bdt

Document Sample
bdt Powered By Docstoc
					                                                                        Black, Derman & Toy
Given:
Maturity Yield Volatility
       1                0.1             0.2
       2              0.11             0.19                We take the market parameters as sacrosanct--we'll build a tree to match t
       3              0.12             0.18                  Volatilities are for the annualized discount rates to this node.
       4             0.125             0.17                It's important to know what these volatilities mean. They refer to the possib
       5              0.13             0.16                    the security of that term. So only the 2-year case corresponds to the sh


Assume: The short rate (I.e., the yield on a one-year T-Bill) follows a binomial process.
             In the spirit of the binomial model, the difference between the nodes in a time period is determined by the standa
             Use discrete-time compounding/discounting in the tree.

So, the price of a 1-Year T-Bill is:           90.909091

Implying the tree: (What are the 2 possible short rates in 1 year?)

                Price of the 2-Yr Bill:        81.162243

                What we know:
                          From sigma(2), the ratio (of the logs) of the rates.
                          The rate at which we discount the time-1 values is 10%. We therefore have 2 equations in 2 unknow
                          Let's use solver to solve for these 2 values:
                                      ru              0.143181               Equation 1:      0.19
                                      rd              0.097916               Equation 2: 81.16224


Price Tree for 2 - Year Zero                               This solves for the second time period of the short-rate tree, using informat
                                       100                 The one-year short rate is 10%, and in 1 year there is a 50% probability tha

                 87.47525                                  This allows us to solve for the price of the note one year from now in the up

81.16224332                            100

                 91.08168                                  This allows us to solve for the price of the note one year from now in the do

                                       100                 Note: All of these trees are in the "Risk-Neutral World." There is not enoug
                                                                        nodes and the probabilities, so we set the up and down probabi

To solve for the 3rd time in the tree, we also need to use the restriction that the tree recombines. (This is consistent with the as

                Price of the 3-Yr Bill:        71.178025

                                              ru            0.194187              Equation 1:            0.18
                                              rd               0.0976             Equation 2:       71.17802

                                              rud = rdu     0.137669
Price Tree for 3 - Year Zero                 Short rate tree (out 3 years)
                                     100

                          83.73897                                   0.194187

              75.07039                                    0.143181
                                     100

71.17802478              87.899054                  0.1              0.137669


              81.52126               100                  0.097916


                         91.107872                                     0.0976

                                     100




Present Value Tree for 3 - Year Coupon Bond (cum- Interest).


                                     110

                         102.11287

               101.325
                                     110

95.50296049              106.68896


              108.7816               110


                         110.21866

                                     110




Price Tree for 3 - Year Coupon Bond (less accrued Interest).
                                      100

                          92.112867

              91.32496
                                      100

95.50296049                96.68896


              98.78155                100


                          100.21866

                                      100




Price Tree for 3 - Year Coupon Bond Call Option.

                                 0

              0.738711


1.765681248               1.6889597


              3.145788


                          5.2186595




Hedge Ratios

Call:         0.322812

Put:          -0.169348
              Price of the 4-Yr Bill:    62.429508

                                        ru           0.217888   Equation 1:          0.17
                                        rd           0.087172   Equation 2:     62.42951



Price Tree for 4 - Year Zero                                    Short rate tree (out 4 years)


                                                         100

                                         82.109384

                           70.456006                     100                                0.194187

               64.56798                  86.165925                              0.143181


62.42950795                77.169704                     100              0.1               0.137669


               72.77694                  89.421184                              0.097916


                             82.63617                    100                                  0.0976

                                         91.981736

                                                         100




              Price of the 5-Yr Bill:    54.275994
                                     ru              0.255246         Equation 1:        0.16
                                     rd              0.086534         Equation 2:   54.27599



        Price Tree for 5 - Year Zero                                                   Short rate tree (out
                                                                100

                                                     79.66567

                                      67.068577                 100

                         58.883897                   83.69831

               55.0976                73.568207                 100                             0.143181


54.27599488              67.089119                   87.06109                            0.1


              64.30958                    79.08217              100                             0.097916


                         74.123873                    89.8146

                                      83.634555                 100

                                                     92.03575

                                                                100
we'll build a tree to match them.
 to this node.
n. They refer to the possible change over the next year in the ytm on
ase corresponds to the short-rate.



