Vasicek CIR

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Vasicek CIR Powered By Docstoc
					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



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                        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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