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Interest Rate One-Factor Equilibrium Models Source: Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. P. 567. Models: Vasicek, O. 1977 "An Equilibrium Characterization of the term structure." Journal of Financial Economics 5: 177-188. Cox, Ingersoll, and Ross. "A Theory of the Term Structure of Interest Rates". Econometrica, 53 (1985). 385-407.

Vasicek Model (discrete version)

Δr=α  b−r  Δtσε  Δt Stochastic process for short-term interest rate r: α : "strength" at which r is pulled back to γ b: long-term equilibrium of short-term rates σ : volatility superimposed (annualized) ε : is a random drawing from a standardized normal distribution, Φ (0,1) Cox, Ingersoll, Ross Model (discrete version)

Δ r=α  b−r  Δ t  r σ ε Δ t Parameters as for the Vasicek model. Because the volatility is proportional to the square root of r, r cannot become negative. As the rates increase, their volatility increases. At the same time, the model has the same mean-reverting or "pull-back" properties as the Vasicek model.

Si 14.0% 12.0% 10.0% 8.0% 6.0% 4.0% 2.0%

Numerical examples (press F9 to generate new random numbers) Vasicek CIR Rate r0 at t=0 8.00% 8.00% Total simulation time (T) 2 2 year(s) "Pullback" α 0.07 0.07 Equilibrium b 6.00% 6.00% Volatility σ 3.00% 10.61% ∆t 0.0067

0.0%

0

0.5

ntice-Hall. P. 567.

ancial Economics 5: 177-188. ica, 53 (1985). 385-407.

Simulation of short-term interest rates

14.0%

12.0%

10.0% 8.0%

Vasicek Cox et al. Equilibrium line b Rate at t=0

6.0% 4.0% 2.0% 0.0%

0

0.5

1 Time

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

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

Vasicek Cox et al. r + ∆r r + ∆r 8.00% 8.00% 1.52 7.98% 7.98% 0.66 7.82% 7.82% -0.47 7.59% 7.60% -1.4 7.50% 7.50% -0.65 7.79% 7.79% 1.39 7.87% 7.87% 0.7 7.72% 7.71% -2.06 7.46% 7.46% -0.41 7.16% 7.17% -0.33 6.95% 6.97% -0.93 6.78% 6.81% 0.08 7.29% 7.28% 0.43 6.91% 6.92% -1.23 6.79% 6.81% -0.86 7.15% 7.14% 1.27 6.91% 6.91% -1.72 6.65% 6.68% 0.28 6.82% 6.83% -0.78 6.69% 6.71% -0.32 6.60% 6.63% -1.8 7.07% 7.05%

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

0.15 0.15 0.16 0.17 0.17 0.18 0.19 0.19 0.2 0.21 0.21 0.22 0.23 0.23 0.24 0.25 0.25 0.26 0.27 0.27 0.28

1.61 0.44 0.1 -1.53 1.67 0.51 -0.47 1.87 -0.43 -0.12 1.32 -1.4 0.73 2.2 -0.42 0.48 0.65 -0.36 -1.96 -1.2 0.09

Period

Vasicek Cox et al. Equilibrium line b Rate at t=0

ε

Time

7.22% 7.40% 7.39% 7.05% 7.12% 7.30% 7.24% 7.18% 7.33% 7.41% 7.68% 8.29% 8.49% 8.30% 8.59% 8.54% 8.53% 8.05% 8.07% 8.18% 8.03%

7.20% 7.37% 7.35% 7.03% 7.10% 7.26% 7.21% 7.15% 7.29% 7.36% 7.63% 8.22% 8.43% 8.23% 8.52% 8.47% 8.46% 7.97% 7.99% 8.10% 7.94%

Equilibrium line b 0 2 Rate at t=0 0 0.05

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 88 89 90 91 92 93 94 95 96

0.29 0.29 0.3 0.31 0.31 0.32 0.33 0.33 0.34 0.35 0.35 0.36 0.37 0.37 0.38 0.39 0.39 0.4 0.41 0.41 0.42 0.43 0.43 0.44 0.45 0.45 0.46 0.47 0.47 0.48 0.49 0.49 0.5 0.51 0.51 0.52 0.53 0.53 0.54 0.55 0.55 0.56 0.57 0.57 0.58 0.59 0.59 0.6 0.61 0.61 0.62 0.63 0.63 0.64