s determined by the standard deviation:                        . 5 × ln( rU / rD )
                                                      st =
                                                                         Dt




ve 2 equations in 2 unknowns ru and rd
             Targets     Criterion
                    0.19    3.78076E-12 Minimize the criterion cell by changing e26 and e27
               81.16224                           subject to the constraint that e26 >= e27


 rt-rate tree, using information in today's yield curve.
 re is a 50% probability that the 1-year rate will be 14.32% and 50% probability that it will be 9.79%.

ne year from now in the up-state



ne year from now in the down-state

World." There is not enough information to separately identify the
 t the up and down probabilities equal to 50%.

his is consistent with the assumption that the volatility at this point does not depend on which node we're in.)


              Targets    Criterion
                    0.18    3.59311E-19 Minimize the criterion cell by changing e45 and e46
               71.17802                           subject to the constraint that e45 >= e46
out 3 years)

               0.750703943            The volatility of the 3-year zero
               1.332083053             refers to the range between the up and down
               1.154159024 0.154159    states in the next period.

                                              Demonstration:           75.07039

                                       Thus, the L54 cell contains the 2-period discount factor for a three-year Zero
                                        in the up-state after 1 period.

               0.815212602
               1.226673873                    By analogy:              81.52126
               1.107553102 0.107553
Price Tree for 3 - Year Coupon Bond Put Option.

                          2.887133

               1.262763


 0.573983306                    0


                     0


                                0
        Targets    Criterion
              0.17    7.37718E-14
         62.42951



out 4 years)



         0.217888                    0.64568
                                    1.548755
                                    1.156985 0.156985

         0.160552
                                                        The distance here is dictated
                                                        by the implied volatility of the 4-year
                                                        Zero.

                                    0.727769
         0.118303                   1.374062
                                    1.111737 0.111737



         0.087172




        Targets     Criterion
               0.16   3.80097E-12
          54.27599



ort rate tree (out 5 years)


                                    0.255246   0.550976
                                               1.814961
                      0.217887587              1.160692 0.160692

          0.194187                  0.194767

                      0.160551578


          0.137669                  0.148619   0.643096
                                               1.554978
                                               1.116686 0.116686
                      0.118303247


            0.0976                  0.113405

                       0.08717235

                                    0.086534
ctor for a three-year Zero
The BDT model assumes that the short-rate process follows a mean-reverting geometric Brownian motion.
         The ability to fit the yield curve exactly by allowing the long-run mean to vary as we move through time.
         The volatility fits the zero-coupon bond volatility at each maturity. (These may be obtained from options markets.)


The volatility equation is derived as follows:



              r0 × es       Dt         This interest rate in the up-state is   rU


   r0


               r0 × e - s    Dt
                                       This interest rate in the up-state is   rD




In BDT, our volatility equation solves for sigma, as a function of Dt, r0, rU, and rD.

Here's the derivation:
                         rU   es          Dt
                            =
                         rD e -s           Dt


Now recall that exponents are additive when the arguments are multiplied (subtractive when divided):



                          es      Dt
                                        = e 2s        Dt

                         e -s     Dt


Thus (taking the log of both sides:)

                            ær     ö
                         ln ç U
                            çr     ÷ = 2s
                                   ÷              Dt
                            è D    ø
Isolating sigma:



                                  1 æ rU         ö             Note that everything in this derivation is true, independent of the prob
                                   ln ç          ÷              This is an important point that suggests that the volatility and nodes
                                  2 ç rD
                                      è
                                                 ÷
                                                 ø
                                                                     or the equivalent risk-neutral world.

                         s =                                   Another way of noting this is by seeing that we do not have to know th
                                                                    with the nodes / volatilities.
                                         Dt                    This is the same reason that under the Black-Scholes model, the imp
                                                                   the real world.
wnian motion.
e move through time.
obtained from options markets.)




n is true, independent of the probability of going up or down.
gests that the volatility and nodes are the same whether we are in the real world

ng that we do not have to know the expected interest rate next period to work

he Black-Scholes model, the implied volatility is the volatility of the stock in

				
DOCUMENT INFO