.35 -0.39 -0.38 -0.39 -0.34 -0.38 -0.39 -0.48 -0.47 -0.55

1.73 1.63 1.71 1.88 2.05 2.11 2.06 2.06 2.17 2.08 1.95 1.85 1.8 1.85 1.91 1.83 1.83 1.85 1.92 1.7 1.64 1.61 1.65 1.61 1.61 1.55 1.49 1.37 1.51 1.62 1.56 1.61 1.8 2.1 2.13 2.26 2.32 2.45 2.39 2.39 2.2 2.18 2.23 2.19 1.96 1.9 1.93 1.93 2.04 1.97 1.97 1.79 1.83 1.69

0.72 0.72 0.73 0.73 0.73 0.73 0.73 0.74 0.74 0.74 0.74 0.75 0.75 0.75 0.75 0.75 0.76 0.76 0.76 0.76 0.77 0.77 0.77 0.77 0.77 0.78 0.78 0.78 0.78 0.79 0.79 0.79 0.79 0.79 0.8 0.8 0.8 0.8 0.8 0.81 0.81 0.81 0.81 0.81 0.82 0.82 0.82 0.82 0.82 0.83 0.83 0.83 0.83 0.83

210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263

0.7 0.7 0.71 0.71 0.71 0.72 0.72 0.72 0.73 0.73 0.73 0.74 0.74 0.74 0.75 0.75 0.75 0.76 0.76 0.76 0.77 0.77 0.77 0.78 0.78 0.78 0.79 0.79 0.79 0.8 0.8 0.8 0.81 0.81 0.81 0.82 0.82 0.82 0.83 0.83 0.83 0.84 0.84 0.84 0.85 0.85 0.85 0.86 0.86 0.86 0.87 0.87 0.87 0.88

0.73 0.87 0.52 0.54 0.54 0.67 0.26 0.97 0.79 0.38 0.73 0.73 0.24 0.85 0.43 0.59 0.11 0.3 0.06 0.87 0.35 0.74 0.29 0.12 0.48 0.79 0.56 0.14 0.69 0.82 0.68 0.42 0.69 0.2 0.96 0.23 0.86 0.22 0.2 0.65 0.6 0.93 0.38 0.84 0.78 0.8 0.43 0.89 0.11 0.5 0.76 0.45 0.24 0.05

-0.35 -1.93 -0.39 0.31 -0.98 -0.78 -0.25 -0.19 1.1 -0.43 -0.21 -0.19 -0.94 -1.46 -0.61 -0.13 0.09 -0.49 0.97 -1.06 0.38 1.03 -0.2 -0.88 -0.53 -0.78 -0.19 0.75 -0.64 -2.3 1.08 0.32 -1.78 -0.81 -0.69 -0.83 -0.18 0.38 0.32 -0.55 2.03 -1.7 0.57 -1.03 -1.19 2.3 0.25 0.39 -0.52 -0.17 0.83 1.64 -0.58 -0.79

-0.57 -0.68 -0.7 -0.69 -0.74 -0.79 -0.8 -0.81 -0.75 -0.78 -0.79 -0.8 -0.85 -0.94 -0.97 -0.98 -0.97 -1 -0.95 -1.01 -0.99 -0.93 -0.94 -0.99 -1.02 -1.06 -1.08 -1.03 -1.07 -1.2 -1.14 -1.12 -1.22 -1.27 -1.31 -1.36 -1.37 -1.35 -1.33 -1.36 -1.24 -1.34 -1.31 -1.37 -1.44 -1.3 -1.29 -1.27 -1.3 -1.31 -1.26 -1.16 -1.2 -1.24

1.66 1.45 1.42 1.47 1.37 1.29 1.28 1.27 1.41 1.37 1.36 1.35 1.26 1.1 1.04 1.04 1.06 1.02 1.15 1.04 1.09 1.23 1.22 1.13 1.08 1 1 1.1 1.04 0.78 0.92 0.97 0.78 0.7 0.63 0.55 0.54 0.6 0.65 0.6 0.85 0.67 0.75 0.64 0.52 0.79 0.84 0.9 0.85 0.84 0.95 1.15 1.1 1.02

0.84 0.84 0.84 0.84 0.84 0.85 0.85 0.85 0.85 0.85 0.86 0.86 0.86 0.86 0.86 0.87 0.87 0.87 0.87 0.87 0.88 0.88 0.88 0.88 0.88 0.89 0.89 0.89 0.89 0.89 0.89 0.9 0.9 0.9 0.9 0.9 0.91 0.91 0.91 0.91 0.91 0.91 0.92 0.92 0.92 0.92 0.92 0.93 0.93 0.93 0.93 0.93 0.93 0.94

264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300

0.88 0.88 0.89 0.89 0.89 0.9 0.9 0.9 0.91 0.91 0.91 0.92 0.92 0.92 0.93 0.93 0.93 0.94 0.94 0.94 0.95 0.95 0.95 0.96 0.96 0.96 0.97 0.97 0.97 0.98 0.98 0.98 0.99 0.99 0.99 1 1

0.57 0.72 0.84 0.13 0.86 0.52 0.95 0.54 0.95 0.64 0.74 0.9 0.87 0.61 0.12 0.07 0.26 0.72 0 0.63 0.56 0.78 0.44 1 0.67 0.54 0.49 0.43 0.99 0.73 0.48 0.56 0.45 0.31 0.69 0.31 0.84

-1.09 0.2 -0.17 0.16 -0.37 0.51 0.18 -0.79 -0.46 0.28 -0.19 0.17 0.89 1.05 -0.79 -0.15 -1.06 -0.87 -0.04 1.09 -0.4 0.26 1.04 1.59 -0.67 -1.02 -0.32 -1.67 -1.27 0.41 -0.63 -0.7 -0.71 0.51 0.85 0.24 -0.27

-1.3 -1.29 -1.3 -1.29 -1.32 -1.29 -1.27 -1.32 -1.35 -1.33 -1.34 -1.33 -1.28 -1.22 -1.27 -1.27 -1.34 -1.39 -1.39 -1.32 -1.35 -1.33 -1.27 -1.18 -1.22 -1.28 -1.3 -1.39 -1.47 -1.44 -1.48 -1.52 -1.56 -1.53 -1.48 -1.47 -1.48

0.91 0.95 0.94 0.97 0.94 1.02 1.05 0.97 0.93 0.98 0.97 1 1.12 1.25 1.18 1.17 1.06 0.98 0.98 1.12 1.09 1.13 1.27 1.46 1.4 1.3 1.27 1.09 0.96 1.02 0.96 0.89 0.82 0.9 1.01 1.05 1.03

0.94 0.94 0.94 0.94 0.95 0.95 0.95 0.95 0.95 0.95 0.96 0.96 0.96 0.96 0.96 0.96 0.97 0.97 0.97 0.97 0.97 0.97 0.98 0.98 0.98 0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.99 0.99 1 1 1

Drift line -T aT + bT aT - bT (Stdev Z) (Stdev X) (Stdev X) 0 0 0 -0.06 0.13 -0.1 -0.08 0.19 -0.14 -0.1 0.24 -0.16 -0.12 0.28 -0.18 -0.13 0.32 -0.19 -0.14 0.36 -0.2 -0.15 0.4 -0.21 -0.16 0.43 -0.22 -0.17 0.47 -0.23 -0.18 0.5 -0.23 -0.19 0.53 -0.24 -0.2 0.56 -0.24 -0.21 0.59 -0.24 -0.22 0.62 -0.25 -0.22 0.65 -0.25 -0.23 0.68 -0.25 -0.24 0.7 -0.25 -0.24 0.73 -0.25 -0.25 0.76 -0.25 -0.26 0.78 -0.25 -0.26 0.81 -0.25 -0.27 0.83 -0.25 -0.28 0.86 -0.25 -0.28 0.89 -0.25 -0.29 0.91 -0.24 -0.29 0.94 -0.24 -0.3 0.96 -0.24 -0.31 0.98 -0.24 -0.31 1.01 -0.24 -0.32 1.03 -0.23 -0.32 1.06 -0.23 -0.33 1.08 -0.23 -0.33 1.1 -0.22 -0.34 1.13 -0.22 -0.34 1.15 -0.22 -0.35 1.17 -0.21 -0.35 1.2 -0.21 -0.36 1.22 -0.21 -0.36 1.24 -0.2 -0.37 1.26 -0.2 -0.37 1.29 -0.19 -0.37 1.31 -0.19 -0.38 1.33 -0.18 -0.38 1.35 -0.18 -0.39 1.37 -0.17 -0.39 1.4 -0.17 -0.4 1.42 -0.16 0.5

0.5

0.5

0 0

1 4

-0.4 -0.4 -0.41 -0.41 -0.42 -0.42 -0.42 -0.43 -0.43 -0.44 -0.44 -0.44 -0.45 -0.45 -0.45 -0.46 -0.46 -0.47 -0.47 -0.47 -0.48 -0.48 -0.48 -0.49 -0.49 -0.49 -0.5 -0.5 -0.5 -0.51 -0.51 -0.51 -0.52 -0.52 -0.52 -0.53 -0.53 -0.53 -0.54 -0.54 -0.54 -0.54 -0.55 -0.55 -0.55 -0.56 -0.56 -0.56 -0.57 -0.57 -0.57 -0.57 -0.58 -0.58

1.44 1.46 1.48 1.5 1.53 1.55 1.57 1.59 1.61 1.63 1.65 1.67 1.69 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 2.02 2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18 2.2 2.22 2.24 2.26 2.28 2.3 2.31 2.33 2.35 2.37 2.39 2.41 2.43 2.45 2.47 2.49 2.51

-0.16 -0.15 -0.15 -0.14 -0.14 -0.13 -0.13 -0.12 -0.12 -0.11 -0.11 -0.1 -0.09 -0.09 -0.08 -0.08 -0.07 -0.06 -0.06 -0.05 -0.05 -0.04 -0.03 -0.03 -0.02 -0.01 -0.01 0 0.01 0.01 0.02 0.03 0.03 0.04 0.05 0.05 0.06 0.07 0.08 0.08 0.09 0.1 0.1 0.11 0.12 0.13 0.13 0.14 0.15 0.16 0.16 0.17 0.18 0.19

-0.58 -0.59 -0.59 -0.59 -0.59 -0.6 -0.6 -0.6 -0.61 -0.61 -0.61 -0.61 -0.62 -0.62 -0.62 -0.62 -0.63 -0.63 -0.63 -0.64 -0.64 -0.64 -0.64 -0.65 -0.65 -0.65 -0.65 -0.66 -0.66 -0.66 -0.66 -0.67 -0.67 -0.67 -0.67 -0.68 -0.68 -0.68 -0.68 -0.69 -0.69 -0.69 -0.69 -0.7 -0.7 -0.7 -0.7 -0.7 -0.71 -0.71 -0.71 -0.71 -0.72 -0.72

2.53 2.55 2.56 2.58 2.6 2.62 2.64 2.66 2.68 2.7 2.72 2.73 2.75 2.77 2.79 2.81 2.83 2.85 2.86 2.88 2.9 2.92 2.94 2.96 2.98 2.99 3.01 3.03 3.05 3.07 3.09 3.1 3.12 3.14 3.16 3.18 3.2 3.21 3.23 3.25 3.27 3.29 3.31 3.32 3.34 3.36 3.38 3.4 3.41 3.43 3.45 3.47 3.49 3.5

0.19 0.2 0.21 0.22 0.22 0.23 0.24 0.25 0.26 0.26 0.27 0.28 0.29 0.3 0.3 0.31 0.32 0.33 0.34 0.34 0.35 0.36 0.37 0.38 0.38 0.39 0.4 0.41 0.42 0.43 0.43 0.44 0.45 0.46 0.47 0.48 0.48 0.49 0.5 0.51 0.52 0.53 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.59 0.6 0.61 0.62 0.63

-0.72 -0.72 -0.73 -0.73 -0.73 -0.73 -0.73 -0.74 -0.74 -0.74 -0.74 -0.75 -0.75 -0.75 -0.75 -0.75 -0.76 -0.76 -0.76 -0.76 -0.77 -0.77 -0.77 -0.77 -0.77 -0.78 -0.78 -0.78 -0.78 -0.79 -0.79 -0.79 -0.79 -0.79 -0.8 -0.8 -0.8 -0.8 -0.8 -0.81 -0.81 -0.81 -0.81 -0.81 -0.82 -0.82 -0.82 -0.82 -0.82 -0.83 -0.83 -0.83 -0.83 -0.83

3.52 3.54 3.56 3.58 3.59 3.61 3.63 3.65 3.67 3.68 3.7 3.72 3.74 3.75 3.77 3.79 3.81 3.83 3.84 3.86 3.88 3.9 3.91 3.93 3.95 3.97 3.98 4 4.02 4.04 4.05 4.07 4.09 4.11 4.12 4.14 4.16 4.18 4.19 4.21 4.23 4.25 4.26 4.28 4.3 4.32 4.33 4.35 4.37 4.39 4.4 4.42 4.44 4.46

0.64 0.65 0.66 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12

-0.84 -0.84 -0.84 -0.84 -0.84 -0.85 -0.85 -0.85 -0.85 -0.85 -0.86 -0.86 -0.86 -0.86 -0.86 -0.87 -0.87 -0.87 -0.87 -0.87 -0.88 -0.88 -0.88 -0.88 -0.88 -0.89 -0.89 -0.89 -0.89 -0.89 -0.89 -0.9 -0.9 -0.9 -0.9 -0.9 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.92 -0.92 -0.92 -0.92 -0.92 -0.93 -0.93 -0.93 -0.93 -0.93 -0.93 -0.94

4.47 4.49 4.51 4.53 4.54 4.56 4.58 4.59 4.61 4.63 4.65 4.66 4.68 4.7 4.71 4.73 4.75 4.77 4.78 4.8 4.82 4.83 4.85 4.87 4.89 4.9 4.92 4.94 4.95 4.97 4.99 5.01 5.02 5.04 5.06 5.07 5.09 5.11 5.13 5.14 5.16 5.18 5.19 5.21 5.23 5.24 5.26 5.28 5.29 5.31 5.33 5.35 5.36 5.38

1.13 1.14 1.15 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.3 1.31 1.32 1.33 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6 1.6 1.61 1.62 1.63

-0.94 -0.94 -0.94 -0.94 -0.95 -0.95 -0.95 -0.95 -0.95 -0.95 -0.96 -0.96 -0.96 -0.96 -0.96 -0.96 -0.97 -0.97 -0.97 -0.97 -0.97 -0.97 -0.98 -0.98 -0.98 -0.98 -0.98 -0.98 -0.99 -0.99 -0.99 -0.99 -0.99 -0.99 -1 -1 -1

5.4 5.41 5.43 5.45 5.46 5.48 5.5 5.51 5.53 5.55 5.56 5.58 5.6 5.62 5.63 5.65 5.67 5.68 5.7 5.72 5.73 5.75 5.77 5.78 5.8 5.82 5.83 5.85 5.87 5.88 5.9 5.92 5.93 5.95 5.97 5.98 6

1.64 1.65 1.66 1.67 1.68 1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.9 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2

Wiener Process Source: Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. P. 221 The figure below illustrates what happens if to the path followed by z (z follows Wiener process) if ∆ t approaches 0. Note that for smaller ∆ t the path becomes quite "jagged" because the size of of a movement in z in ∆ t is proportional to sqrt(∆ t). If ∆ t is small, sqrt(∆ t) is much larger. This leads to two intriguing properties of the Wiener process: - the expected length of the path followed by z in any time interval is infinite. - the expected number of times z equals any particular value in any time interval is infinite.