0.14 0.78 0.37 0.68 0.47 0.49 -0.35 1.27 -1.53 -0.26 1.01 0.62 1.53 1.32 0.48 0.33 -0.75 0.4 1.81 0.46 0.15 -1.69 -0.14 0.12 -0.3 -1.12 -0.55 0.65 -1.01 -0.24 -0.59 0.56 -0.89 1.41 -0.08 1.1 0.27 0.95 0.71 0.44 -0.75 0.27 -0.24 -0.98 0.03 -1.35 0.82 0.64 0.66 0.7 -0.21 -0.51 0.86 1.08

8.03% 8.19% 8.24% 8.20% 8.52% 8.67% 8.46% 8.25% 7.90% 7.67% 8.05% 8.30% 8.40% 7.91% 7.88% 7.83% 7.79% 8.05% 7.80% 7.41% 7.55% 7.30% 7.55% 7.94% 7.49% 7.34% 7.29% 6.92% 7.12% 6.82% 6.71% 6.59% 6.84% 7.15% 7.33% 7.50% 7.57% 8.02% 8.42% 8.55% 8.29% 8.58% 8.61% 8.81% 9.14% 8.55% 8.36% 8.26% 8.53% 8.45% 8.38% 8.44% 8.46% 8.28%

7.95% 8.11% 8.16% 8.12% 8.44% 8.59% 8.38% 8.16% 7.81% 7.59% 7.96% 8.20% 8.30% 7.80% 7.78% 7.73% 7.69% 7.94% 7.70% 7.30% 7.44% 7.20% 7.44% 7.81% 7.37% 7.23% 7.18% 6.83% 7.02% 6.73% 6.63% 6.52% 6.75% 7.03% 7.20% 7.36% 7.43% 7.86% 8.26% 8.39% 8.12% 8.42% 8.45% 8.66% 9.00% 8.37% 8.17% 8.07% 8.34% 8.27% 8.20% 8.25% 8.27% 8.10%

97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

0.65 0.65 0.66 0.67 0.67 0.68 0.69 0.69 0.7 0.71 0.71 0.72 0.73 0.73 0.74 0.75 0.75 0.76 0.77 0.77 0.78 0.79 0.79 0.8 0.81 0.81 0.82 0.83 0.83 0.84 0.85 0.85 0.86 0.87 0.87 0.88 0.89 0.89 0.9 0.91 0.91 0.92 0.93 0.93 0.94 0.95 0.95 0.96 0.97 0.97 0.98 0.99 0.99 1

1.05 -0.89 2.41 -1.07 1.47 -1.04 0.12 -0.82 1.09 0.62 -1.14 -0.45 0.25 0.63 -1.01 -0.36 -0.24 -0.3 1.82 0.87 0.11 -0.05 1.27 -0.42 -0.78 0.36 -1.3 0.34 1.62 1.29 -0.02 0.84 -1.35 -0.04 1.57 -2.09 0.34 -0.03 -0.76 -0.01 -0.77 -0.38 0.93 0.91 -1.36 2.05 -0.92 -0.02 -0.35 -1.06 -0.52 1.32 0.31 -0.85

8.30% 8.34% 8.35% 8.77% 8.85% 8.73% 8.36% 8.77% 8.85% 9.19% 9.44% 9.37% 9.47% 9.45% 9.36% 9.44% 9.20% 9.19% 9.20% 9.34% 9.20% 9.16% 9.30% 9.30% 9.48% 9.53% 9.83% 9.60% 9.36% 9.32% 9.26% 9.39% 9.95% 10.30% 10.40% 10.47% 10.29% 10.44% 10.48% 10.74% 10.83% 10.78% 10.61% 10.50% 10.79% 10.85% 11.25% 10.84% 10.53% 10.50% 10.68% 10.61% 10.91% 10.89%

8.12% 8.16% 8.16% 8.59% 8.68% 8.55% 8.17% 8.58% 8.66% 9.01% 9.28% 9.20% 9.32% 9.30% 9.20% 9.28% 9.02% 9.02% 9.03% 9.17% 9.03% 8.98% 9.13% 9.13% 9.33% 9.38% 9.71% 9.45% 9.19% 9.14% 9.08% 9.22% 9.82% 10.21% 10.32% 10.40% 10.20% 10.37% 10.41% 10.71% 10.82% 10.76% 10.56% 10.43% 10.76% 10.84% 11.30% 10.82% 10.45% 10.42% 10.62% 10.55% 10.89% 10.87%

