# Vasicek CIR

Document Sample

```					Vasicek Model Parameters
t                         0                 see formula
r0                2.000%
T                   5.000
a                   1.075
140.000
b                 3.939%
120.000
σ                 2.405%
dr             #NAME?                                                        100.000
80.000

Price
CIR Model Parameters                                                          60.000
t                     0                                                       40.000
r                   6%
20.000
T                     5
0.000
a                   6.5                                                                 0        2
b                   5%
σ                  10%
dr            #NAME?

Name     Maturity         Next Coupon Settlement      YTM      Coupon(c)            Price
01M     2007/01/04         2007/01/04   2006/11/30    2.000%     0.00%              99.811
02M     2007/02/03         2007/02/03   2006/11/30    2.000%     0.00%              99.651
03M     2007/03/05         2007/03/05   2006/11/30    2.000%     0.00%              99.475
06M     2007/06/03         2007/06/03   2006/11/30    2.160%     0.00%              98.914
09M     2007/09/01         2007/09/01   2006/11/30    2.275%     0.00%              98.316
12M     2007/11/30         2007/11/30   2006/11/30    2.415%     0.00%              97.642
1049    2015/08/12         2007/08/12   2006/11/30    3.875%     4.50%             105.874
1047    2020/12/01         2006/12/01   2006/11/30    3.875%     5.00%             116.970
1043    2009/01/28         2007/01/28   2006/11/30    3.250%     5.00%             107.780
1048    2009/12/01         2006/12/01   2006/11/30    3.430%     4.00%             105.589
1041    2014/05/05         2007/05/05   2006/11/30    3.810%     6.75%             122.532
1046    2012/10/08         2007/10/08   2006/11/30    3.730%     5.50%             109.941
1045    2011/03/15         2007/03/15   2006/11/30    3.605%     5.25%             110.134
1040    2008/05/05         2007/05/05   2006/11/30    3.030%     6.50%             108.465
1037    2007/08/15         2007/08/15   2006/11/30    2.685%     8.00%             105.984
1034    2009/04/20         2007/04/20   2006/11/30    3.345%     9.00%             118.244

Formulas CIR Interest Rate Model
with constants

dr  a b  r dt   r dz
b: long-term equilibrium of mean reverti
Interest rate process:                                             a: "pull-back" factor - speed of adjustm
dr  a b  r dt   r dz                          : spot rate volatility
Value of zero=coupon bond:                                                 dz standard Wiener process

with       P (t , T )  A(t , T )e  B ( t ,T ) r ( t )

B(t,T) 

2 e Tt 1                                               
A (t , T )  
2 e    a   T  t 
2           2 ab    2


 a e Tt 1 2                                                    
   a  e
 T  t 
 1  2 

Long-term distribution of r (Steady State Probability Density Function) is gamma distributed

k                                                          G(.) is Gamma Function
 2a                                                                Excel worksheetfunction is GAMMALN(.) wh
 2                            k

P     r k 1e 2 ar  2   2a  r k 1e 2 ar  2 ln G k 
 2
Gk 
Mean & standard deviation gamma distribu
 
        2
2ab
with k  2
G Mean   k                              b
                                                                                               2a
   2
G Stdev  
b
k                        
2a           2a
Gamma distribution in Excel notation

with
f x,a, b   a
1
xa1ex b                                            a = k

b Ga                                                      b = (2)/2a

CIR volatitility
Infinitely-long Rate (Y)of zero rate Y(t,T)
where r0 spotrate at t=0
Y  2ab                                                     B(t , T )
a                      Y t ,T )      r0                              a: "pull-back" factor - speed

T  t                      : spot rate volatility

Market Price & Model Price

Price   Model Price

2         4         6         8           10   12      14      16
Time

Clean Price Model Price            Error             Time
99.811        #NAME?          #NAME?            0.094444
99.651        #NAME?          #NAME?               0.175
99.475        #NAME?          #NAME?            0.263889
98.914        #NAME?          #NAME?            0.508333
98.316        #NAME?          #NAME?            0.752778
97.642        #NAME?          #NAME?                   1
104.524       #NAME?          #NAME?                 8.7
111.984       #NAME?          #NAME?            14.00278
103.585       #NAME?          #NAME?            2.161111
101.600       #NAME?          #NAME?            3.002778
118.688       #NAME?          #NAME?            7.430556
109.146       #NAME?          #NAME?            5.855556
106.415       #NAME?          #NAME?            4.291667
104.764       #NAME?          #NAME?            1.430556
103.651       #NAME?          #NAME?            0.708333
112.744       #NAME?          #NAME?            2.388889
#NAME?

