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Short-rate tree calibration under BDT:
Step 0. Set the
risk-neutral probability of an up move, p, to 50%
Step 1. For each input
spot rate ,
iteratively :
adjust the rate at the top-most node at the current time-step, i;
find all other rates in the time-step, where these are linked to the node immediately above (ru ; rd being the node in question) via
ln
(
r
u
/
r
d
)
/
2
=
σ
i
Δ
t
{\displaystyle \ln(r_{u}/r_{d})/2=\sigma _{i}{\sqrt {\Delta t}}}
(this node-spacing being consistent with p = 50%; Δt being the length of the time-step);
discount recursively through the tree using the rate at each node, i.e. via "backwards induction", from the time-step in question to the first node in the tree (i.e. i=0);
repeat until the discounted value at the first node in the tree equals the
zero-price corresponding to the given
spot interest rate for the i-th time-step.
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 Black–Derman–Toy 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 factor—the short rate—determines 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]
History
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]
Formulae
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]
d
ln
(
r
)
=
θ
t
+
σ
t
′
σ
t
ln
(
r
)
d
t
+
σ
t
d
W
t
{\displaystyle d\ln(r)=\left[\theta _{t}+{\frac {\sigma '_{t}}{\sigma _{t}}}\ln(r)\right]dt+\sigma _{t}\,dW_{t}}
where,
r
{\displaystyle r\,}
= the instantaneous short rate at time t
θ
t
{\displaystyle \theta _{t}\,}
= value of the underlying asset at option expiry
σ
t
{\displaystyle \sigma _{t}\,}
= instant short rate volatility
W
t
{\displaystyle W_{t}\,}
= a standard
Brownian motion under a
risk-neutral probability measure;
d
W
t
{\displaystyle dW_{t}\,}
its
differential .
For constant (time independent) short rate volatility,
σ
{\displaystyle \sigma \,}
, the model is:
d
ln
(
r
)
=
θ
t
d
t
+
σ
d
W
t
{\displaystyle d\ln(r)=\theta _{t}\,dt+\sigma \,dW_{t}}
One reason that the model remains popular, is that the "standard"
Root-finding algorithms —such as
Newton's method (the
secant method ) or
bisection —are very easily applied to the calibration.
[6] Relatedly, the model was originally described in
algorithmic language, and not using
stochastic calculus or
martingales .
[7]
References
Notes
Articles
Benninga, S.; Wiener, Z. (1998).
"Binomial Term Structure Models" (PDF) . Mathematica in Education and Research : vol.7 No. 3.
Black, F.; Derman, E.; Toy, W. (January–February 1990).
"A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options" (PDF) . Financial Analysts Journal : 24–32. Archived from
the original (PDF) on 2008-09-10.
Boyle, P. ; Tan, K.; Tian, W. (2001).
"Calibrating the Black–Derman–Toy model: some theoretical results" (PDF) . Applied Mathematical Finance : 8, 27–48. Archived from
the original (PDF) on 2012-04-22.
Hull, J. (2008).
"The Black, Derman, and Toy Model" (PDF) . Technical Note No. 23, Options, Futures, and Other Derivatives. Archived from
the original (PDF) on 2011-01-29. Retrieved 2011-04-08 .
Klose, C.; Li C. Y. (2003).
"Implementation of the Black, Derman and Toy Model" (PDF) . Seminar Financial Engineering, University of Vienna.
External links