Numerical example (press F9 to generate new random numbers) Time (T) 1

Wiener Process with Dt -> 0 1.40 1.20 1.00 0.80 0.60 0.40

Delta t =0.00333

0.20 0.00 -0.20 -0.40 0

50 100 150 200 250 300 350 Time

Delta t =0.0100

Delta t =0.1000

if ∆ t approaches 0. z in ∆ t is proportional

0

Wiener Process

Δz=ε  Δt where ε is a random drawing from a standardized normal distribution, Φ (0,1)

Delta t =0.1000

Delta t =0.00333 Delta t =0.0100 ∆t ∆t 0.0033 0.0100 Trimmed range for chart Z + ∆Z Period Time Modulo 3 0 0 0 0 0 #VALUE! #VALUE! 1 0.0033 -0.02 #VALUE! #VALUE! 2 0.0067 0.03 #VALUE! #VALUE! 3 0.0100 0.06 0.01 0 #VALUE! #VALUE! 4 0.0133 0.19 #VALUE! #VALUE! 5 0.0167 0.15 #VALUE! #VALUE! 6 0.0200 0.12 0.02 0.02 #VALUE! #VALUE! 7 0.0233 0.17 #VALUE! #VALUE! 8 0.0267 0.09 #VALUE! #VALUE! 9 0.0300 0.09 0.03 0.03 #VALUE! #VALUE! 10 0.0333 0.05 #VALUE! #VALUE! 11 0.0367 0.01 #VALUE! #VALUE! 12 0.0400 0.03 0.04 -0.01 #VALUE! #VALUE! 13 0.0433 0.07 #VALUE! #VALUE! 14 0.0467 -0.01 #VALUE! #VALUE! 15 0.0500 -0.11 0.05 0.01 #VALUE! #VALUE! 16 0.0533 -0.14 #VALUE! #VALUE! 17 0.0567 -0.05 #VALUE! #VALUE! 18 0.0600 0.07 0.06 -0.02 #VALUE! #VALUE! 19 0.0633 0.08 #VALUE! #VALUE! 20 0.0667 0.01 #VALUE! #VALUE! 21 0.0700 -0.01 0.07 -0.05 #VALUE! #VALUE! 22 0.0733 0 #VALUE! #VALUE! 23 0.0767 -0.04 #VALUE! #VALUE! 24 0.0800 -0.18 0.08 -0.02 #VALUE! #VALUE! 25 0.0833 -0.11 #VALUE! #VALUE! 26 0.0867 -0.11 #VALUE! #VALUE! 27 0.0900 -0.15 0.09 -0.05 #VALUE! #VALUE! 28 0.0933 -0.14 #VALUE! #VALUE! 29 0.0967 -0.18 #VALUE! #VALUE! 30 0.1000 -0.12 0.1 -0.01 #VALUE! #VALUE! 31 0.1033 -0.01 #VALUE! #VALUE! 32 0.1067 0.01 #VALUE! #VALUE! 33 0.1100 0.03 0.11 0.04 #VALUE! #VALUE! 34 0.1133 0.06 #VALUE! #VALUE! 35 0.1167 0.08 #VALUE! #VALUE! 36 0.1200 0.15 0.12 0.11 #VALUE! #VALUE!

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

0.1233 0.1267 0.1300 0.1333 0.1367 0.1400 0.1433 0.1467 0.1500 0.1533 0.1567 0.1600 0.1633 0.1667 0.1700 0.1733 0.1767 0.1800 0.1833 0.1867 0.1900 0.1933 0.1967 0.2000 0.2033 0.2067 0.2100 0.2133 0.2167 0.2200 0.2233 0.2267 0.2300 0.2333 0.2367 0.2400 0.2433 0.2467 0.2500 0.2533 0.2567 0.2600 0.2633 0.2667 0.2700 0.2733 0.2767 0.2800 0.2833 0.2867 0.2900

0.26 0.24 0.36 0.35 0.38 0.33 0.27 0.25 0.22 0.17 0.2 0.16 0.11 0.07 0.05 0.06 0.13 0.14 0.14 0.19 0.04 0.09 0.07 0.03 0.05 0.02 0.09 0.06 -0.05 -0.05 -0.11 -0.26 -0.19 -0.25 -0.25 -0.25 -0.27 -0.25 -0.17 -0.14 -0.09 -0.07 0 0.13 0.2 0.16 0.16 0.21 0.27 0.16 0.17

0.13

0.12

0.14

0.04

0.15

-0.02

0.16

-0.05

0.17

-0.05

0.18

-0.05

0.19

-0.1

0.2

-0.15

0.21

-0.13

0.22

-0.13

0.23

-0.2

0.24

-0.26

0.25

-0.26

0.26

-0.25

0.27

-0.2

0.28

-0.07

0.29

-0.08

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138

0.2933 0.2967 0.3000 0.3033 0.3067 0.3100 0.3133 0.3167 0.3200 0.3233 0.3267 0.3300 0.3333 0.3367 0.3400 0.3433 0.3467 0.3500 0.3533 0.3567 0.3600 0.3633 0.3667 0.3700 0.3733 0.3767 0.3800 0.3833 0.3867 0.3900 0.3933 0.3967 0.4000 0.4033 0.4067 0.4100 0.4133 0.4167 0.4200 0.4233 0.4267 0.4300 0.4333 0.4367 0.4400 0.4433 0.4467 0.4500 0.4533 0.4567 0.4600

0.23 0.28 0.37 0.29 0.28 0.25 0.36 0.39 0.36 0.36 0.42 0.4 0.38 0.25 0.29 0.37 0.35 0.34 0.4 0.28 0.24 0.23 0.32 0.27 0.25 0.2 0.24 0.18 0.07 0.11 0.15 0.21 0.22 0.25 0.35 0.52 0.63 0.71 0.74 0.71 0.69 0.7 0.72 0.76 0.77 0.76 0.78 0.88 0.94 0.94 1.05

0.3

-0.15

0.31

-0.16

0.32

-0.16

0.33

-0.17

0.34

-0.16

0.35

-0.16

0.36

-0.15

0.37

-0.16

0.38

-0.2

0.39

-0.23

0.4

-0.38

0.41

-0.38

0.42

-0.35

0.43

-0.38

0.44

-0.42

0.45

-0.4

0.46

-0.35

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189

0.4633 0.4667 0.4700 0.4733 0.4767 0.4800 0.4833 0.4867 0.4900 0.4933 0.4967 0.5000 0.5033 0.5067 0.5100 0.5133 0.5167 0.5200 0.5233 0.5267 0.5300 0.5333 0.5367 0.5400 0.5433 0.5467 0.5500 0.5533 0.5567 0.5600 0.5633 0.5667 0.5700 0.5733 0.5767 0.5800 0.5833 0.5867 0.5900 0.5933 0.5967 0.6000 0.6033 0.6067 0.6100 0.6133 0.6167 0.6200 0.6233 0.6267 0.6300

0.96 1.06 1.14 1.11 1.08 1.01 0.87 0.87 0.81 0.85 0.8 0.66 0.63 0.7 0.8 0.78 0.84 0.91 0.8 0.74 0.7 0.72 0.75 0.64 0.59 0.53 0.56 0.5 0.61 0.56 0.5 0.4 0.34 0.24 0.21 0.21 0.15 0.13 0.22 0.19 0.11 0.06 0.03 0.01 -0.09 -0.12 -0.09 -0.02 -0.08 -0.03 -0.16

0.47

-0.36

0.48

-0.31

0.49

-0.29

0.5

-0.33

0.51

-0.35

0.52

-0.4

0.53

-0.39

0.54

-0.35

0.55

-0.37

0.56

-0.29

0.57

-0.32

0.58

-0.31

0.59

-0.31

0.6

-0.2

0.61

-0.22

0.62

-0.21

0.63

-0.27

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240

0.6333 0.6367 0.6400 0.6433 0.6467 0.6500 0.6533 0.6567 0.6600 0.6633 0.6667 0.6700 0.6733 0.6767 0.6800 0.6833 0.6867 0.6900 0.6933 0.6967 0.7000 0.7033 0.7067 0.7100 0.7133 0.7167 0.7200 0.7233 0.7267 0.7300 0.7333 0.7367 0.7400 0.7433 0.7467 0.7500 0.7533 0.7567 0.7600 0.7633 0.7667 0.7700 0.7733 0.7767 0.7800 0.7833 0.7867 0.7900 0.7933 0.7967 0.8000

-0.17 -0.18 -0.15 -0.15 -0.14 -0.06 -0.13 -0.09 -0.02 -0.05 0.03 0.11 0.1 0.19 0.15 0.24 0.17 0.18 0.25 0.24 0.26 0.17 0.2 0.12 0.05 0.13 0.19 0.31 0.23 0.25 0.39 0.39 0.29 0.33 0.42 0.44 0.46 0.38 0.37 0.37 0.35 0.33 0.32 0.38 0.34 0.37 0.42 0.4 0.36 0.39 0.43

0.64

-0.28

0.65

-0.35

0.66

-0.4

0.67

-0.44

0.68

-0.5

0.69

-0.44

0.7

-0.39

0.71

-0.35

0.72

-0.33

0.73

-0.24

0.74

-0.33

0.75

-0.33

0.76

-0.33

0.77

-0.36

0.78

-0.3

0.79

-0.2

0.8

-0.18

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291

0.8033 0.8067 0.8100 0.8133 0.8167 0.8200 0.8233 0.8267 0.8300 0.8333 0.8367 0.8400 0.8433 0.8467 0.8500 0.8533 0.8567 0.8600 0.8633 0.8667 0.8700 0.8733 0.8767 0.8800 0.8833 0.8867 0.8900 0.8933 0.8967 0.9000 0.9033 0.9067 0.9100 0.9133 0.9167 0.9200 0.9233 0.9267 0.9300 0.9333 0.9367 0.9400 0.9433 0.9467 0.9500 0.9533 0.9567 0.9600 0.9633 0.9667 0.9700

0.51 0.55 0.54 0.43 0.49 0.46 0.53 0.59 0.58 0.58 0.64 0.72 0.72 0.74 0.8 0.75 0.71 0.8 0.81 0.85 0.83 0.91 0.86 0.81 0.76 0.82 0.69 0.69 0.62 0.65 0.62 0.49 0.49 0.52 0.42 0.4 0.28 0.36 0.31 0.3 0.3 0.17 0.15 0.33 0.34 0.3 0.25 0.3 0.25 0.3 0.31

0.81

-0.24

0.82

-0.25

0.83

-0.18

0.84

-0.13

0.85

-0.13

0.86

-0.19

0.87

-0.15

0.88

-0.17

0.89

-0.23

0.9

-0.19

0.91

-0.19

0.92

-0.17

0.93

-0.15

0.94

-0.19

0.95

-0.2

0.96

-0.17

0.97

-0.05

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

LAST

292 293 294 295 296 297 298 299 300

0.9733 0.9767 0.9800 0.9833 0.9867 0.9900 0.9933 0.9967 1.0000

0.31 0.35 0.38 0.41 0.51 0.52 0.55 0.55 0.57

0.98

0.99

1

#VALUE! #VALUE! -0.12 #VALUE! #VALUE! #VALUE! -0.15 #VALUE! #VALUE! #VALUE! -0.15 #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

Delta t =0.1000 ∆t 0.1000 Trimmed range for chart Modulo 30 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

Normal Variates

0.32 -0.16 -0.26 0.58 0.33 0.28 0.07 -0.21 0.32 -1.53 0.88 -1.19 1.39 -0.34 0.77 -0.76 -0.78 -0.35 0.43 -1.17 -0.11 2.08 0.23 -1.21 -1.4 0.06 1.36 -0.51 -0.51 1.19 -0.01 1.46 -0.47 1.25 0.37 1.64

-0.03

-0.03

-0.26

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

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0.02 -0.94 -0.34 -1.11 -0.71 -0.91 -0.67 -1.01 0.12 -1.85 -0.31 0.73 -0.26 -1.7 1.22 1.05 -1.95 -0.05 1.81 -1.25 -2.55 1.16 -0.23 -0.5 0.71 1.33 -0.71 -1.41 0.85 1.19 -1.89 -0.5 -1.58 -0.26 -0.14 -0.54 0.2 0.51 -0.13 -1.14 0.6 0.95 1.25 -1.16 1.14 0.78 1.97 1.27 -0.63 1.56 -2.67

