Black-Derman-Toy BDT
Tags: #Financial #EconomicsEquation
$$\text{First Node: 1-year bond price} \\ P_{0} = \frac{1}{1 + R_{0}} \\ \text{Second Node} \\ P_{1} = \frac{1}{1+R_{0}} [\frac{1}{2} P(1,2,r_{u}) + \frac{1}{2} P(1,2,r_{d})] \\ = \frac{1}{1+R_{0}} [\frac{1}{2(1 + R_{1}e^{2\sigma_{1}})} + \frac{1}{2(1 + R_{1})}] \\ R_{0} = \frac{1}{2} \ln (\frac{R_{1} e^{2\sigma_{1}} }{R_{1}})$$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
Equation
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 .
: 1-year interest rate.
: 1-year bond price
: year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.
: Observed year-1 price of a 1-year bond
Related Documents
- Black-Derman-Toy Model Developed by Team at Goldman Sachs
- A Binomial Interest Rate Model and the Black-Derman-Toy Model
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