# bdt

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```					                                                                        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

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