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

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-0.62 0.91 -0.22 0.2 0.2 -0.46 0.9 0.41 -1.16 0.52 0.49 -0.17 0.69 -1.36 1.16 -1.63 0.43 0.18 0 0.82 -0.52 -0.44 0.49 -0.19 -0.83 -0.08 -0.25 -1.27 0.62 -0.41 -1.25 -3.12 -1 0.05 0.2 0.64 1 -0.27 -0.26 1.8 -1.33 -1.37 -0.26 1.03 -0.95 -0.6 1.28 0.98 1.03 -0.25 0.33

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

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-1.04 0.61 -0.48 0.19 1.5 0.79 1.93 -1.82 0.05 -0.49 -1.08 0.05 -0.07 0.41 -1.29 -0.81 0 -0.02 0.27 -0.09 0.21 0.5 1.08 -0.17 1.09 -1.1 -0.44 0.52 2.19 0.62 -0.44 -1.1 -0.1 0.37 0.55 -0.27 -0.17 0.47 -0.18 1.64 2.61 0.17 -0.08 -0.08 -0.75 0.17 0.46 0.6 -1.57 -0.57 -0.88

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

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0.36 1.95 -1.65 -1.45 0.98 -0.89 -0.24 0.09 -0.95 -1.1 1.11 -0.51 0.25 0.03 -2.16 2.4 1.58 0.52 -0.2 1.24 0.5 2.29 -0.35 -0.99 -0.79 1.47 0.99 1.57 2.24 -1.17 -1.13 0.33 -1.31 -0.92 0.54 1.5 -0.35 -1.23 -0.08 0.09 -0.69 -0.42 1.01 1 1.03 1.07 0.97 0.81 0.14 -0.66 0.48

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

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-0.68 -0.7 -1.7 -0.23 -0.1 1.94 -0.68 -0.25 1.76 -0.33 0.08 0.23 -0.09 0.55 -0.54 -0.26 -1.45 -0.24 1.12 0.28 0.38 -0.83 -0.12 -0.02 -1.63 -0.49 -0.45 1 1.12 0.2 0.06 -1.19 0.93 0.18 0.07 -0.58 0.11 0.76 0.47 0.01 -1.06 -0.8 -1.31 0.28 1.41 -0.28 -2.38 2.56 0.82 1.16 0.27

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

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-0.15 -1.4 -1.57 1.49 -1.11 -0.33 -0.65 1.97 -0.71

Geometric Brownian Motion Source: Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. P. 226

ΔS S

= μΔt σε

 Δt

The variable ∆ S is the change in the stock price, S, in a small interval of time ∆ t. are independent. The parameters: µ is the expected rate of return per unit of time (usually per year) σ is the volatility of the stock price (σ and µ are assumed contant) ε is a random drawing from a standardized normal distribution, Φ (0,1) Numerical example (press F9 to generate new random numbers) Stock price to t=0 Time (T) Expected return (µ) Volatility (σ)

$2.00 1 year(s) 20.00% 40.00% 0.0033

∆T

Simulation of Stock Price Paths $7.00 $6.00 $5.00 $4.00 S -1

$3.00 $2.00 $1.00 $0

0.2

0.4

0.6

Time

0.8

1

1.2

S -2

S -3

S -4

S -5

S -6

ntice-Hall. P. 226

Period

independent.

S -3

S -4

S -5

S -6

Random Numbers Uniform Normal

Time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

0 0 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.06 0.06 0.07 0.07 0.07 0.08 0.08 0.08 0.09 0.09 0.09 0.1 0.1 0.1 0.11 0.11 0.11 0.12 0.12 0.12 0.13 0.13 0.13 0.14 0.14 0.14 0.15 0.15 0.15 0.16 0.16

0.64 0.28 0.15 0.21 0.22 0 0.67 0.11 0.58 0.9 0.73 0.52 0.2 0.01 0.53 0.62 0.7 0.28 0.58 0.34 0.4 0.31 0.81 0.51 0.67 0.93 0.71 0.56 0.47 0.02 0.89 0.72 0.8 0.26 0.33 0.22 0.93 0.14 0.96 0.95 0.03 0.92 0.15 0.9 0.19 0.54 0.58 0

S -1 S -2 S -3 S + ∆S S + ∆S S + ∆S $2.00 $2.00 $2.00 -0.17 $1.99 $2.05 $2.04 0.73 $2.03 $2.06 $2.06 0.47 $2.05 $2.02 $2.10 1.38 $2.12 $2.02 $2.11 -0.12 $2.11 $1.96 $2.04 0.6 $2.14 $1.96 $1.98 -0.64 $2.11 $1.90 $1.99 -0.06 $2.11 $1.83 $1.97 0.38 $2.13 $1.85 $2.00 -0.14 $2.13 $1.83 $1.93 1.39 $2.20 $1.91 $1.99 -1.35 $2.13 $1.88 $1.99 -0.09 $2.13 $1.87 $2.02 -0.59 $2.10 $1.88 $1.97 -1.35 $2.04 $1.92 $1.94 -1.59 $1.96 $1.96 $1.98 0.01 $1.96 $1.96 $2.03 -0.55 $1.94 $1.98 $2.10 1.14 $1.99 $1.94 $2.10 0.44 $2.01 $1.88 $2.16 -1.22 $1.96 $1.92 $2.21 -0.18 $1.95 $1.90 $2.14 -0.96 $1.91 $1.90 $2.13 1.1 $1.96 $1.91 $2.06 -1.01 $1.91 $1.84 $2.12 1.12 $1.97 $1.86 $2.15 0.88 $2.01 $1.85 $2.11 0.61 $2.04 $1.87 $2.16 0.2 $2.05 $1.85 $2.30 -0.52 $2.02 $1.86 $2.22 -0.27 $2.01 $1.96 $2.29 -2.3 $1.91 $1.93 $2.25 -1.48 $1.84 $1.88 $2.29 -1.46 $1.78 $1.89 $2.26 1.45 $1.84 $1.87 $2.24 0.05 $1.85 $1.84 $2.15 0.83 $1.88 $1.90 $2.14 -0.88 $1.85 $1.90 $2.08 2.72 $1.96 $1.97 $2.06 -0.14 $1.96 $2.06 $2.14 0.34 $1.98 $2.11 $2.21 1.36 $2.04 $2.13 $2.17 0.09 $2.04 $2.11 $2.15 -1.3 $1.98 $2.13 $2.23 -0.87 $1.95 $2.22 $2.20 0.31 $1.96 $2.18 $2.19 -1.02 $1.92 $2.12 $2.18 0.51 $1.94 $2.10 $2.24

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102

0.16 0.17 0.17 0.17 0.18 0.18 0.18 0.19 0.19 0.19 0.2 0.2 0.2 0.21 0.21 0.21 0.22 0.22 0.22 0.23 0.23 0.23 0.24 0.24 0.24 0.25 0.25 0.25 0.26 0.26 0.26 0.27 0.27 0.27 0.28 0.28 0.28 0.29 0.29 0.29 0.3 0.3 0.3 0.31 0.31 0.31 0.32 0.32 0.32 0.33 0.33 0.33 0.34 0.34

0.1 0.44 0.38 0.31 0.53 0.86 0.81 0.05 0.5 0.89 0.75 0.56 0.35 0.99 0.22 0.82 0.06 0.28 0.68 0.21 0.01 0.02 0.22 0.62 0.15 0.96 0.95 0.3 0.56 0.29 0.5 0.28 0.08 0.28 0.19 0.33 0.56 0.68 0.75 0.27 0.12 0.06 0.63 0.19 0.74 0.48 0.25 0.1 0.31 0.57 0.08 0.85 0.9 0.45

1.36 -1.95 -0.35 -1.99 -0.11 0.09 1.21 0.28 -0.24 0.52 -0.41 2.14 3.8 0.62 2.55 -1.48 0.86 1.41 0.23 0.26 -1.23 0.22 -1.67 0.66 0 0.21 1 0.51 -0.83 2.49 -0.28 -1.19 -2.07 0.69 -1.08 -0.07 -0.56 -1.91 -1.37 0.5 0.6 -0.19 0.46 0.6 -1.01 0.44 0.83 1.77 0.23 -0.29 0.16 0.53 1.96 0.28

$2.00 $1.91 $1.90 $1.81 $1.81 $1.81 $1.87 $1.88 $1.87 $1.89 $1.88 $1.97 $2.15 $2.18 $2.31 $2.23 $2.28 $2.35 $2.37 $2.38 $2.32 $2.33 $2.24 $2.28 $2.28 $2.29 $2.35 $2.37 $2.33 $2.47 $2.45 $2.39 $2.27 $2.31 $2.25 $2.25 $2.23 $2.13 $2.06 $2.09 $2.12 $2.11 $2.13 $2.17 $2.12 $2.14 $2.18 $2.27 $2.29 $2.27 $2.28 $2.31 $2.42 $2.43

$2.06 $2.02 $1.98 $1.93 $1.91 $1.87 $1.85 $1.91 $1.91 $1.91 $1.92 $1.82 $1.81 $1.84 $1.81 $1.75 $1.81 $1.87 $1.87 $1.85 $1.79 $1.81 $1.91 $2.00 $1.96 $1.87 $1.81 $1.80 $1.73 $1.70 $1.68 $1.66 $1.66 $1.62 $1.59 $1.57 $1.54 $1.54 $1.48 $1.46 $1.49 $1.45 $1.42 $1.42 $1.44 $1.51 $1.52 $1.49 $1.56 $1.54 $1.53 $1.52 $1.47 $1.44

$2.28 $2.33 $2.35 $2.31 $2.32 $2.29 $2.25 $2.24 $2.24 $2.29 $2.18 $2.14 $2.13 $2.13 $2.18 $2.29 $2.27 $2.24 $2.27 $2.23 $2.27 $2.32 $2.28 $2.35 $2.29 $2.25 $2.19 $2.12 $2.12 $2.18 $2.25 $2.26 $2.27 $2.28 $2.32 $2.36 $2.42 $2.37 $2.34 $2.34 $2.38 $2.29 $2.30 $2.33 $2.42 $2.37 $2.35 $2.41 $2.47 $2.43 $2.34 $2.28 $2.26 $2.31

103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156

0.34 0.35 0.35 0.35 0.36 0.36 0.36 0.37 0.37 0.37 0.38 0.38 0.38 0.39 0.39 0.39 0.4 0.4 0.4 0.41 0.41 0.41 0.42 0.42 0.42 0.43 0.43 0.43 0.44 0.44 0.44 0.45 0.45 0.45 0.46 0.46 0.46 0.47 0.47 0.47 0.48 0.48 0.48 0.49 0.49 0.49 0.5 0.5 0.5 0.51 0.51 0.51 0.52 0.52

0.12 0.07 0.25 0.61 0.46 0.35 0.56 0.6 0.75 0.08 0.89 0.36 0.81 0.81 0.03 0.23 0.68 0.5 0.74 0.05 0.6 0.14 0.15 0.87 0.41 0.07 0.42 0.55 0.3 0.58 0.82 0.17 0.67 0.59 0.54 0.96 0.21 0.84 0.47 0.85 0.05 0.97 0.97 0.86 0.45 0.77 0.05 0.97 0.04 0.98 0.85 0.25 0.31 0.19

-0.14 -0.06 -0.87 -0.56 -0.95 -0.26 -0.58 0.15 0.03 -0.53 -0.49 0.4 0.75 0.26 0.47 1.07 -0.54 -0.23 -0.57 1.89 -1.11 -0.15 0.36 1.47 -0.25 -0.73 -0.48 1.8 1.51 -0.55 0.19 -0.31 0.73 0.73 0.43 1.55 -0.98 1.61 0 0.46 -0.69 0.89 -0.44 -2.1 -0.22 -2.27 1.15 0.55 -0.2 -1 0.46 0.13 0.25 -0.47

$2.43 $2.43 $2.38 $2.35 $2.30 $2.29 $2.26 $2.27 $2.27 $2.25 $2.22 $2.24 $2.28 $2.30 $2.33 $2.38 $2.36 $2.34 $2.32 $2.42 $2.36 $2.35 $2.37 $2.45 $2.44 $2.40 $2.38 $2.48 $2.57 $2.53 $2.55 $2.53 $2.58 $2.62 $2.65 $2.75 $2.68 $2.79 $2.79 $2.82 $2.78 $2.84 $2.81 $2.67 $2.66 $2.52 $2.59 $2.63 $2.62 $2.56 $2.59 $2.60 $2.61 $2.59

$1.52 $1.52 $1.54 $1.57 $1.53 $1.55 $1.58 $1.59 $1.59 $1.66 $1.62 $1.58 $1.57 $1.50 $1.50 $1.53 $1.53 $1.61 $1.64 $1.68 $1.70 $1.64 $1.69 $1.71 $1.74 $1.81 $1.83 $1.78 $1.83 $1.90 $1.90 $1.92 $1.96 $1.98 $1.92 $1.88 $1.89 $1.84 $1.83 $1.72 $1.76 $1.86 $1.88 $1.85 $1.89 $1.92 $2.00 $2.04 $2.01 $1.96 $1.92 $1.87 $1.82 $1.90

$2.30 $2.30 $2.44 $2.40 $2.36 $2.35 $2.40 $2.37 $2.34 $2.35 $2.35 $2.23 $2.26 $2.26 $2.29 $2.27 $2.21 $2.25 $2.22 $2.18 $2.16 $2.17 $2.24 $2.32 $2.31 $2.28 $2.30 $2.32 $2.30 $2.26 $2.33 $2.50 $2.57 $2.58 $2.56 $2.53 $2.61 $2.52 $2.61 $2.68 $2.67 $2.58 $2.65 $2.66 $2.68 $2.73 $2.71 $2.71 $2.80 $2.89 $2.94 $2.99 $2.97 $3.00

157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210

0.52 0.53 0.53 0.53 0.54 0.54 0.54 0.55 0.55 0.55 0.56 0.56 0.56 0.57 0.57 0.57 0.58 0.58 0.58 0.59 0.59 0.59 0.6 0.6 0.6 0.61 0.61 0.61 0.62 0.62 0.62 0.63 0.63 0.63 0.64 0.64 0.64 0.65 0.65 0.65 0.66 0.66 0.66 0.67 0.67 0.67 0.68 0.68 0.68 0.69 0.69 0.69 0.7 0.7

