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Pricing Interest Rate Derivative

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Pricing Interest Rate Derivative Powered By Docstoc
					Remember the idea behind the BDT approach is that we match the model to the market prices of Z
  from exchange-traded options. These prices are used to build a binomial tree of the short-term
  such the model fits the current yield curve exactly. This approach is often called an arbitrage-fr
The market data obviously includes risk premia associated with longer-term lending. Thus when
  of the short rate in the equivalent risk-neutral world. As such our tree can be used to price any
Of course, at this point, this is a circular exercise (it would be reasonable to ask why we need a ra
  directly off Bloomberg). But, as noted, the model can deliver the price of any interest rate deriv
  Salomon Brothers seeking a quote on an interest rate collar. Salomon can use this model--calib
  fair value of such a derivative, and ascertain the risk of the position, and ultimately hedge the p


Consider the following information obtained from Bloomberg.
           Maturity       YTM (%)     Volatility
           (Years)                    Of Yields (%)           P0(T)                       Pricing
                      1         0.1                            0.90909091
                      2       0.11          0.19               0.81162243                 4.53697E-09
                      3       0.12          0.18               0.71178025                 7.15713E-07
                      4      0.125          0.17               0.62429508                 4.33964E-08
                      5       0.13          0.16               0.54275994                 2.36621E-07

Short Rate Tree                                                                           Zero price tree given Short Rate Tree
                                                   0.255245
                                       0.217889                                           P11(2),P10(2)
                          0.194186                 0.194767                                               0.837391
             0.14318                   0.160553                                           0.750704802
    0.10                  0.137668                 0.148618                                               0.878991
            0.097916                   0.118304                                           0.815213318
                          0.097599                 0.113404                                               0.911079
                                       0.087173
                                                   0.086534
                                                                                          P11(4),P10(4)

                                                                                                          0.588839
                                                                                          0.550976335
                                                                            0.542760173                   0.670891
                                                                                          0.643096045
                                                                                                          0.741239




A Caplet is an interest-rate derivative that pays off the following for every dollar of notional princip
The contract specifies the interest rate used, example 3-Month LIBOR, and its expiration date, alo
with the difference between the yield setting and the expiration.
Example: $10,000,000 caplet on 3-Month LIBOR struck at 3%, expiring on December 31, 2005.
         If on September 30, 2005, 3-Month LIBOR is 3.2%, then the holder of this caplet would
         receive {10,000,000 (.032 - .03) 92 } / 360, or     $5,111.11 on December 31, 2005.
   (The strike rate is also called the cap rate.)
A cap is a portfolio of caplets, the time between resets is called the tenor. (The compounding cor
 Note that a caplet is a call option on the interest rate.
A floorlet is analogous to a caplet, but it pays off only if the prevaling rate is less than the floor rat
A floor is a portfolio of floorlets.

Floors, caps and swaps are connected to one-another through a put-call parity relationship.
Consider a swap, cap and floor with the exact same terms (expiration date, tenor, strike rate).
Value of Swap = Cap Price - Floor Price.   (A long position in the cap and a short position in the
                                           The swap here is a "plain vanilla" (floating for fixed) s
                                              the swap (fixed) rate (using the same terms as the c
Finally, a collar is a cap and a floor.



Example:
Consider a cap with a total life of 5 years, a tenor of 1 year, a notional principal of $10,000,000, and a cap
the price of the cap given that the underlying LIBOR tree is the same as the one you obtained in (a).

Cap Tree                                                     Caplet Prices
                                             519777.6
                                  458768.3
                       234331.6              39896.38                                          103401.3
            105795.1              17188.53                                       48529.33082
 49652.46              7554.283                    0            23622.5523                     7554.283
            3440.284                    0                                        3440.284244
                             0                     0                                                 0
                                        0
                                                   0


                                                                                               95879.62
                                                                                 41935.47269
                                                                19061.5785                           0
                                                                                          0
                                                                                                     0


                                                                                               35050.65
                                                                                 15330.32166
                                                              6968.328027                            0
                                                                                          0
                                                                                                     0

                                                                                          0
                                                                             0
                                                                                          0
                                                               Sum of Caplet prices=

Cost of Tightening the cap rate to 18%:         38498.39




Consider a floor with a total life of 5 years, a tenor of 1 year, a notional principal of $10,000,000, and a flo
the price of the floor given that the underlying 1-year LIBOR tree is the same as the one you obtained in (


Floor Tree                                                     Floorlet Prices
                                                      0
                                            0
                                 0                    0                                              0
                     0                      0                                             0
 10167.97                        0                    0         3379.085198                          0
             22369.54                       0                                    7433.987436
                         49119.73                     0                                        16323.78
                                     107827.5
                                                77915.38