151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204

1.01 1.01 1.02 1.03 1.03 1.04 1.05 1.05 1.06 1.07 1.07 1.08 1.09 1.09 1.1 1.11 1.11 1.12 1.13 1.13 1.14 1.15 1.15 1.16 1.17 1.17 1.18 1.19 1.19 1.2 1.21 1.21 1.22 1.23 1.23 1.24 1.25 1.25 1.26 1.27 1.27 1.28 1.29 1.29 1.3 1.31 1.31 1.32 1.33 1.33 1.34 1.35 1.35 1.36

-0.69 0.99 0.99 -0.4 -0.07 0.8 -0.69 1.82 0.22 -0.43 -0.12 0.07 -0.38 -1.37 -2.1 0.19 0.2 -0.71 1.47 -0.24 -1.58 -2.16 -0.3 -0.72 0.02 0.66 -0.31 0.87 0.39 -3.14 -2.53 1.27 0.99 1.02 -0.71 -0.49 0.34 -0.04 -0.59 -0.77 0.9 0.61 0.67 -0.96 0.89 0.73 0.66 -0.27 -2.53 0.45 0.84 -1.53 0.49 -0.9

11.22% 11.32% 10.94% 11.14% 11.38% 11.20% 11.01% 10.84% 11.21% 11.02% 10.78% 10.84% 11.07% 11.01% 10.90% 10.84% 10.67% 11.19% 11.42% 11.42% 11.29% 11.42% 11.11% 10.95% 11.12% 10.91% 11.05% 11.06% 11.12% 11.19% 12.13% 12.14% 11.98% 11.60% 11.48% 11.16% 11.03% 11.05% 10.94% 10.77% 10.93% 10.93% 11.18% 11.22% 11.54% 11.31% 11.82% 11.58% 11.33% 11.54% 11.64% 11.62% 11.62% 11.80%

11.25% 11.37% 10.92% 11.16% 11.43% 11.22% 11.00% 10.79% 11.23% 11.00% 10.72% 10.79% 11.06% 10.98% 10.86% 10.79% 10.60% 11.20% 11.46% 11.46% 11.31% 11.47% 11.10% 10.90% 11.10% 10.86% 11.02% 11.04% 11.11% 11.18% 12.30% 12.31% 12.12% 11.65% 11.51% 11.12% 10.97% 10.99% 10.87% 10.66% 10.85% 10.86% 11.14% 11.20% 11.57% 11.30% 11.91% 11.61% 11.31% 11.56% 11.68% 11.65% 11.66% 11.87%

205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258

1.37 1.37 1.38 1.39 1.39 1.4 1.41 1.41 1.42 1.43 1.43 1.44 1.45 1.45 1.46 1.47 1.47 1.48 1.49 1.49 1.5 1.51 1.51 1.52 1.53 1.53 1.54 1.55 1.55 1.56 1.57 1.57 1.58 1.59 1.59 1.6 1.61 1.61 1.62 1.63 1.63 1.64 1.65 1.65 1.66 1.67 1.67 1.68 1.69 1.69 1.7 1.71 1.71 1.72

-0.53 -0.86 1.46 0.46 -2.73 0.2 0.46 -2.37 -0.06 0.05 1.06 0.58 0.95 -0.03 -0.83 -1.24 0.55 -2.07 0.98 -0.11 -0.89 0.38 -0.76 1.48 -1.7 -0.74 0.26 1.01 -0.55 -0.55 -1.94 0.23 -0.03 1.69 -0.62 -0.02 0.09 1.61 0 -2.24 -0.09 -0.38 0.62 -0.54 1 1.5 -0.25 0.12 1.67 -0.72 -2.54 -1.08 1.2 -0.74

11.94% 11.78% 12.28% 12.24% 12.43% 12.47% 12.05% 12.23% 11.91% 11.79% 12.03% 11.98% 11.91% 11.65% 12.05% 12.21% 12.13% 12.26% 12.09% 11.90% 11.75% 11.50% 11.44% 11.28% 11.16% 11.20% 10.82% 11.33% 10.86% 10.90% 11.06% 11.23% 11.46% 11.61% 11.39% 11.46% 11.34% 11.15% 11.09% 11.03% 11.05% 10.89% 11.11% 11.48% 11.41% 11.59% 11.71% 11.89% 11.87% 12.24% 11.35% 11.52% 11.19% 11.12%