rm equilibrium of mean reverting spot rate process
ack" factor - speed of adjustment
te volatility
d Wiener process

 a   T  t                2 ab    2

e                      2


e  T  t   1  2 
 

mma Function
sheetfunction is GAMMALN(.) which LN of G(.)

ndard deviation gamma distribution

        2
n k                           b
2a
   2
v
b
k                        
2a           2a

where r0 spotrate at t=0
a: "pull-back" factor - speed of adjustment
: spot rate volatility
Parameters of CIR Term Structure
t                       0r                      5.0019996          Periods          A
r0                2.000% B                     0.19996002             0.001             1
T                   5.000 A                    0.74980207                  0.5   0.981195
a                   5.000 Y∞                     0.059988                    1   0.953063
b                 6.000% discount factor       0.74680944                    2   0.897643
σ                10.000% zero coupon rate      0.05838904                    3   0.845379
dr            0.002777778 zero coupon vol(σ)   0.00056557                    4   0.796158
solution with VB   #NAME?                         5 0.749802
6 0.706145
7         0.665031
8          0.62631
Long Term Distribution                                                9         0.589843
r                    0.06                                             10           0.5555
K                      60                                             15         0.411549
P∞             51.431745                                              20         0.304902
P                6.000%                                              25          0.22589
 P          0.00774597                                              30         0.167354
35         0.123986
Spot Rates       Probability                            40         0.091857
-4.5   0.02514315       3.65769E-06                            45         0.068053
-4   0.02901613       0.000356585                            50         0.050418
-3.5   0.03288912       0.012036351
-3    0.0367621       0.178245108
-2.5   0.04063508       1.366318831
-2   0.04450807       6.112990257
-1.5   0.04838105       17.46607956                         Probability Distribution
-1   0.05225403       34.15130678
60
-0.5   0.05612702        48.2467805
0         0.06          51.431745      50
0.5   0.06387298       42.85555025
1   0.06774597       28.73426594       40

1.5   0.07161895       15.88219916
30
2   0.07549193       7.385478465
2.5   0.07936492       2.939790208       20
3    0.0832379        1.01662671
3.5   0.08711088       0.309366358       10

4   0.09098387       0.083769156
0
4.5   0.09485685       0.020380886             0    0.02       0.04             0.06
-10
-10

Probability

spot rates, forward rates and YTM

0.18
0.16
0.14
0.12
spot rates

0.1
0.08
0.06
0.04
0.02
0
0   10         20            30             40
maturity

spot rates    forward rates   YTM
B                               Spot
Discount F Zero Coupon Rates Rates Forward Rates
0.007127    0.999857327   0.142683055                       0.153364191   0.153364191
0.061908      0.9799813   0.040443579                       0.041272559   0.041059236
0.169047    0.949845812   0.051455611                       0.052802452   0.064460015
0.199714    0.894064966   0.055988419                       0.057585435   0.062390148
0.199958    0.842005122   0.057323061                       0.058997876   0.061828418
0.19996     0.79298005   0.057989304                       0.059703661   0.061823841
0.19996 0.746809444     0.058389044                       0.060127352    0.06182381
0.19996 0.703327085     0.058655538                       0.060409906    0.06182381
0.19996 0.662376449      0.05884589                       0.060611777    0.06182381
0.19996 0.62381013      0.058988654                       0.060763206    0.06182381
0.19996 0.587489303     0.059099693                       0.060880998    0.06182381
0.19996   0.553283227   0.059188524                       0.060975242    0.06182381
0.19996   0.409906879   0.059455018                       0.061258022    0.06182381
0.19996   0.303684698   0.059588265                       0.061399441    0.06182381
0.19996   0.224988651   0.059668213                       0.061484301    0.06182381
0.19996   0.166685689   0.059721511                       0.061540878    0.06182381
0.19996 0.123491202     0.059759582                       0.061581292    0.06182381
0.19996 0.09149002      0.059788135                       0.061611604    0.06182381
0.19996 0.067781539     0.059810343                        0.06163518    0.06182381
0.19996 0.050216811     0.059828109                       0.061654042    0.06182381

y Distribution                                                                                  Discount Function

1.2

1
Discount Facotors

0.8

0.6

0.4

0.2
0.06           0.08        0.1
0
0          10       20           30          40
0   10       20       30          40
Yields          maturity

Zero Coupon Yield

0.16
0.14
0.12
zero coupon rate

0.1
0.08
0.06
0.04
0.02
0
0    10      20        30         40

period           Zero Coupon Rates

50   60
50   60
50               60

50              60

Zero Coupon Rates

```
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