Black-Derman-Toy BDT Equation Formula Latex Code and Summary

Black-Derman-Toy BDT

Tags: #Financial #Economics

Equation

$$\text{First Node: 1-year bond price} \\ P_{0} = \frac{1}{1 + R_{0}} \\ \text{Second Node} \\ P_{1} = \frac{1}{1+R_{0}} [\frac{1}{2} P(1,2,r_{u}) + \frac{1}{2} P(1,2,r_{d})] \\ = \frac{1}{1+R_{0}} [\frac{1}{2(1 + R_{1}e^{2\sigma_{1}})} + \frac{1}{2(1 + R_{1})}] \\ R_{0} = \frac{1}{2} \ln (\frac{R_{1} e^{2\sigma_{1}} }{R_{1}})$$

Latex Code

                                 \text{First Node: 1-year bond price} \\
            P_{0} = \frac{1}{1 + R_{0}} \\
            \text{Second Node} \\
            P_{1} = \frac{1}{1+R_{0}} [\frac{1}{2} P(1,2,r_{u}) + \frac{1}{2} P(1,2,r_{d})] \\
            = \frac{1}{1+R_{0}} [\frac{1}{2(1 + R_{1}e^{2\sigma_{1}})} + \frac{1}{2(1 + R_{1})}] \\
            R_{0} = \frac{1}{2} \ln (\frac{R_{1} e^{2\sigma_{1}} }{R_{1}})

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Introduction

$\text{First Node: 1-year bond price} \\ P_{0} = \frac{1}{1 + R_{0}} \\ \text{Second Node} \\ P_{1} = \frac{1}{1+R_{0}} [\frac{1}{2} P(1,2,r_{u}) + \frac{1}{2} P(1,2,r_{d})] \\ = \frac{1}{1+R_{0}} [\frac{1}{2(1 + R_{1}e^{2\sigma_{1}})} + \frac{1}{2(1 + R_{1})}] \\ R_{0} = \frac{1}{2} \ln (\frac{R_{1} e^{2\sigma_{1}} }{R_{1}})$

Latex Code

            \text{First Node: 1-year bond price} \\
            P_{0} = \frac{1}{1 + R_{0}} \\
            \text{Second Node} \\
            P_{1} = \frac{1}{1+R_{0}} [\frac{1}{2} P(1,2,r_{u}) + \frac{1}{2} P(1,2,r_{d})] \\
            = \frac{1}{1+R_{0}} [\frac{1}{2(1 + R_{1}e^{2\sigma_{1}})} + \frac{1}{2(1 + R_{1})}] \\
            R_{0} = \frac{1}{2} \ln (\frac{R_{1} e^{2\sigma_{1}} }{R_{1}})

Explanation

Latex code for the Black-Derman-Toy BDT model. The BDT model is a commonly used interest rate model. The basic idea of the BDT model is to compute a binomial tree of short-term interest rates, with a flexible enough structure to match the data. Black, Derman, and Toy describe their tree as driven by the short-term rate, which they assume is lognormally distributed. Constructing the black-Derman-Toy tree, the first node is given by the prevailing 1-year rate R0. The yield volatility for period-3 is $P(r_{u}, h, 3h)$ .

$R_{0}$ : 1-year interest rate.
$P_{0}$ : 1-year bond price
$P(1, 2, r_{u}), P(1, 2, r_{d})$ : year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.
$P_{1}$ : Observed year-1 price of a 1-year bond

Discussion

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Roy Butler

I'm ready to tackle this test head-on.

2023-06-28 00:00
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Tina White reply to Roy Butler

Nice~

2023-07-03 00:00:00.0
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Timothy Perez

The pressure is on to pass this exam.

2023-12-24 00:00
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Jack Reed reply to Timothy Perez

Nice~

2024-01-17 00:00:00.0
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Carol Lee

Longing to see a pass when I get my exam results.

2023-11-11 00:00
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Diana George reply to Carol Lee

Nice~

2023-12-03 00:00:00.0
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