12.04% 11.85% 12.46% 12.40% 12.64% 12.70% 12.16% 12.40% 12.00% 11.84% 12.15% 12.07% 12.00% 11.68% 12.16% 12.35% 12.26% 12.42% 12.21% 11.98% 11.80% 11.49% 11.42% 11.23% 11.09% 11.13% 10.69% 11.28% 10.71% 10.76% 10.96% 11.14% 11.43% 11.60% 11.34% 11.42% 11.28% 11.06% 10.98% 10.92% 10.94% 10.74% 11.00% 11.44% 11.36% 11.58% 11.71% 11.94% 11.91% 12.36% 11.26% 11.45% 11.06% 10.98%

259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300

1.73 1.73 1.74 1.75 1.75 1.76 1.77 1.77 1.78 1.79 1.79 1.8 1.81 1.81 1.82 1.83 1.83 1.84 1.85 1.85 1.86 1.87 1.87 1.88 1.89 1.89 1.9 1.91 1.91 1.92 1.93 1.93 1.94 1.95 1.95 1.96 1.97 1.97 1.98 1.99 1.99 2

-1.47 1.48 -0.11 0.45 0.75 0.67 1.83 0.84 -0.41 -0.12 0.68 0.27 1.59 -0.56 0.14 -0.14 -0.52 -2.09 0.83 -1.86 -0.49 0.35 1.81 0.56 -0.89 1.39 1.64 -0.87 -1.58 0.97 -0.2 -1.14 0.72 1.17 0.85 0.1 0.33 0.74 -0.42 1.87 -0.76 -1.34

10.57% 10.51% 10.94% 10.59% 10.52% 11.07% 11.16% 10.89% 11.05% 11.03% 11.31% 11.65% 11.34% 11.13% 10.86% 10.87% 10.93% 11.04% 11.21% 11.16% 11.23% 10.98% 10.76% 10.63% 10.99% 11.09% 10.91% 11.28% 11.22% 11.10% 11.00% 10.95% 11.10% 10.96% 10.96% 10.67% 10.52% 10.27% 10.11% 9.59% 9.68% 9.40%

10.34% 10.28% 10.76% 10.36% 10.27% 10.90% 11.00% 10.69% 10.87% 10.86% 11.18% 11.58% 11.21% 10.95% 10.65% 10.65% 10.72% 10.85% 11.05% 10.99% 11.07% 10.78% 10.53% 10.38% 10.79% 10.91% 10.70% 11.12% 11.06% 10.92% 10.79% 10.74% 10.91% 10.75% 10.75% 10.41% 10.25% 9.96% 9.78% 9.21% 9.30% 9.00%

Equilibrium line b 6.00% 6.00% Rate at t=0 8.00% 8.00%

Term structure in Vasicek Model

Have a a look at the formulas

t Rate r0 at t=0

0

Vasicek Te

8.0%

16

9%

2.0

20

8%

"Pullback" a

0.15

15

7%

Equilibrium b

6.0%

60

6%

Instanteanous StDev. of short rate (σ)

2.0%

20

Maturity time (T)

5% 4%

Results: B in Vasicek Model (Hull) A in Vasicek Model (Hull) Infinitely-long Rate (Y∞) Vasicek Discount Factor Solution with VBA Function Vasicek Zero Rate Vasicek volatitility of zero rate σY(t,T)

0.00 1

3%

5.11% 0.999920 #VALUE! 0.004%

1%

2% 0% 0

10

15

Mean of P∞

6.00%

StDev of P∞

3.65%

Vasicek 1.0 1.0 1.0

Vasicek Model: Steady State Probability Density Function for Spotrate r 2

2

Time to matur

0.001%

Long-term distribution of r (Steady State Probability Density Function) r 5.00% P∞ 0.000 10.523

2

5

1.0 1.0

2

1.0 1.0 0 1

-15%

-10%

-5%

0%

1

5

10

1

5% 10% Spotrate (r)

15

Time to matu

15%

20%

25%

Formulas

Interest rate process: Value of zero=coupon bond: with

d r=a  b−r  d tσ d z P t ,T = At ,T e −B t ,T r  t 

[

with constants b: long-term equilibrium of mea a: "pull-back" factor - speed of a σ: spot rate volatility dz standard Wiener process