0.74 0.38 0.17 0.54 0.02 0.94 0.86 0.61 0.34 0.75 0.41 0.88 0.09 0.27 0.25 0.5 0.88 0.43 0.13 0.94 0.72 0.93 0.33 0.97 0.83 0.93 0.28 0.7 0.46 0.87 0.11 0.93 0.71 0.61 0.19 0.45 0.34 0.72 0.2 0.12 0.23 0.94 0.13 0.89 0.5 0.18 0.46 0.19 0.82 0.46 0.06 0.33 0.22 0.04

-0.04 3 0.13 0.84 1.79 -0.03 -1.34 0.1 1.12 1.07 -0.51 0.09 0.83 -0.08 -0.05 -0.53 -1.06 0.6 -1.23 -0.06 0.65 0.06 -0.03 0.3 -0.73 1.31 0.76 1.35 -0.1 -0.32 -0.74 1.56 -0.32 0.79 0.64 -0.42 -0.61 0.95 1.15 -1.08 0.49 -0.92 0.45 -0.04 0.38 -1.02 0.78 0.81 1.18 1.23 -0.62 0.32 -0.22 0.68

$2.59 $2.77 $2.78 $2.83 $2.95 $2.95 $2.86 $2.87 $2.95 $3.02 $2.99 $3.00 $3.06 $3.05 $3.05 $3.01 $2.94 $2.99 $2.90 $2.90 $2.95 $2.95 $2.95 $2.97 $2.93 $3.02 $3.07 $3.17 $3.16 $3.14 $3.09 $3.20 $3.18 $3.24 $3.29 $3.26 $3.22 $3.29 $3.38 $3.30 $3.34 $3.27 $3.31 $3.31 $3.34 $3.26 $3.32 $3.39 $3.48 $3.58 $3.53 $3.56 $3.54 $3.60

$1.90 $1.86 $1.83 $1.80 $1.77 $1.74 $1.69 $1.67 $1.69 $1.73 $1.74 $1.81 $1.76 $1.78 $1.80 $1.82 $1.82 $1.76 $1.76 $1.72 $1.74 $1.73 $1.74 $1.78 $1.73 $1.74 $1.74 $1.79 $1.70 $1.77 $1.79 $1.80 $1.87 $1.92 $1.95 $1.97 $2.05 $2.07 $2.01 $2.10 $2.03 $1.93 $1.95 $1.94 $1.94 $2.02 $2.00 $1.92 $1.89 $1.87 $1.88 $1.89 $1.91 $1.91

$3.00 $2.95 $2.88 $3.02 $2.97 $2.96 $2.97 $3.00 $3.07 $3.12 $3.07 $3.05 $3.08 $3.16 $3.07 $3.04 $3.03 $3.08 $3.07 $3.02 $3.09 $3.14 $3.24 $3.26 $3.21 $3.30 $3.28 $3.28 $3.39 $3.30 $3.34 $3.27 $3.33 $3.35 $3.39 $3.35 $3.40 $3.43 $3.46 $3.48 $3.43 $3.53 $3.61 $3.66 $3.66 $3.57 $3.45 $3.50 $3.57 $3.64 $3.74 $3.85 $3.88 $4.02

211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264

0.7 0.71 0.71 0.71 0.72 0.72 0.72 0.73 0.73 0.73 0.74 0.74 0.74 0.75 0.75 0.75 0.76 0.76 0.76 0.77 0.77 0.77 0.78 0.78 0.78 0.79 0.79 0.79 0.8 0.8 0.8 0.81 0.81 0.81 0.82 0.82 0.82 0.83 0.83 0.83 0.84 0.84 0.84 0.85 0.85 0.85 0.86 0.86 0.86 0.87 0.87 0.87 0.88 0.88

0.77 0.23 0.37 0.44 0.16 0.98 0.78 0.65 0.54 0.15 0.1 0.56 0.99 0.72 0.49 0.5 0.86 0.81 0.29 0.43 0.12 0.85 0.82 0.47 0.01 0.25 0.65 0.27 0.43 0.36 0.1 0.41 0.54 0.83 0.75 0.44 0.88 0.85 0.01 0.89 0.64 0.37 0.63 0.41 0.42 0.17 0.14 0.21 0.18 0.35 0.15 0.7 0.85 0.42

-0.7 0.37 0.41 -2.05 0.14 -1.19 -0.27 0.8 -1.58 0.73 0.18 0.83 -1.25 1.01 0.32 2.04 2.11 -0.49 -1 0.4 0.83 0.87 0.88 0.2 0.25 0.5 -0.08 1.11 -0.48 1.18 0.3 0.15 0.06 -0.59 0.2 -1.4 -0.27 1.95 1.12 -0.12 0.65 -0.15 -0.69 1.03 -0.58 1.13 0.94 -0.63 -0.91 2.21 1.41 2.04 0.82 0.62

$3.55 $3.58 $3.62 $3.45 $3.46 $3.37 $3.35 $3.41 $3.29 $3.35 $3.37 $3.43 $3.33 $3.41 $3.44 $3.61 $3.78 $3.74 $3.66 $3.70 $3.77 $3.85 $3.93 $3.95 $3.97 $4.02 $4.02 $4.12 $4.08 $4.19 $4.23 $4.24 $4.25 $4.20 $4.22 $4.09 $4.06 $4.25 $4.36 $4.35 $4.42 $4.41 $4.34 $4.45 $4.39 $4.51 $4.61 $4.55 $4.45 $4.68 $4.84 $5.07 $5.17 $5.25

$1.84 $1.81 $1.83 $1.88 $1.91 $1.87 $1.83 $1.84 $1.87 $1.91 $1.92 $1.96 $1.92 $1.93 $1.97 $2.00 $2.01 $1.97 $1.99 $2.05 $2.14 $2.15 $2.15 $2.12 $2.15 $2.11 $2.05 $2.11 $2.07 $2.09 $2.12 $2.14 $2.13 $2.19 $2.26 $2.21 $2.25 $2.18 $2.18 $2.23 $2.33 $2.31 $2.39 $2.43 $2.43 $2.51 $2.57 $2.59 $2.62 $2.63 $2.59 $2.60 $2.55 $2.57

$4.02 $4.00 $4.07 $4.14 $3.90 $3.87 $4.02 $4.02 $3.92 $3.97 $4.03 $4.20 $4.24 $4.25 $4.18 $4.16 $4.19 $4.10 $4.25 $4.31 $4.27 $4.15 $4.11 $4.24 $4.29 $4.28 $4.47 $4.37 $4.49 $4.56 $4.63 $4.57 $4.65 $4.42 $4.49 $4.69 $4.53 $4.74 $4.64 $4.73 $4.75 $4.63 $4.66 $4.65 $4.61 $4.62 $4.57 $4.55 $4.68 $4.76 $4.83 $4.69 $4.45 $4.53

265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300

0.88 0.89 0.89 0.89 0.9 0.9 0.9 0.91 0.91 0.91 0.92 0.92 0.92 0.93 0.93 0.93 0.94 0.94 0.94 0.95 0.95 0.95 0.96 0.96 0.96 0.97 0.97 0.97 0.98 0.98 0.98 0.99 0.99 0.99 1 1

0.88 0.24 0.76 0.08 0.47 0.36 0.12 0.84 0.94 0.96 0.72 0.97 0.16 0.17 0.84 0.19 0.09 0.2 0.89 0.8 0.32 0.88 0.65 0.78 0.47 0.9 0.43 0.21 0.94 0.47 0.21 0.06 0.24 0.51 0.96 0.05

0.14 -0.28 -0.19 -1.98 -0.65 0.65 1.31 1.15 -0.52 -0.18 -1.03 1.17 0.03 0.12 1.05 -0.31 2.68 0.25 0.91 -0.7 -0.14 -1.28 -1.21 -0.6 -0.35 -0.54 -0.65 -0.57 -0.6 -1.63 -2.09 0.96 -0.14 -0.14 1.06 0.53

$5.27 $5.24 $5.22 $4.98 $4.91 $4.99 $5.14 $5.28 $5.22 $5.20 $5.09 $5.23 $5.23 $5.25 $5.38 $5.35 $5.68 $5.72 $5.84 $5.75 $5.74 $5.57 $5.42 $5.35 $5.31 $5.25 $5.17 $5.11 $5.04 $4.85 $4.62 $4.73 $4.72 $4.70 $4.82 $4.88

$2.47 $2.43 $2.41 $2.42 $2.46 $2.38 $2.35 $2.44 $2.49 $2.45 $2.37 $2.35 $2.28 $2.23 $2.25 $2.10 $2.05 $2.13 $2.20 $2.26 $2.27 $2.27 $2.31 $2.37 $2.40 $2.48 $2.30 $2.25 $2.19 $2.25 $2.21 $2.15 $2.18 $2.19 $2.33 $2.40

$4.46 $4.38 $4.12 $4.16 $4.13 $4.26 $4.14 $4.19 $4.32 $4.33 $4.35 $4.42 $4.31 $4.36 $4.35 $4.36 $4.44 $4.50 $4.51 $4.59 $4.70 $4.60 $4.54 $4.53 $4.39 $4.26 $4.18 $4.09 $3.96 $4.10 $4.11 $3.96 $4.03 $4.07 $4.04 $3.99

S -4 S -5 S -6 Drift line S + ∆S S + ∆S S + ∆S 0 $2.00 $2.00 $2.00 0 $1.95 $1.98 $2.05 $1.93 $1.96 $2.05 $1.93 $1.96 $2.03 $1.93 $1.96 $2.05 $1.95 $1.97 $1.98 $2.01 $1.97 $2.04 $2.02 $1.93 $2.12 $2.04 $1.85 $2.13 $2.01 $1.89 $2.15 $1.95 $1.88 $2.17 $1.95 $1.93 $2.11 $2.03 $1.86 $2.16 $2.04 $1.93 $2.21 $1.96 $1.98 $2.17 $2.05 $2.06 $2.16 $2.01 $2.00 $2.19 $1.98 $2.00 $2.21 $1.98 $2.04 $2.20 $2.05 $2.13 $2.24 $2.02 $2.14 $2.23 $2.03 $2.10 $2.20 $1.92 $2.04 $2.18 $1.91 $2.07 $2.23 $1.86 $2.08 $2.22 $1.86 $2.04 $2.21 $1.74 $1.95 $2.25 $1.69 $2.03 $2.25 $1.64 $2.05 $2.30 $1.69 $2.00 $2.20 $1.69 $2.05 $2.20 $1.64 $2.08 $2.10 $1.70 $2.06 $2.16 $1.67 $2.05 $2.28 $1.64 $2.00 $2.26 $1.62 $2.00 $2.27 $1.64 $2.07 $2.19 $1.66 $2.09 $2.16 $1.59 $2.00 $2.16 $1.58 $1.96 $2.18 $1.59 $2.01 $2.26 $1.58 $2.02 $2.19 $1.63 $2.03 $2.23 $1.63 $2.11 $2.26 $1.60 $2.10 $2.34 $1.56 $2.14 $2.22 $1.53 $2.10 $2.17 $1.53 $2.06 $2.20 $1.54 $2.08 $2.21

1 0

$1.55 $1.58 $1.55 $1.60 $1.55 $1.61 $1.57 $1.55 $1.52 $1.48 $1.44 $1.40 $1.34 $1.34 $1.35 $1.31 $1.34 $1.31 $1.30 $1.37 $1.30 $1.33 $1.33 $1.35 $1.33 $1.32 $1.28 $1.29 $1.32 $1.34 $1.29 $1.30 $1.32 $1.37 $1.41 $1.39 $1.43 $1.37 $1.35 $1.30 $1.30 $1.34 $1.38 $1.41 $1.42 $1.38 $1.37 $1.37 $1.37 $1.42 $1.42 $1.40 $1.40 $1.42

$2.12 $2.21 $2.20 $2.28 $2.36 $2.36 $2.36 $2.37 $2.32 $2.35 $2.32 $2.29 $2.27 $2.27 $2.17 $2.13 $2.19 $2.12 $2.04 $2.03 $2.11 $2.06 $2.00 $2.02 $2.06 $2.05 $2.05 $2.00 $1.93 $1.92 $1.96 $2.02 $2.06 $2.15 $2.23 $2.29 $2.24 $2.23 $2.26 $2.26 $2.25 $2.27 $2.28 $2.31 $2.31 $2.33 $2.37 $2.47 $2.47 $2.46 $2.42 $2.40 $2.38 $2.40

$2.22 $2.22 $2.11 $2.17 $2.21 $2.27 $2.24 $2.19 $2.21 $2.25 $2.36 $2.39 $2.35 $2.22 $2.22 $2.19 $2.18 $2.10 $2.08 $2.10 $2.10 $2.25 $2.26 $2.28 $2.32 $2.30 $2.30 $2.35 $2.35 $2.31 $2.30 $2.20 $2.22 $2.22 $2.29 $2.28 $2.26 $2.21 $2.22 $2.31 $2.39 $2.41 $2.43 $2.38 $2.40 $2.34 $2.34 $2.47 $2.43 $2.47 $2.49 $2.53 $2.58 $2.53

$1.37 $1.34 $1.31 $1.30 $1.30 $1.32 $1.33 $1.36 $1.37 $1.41 $1.44 $1.45 $1.41 $1.41 $1.39 $1.37 $1.31 $1.36 $1.36 $1.33 $1.32 $1.31 $1.33 $1.38 $1.37 $1.38 $1.39 $1.39 $1.36 $1.33 $1.32 $1.34 $1.36 $1.37 $1.42 $1.41 $1.43 $1.42 $1.44 $1.46 $1.42 $1.42 $1.43 $1.42 $1.38 $1.34 $1.38 $1.43 $1.43 $1.48 $1.46 $1.49 $1.53 $1.51

$2.44 $2.46 $2.39 $2.25 $2.19 $2.15 $2.20 $2.27 $2.25 $2.21 $2.25 $2.22 $2.30 $2.30 $2.27 $2.25 $2.24 $2.19 $2.16 $2.10 $2.16 $2.06 $2.00 $1.99 $2.03 $2.01 $1.97 $1.99 $1.94 $1.93 $1.91 $1.90 $1.93 $1.90 $1.93 $1.91 $1.93 $1.92 $1.90 $1.86 $1.84 $1.87 $1.77 $1.79 $1.80 $1.80 $1.84 $1.86 $1.86 $1.86 $1.83 $1.90 $1.86 $1.86