                                                                                                     0
                                                                                          0
                                                                6788.886095                          0
                                                                                 14935.54941
                                                                                               32795.95


                                                                                                     0
                                                                                          0
                                                                             0                       0
                                                                                          0
                                                                                                     0

                                                                                          0
                                                                             0
                                                                                          0



                                                               Sum of Floorlet prices=
Cost of Tightening the floor rate to 10%:       19490.13
 10167.97 29658.1
 he market prices of Zero-Coupon Treasury Securities and Implied Volatilities
 ree of the short-term interest rate. This tree is calibrated to these data--as
 called an arbitrage-free modeling approach.
 ending. Thus when we construct our tree, we have a picture of the dynamics
be used to price any security.
ask why we need a rather complicated model to produce prices that we can read
ny interest rate derivative security. So, for example, a customer might call
  use this model--calibrated to prices on liquid securities--to determine the
 timately hedge the position.




           Vol                      Criterion

               -9E-08                8.09E-15
             1.17E-07                5.26E-13
             -3.4E-08                3.07E-15
             -8.8E-08                6.37E-14

ee given Short Rate Tree

                        P11(3),P10(3)        0.821093
                                     0.70456
                          0.64568            0.861658
                                   0.771697
                         0.727769            0.894211
                                   0.826362
                                             0.919817

                         0.796657
            0.670685
                         0.836983
            0.735682
                         0.870611
            0.790821
                         0.898146
            0.836345
                         0.920358

 ar of notional principal:           max( rt - r ,0) × d               Notional Principal:
s expiration date, along                   360                         Strike
                                                                       Prevaling LIBOR
cember 31, 2005.                                                       Tenor
this caplet would
mber 31, 2005.
he compounding corresponds to the tenor--note the last example has quarterly compounding.)

ess than the floor rate.


ty relationship.
 nor, strike rate).
 short position in the floor produces the same cash flows as a long position in the swap.)
" (floating for fixed) swap--where the holder receives the prevaling (floating) rate and pays
e same terms as the cap/floor).




10,000,000, and a cap rate of 19%. Find
u obtained in (a).                    Princ         10000000
                                           Cap           0.19

                       519777.6
            229772.1
                       39896.38
            17188.53
                             0             5-Year
                  0
                             0
                  0
                             0

            228996.2

                  0
                                  4-Year
                  0

                  0




                       3-Year



           2-Year
             49652




$10,000,000, and a floor rate of 9.5%. Find
e one you obtained in (a).              Princ       10000000
                                           Floor        0.095


                             0
                   0
                             0
                   0
                             0             5-Year
                   0
                             0
            35833.94
                       77915.38

                   0

                   0
                                  4-Year
                   0

             71993.6




                       3-Year



           2-Year


             10168
$10,000,000.00
          0.03
         0.032
            92
pounding.)
Remember the idea behind the BDT approach is that we match the model to the market prices of Z
  from exchange-traded options. These prices are used to build a binomial tree of the short-term
  such the model fits the current yield curve exactly. This approach is often called an arbitrage-fr
The market data obviously includes risk premia associated with longer-term lending. Thus when
  of the short rate in the equivalent risk-neutral world. As such our tree can be used to price any
Of course, at this point, this is a circular exercise (it would be reasonable to ask why we need a ra
  directly off Bloomberg). But, as noted, the model can deliver the price of any interest rate deriv
  Salomon Brothers seeking a quote on an interest rate collar. Salomon can use this model--calib
  fair value of such a derivative, and ascertain the risk of the position, and ultimately hedge the p


Consider the following information obtained from Bloomberg.
                   Maturity       YTM (%)     Volatility
                   (Years)                    Of Yields (%)                P0(T)                       Pricing
                              1         0.1                                 0.90909091
                              2       0.11          0.19                    0.81162243                 4.53697E-09
                              3       0.12          0.18                    0.71178025                 7.15713E-07
                              4      0.125          0.17                    0.62429508                 4.33964E-08
                              5       0.13          0.16                    0.54275994                 2.36621E-07

Short Rate Tree                                                                                        Zero price tree given Sho
                                                           0.255244816
                                               0.217889                                                P11(2),P10(2)
                                  0.194186                 0.194766687
                     0.14318                   0.160553                                                0.750704802
            0.10                  0.137668                 0.148618346
                    0.097916                   0.118304                                                0.815213318
                                  0.097599                    0.11340447
                                               0.087173
                                                           0.086534227
                                                                                                       P11(4),P10(4)


                                                                                                       0.550976335
                                                                                         0.542760173
                                                                                                       0.643096045