A t , T =exp  B  t , T −T t 



2

a b−

σ2

a2

2

1−e−a  T−t  B  t , T = a

[

A t , T =exp  B  t , T −T t 

Long-term distribution of r (Steady State Probability Density Function)



P ∞=

−

a 1 e π σ2

a  r−b  2 σ2

Infinitely-long Rate (Y∞)

Y ∞ =b−

σ2

Thus P∞ is normally distributed with

λσ  2a 2 a

 

2

a b−

P∞~ Ν b ,

σ2

2

a2

σ  2a



CIR volatitility of zero rate σY(t,T)

1−e −a T −t  Bt ,T  σ Y  t ,T =σ =σ a  T −t   T −t 

Back to Top Source: Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. p. 567. Model: Vasicek, O. 1977 "An Equilibrium Characterization of the term structure." Journal of Financial Economics 5: 177-188.

Steady-state probability density function formula for Vasicek model from Wilmott, Paul. Paul Wilmott on Quantitative Financ Formula infinitely long rate from Holden, Craig W. Spreadsheet Modeling in Investments. Prentice Hall. 2002 edition. p.49 Formula volatility zero rates from Jackson, M. Staunton, M. "Advanced Modelling in Finance using Excel and VBA", Wiley F

Data Table

Vasicek Term Structure of Interest

1 Vasicek Zero Rate Long-term equilibrium rate r at t=0 Infinitely long rate

5

10

15

20

25

30

35

Time to maturity Vasicek Discount Function Vasicek Discount Factor

5

10

15

20

25

30

35

Time to maturity

with constants b: long-term equilibrium of mean reverting spot rate process a: "pull-back" factor - speed of adjustment σ: spot rate volatility dz standard Wiener process

−T t 



2

a b−

σ2

a2

2



σ 2 B  t , T 2 − 4a

]

0

1

0.5

1

1

1

2

1

3 4 5 6

1 1 1 1

7 8 9 10

1 1 1 1

15 20 25 30

1 1 1 1

-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-10.43% -8.61% -6.78% -4.95% -3.13% -1.30% 0.52% 2.35% 4.17% 6.00% 7.83% 9.65% 11.48% 13.30% 15.13% 16.95% 18.78% 20.61% 22.43%

−T t 

 

2

a b−

P∞~ Ν b ,

σ2 2

a2

σ  2a

Bt ,T  =σ  T −t 



σ 2 B  t , T 2 − 4a

]



where r0 spotrate at t=0 a: "pull-back" factor - speed of adjustment σ: spot rate volatility

p. 567.

cial Economics 5: 177-188.

ul Wilmott on Quantitative Finance, Volume 2, p. 563, John Wiley 2000. Prentice Hall. 2002 edition. p.49 ce using Excel and VBA", Wiley Finance (2001). p. 238 and file "Bond1.xls".

Long-term equilibrium rate 8.000%

0

6.00%

0.016%

30

6.00%

0

8.00%

0.008% 0.004% 0.003% 0.002% 0.002% 0.001% 0.001% 0.001% 0.001% 0.001%

r at t=0

Infinitely long rate 0 30

5.11% 5.11%

Mean 6.00% 6.00%

0.000

+ StDev 9.65% 9.65%

0.000

- StDev 2.35% 2.35%

0.000

0.001% 0.000% 0.000% 0.000%

0.00044 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Term structure CIR Model

Have a a look at the formulas

t (nowyear) Rate r0 at t=0

0

CIR Te

8.0%

16

9%

2.0

20

8%

"Pullback" a

0.15

15

7%

Equilibrium b

6.0%

60

6%

Instanteanous StDev. of short rate (σ)

5.0%

50

Maturity time (T)

Results: γ in CIR Model (Hull) B in CIR Model (Hull) A in CIR Model (Hull) Infinitely-long Rate (Y∞)

5% 4%

0.16583 0.0010 1.0000

3% 2%

5.70% 0.999920 #VALUE! 0.004%

CIR Discount Factor Solution with VBA Function CIR Zero Rate CIR volatitility of zero rate σY(t,T)