$2.51 $2.52 $2.53 $2.60 $2.59 $2.75 $2.66 $2.72 $2.60 $2.64 $2.61 $2.70 $2.64 $2.65 $2.58 $2.62 $2.63 $2.52 $2.50 $2.45 $2.41 $2.52 $2.49 $2.52 $2.63 $2.64 $2.64 $2.65 $2.62 $2.58 $2.64 $2.66 $2.71 $2.69 $2.70 $2.64 $2.62 $2.64 $2.51 $2.50 $2.47 $2.44 $2.42 $2.41 $2.48 $2.47 $2.43 $2.43 $2.42 $2.44 $2.45 $2.49 $2.50 $2.58

$1.51 $1.54 $1.54 $1.52 $1.49 $1.51 $1.49 $1.53 $1.54 $1.58 $1.57 $1.57 $1.51 $1.52 $1.51 $1.51 $1.45 $1.45 $1.45 $1.45 $1.46 $1.38 $1.33 $1.26 $1.27 $1.29 $1.30 $1.31 $1.31 $1.33 $1.41 $1.45 $1.44 $1.44 $1.42 $1.42 $1.41 $1.42 $1.41 $1.35 $1.34 $1.28 $1.33 $1.38 $1.39 $1.39 $1.41 $1.43 $1.45 $1.38 $1.37 $1.38 $1.34 $1.29

$1.82 $1.83 $1.80 $1.72 $1.74 $1.73 $1.76 $1.73 $1.76 $1.71 $1.74 $1.75 $1.73 $1.79 $1.84 $1.90 $1.87 $1.87 $1.92 $1.94 $1.91 $1.95 $2.00 $2.02 $2.08 $2.16 $2.13 $2.16 $2.10 $2.18 $2.14 $2.04 $2.03 $2.08 $2.02 $2.05 $2.05 $2.04 $2.05 $1.98 $2.01 $1.97 $1.98 $1.97 $1.99 $2.01 $1.98 $1.94 $1.91 $1.97 $1.98 $2.00 $1.96 $1.96

$2.57 $2.58 $2.57 $2.63 $2.57 $2.61 $2.53 $2.66 $2.64 $2.72 $2.70 $2.76 $2.68 $2.72 $2.71 $2.64 $2.57 $2.61 $2.59 $2.59 $2.54 $2.51 $2.58 $2.57 $2.61 $2.59 $2.49 $2.47 $2.36 $2.38 $2.35 $2.34 $2.29 $2.32 $2.29 $2.26 $2.26 $2.24 $2.21 $2.26 $2.22 $2.25 $2.26 $2.25 $2.23 $2.16 $2.15 $2.15 $2.20 $2.26 $2.35 $2.24 $2.19 $2.24

$1.31 $1.32 $1.34 $1.34 $1.31 $1.28 $1.33 $1.33 $1.38 $1.41 $1.41 $1.42 $1.41 $1.40 $1.40 $1.46 $1.47 $1.46 $1.42 $1.39 $1.36 $1.30 $1.25 $1.27 $1.29 $1.30 $1.30 $1.34 $1.34 $1.33 $1.35 $1.32 $1.35 $1.37 $1.37 $1.34 $1.31 $1.33 $1.35 $1.33 $1.29 $1.30 $1.27 $1.26 $1.29 $1.30 $1.26 $1.30 $1.28 $1.32 $1.32 $1.31 $1.29 $1.36

$1.90 $1.87 $1.88 $1.91 $1.89 $1.94 $1.95 $1.98 $1.94 $1.96 $2.01 $2.04 $2.01 $2.01 $2.00 $1.97 $1.97 $1.96 $2.01 $1.98 $1.96 $1.90 $1.91 $1.78 $1.79 $1.75 $1.72 $1.77 $1.75 $1.80 $1.79 $1.77 $1.78 $1.84 $1.89 $1.90 $1.87 $1.84 $1.84 $1.84 $1.90 $1.91 $1.91 $1.93 $1.96 $1.97 $2.00 $2.08 $2.09 $2.12 $2.04 $2.07 $1.97 $2.10

$2.24 $2.30 $2.25 $2.25 $2.21 $2.17 $2.12 $2.11 $2.13 $2.11 $2.06 $2.12 $2.14 $2.15 $2.11 $2.19 $2.18 $2.18 $2.20 $2.07 $2.12 $2.24 $2.17 $2.14 $2.15 $2.21 $2.16 $2.15 $2.22 $2.20 $2.17 $2.16 $2.12 $2.11 $2.18 $2.17 $2.19 $2.12 $2.16 $2.13 $2.12 $2.21 $2.16 $2.07 $2.08 $2.07 $2.18 $2.26 $2.32 $2.34 $2.42 $2.45 $2.47 $2.62

$1.36 $1.35 $1.33 $1.29 $1.29 $1.37 $1.35 $1.42 $1.39 $1.38 $1.39 $1.35 $1.34 $1.36 $1.36 $1.36 $1.46 $1.46 $1.41 $1.42 $1.38 $1.33 $1.28 $1.31 $1.36 $1.38 $1.37 $1.37 $1.36 $1.33 $1.24 $1.27 $1.29 $1.26 $1.30 $1.30

$2.07 $2.08 $2.01 $2.00 $2.00 $1.99 $1.99 $1.98 $2.04 $2.04 $2.01 $1.92 $1.92 $1.92 $1.95 $2.03 $1.91 $1.87 $1.88 $1.90 $1.91 $1.90 $1.82 $1.80 $1.75 $1.75 $1.80 $1.81 $1.79 $1.83 $1.85 $1.72 $1.71 $1.74 $1.77 $1.68

$2.52 $2.44 $2.51 $2.47 $2.48 $2.52 $2.53 $2.59 $2.54 $2.58 $2.54 $2.46 $2.40 $2.47 $2.42 $2.51 $2.45 $2.47 $2.54 $2.51 $2.53 $2.53 $2.56 $2.58 $2.56 $2.57 $2.53 $2.50 $2.47 $2.52 $2.49 $2.47 $2.55 $2.49 $2.51 $2.50

Stock Price Paths with Jumps Source: Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. Brownian motion: p. 226, Jump Diffusion (Merton): p. 446

Geometric Brownian Motion with Jump Term Δ S

=μ Δ t σ ε  Δ t κ Δ q S 0 w ith  p ro b a b ility  1− λ Δ t 1 w ith  p ro b a b ility  λ Δ t ¿ ¿ ¿ Δ q =¿ {¿ ¿ ¿ ¿ ¿

The variable ∆ S is the change in the stock price, S, in a small interval of time ∆ t. are independent. The parameters: µ is the expected rate of return per unit of time (usually per year) σ is the volatility of the stock price (σ and µ are assumed contant) ε is a random drawing from a standardized normal distribution, Φ (0,1) κ Average jump size measured as a proportional increase in asset price λ Rate at which jumps happen per time unit (year), also called INTENSITY of Poisson process Numerical example (press F9 to generate new random numbers) Stock price to t=0 $2.00 Time (T) 11 year(s) Drift (µ) 20.00% Volatility (σ) 40.00% Intensity (λ) Lambda 0.50 number of jumps per year Jump size (κ) Kappa -5.0% as % of previous stock price % stock price after jump (J) 95.0% as % of previous stock price γ Gamma = ln(1+κ) -0.05 Drift of ln(J) Standard deviation log of J (σ') J (-σ') J (+σ') 50.00% ∆T 0.0220 0.58 1.57

Simulation of Stock Price Paths with Jump Term (Dynamic Jump Size) $8.00 $6.00 $4.00

Pure Brownian Motion

$2.00 $-$2.00 -$4.00 0

2

4 Time

6

8

10

12

Brownian motion with stochastic jumps (ln J normally distributed)

Jump

$-$2.00 -$4.00 0

2

4 Time

6

8

10

12

ice-Hall.

FIXED JUMP SIZE Pure Brownian Brownian Motionmotion Jump with fixed jumps Period

dependent.

rocess

mp Term

Brownian motion with stochastic jumps (ln J normally distributed)

Jump

Time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 $6.00 30 31 $5.00 32 33 $4.00 34 35 $3.00 36 37 $2.00 38 39 $1.00 40 41 $42 -$1.00 43 44 -$2.00 45

Normal Variates Rand() 0 -0.51 0.02 1.82 0.39 0.04 -1.08 0.39 0.07 -0.67 0.54 0.09 -1.04 0.15 0.11 1.2 0.68 0.13 0.01 0.49 0.15 -0.67 0.24 0.18 -2.33 0.42 0.2 -0.26 0.99 0.22 -0.18 0.57 0.24 1.69 0.32 0.26 0.12 0.51 0.29 -2.62 0.18 0.31 1.09 0.68 0.33 1.02 0.78 0.35 0.32 0.34 0.37 -0.47 0.53 0.4 0.08 0.78 0.42 -0.72 0.04 0.44 0.5 0.33 0.46 -1.27 0.78 0.48 -0.12 0.74 0.51 2.37 0.65 0.53 -0.22 0.97 0.55 -0.65 0.52 0.57 -0.57 0.57 Simulation 0.59 0.76 0.73 0.62 -1 0.97 0.64 -0.1 0.59 0.66 -0.35 0.03 0.68 0.24 0.47 0.7 -0.19 0.98 0.73 0.01 0.42 0.75 1.02 0.01 0.77 0.18 0.12 0.79 -0.28 0.1 0.81 -0.39 0.5 0.84 0.92 0.36 0.86 -0.26 0.52 0.88 0.66 0.49 0.9 -0.68 0.93 0.92 -0.38 0.84 0.95 -0.54 1 0.97 -0.66 0.11 0.99 -0.79 0.52

0

2

4

Time

6

of

8

$2.00 $2.00 $2.13 $2.13 $$2.28 $2.28 $$1.99 $1.99 $$1.82 $1.82 $$1.74 $1.74 $$1.80 $1.80 $$1.88 $1.88 $$2.08 $1.99 -$0.094 $2.12 $2.03 -$0.094 $2.07 $1.98 -$0.094 $2.02 $1.93 -$0.094 $1.90 $1.72 -$0.191 $1.86 $1.68 -$0.191 $1.76 $1.59 -$0.191 $1.78 $1.61 -$0.191 $1.71 $1.55 -$0.191 $1.49 $1.35 -$0.191 $1.39 $1.26 -$0.191 $1.39 $1.26 -$0.191 $1.47 $1.33 -$0.191 $1.54 $1.39 -$0.191 $1.67 $1.51 -$0.191 $1.45 $1.31 -$0.191 $1.51 $1.37 -$0.191 $1.61 $1.46 -$0.191 $1.49 $1.27 -$0.264 Stock Price $1.47 $1.26 Paths -$0.264with Jump Ter (Fixed$1.37 Jump-$0.264 Size) $1.61 $1.78 $1.53 -$0.264 $1.75 $1.50 -$0.264 $1.76 $1.50 -$0.264 $1.85 $1.58 -$0.264 $1.78 $1.52 -$0.264 $1.79 $1.53 -$0.264 $1.91 $1.64 -$0.264 $1.90 $1.63 -$0.264 Brownian Browni $1.81 $1.55 Pure -$0.264 with fix $1.85 $1.58 Motion -$0.264 $1.83 $1.56 -$0.264 $1.92 $1.64 -$0.264 $2.12 $1.81 -$0.264 $2.19 $1.87 -$0.264 $2.39 $2.04 -$0.264 $2.32 $1.98 -$0.264 $2.48 $2.12 -$0.264

10 12

$-$1.00 -$2.00 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

1.01 0 1.03 1.06 1.08 1.1 1.12 1.14 1.17 1.19 1.21 1.23 1.25 1.28 1.3 1.32 1.34 1.36 1.39 1.41 1.43 1.45 1.47 1.5 1.52 1.54 1.56 1.58 1.61 1.63 1.65 1.67 1.69 1.72 1.74 1.76 1.78 1.8 1.83 1.85 1.87 1.89 1.91 1.94 1.96 1.98 2 2.02 2.05 2.07 2.09 2.11 2.13 2.16 2.18

2

-1.08 4 6 0.77 8 1.17 0.46 Time 0.04 0.08 0.44 0.55 -1.2 0.5 -0.75 0.38 0.75 0.33 0.17 0.24 -0.78 0.03 -0.89 0.3 0.93 0.76 0 0.6 1.76 0.03 2.11 0.73 1.75 0.18 -0.45 0.06 0.2 0.2 -0.3 0.16 0.95 0.48 -0.19 0.21 1.7 0.28 -0.78 0.58 0.74 0.71 0.77 0.65 0.3 0.1 0.04 0.2 -1.92 0.58 1.89 0.93 0.89 0.2 -0.58 0.69 -0.89 0.45 1.68 0.97 -0.99 0.15 0.58 0.49 0.32 0.52 -0.59 0.64 1.6 0.87 -0.26 0.85 1.01 0.88 -0.99 0.9 0.21 0.16 0.43 0.64 0.42 0.49 0.11 0.19 0.39 0.38 0.31 0.67 1.12 0.25 -0.75 0.57 -2.47 0.83 0.52 0.74 0.39 0.78 1.78 0.12 1.38 0.32 -0.2 0.5

$2.48 10 $2.49 $2.68 $2.72 $3.14 $3.24 $3.18 $3.44 $3.55 $3.31 $3.46 $3.62 $3.64 $3.77 $3.92 $4.19 $3.97 $4.00 $3.72 $3.83 $3.79 $3.82 $4.07 $3.60 $3.93 $4.02 $4.48 $4.59 $4.65 $4.48 $4.50 $3.92 $4.11 $4.05 $4.13 $4.32 $3.93 $4.34 $4.47 $4.65 $4.26 $4.23 $4.25 $4.31 $4.02 $3.86 $3.84 $3.72 $3.89 $3.93 $4.32 $4.07 $3.74 $3.93

