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Black–Derman–Toy Model - A Finance Concept

Black–Derman–Toy Model : Everything You Should Know

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Black-Derman-Toy (BDT) model is used to analyze the binomial tree having short-term interest rates with an elastic structure. The structure of the BDT model is designed flexible enough to match the data collected from the binomial tree. According to Fischer Black, Emanuel Derman, and Bill Toy, the Balck-Derman-Toy model tree is driven by the short-term rate. As per Black, Derman, and Toy, the short-term rate is assumed to be lognormally distributed. Here, you will learn Black-Derman-Toy (BDT) formulae and modeling other binomial pricing options.

Definition of Black-Derman-Toy Model

Black-Derman-Toy (BDT) model is an essential tool in mathematical finance, used to price the bond options, swaptions, and other interest rates derivatives. This one-factor model stands tall among the most popular yield-based models used to determine the price of the bonds and interest-rate options. It is a single stochastic factor interpreting the future evaluation of all interest rates. Thus, it was the first model to collectively determine the mean-reverting behavior of the short-term rate with a lognormal distribution.

In the year 1991, Fischer Black and Piotr Karasinski concluded this model as arbitrage-free and coherent regarding the interest rates. A BDT model uses a random variable predicting the behavior of short-term rates responsible for the modeling of long-term interest rates. The long-term interest rates are compelled to suit the current yield curve and its volatility. Therefore, the short-term rate volatility is potentially time-dependent.

Binomial Options Pricing Model

This model was established by John Carrington Cox, Steve Ross, and Mark Edward Rubinstein in 1979 to price options effectively. So, the Binomial Options Pricing model works on a repetitive procedure between the valuation date and the option's expiration date. It is crucial to repeat the process over and over again to allow the specification of nodes or points in time.

Fundamentally, this model follows a lattice-based structure of the different price for a given period of the financial assistant. As compared to complex models like the Black Scholes model, the Binomial Options Pricing model is considered to be mathematically natural. Thus, being simple and easy to use, it reduces the probability of price variations and withdraws the chances for arbitrage.

Black-Derman-Toy Model Formulae

A BDT model uses a binomial lattice to configure the model structure satisfying both the yield curve and volatility structure. It is crucial to set the components to measure various complex interest-rate sensitive securities and its derivatives accurately. To do so, the BDT model formula was derived which can be given as:

where,

r = the instantaneous short rate at time t

 Theta(t) = value of the underlying asset at option expiry

 Sigma(t) = instant short rate volatility

W = a standard Brownian motion under a risk-neutral probability measure

dW = differential of W

Also, to calculate the constant short-term interest rate, the formula is given as follows:

Black-Derman-Toy Model Vs Black-Karasinski Model

The primary purpose of the Black-Derman-Toy model is to monitor the binomial tree with short-term interest rates and a flexible structure. This model is dedicated to accurately evaluate price bond option, swaptions, and other interest rate derivatives. So, it works collectively to determine the mean-reverting behavior of short-term rate with a lognormal distribution. The method uses the yield based model to design a flexible structure regulating the binomial tree.

Fischer Black and Piotr Karasinski established the Black-Karasinski model in 1991 for the pricing of foreign interest rate derivatives. The function of this tool is to calibrate the interest parameters to its current structure and the volatilities of caps. It plays a crucial role in the structure of interest rates or short-term interest rates to normalize the flow of interest. Also, it uses the binomial trees for the calibration as well as for pricing.

The Black-Karasinski model follows the single source of stochastic factors to predict the flow of short-term interest rates. Thus, the model can successfully adjust to today's zero-coupon bond prices in its most simple form and floors of European swaptions.

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References:

  1. https://janroman.dhis.org/calc/BDT.pdf
  2. https://web.ma.utexas.edu/users/mcudina/Lecture24_4and5.pdf
  3. https://en.wikipedia.org/wiki/Black%E2%80%93Karasinski_model
  4. https://www.investopedia.com/terms/b/binomialoptionpricing.asp
  5. https://en.wikipedia.org/wiki/Binomial_options_pricing_model
  6. https://www.simplilearn.com/binomial-option-pricing-model-article
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