1% 0% 0

StDev of P∞

15

Time to ma

CIR 1.0 1.0 1.0

2.24%

1.0

CIR Model: Steady State Probability Density Function for Spotrate r 2

10

0.001%

Long-term distribution of r (Steady State Probability Density Function) r 6.00% k = 2ab/σ2 7.20 P∞ 17.636 17.636 Mean of P∞ 6.00%

2

5

2

1.0 1.0 1.0 0

5

10

15 Time to

0%

2%

1

1

4%

6%

1

8% 10% Spotrate (r)

12%

14%

16%

18%

Formulas CIR Interest Rate Model

Interest rate process: Value of zero=coupon bond: with

d r=a  b−r  d t σ  r d z P  t ,T = At , T  e−B  t ,T r  t 

with constants b: long-term equilibrium of m a: "pull-back" factor - speed σ: spot rate volatility dz standard Wiener process

B  t , T =

2  e γ T −t −1 

 γa   e γ  T −t  −1 2γ

[

A t , T =

2γ

 γa 

Long-term distribution of r (Steady State Probability Density Function) is gamma distributed

 

k

2a k 2 σ2 2a k −1 −2 ar/ σ 2 −ln  Γ  k   k −1 −2 ar/σ P ∞= r e = 2 r e Γ k σ 2ab with k= 2 σ

 

Γ(.) is Gamma Function Excel worksheetfunction is G

Mean & standard deviation g

σ2 Γ  Mean= k 2a σ Γ  Stdev =  k 2

Gamma distribution in Excel notation

1   f x ,α , β = α x α−1 e−x/ β β Γ  α

with α=k β = (σ2)/2a

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Infinitely-long Rate (Y∞)

Y ∞=

2 ab

 aγ 

CIR volatitility of zero rate σY(t,T)

B t , T  σ Y  t ,T  =σ  r0  T −t 

Source: Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. p. 570. Model: Cox, C.J. Ingersoll, J.E. Ross, S.A. (1985) . "A Theory of the Term Structure of Interest Rates". Econometrica, 53 (1985),

Steady-state probability density function formula for Vasicek model from Wilmott, Paul. Paul Wilmott on Quantitative Finan

Formula infinitely long rate, volatility zero rates from Jackson, M. Staunton, M. "Advanced Modelling in Finance using Exc "Bond1.xls".

Data Table

CIR Term Structure of Interest

1 CIR Zero Rate Long-term equilibrium rate r at t=0 Infinitely long rate

5

10

15

20

25

30

35

Time to maturity CIR Discount Function CIR Discount Factor

5

10

15

20

25

30

Time to maturity

with constants b: long-term equilibrium of mean reverting spot rate process a: "pull-back" factor - speed of adjustment σ: spot rate volatility dz standard Wiener process

35

0

1

0.5

1

1

1

2

1

3 4 5 6

1 1 1 1

7 8 9 10

1 1 1 1

15 20 25 30

1 1 1 1

-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0.00% 0.00% 0.00% 0.00% 0.41% 1.53% 2.65% 3.76% 4.88% 6.00% 7.12% 8.24% 9.35% 10.47% 11.59% 12.71% 13.83% 14.94% 16.06%

[

, T =

2γe a γ  T −t / 2

 γa   e γ T −t −1  2γ

]

2 ab/σ 2

Γ(.) is Gamma Function Excel worksheetfunction is GAMMALN(.) which LN of Γ(.) Mean & standard deviation gamma distribution

σ2 Γ  Mean= k =b 2a σ2 b Γ  Stdev =  k = σ 2a 2a



where r0 spotrate at t=0 a: "pull-back" factor - speed of adjustment σ: spot rate volatility

570.

es". Econometrica, 53 (1985), p. 385-407 Wilmott on Quantitative Finance, Volume 2, p. 563, John Wiley 2000.

Modelling in Finance using Excel and VBA", Wiley Finance (2001). p. 238 and file

Long-term equilibrium rate 8.000%

0

6.00%

0.016%

30

6.00%

0

8.00%

0.008% 0.004% 0.003% 0.002% 0.002% 0.001% 0.001% 0.001% 0.001% 0.001%

r at t=0

Infinitely long rate 0 30

5.70% 5.70%

Mean 6.00% 6.00%

21.163

+ StDev 8.24% 8.24%

21.163

- StDev 3.76% 3.76%

21.163

0.001% 0.000% 0.000% 0.000%

17.63607 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64 17.64