Short-rate tree calibration under BDT:
Step 0. Set the
risk-neutral probability of an up move, p, to 50%
Step 2. Once solved, retain these known short rates, and proceed to the next time-step (i.e. input spot-rate), "growing" the tree until it incorporates the full input yield-curve. |
In mathematical finance, the BlackâDermanâToy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) § Interest rate derivatives. It is a one-factor model; that is, a single stochastic factorâthe short rateâdetermines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the log-normal distribution, [1] and is still widely used. [2] [3]
The model was introduced by Fischer Black, Emanuel Derman, and Bill Toy. It was first developed for in-house use by Goldman Sachs in the 1980s and was published in the Financial Analysts Journal in 1990. A personal account of the development of the model is provided in Emanuel Derman's memoir My Life as a Quant. [4]
Under BDT, using a binomial lattice, one calibrates the model parameters to fit both the current term structure of interest rates ( yield curve), and the volatility structure for interest rate caps (usually as implied by the Black-76-prices for each component caplet); see aside. Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and interest rate derivatives.
Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous stochastic differential equation: [1] [5]
For constant (time independent) short rate volatility, , the model is:
One reason that the model remains popular, is that the "standard" Root-finding algorithmsâsuch as Newton's method (the secant method) or bisectionâare very easily applied to the calibration. [6] Relatedly, the model was originally described in algorithmic language, and not using stochastic calculus or martingales. [7]
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Articles
Short-rate tree calibration under BDT:
Step 0. Set the
risk-neutral probability of an up move, p, to 50%
Step 2. Once solved, retain these known short rates, and proceed to the next time-step (i.e. input spot-rate), "growing" the tree until it incorporates the full input yield-curve. |
In mathematical finance, the BlackâDermanâToy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) § Interest rate derivatives. It is a one-factor model; that is, a single stochastic factorâthe short rateâdetermines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the log-normal distribution, [1] and is still widely used. [2] [3]
The model was introduced by Fischer Black, Emanuel Derman, and Bill Toy. It was first developed for in-house use by Goldman Sachs in the 1980s and was published in the Financial Analysts Journal in 1990. A personal account of the development of the model is provided in Emanuel Derman's memoir My Life as a Quant. [4]
Under BDT, using a binomial lattice, one calibrates the model parameters to fit both the current term structure of interest rates ( yield curve), and the volatility structure for interest rate caps (usually as implied by the Black-76-prices for each component caplet); see aside. Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and interest rate derivatives.
Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous stochastic differential equation: [1] [5]
For constant (time independent) short rate volatility, , the model is:
One reason that the model remains popular, is that the "standard" Root-finding algorithmsâsuch as Newton's method (the secant method) or bisectionâare very easily applied to the calibration. [6] Relatedly, the model was originally described in algorithmic language, and not using stochastic calculus or martingales. [7]
Notes
Articles