12$2.12 $2.13 $2.29 $2.33 $2.68 $2.77 $2.72 $2.94 $3.03 $2.83 $2.96 $3.10 $3.11 $3.06 $3.19 $3.41 $3.23 $3.26 $3.03 $3.11 $3.08 $3.11 $3.15 $2.79 $3.04 $3.12 $3.47 $3.56 $3.60 $3.47 $3.49 $3.04 $3.18 $3.14 $3.20 $3.35 $3.04 $3.36 $3.46 $3.61 $3.30 $3.28 $3.29 $3.34 $3.12 $2.99 $2.98 $2.89 $3.02 $3.05 $3.35 $3.16 $2.90 $3.05

-$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.264 -$0.419 -$0.419 -$0.419 -$0.419 -$0.419 -$0.419 -$0.419 -$0.419 -$0.419 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153

2.2 2.22 2.24 2.27 2.29 2.31 2.33 2.35 2.38 2.4 2.42 2.44 2.46 2.49 2.51 2.53 2.55 2.57 2.6 2.62 2.64 2.66 2.68 2.71 2.73 2.75 2.77 2.79 2.82 2.84 2.86 2.88 2.9 2.93 2.95 2.97 2.99 3.01 3.04 3.06 3.08 3.1 3.12 3.15 3.17 3.19 3.21 3.23 3.26 3.28 3.3 3.32 3.34 3.37

0.65 0.09 1.55 0.73 0.01 0.68 -1.64 0.5 0.54 -0.8 -0.23 -0.44 -0.03 -0.37 0.6 -0.44 0.04 -0.5 2.84 -0.91 1.04 0.38 0.85 0.58 1.15 0.88 -0.21 0.06 0.75 -0.43 1.61 0.03 1.12 1.21 -0.58 -0.32 0.35 -0.43 -1.52 -0.4 0.11 -0.06 0.45 -1.87 0.96 -0.25 -0.36 -0.39 0.55 -0.36 0.07 0.15 -2.51 -0.44

0.76 0.41 0.7 0.34 0.35 0.89 0.04 0.79 0.86 0.18 0.29 0.38 0.83 0.16 0.24 0.71 0.05 0.39 0.35 0.54 0.58 0.73 0.22 0.83 0.3 0.05 0.57 0.09 0.17 0.88 0.58 0.93 0.3 0.28 0.27 0.64 0.17 0.31 0.44 0.04 0.49 0.72 0.42 0.32 0.88 0.66 0.03 0.93 0.05 0.38 0.47 0.63 0.11 0.69

$3.73 $3.05 $3.02 $2.84 $2.91 $3.24 $3.20 $3.00 $3.38 $3.19 $3.21 $3.26 $3.55 $3.60 $3.63 $3.69 $3.60 $3.66 $3.14 $3.32 $3.14 $2.93 $2.61 $2.66 $2.63 $2.76 $2.60 $2.55 $2.78 $2.66 $2.50 $2.32 $2.22 $2.25 $2.16 $2.30 $2.32 $2.37 $2.67 $2.68 $2.81 $2.82 $2.58 $2.68 $2.44 $2.48 $2.41 $2.37 $2.26 $2.20 $2.08 $2.04 $2.02 $1.93

$2.89 $2.36 $2.34 $2.20 $2.26 $2.51 $2.48 $2.33 $2.62 $2.47 $2.49 $2.53 $2.75 $2.79 $2.82 $2.86 $2.79 $2.84 $2.44 $2.57 $2.43 $2.27 $2.02 $2.06 $2.04 $2.14 $2.02 $1.98 $2.16 $2.07 $1.94 $1.80 $1.72 $1.74 $1.67 $1.78 $1.80 $1.84 $2.07 $2.08 $2.18 $2.19 $2.00 $2.08 $1.89 $1.93 $1.78 $1.75 $1.66 $1.62 $1.53 $1.50 $1.49 $1.42

-$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.575 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671

154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207

3.39 3.41 3.43 3.45 3.48 3.5 3.52 3.54 3.56 3.59 3.61 3.63 3.65 3.67 3.7 3.72 3.74 3.76 3.78 3.81 3.83 3.85 3.87 3.89 3.92 3.94 3.96 3.98 4 4.03 4.05 4.07 4.09 4.11 4.14 4.16 4.18 4.2 4.22 4.25 4.27 4.29 4.31 4.33 4.36 4.38 4.4 4.42 4.44 4.47 4.49 4.51 4.53 4.55

-0.58 1.18 -1.07 0.52 -0.24 1.37 -1.82 -0.38 -0.18 1.3 -0.71 0.55 -0.59 1.16 -1.74 -0.41 -0.77 0.21 0.9 1.25 0.29 0.38 -0.54 1.96 2.57 -2.67 -0.43 0.06 0.16 -0.41 1.05 1.02 -0.77 2.6 0.11 0.32 1.34 0.2 2.04 -0.42 -0.04 -0.78 -1.69 0.71 -1.31 -1.36 -1.27 -0.48 0.44 1.4 -0.8 -0.57 0.17 0.02

0.46 0.41 0.74 0.02 0.5 0.9 0.91 0.08 0.83 0.2 0.36 0.11 0.85 0.53 0.41 0.28 0.57 0.91 0.01 0.99 0.22 0.89 0.64 0.25 0.81 0.69 0.63 0.29 0.32 0.74 0.97 0.77 0.15 0.71 0.79 0.65 0.61 0.7 0.73 0.44 0.9 0.09 0.55 0.75 0.62 0.97 0.04 0.18 0.87 0.04 0.17 0.1 0.93 0.81

$2.13 $2.25 $2.51 $2.33 $2.20 $2.37 $2.32 $2.25 $2.55 $2.58 $2.75 $2.63 $2.60 $2.56 $2.72 $2.67 $2.86 $2.92 $3.11 $3.40 $3.33 $3.61 $3.64 $3.71 $3.60 $3.73 $4.05 $4.03 $3.68 $3.60 $3.68 $3.22 $3.04 $3.10 $2.89 $2.82 $2.82 $2.80 $2.86 $2.86 $3.16 $3.16 $3.27 $3.20 $3.40 $3.43 $3.54 $3.67 $3.45 $3.49 $3.58 $3.58 $3.57 $3.37

$1.57 $1.66 $1.85 $1.71 $1.62 $1.75 $1.70 $1.66 $1.88 $1.89 $2.02 $1.94 $1.91 $1.88 $2.00 $1.96 $2.10 $2.15 $2.29 $2.50 $2.45 $2.65 $2.67 $2.73 $2.64 $2.74 $2.98 $2.96 $2.70 $2.64 $2.71 $2.37 $2.24 $2.28 $2.12 $2.07 $2.07 $2.06 $2.11 $2.10 $2.33 $2.33 $2.40 $2.35 $2.50 $2.52 $2.61 $2.70 $2.54 $2.57 $2.63 $2.63 $2.62 $2.48

-$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671 -$0.671

208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261

4.58 4.6 4.62 4.64 4.66 4.69 4.71 4.73 4.75 4.77 4.8 4.82 4.84 4.86 4.88 4.91 4.93 4.95 4.97 4.99 5.02 5.04 5.06 5.08 5.1 5.13 5.15 5.17 5.19 5.21 5.24 5.26 5.28 5.3 5.32 5.35 5.37 5.39 5.41 5.43 5.46 5.48 5.5 5.52 5.54 5.57 5.59 5.61 5.63 5.65 5.68 5.7 5.72 5.74

-0.64 0.15 0.03 0.24 -1.38 -1.44 1.51 1.54 1.4 -1.02 1.5 -0.09 0.78 1 -1.71 0.71 -0.92 0.06 2.03 -0.76 -0.57 -1.45 -0.49 -0.28 -0.28 2.1 -0.5 0.26 -0.62 1.14 -1.24 0.07 -0.17 0.3 -1.16 0.04 0.5 -1.58 -0.11 0.28 -0.8 -0.28 -1.44 2.51 -0.74 -1.18 0.69 -0.23 0.37 0.62 0.34 1.48 0.87 1.61

0.35 0.74 0.5 0.98 0.03 0.82 0.72 1 0.59 0.87 0.71 0.38 0.51 0.32 0.08 0.24 0.76 0.99 0.33 0.31 0.74 0.94 0.28 0.77 0.12 0.15 0.82 0.29 0.24 0.75 0.1 0.59 0.49 0.6 0.57 0.52 0.42 0.29 0.51 0.01 0.15 0.22 0.39 0.67 0.54 0.47 0.91 0.3 0.46 0.23 0.61 0.2 0.17 0.89

$3.11 $3.23 $3.46 $3.97 $3.75 $4.10 $4.01 $4.22 $4.64 $4.64 $4.58 $4.18 $4.39 $4.70 $4.71 $4.66 $4.56 $4.54 $4.86 $4.28 $4.49 $4.65 $4.70 $4.70 $4.65 $4.81 $4.23 $3.82 $3.59 $3.60 $3.44 $3.30 $3.15 $2.89 $2.77 $2.73 $2.35 $2.37 $2.24 $2.48 $2.13 $2.15 $1.88 $1.98 $1.96 $1.97 $1.88 $1.96 $2.15 $1.96 $2.04 $2.12 $2.22 $2.16

$2.28 $2.26 $2.31 $2.65 $2.50 $2.74 $2.68 $2.81 $3.10 $3.10 $3.06 $2.79 $2.93 $3.13 $3.14 $3.11 $3.04 $3.03 $3.24 $2.85 $2.99 $3.10 $3.14 $3.14 $3.11 $3.21 $2.82 $2.55 $2.40 $2.40 $2.29 $2.20 $2.10 $1.93 $1.85 $1.82 $1.57 $1.58 $1.50 $1.65 $1.42 $1.43 $1.26 $1.32 $1.31 $1.32 $1.25 $1.30 $1.43 $1.31 $1.36 $1.42 $1.48 $1.44

-$0.671 -$0.785 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898

262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315

5.76 5.79 5.81 5.83 5.85 5.87 5.9 5.92 5.94 5.96 5.98 6.01 6.03 6.05 6.07 6.09 6.12 6.14 6.16 6.18 6.2 6.23 6.25 6.27 6.29 6.31 6.34 6.36 6.38 6.4 6.42 6.45 6.47 6.49 6.51 6.53 6.56 6.58 6.6 6.62 6.64 6.67 6.69 6.71 6.73 6.75 6.78 6.8 6.82 6.84 6.86 6.89 6.91 6.93

-0.47 -0.85 1.46 0.32 0.83 -0.86 0 1.31 0.6 1.49 0.05 1.03 -0.13 -0.8 1.31 1.31 0.93 -1.18 -0.54 1.28 -1.22 0.06 -2.11 1.11 1.51 0.42 0.25 0.17 0.23 -0.23 0.04 1.35 0.26 -0.14 0.09 -0.25 0.36 -1.82 -0.51 -0.35 1.83 0.94 -0.8 -0.21 -1.76 -1.19 -0.47 1.09 -0.72 0.29 0.71 -0.39 -1.09 0.77

0.97 0.29 0.03 0.79 0.58 0.28 0.53 0.69 0.87 0.02 0.29 0.44 0.54 0.71 0.73 0.05 0.72 0.88 0.27 0.11 0.54 0.81 0.58 0.45 0.1 0.03 0.68 0.71 0.23 0.85 0.6 0.2 0.14 0.63 0.99 0.72 0.91 0.52 0.41 0.78 0.54 0.7 0.22 0.07 0.41 0.95 0.12 0.13 0.82 0.39 0.23 0.37 0.2 0.81

$2.06 $1.95 $2.18 $2.08 $2.00 $1.87 $1.93 $1.91 $1.87 $2.01 $2.02 $2.05 $1.97 $1.98 $1.84 $1.82 $1.82 $1.69 $1.85 $1.85 $1.99 $1.95 $2.19 $2.02 $1.85 $2.01 $1.89 $2.00 $2.06 $2.22 $2.07 $2.20 $2.02 $1.86 $1.67 $1.62 $1.55 $1.61 $1.67 $1.69 $1.72 $1.64 $1.52 $1.62 $1.68 $1.61 $1.55 $1.46 $1.42 $1.36 $1.39 $1.32 $1.33 $1.36

$1.37 $1.30 $1.46 $1.39 $1.33 $1.24 $1.29 $1.27 $1.25 $1.28 $1.29 $1.30 $1.26 $1.26 $1.17 $1.16 $1.16 $1.07 $1.18 $1.18 $1.27 $1.24 $1.39 $1.29 $1.18 $1.28 $1.21 $1.28 $1.31 $1.42 $1.32 $1.40 $1.28 $1.19 $1.06 $1.03 $0.99 $1.02 $1.06 $1.07 $1.09 $1.04 $0.97 $1.03 $1.07 $1.02 $0.99 $0.93 $0.91 $0.86 $0.88 $0.84 $0.85 $0.87

-$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.898 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961 -$0.961

316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369

6.95 6.97 7 7.02 7.04 7.06 7.08 7.11 7.13 7.15 7.17 7.19 7.22 7.24 7.26 7.28 7.3 7.33 7.35 7.37 7.39 7.41 7.44 7.46 7.48 7.5 7.52 7.55 7.57 7.59 7.61 7.63 7.66 7.68 7.7 7.72 7.74 7.77 7.79 7.81 7.83 7.85 7.88 7.9 7.92 7.94 7.96 7.99 8.01 8.03 8.05 8.07 8.1 8.12

-0.79 -1.45 -0.15 0.9 0.36 -1.82 -0.76 -1.01 1.6 0.94 0.25 -0.05 -0.75 -0.71 0.04 0.08 0.26 -0.05 -0.36 0.9 1.27 -0.26 1.5 -0.78 -0.64 -0.95 0.96 -1.98 0.05 1.88 0.86 0.62 -1.72 -0.69 0.12 0.48 -0.58 0.74 1 -0.75 0.24 -0.15 0.54 0.93 0.47 -0.76 -0.37 -0.57 0.54 0.57 -1.28 0.26 -1.21 -1.83