Here we can use the machinery of caps and floors to compute the swap rate. If you strike a cap a
 a swap. (Long one short other).
So if use Solver to find the strike rate that makes the value of the floor equal to that of the cap, we

Example:
Cap
Cap Tree                                                          Caplet Prices
                                                    1236769.225
                                         1801369
                              1980817               793181.5322
                   1668123               1045844                                    396058.7056
     1002506.774              969505.6              423276.7698    277964.1235
                   537391.8              406755.1                                   215462.3661
                               210516               120391.7358
                                         55369.17
                                                             0



                                                                                    404569.7952
                                                                   261689.0627
                                                                                    171146.1428




                                                                                    489772.5716
                                                                   291161.7394
                                                                                    150783.2551


                                                                                    377722.0662
                                                                   171691.8483
                                                                                             0



                                                                  Sum of Caplet prices=




Floor Tree                                                        Floorlet Prices
                                                             0
                                               0
                                    0                        0
                         0                     0                                             0
     29658.10344                    0                        0     5374.818643
                   65247.83                    0                                    11824.60101
                              101585.2                       0
                                         174982.4
                                                    123933.2596
                                                                                                                   0
                                                                                       11125.75149
                                                                                                      24476.65327




                                                                                                                   0
                                                                                       4527.960371
                                                                                                      9961.512817


                                                                                                                   0
                                                                                       8629.572939
                                                                                                      18985.06047



                                                                                      Sum of Floorlet prices=
Criterion:
                    Difference between floor and cap: -972848.67
                    Squared Difference:              9.46435E+11

cap - floor          972848.7

Verification.
                    Valuation of a fixed for floating swap at this swap rate:

                                                        1236769.225
                                             1801369
                                 1980817                793181.5322
                     1668123                 1045844
    $972,848.6704                969505.6               423276.7698
                     472143.9                406755.1
                                 108930.7               120391.7358
                                              -119613
                                                          -123933.26



Swaption
                    Valuation of a swaption to enter a swap that pays fixed for floating swap at t


Example: 2-Year option on 3-Year Swap, struck at                           0.136159

                    Refer to the expiration date in the swap tree if the value in a node is positive, we would exercise the swapti
                                That is, the swaption is an option to enter the swap tree at time 2.
                        1980817
              1290401
809786.3951             969505.6
              491129
                        108930.7
he market prices of Zero-Coupon Treasury Securities and Implied Volatilities
 ee of the short-term interest rate. This tree is calibrated to these data--as
called an arbitrage-free modeling approach.
ending. Thus when we construct our tree, we have a picture of the dynamics
be used to price any security.
 sk why we need a rather complicated model to produce prices that we can read
ny interest rate derivative security. So, for example, a customer might call
 use this model--calibrated to prices on liquid securities--to determine the
 imately hedge the position.




                          Vol                            Criterion

                           -8.98459E-08                   8.09E-15
                            1.16887E-07                   5.26E-13
                           -3.44947E-08                   3.07E-15
                           -8.80206E-08                   6.37E-14

ero price tree given Short Rate Tree

11(2),P10(2)                                 P11(3),P10(3)        0.821093
               0.837391                                   0.70456
                                               0.64568            0.861658
               0.878991                                 0.771697
                                              0.727769            0.894211
               0.911079                                 0.826362
                                                                  0.919817

11(4),P10(4)                                  0.796657
                            0.670685333
               0.588839                       0.836983
                                0.73568161
               0.670891                       0.870611
                            0.790821252
               0.741239                       0.898146
                            0.836345132
                                              0.920358

If you strike a cap and a floor at the same price, you have

o that of the cap, we have the swap rate.




                                                                     Princ   10000000
                                                                     Cap           0.1
                                        1236769
                          833388.935
          568368.3                     793181.5
                         524085.7226
          337164.8                     423276.8            5-Year
                         243077.2055
          135954.2                     120391.7
                         55369.16861
                                              0

                         967979.6378
          623746.3
                          521758.224
          301246.2                                4-Year
                         163677.9291
              74561.8
                                  0

          788702.3

          331094.6                     3-Year
                   0



                        2-Year


let prices=              1002507

                                                           Princ    10000000
                                                           Floor          0.1


                                              0
                                  0
                   0                          0
                                  0
                   0                          0            5-Year
                                  0
          25964.83                            0
                          56997.9451
                                       123933.3

                                  0
                   0
                                      0
                      0                             4-Year
                                      0
               53746.6
                           117984.4409

                      0

                      0                   3-Year
               21873.8



                          2-Year


rlet prices=                29658.1




r floating swap at this swap rate:



would exercise the swaption in that node, otherwise we would not.

				
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