0.81 0.3 0.84 0.49 0.01 0.07 0.34 0.61 0.27 0.48 0.24 0.26 0.2 0.15 0.78 0.61 0.93 0.32 0.31 0.16 0.39 0.72 0.1 0.52 0.84 0.92 0.91 0.08 0.29 0.11 0.89 0.1 0.41 0.73 0.6 0.42 0.81 0.94 0.03 0.08 0.42 0.28 0.34 0.62 0.43 0.12 0.24 0.36 0.45 0.55 0.52 0.84 0.27 0.62

$1.32 $1.35 $1.36 $1.35 $1.34 $1.26 $1.24 $1.25 $1.44 $1.42 $1.45 $1.29 $1.34 $1.41 $1.47 $1.45 $1.43 $1.40 $1.43 $1.42 $1.35 $1.32 $1.36 $1.35 $1.35 $1.50 $1.53 $1.42 $1.34 $1.26 $1.30 $1.30 $1.35 $1.30 $1.39 $1.31 $1.29 $1.23 $1.27 $1.25 $1.27 $1.23 $1.22 $1.19 $1.09 $1.03 $1.06 $1.05 $1.06 $1.03 $1.10 $1.14 $1.16 $1.17

$0.84 $0.81 $0.82 $0.82 $0.81 $0.77 $0.75 $0.75 $0.87 $0.86 $0.88 $0.78 $0.81 $0.85 $0.89 $0.88 $0.86 $0.85 $0.86 $0.86 $0.82 $0.80 $0.82 $0.82 $0.81 $0.91 $0.93 $0.86 $0.81 $0.76 $0.79 $0.79 $0.82 $0.79 $0.84 $0.79 $0.78 $0.74 $0.77 $0.76 $0.77 $0.75 $0.74 $0.72 $0.66 $0.62 $0.64 $0.64 $0.64 $0.62 $0.67 $0.69 $0.70 $0.71

-$0.961 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003

370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423

8.14 8.16 8.18 8.21 8.23 8.25 8.27 8.29 8.32 8.34 8.36 8.38 8.4 8.43 8.45 8.47 8.49 8.51 8.54 8.56 8.58 8.6 8.62 8.65 8.67 8.69 8.71 8.73 8.76 8.78 8.8 8.82 8.84 8.87 8.89 8.91 8.93 8.95 8.98 9 9.02 9.04 9.06 9.09 9.11 9.13 9.15 9.17 9.2 9.22 9.24 9.26 9.28 9.31

0.91 -0.32 -0.83 0.38 -0.26 0.59 0.3 -0.84 -0.11 0.42 -0.13 -2.75 -0.4 0.61 0.75 -0.89 1.68 -0.33 0.33 0.41 -0.86 -0.5 1.19 0.13 0.24 0.22 -0.63 0.51 -0.89 -0.31 0.61 -2.52 0.68 1.49 0.4 -1.04 0.4 -0.6 -0.38 -1.23 1.53 0.84 -1.22 -0.57 0.07 1.2 -0.07 -0.05 -0.66 -1.28 -1.08 -0.13 -0.23 -2.01

0.36 0.11 0.54 0.27 0.19 0.84 0.38 0.08 0.94 0.79 0.81 0.54 0.21 0.62 0.47 0.24 0.7 0.89 0.52 0.04 0.51 0.95 0.17 0.75 0.31 0.61 0.3 0.83 0.45 0.57 0.45 0.81 0.68 0.99 0.08 0.87 0.83 0.46 0.94 0.77 0.25 0.76 0.3 0.45 0.37 0.78 0.7 0.07 0.67 0.21 0.12 0.18 0.16 0.29

$1.10 $0.98 $0.99 $1.03 $0.96 $0.99 $1.07 $1.13 $1.06 $1.08 $0.98 $1.10 $1.15 $1.15 $1.09 $1.08 $1.15 $1.20 $1.24 $1.20 $1.08 $1.13 $1.12 $1.17 $1.11 $1.21 $1.05 $0.96 $0.98 $0.99 $1.01 $1.06 $1.08 $1.00 $1.00 $1.05 $1.10 $1.08 $1.10 $1.18 $1.35 $1.38 $1.47 $1.54 $1.41 $1.24 $1.18 $1.28 $1.35 $1.44 $1.36 $1.45 $1.48 $1.50

$0.66 $0.59 $0.60 $0.62 $0.58 $0.60 $0.65 $0.69 $0.64 $0.65 $0.59 $0.66 $0.70 $0.69 $0.66 $0.65 $0.69 $0.73 $0.75 $0.72 $0.66 $0.68 $0.68 $0.71 $0.67 $0.73 $0.64 $0.58 $0.59 $0.60 $0.61 $0.64 $0.65 $0.61 $0.61 $0.63 $0.67 $0.65 $0.66 $0.71 $0.81 $0.83 $0.89 $0.93 $0.85 $0.75 $0.71 $0.74 $0.74 $0.79 $0.75 $0.80 $0.81 $0.83

-$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.003 -$1.039 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076

424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477

9.33 9.35 9.37 9.39 9.42 9.44 9.46 9.48 9.5 9.53 9.55 9.57 9.59 9.61 9.64 9.66 9.68 9.7 9.72 9.75 9.77 9.79 9.81 9.83 9.86 9.88 9.9 9.92 9.94 9.97 9.99 10.01 10.03 10.05 10.08 10.1 10.12 10.14 10.16 10.19 10.21 10.23 10.25 10.27 10.3 10.32 10.34 10.36 10.38 10.41 10.43 10.45 10.47 10.49

3.02 -2.63 0.28 -0.63 0.52 0.83 0.36 -0.18 0.85 -0.27 -0.36 -0.11 0.11 -2.23 0.62 1.25 -1.17 0.44 0.5 -0.73 1.68 -0.77 -1.18 -0.19 0.52 -0.32 0.06 0.99 0.92 1.51 1.09 0.92 1.55 -0.07 -1.38 0.35 -0.62 0.6 -1.52 -1.48 -1.19 -0.19 0.07 0.41 -0.16 -0.64 0.14 0.15 1.47 -0.69 0.81 1.19 -0.09 1.33

0.93 0.47 0.9 0.23 0.3 0.36 0.8 0.74 0.17 0.48 0.74 0.25 0.34 0.56 0.71 0.29 0.33 0.95 0.04 0.63 0.41 0.42 0.41 0.1 0.49 0.08 0.32 0.61 0.26 0.47 0.9 0.19 0.94 0.8 0.43 0.24 0.15 0.22 0.98 0.32 0.7 0.72 0.57 0.05 0.28 0.28 0.34 0.61 0.23 0.38 0.24 0.64 0.8 0.65

$1.52 $1.60 $1.61 $1.64 $1.74 $1.63 $1.54 $1.53 $1.64 $1.72 $1.73 $1.70 $1.72 $1.67 $1.73 $1.62 $1.53 $1.59 $1.66 $1.47 $1.46 $1.55 $1.40 $1.43 $1.49 $1.58 $1.58 $1.71 $1.76 $1.62 $1.63 $1.56 $1.67 $1.49 $1.57 $1.62 $1.53 $1.68 $1.57 $1.39 $1.45 $1.49 $1.46 $1.42 $1.55 $1.43 $1.41 $1.31 $1.37 $1.29 $1.21 $1.16 $1.28 $1.21

$0.84 $0.88 $0.89 $0.90 $0.95 $0.89 $0.85 $0.84 $0.90 $0.95 $0.95 $0.94 $0.94 $0.92 $0.95 $0.89 $0.84 $0.87 $0.91 $0.81 $0.76 $0.81 $0.73 $0.75 $0.78 $0.82 $0.82 $0.89 $0.92 $0.85 $0.85 $0.82 $0.87 $0.78 $0.82 $0.84 $0.80 $0.88 $0.82 $0.73 $0.76 $0.78 $0.76 $0.74 $0.81 $0.75 $0.74 $0.68 $0.72 $0.67 $0.63 $0.61 $0.67 $0.63

-$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.076 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116

478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500

10.52 10.54 10.56 10.58 10.6 10.63 10.65 10.67 10.69 10.71 10.74 10.76 10.78 10.8 10.82 10.85 10.87 10.89 10.91 10.93 10.96 10.98 11

-0.5 -0.97 -0.58 0.96 -2.62 -1.26 0.73 1.11 1.44 0.54 -0.42 -2.36 -0.4 0.26 0.63 -0.23 0.45 1.04 0.96 -0.85 0.03 -0.62 1.32

0.74 0.29 0.73 0.05 0.9 0.99 0.53 0.79 0.18 0.47 0.59 0.61 0.71 0.74 0.83 0.69 0.06 0.53 0.41 0.63 0.58 0.7 0.92

$1.29 $1.33 $1.51 $1.48 $1.51 $1.55 $1.54 $1.43 $1.52 $1.48 $1.39 $1.42 $1.58 $1.53 $1.56 $1.43 $1.40 $1.36 $1.41 $1.46 $1.49 $1.52 $1.60

$0.67 $0.70 $0.79 $0.77 $0.79 $0.81 $0.80 $0.75 $0.79 $0.77 $0.72 $0.74 $0.82 $0.80 $0.81 $0.75 $0.73 $0.71 $0.74 $0.76 $0.78 $0.80 $0.84

-$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116 -$1.116

LOGNORMALLY DISTRIBUTED JUMP SIZE Brownian motion with stochastic jumps (ln J normally distributed) Normal Variates 0 $2.00 -0.51 $2.13 $0.7350 -0.05 -1.1 $2.28 $2.3604 -0.53 $1.99 $0.5539 0.13 $1.82 $0.6783 0.74 $1.74 $0.5637 -1.3 $1.80 $1.7276 -0.83 $1.88 $0.9544 0.02 $2.72 $0.64 0.6781 -0.29 $2.78 $0.64 0.2969 1.5 $2.71 $0.64 0.8339 0.16 $2.65 $0.64 0.8669 -0.57 $1.81 -$0.03 2.2124 1.42 $1.77 -$0.03 1.0103 -1.33 $1.68 -$0.03 0.2566 -1.41 $1.69 -$0.03 1.6394 -1.27 $1.63 -$0.03 1.5797 0.16 $1.42 -$0.03 1.1139 0.44 $1.33 -$0.03 0.7494 0.55 $1.33 -$0.03 0.9864 -0.6 $1.40 -$0.03 0.6641 -0.23 $1.47 -$0.03 1.2210 0.1 $1.59 -$0.03 0.5036 1.85 $1.38 -$0.03 0.8925 -0.07 $1.44 -$0.03 3.1044 -1.46 $1.54 -$0.03 0.8512 -0.3 $2.39 $0.94 0.6871 e Paths with -1.27 Jump Term $2.36 $0.94 0.7151 mp Size) 0.39 $2.58 $0.94 1.3865 -1.43 $2.87 $0.94 0.5750 -2.38 $2.82 $0.94 0.9015 0.17 $2.82 $0.94 0.7986 -0.3 $2.98 $0.94 1.0714 -1.06 $2.86 $0.94 0.8631 1.1 $2.87 $0.94 0.9555 1.5 $3.08 $0.94 1.5792 1.37 $3.06 $0.94 1.0404 Pure Brownian 1.75 Brownian Jump $2.91 motion $0.94 0.8248 Motion fixed jumps -1.09 with$2.98 $0.94 0.7824 -0.19 $2.94 $0.94 1.5078 -0.4 $3.09 $0.94 0.8356 -1.48 $3.40 $0.94 1.3230 2.18 $3.52 $0.94 0.6776 0.34 $3.84 $0.94 0.7870 2.4 $3.72 $0.94 0.7257 -1.43 $3.99 $0.94 0.6823

0.55 -1.31 0.36 -0.31 0.85 1.36 0.81 -0.4 1.18 -0.64 -0.21 -0.64 -0.35 -1.5 -0.42 -0.32 0.34 0.58 0.04 -0.01 0.4 -0.17 -0.11 0.81 1.05 0.86 1.08 0.99 -0.16 1.05 1.38 -1.06 1.62 0.14 0.06 0.68 -0.08 -0.48 -1.32 -0.39 0.19 0.02 0.27 1.57 0.19 1.6 -0.48 -0.81 0.43 0.96 0.54 -0.47 -0.63 -0.98

$3.98 $4.01 $4.31 $4.38 $5.05 $5.20 $5.12 $5.53 $5.71 $5.32 $5.56 $5.82 $5.85 $5.80 $6.03 $6.46 $6.12 $6.17 $5.73 $5.90 $5.83 $5.89 $6.23 $5.51 $6.01 $6.15 $6.86 $7.03 $7.11 $6.86 $6.89 $6.00 $6.29 $6.20 $6.32 $6.62 $6.01 $6.65 $6.84 $7.13 $6.53 $6.48 $6.50 $6.60 $6.15 $5.90 $5.88 $5.70 $5.96 $6.02 $6.62 $6.23 $5.73 $6.02

$0.94 $0.94 $0.94 $0.94 $0.94 $0.94 $0.94 $0.94 $0.94 $0.94 $0.94 $0.94 $0.94 $0.69 $0.69 $0.69 $0.69 $0.69 $0.69 $0.69 $0.69 $0.69 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65

0.6410 0.5542 1.7047 0.9906 1.1845 0.5225 0.6517 1.3806 1.0326 0.6420 0.6076 1.5151 0.9518 2.2923 2.7226 2.2771 0.7574 1.0474 0.8194 1.5301 0.8617 2.2282 0.6441 1.3783 1.3933 1.1028 0.9714 0.3646 2.4431 1.4814 0.7117 0.6083 2.1995 0.5786 1.2707 1.1125 0.7062 2.1133 0.8333 1.5710 0.5798 1.0540 1.1797 1.1726 1.0030 1.1517 1.1111 1.6623 0.6537 0.2759 1.2316 1.1519 2.3154 1.8971

-1.23 -1.19 1.86 1.89 1.71 -0.24 0.93 1.14 0.12 0.74 -0.18 -1.45 0.05 1.25 -0.29 0.29 -0.7 1.79 -1.15 1.03 1.31 0.53 0.82 -0.77 1.34 -0.1 -1.63 0.3 0.77 -0.48 0.78 1.22 -0.17 0.68 1.08 -0.28 -1.02 0.48 -0.64 0.53 -0.12 0.49 0.76 1.96 0.21 -0.98 0.23 0.92 -1.15 0.57 0.44 -1.87 -0.21 -0.09

$5.71 $4.67 $4.63 $4.35 $4.46 $4.96 $4.90 $4.60 $5.17 $4.89 $4.92 $4.99 $5.44 $5.51 $5.56 $5.66 $5.51 $5.61 $4.82 $5.08 $4.81 $4.49 $4.00 $4.08 $4.03 $4.23 $3.98 $3.91 $4.26 $4.08 $3.83 $3.55 $3.40 $3.44 $3.30 $3.52 $3.55 $3.63 $4.09 $4.10 $4.31 $4.32 $3.96 $4.10 $3.74 $3.80 $3.96 $3.89 $3.70 $3.60 $3.41 $3.34 $3.32 $3.16

$0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.65 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91

0.8606 1.3120 0.9914 2.0621 1.3711 0.9554 1.3359 0.4179 1.2187 1.2451 0.6357 0.8483 0.7621 0.9352 0.7894 1.2834 0.7627 0.9707 0.7399 3.9282 0.6015 1.6014 1.1482 1.4557 1.2719 1.6884 1.4736 0.8571 0.9802 1.3805 0.7679 2.1280 0.9660 1.6652 1.7361 0.7118 0.8087 1.1340 0.7646 0.4447 0.7773 1.0027 0.9206 1.1900 0.3721 1.5321 0.8371 0.7948 0.7810 1.2504 0.7924 0.9839 1.0252 0.2708

-0.67 -0.45 1.44 -0.52 1.59 0.54 0.3 0.6 0.24 -1.65 -0.06 -0.11 -0.15 0.33 -1.09 0.52 -0.43 0.23 -0.29 -1.24 0.17 1.87 -0.61 -1 0.81 -0.26 1.14 -0.09 -0.18 -0.55 1.46 0.46 0.3 1.05 2.06 0.15 0.15 0.25 -0.56 -1.02 0.38 0.72 0.28 -1.36 -0.27 0.66 0.79 0.6 -0.41 -0.94 0.97 1.32 -1.07 -1.26

$3.49 $3.69 $4.11 $3.81 $3.61 $3.89 $3.80 $3.69 $4.18 $4.22 $4.50 $4.31 $4.25 $4.19 $4.45 $4.38 $4.68 $4.79 $5.10 $5.58 $5.45 $5.91 $5.96 $6.09 $5.89 $6.11 $6.63 $6.60 $6.02 $5.89 $6.03 $5.28 $4.98 $5.07 $4.73 $4.61 $4.62 $4.59 $4.69 $4.69 $5.18 $5.18 $5.35 $5.24 $5.57 $5.61 $5.81 $6.02 $5.66 $5.72 $5.87 $5.86 $5.85 $5.53

$0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91 $0.91

0.7638 0.7107 1.7137 0.5551 1.2336 0.8434 1.8800 0.3827 0.7863 0.8668 1.8224 0.6659 1.2512 0.7071 1.6970 0.3986 0.7750 0.6470 1.0566 1.4923 1.7781 1.0990 1.1476 0.7258 2.5372 3.4285 0.2504 0.7663 0.9776 1.0302 0.7726 1.6093 1.5789 0.6465 3.4891 1.0046 1.1165 1.8530 1.0496 2.6317 0.7693 0.9296 0.6434 0.4083 1.3526 0.4932 0.4811 0.5034 0.7478 1.1853 1.9103 0.6360 0.7157 1.0324

-1.52 1.51 0 1.54 -0.26 1.46 -0.75 -0.45 0.27 1.01 -0.92 0.22 -0.24 0.64 -0.9 0.5 1.24 0.96 -0.11 0 1.39 1.03 -0.68 0.54 0.19 0.22 1.16 -0.23 -0.02 -2.05 0.04 0.15 1.69 -2.15 -0.01 -0.38 1.58 0.59 0.46 0.12 0.17 1.06 -1.1 1.91 0.24 -0.47 0.42 -0.03 -1.04 -1.18 2.26 -1.5 1.87 -0.72

$5.09 $2.59 $3.51 $4.02 $3.81 $4.16 $4.07 $4.28 $4.71 $4.71 $4.64 $4.24 $4.45 $4.76 $4.77 $4.72 $4.62 $4.60 $4.93 $4.34 $4.55 $4.72 $4.77 $4.77 $4.72 $4.88 $4.29 $3.87 $3.65 $3.65 $3.48 $3.34 $3.19 $2.93 $2.81 $2.77 $2.38 $2.41 $2.27 $2.51 $2.16 $2.18 $1.91 $2.01 $1.99 $2.00 $1.90 $1.98 $2.18 $1.98 $2.07 $2.15 $2.25 $2.19

$0.91 -$1.80 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06

0.9575 0.6896 1.0257 0.9621 1.0707 0.4764 0.4616 2.0193 2.0486 1.9156 0.5717 2.0112 0.9082 1.4013 1.5699 0.4030 1.3541 0.5999 0.9814 2.6166 0.6507 0.7162 0.4601 0.7428 0.8240 0.8258 2.7201 0.7381 1.0793 0.6971 1.6799 0.5103 0.9849 0.8740 1.1025 0.5322 0.9711 1.2199 0.4320 0.8982 1.0940 0.6374 0.8247 0.4630 3.3344 0.6577 0.5259 1.3411 0.8478 1.1432 1.2973 1.1283 1.9899 1.4652

0.74 0.11 0.93 0.38 1.7 -0.48 0.44 -0.07 1.15 0.31 -0.03 -0.33 1.95 -0.18 -1.31 0.39 2.02 0.42 0.02 -1.21 0.34 -1.27 -0.17 -0.53 0.23 0.2 -0.21 0.2 0.38 -0.29 0.89 -0.2 1.49 0.35 -1.49 1.22 1.67 0.63 -0.36 0.94 -0.36 1.02 -0.85 -0.43 -0.6 -0.55 2.16 -0.66 1.69 -0.02 -0.34 0.22 0.24 0.83

$2.09 $1.98 $2.21 $2.11 $2.02 $1.89 $1.95 $1.94 $1.90 $1.62 $1.62 $1.65 $1.59 $1.59 $1.48 $1.46 $1.46 $1.36 $1.48 $1.49 $1.60 $1.57 $1.76 $1.62 $1.49 $1.62 $1.52 $1.61 $1.65 $1.79 $1.67 $1.77 $1.62 $1.50 $1.34 $1.30 $1.24 $1.29 $1.34 $1.35 $1.38 $1.32 $1.23 $1.31 $1.35 $1.29 $1.24 $1.18 $1.15 $1.09 $1.12 $1.06 $1.07 $1.10

-$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.06 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49 -$1.49

2.1210 0.7502 0.6198 1.9739 1.1144 1.4401 0.6193 0.9477 1.8254 1.2807 2.0013 0.9757 1.5888 0.8915 0.6378 1.8330 1.8286 1.5127 0.5254 0.7240 1.7988 0.5173 0.9796 0.3316 1.6561 2.0215 1.1708 1.0786 1.0364 1.0678 0.8447 0.9680 1.8620 1.0835 0.8845 0.9940 0.8380 1.1354 0.3818 0.7364 0.7985 2.3670 1.5204 0.6373 0.8539 0.3943 0.5252 0.7523 1.6423 0.6631 1.0979 1.3576 0.7816 0.5514

1.21 -0.9 -0.31 -0.51 0.71 -1.93 0.5 0.19 -0.13 0.32 -0.8 0.04 0.03 -0.98 -0.67 1.13 2.36 0.28 0.57 -0.88 1.59 2.34 -0.05 1.46 -0.69 -0.16 -0.2 0.28 -1.98 -2.19 -0.23 1.35 -0.85 0.79 -0.79 1.75 0.89 1.33 0.09 -0.65 0.08 0.66 0.75 -1.71 1.34 -1.97 1.37 1.29 0.34 0.33 -1.34 0.2 0.31 0.18

$1.06 $0.77 $0.78 $0.78 $0.77 $0.73 $0.71 $0.72 $0.83 $0.82 $0.83 $0.74 $0.77 $0.81 $0.85 $0.84 $0.82 $0.80 $0.82 $0.82 $0.78 $0.76 $0.78 $0.78 $0.77 $0.86 $0.88 $0.81 $0.77 $0.72 $0.75 $0.75 $0.78 $0.75 $0.80 $0.75 $0.74 $0.71 $0.73 $0.72 $0.73 $0.71 $0.70 $0.68 $0.63 $0.59 $0.61 $0.61 $0.61 $0.59 $0.63 $0.66 $0.67 $0.67

-$1.49 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79

1.3977 0.6402 0.4590 0.8832 1.4878 1.1394 0.3826 0.6483 0.5741 2.1151 1.5190 1.0784 0.9280 0.6534 0.6667 0.9686 0.9893 1.0832 0.9283 0.7929 1.4896 1.7883 0.8349 2.0158 0.6430 0.6901 0.5894 1.5348 0.3525 0.9717 2.4266 1.4568 1.2965 0.4015 0.6735 1.0093 1.2077 0.7101 1.3758 1.5662 0.6517 1.0736 0.8825 1.2471 1.5120 1.2016 0.6483 0.7899 0.7144 1.2418 1.2617 0.5007 1.0797 0.5185

0.02 1.1 -2.51 1.45 -0.06 -1.88 1.56 1.18 1.56 -1.08 0.43 -1.01 -1.29 -0.05 -1.52 0.34 0.64 0.24 -0.3 0.03 0.36 -0.55 0.1 0.14 -0.87 -0.94 -0.89 -0.58 0.68 0.87 1.05 -0.65 0.44 1.07 -0.9 -1.05 1.22 -1.15 -1.92 0.95 -0.63 0.51 2.2 -0.35 -0.93 -1.64 2.8 1.37 0.38 -0.31 -0.18 -0.56 0.44 1.84

$0.63 $0.56 $0.57 $0.59 $0.55 $0.57 $0.62 $0.65 $0.61 $0.62 $0.56 $0.63 $0.66 $0.66 $0.62 $0.62 $0.66 $0.69 $0.71 $0.69 $0.62 $0.65 $0.65 $0.67 $0.64 $0.69 $0.61 $0.55 $0.56 $0.57 $0.58 $0.61 $0.62 $0.58 $0.58 $0.60 $0.63 $0.62 $0.63 $0.68 $0.77 $0.79 $0.85 $0.88 $0.81 $0.72 $0.68 $0.46 $0.90 $0.96 $0.91 $0.97 $0.99 $1.01

-$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$1.79 -$2.07 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65

0.3813 1.4971 0.8093 0.6279 1.1515 0.8348 1.2748 1.1060 0.6256 0.8975 1.1707 0.8893 0.2398 0.7797 1.2861 1.3825 0.6099 2.1973 0.8049 1.1201 1.1677 0.6167 0.7413 1.7201 1.0113 1.0702 1.0595 0.6935 1.2248 0.6096 0.8128 1.2872 0.2698 1.3348 2.0024 1.1608 0.5655 1.1579 0.7047 0.7851 0.5126 2.0421 1.4443 0.5166 0.7134 0.9823 1.7316 0.9162 0.9263 0.6836 0.5008 0.5535 0.8910 0.8479

1.01 1.08 -1.08 -1.9 -1.07 1.23 0.98 2.55 -1.04 0.02 1.05 -0.43 0.39 0.65 -0.11 0.47 0.18 0.59 -0.32 0.14 -1.38 0.13 0.27 -1.4 -0.08 -0.66 -0.1 1.25 0.1 -1.13 1.09 -0.29 2.3 -2.94 -0.22 -1.12 1.24 -0.68 -1.14 -1.75 0.7 2.03 -0.33 -0.23 0.58 0.95 -1.33 -0.55 0.12 -0.08 1.02 0.34 -2.08 -0.12

$1.02 $1.07 $1.08 $1.09 $1.16 $1.09 $1.03 $1.02 $1.09 $1.15 $1.16 $1.14 $1.15 $1.12 $1.16 $1.08 $1.03 $1.06 $1.11 $0.98 $0.34 $0.36 $0.33 $0.34 $0.35 $0.37 $0.37 $0.40 $0.41 $0.38 $0.38 $0.37 $0.39 $0.35 $0.37 $0.38 $0.36 $0.40 $0.37 $0.33 $0.34 $0.35 $0.34 $0.33 $0.36 $0.33 $0.33 $0.31 $0.32 $0.30 $0.28 $0.27 $0.30 $0.28

-$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$1.65 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29

0.3484 4.3029 0.2550 1.0907 0.6922 1.2323 1.4374 1.1377 0.8664 1.4526 0.8290 0.7930 0.9010 1.0019 0.3111 1.2964 1.7723 0.5298 1.1838 1.2221 0.6598 2.1996 0.6450 0.5253 0.8638 1.2335 0.8082 0.9799 1.5594 1.5077 2.0236 1.6365 1.5040 2.0588 0.9154 0.4772 1.1329 0.6957 1.2810 0.4448 0.4521 0.5248 0.8626 0.9822 1.1676 0.8768 0.6882 1.0165 1.0215 1.9839 0.6744 1.4234 1.7226 0.9103

0.57 -0.04 0.54 -0.94 -0.3 -0.68 -0.52 -0.7 0.34 -0.55 -0.69 -1.69 -0.19 -1.1 -0.54 0.13 -0.93 -1.63 0.07 0.12 -0.09 -0.68 -0.31

$0.30 $0.31 $0.35 $0.35 $0.35 $0.36 $0.36 $0.34 $0.36 $0.35 $0.33 $0.33 $0.37 $0.36 $0.37 $0.33 $0.33 $0.32 $0.33 $0.34 $0.35 $0.36 $0.38

-$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29 -$2.29

1.8497 0.7388 0.5840 0.7118 1.5377 0.2566 0.5063 1.3679 1.6531 1.9511 1.2457 0.7693 0.2924 0.7769 1.0827 1.2990 0.8460 1.1869 1.6018 1.5362 0.6220 0.9644 0.6972