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/*
Copyright (C) 2000-2003 Sadruddin Rejeb
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \defgroup shortrate Short-rate modelling framework
This framework (corresponding to the ql/models/shortrate directory)
implements some single-factor and two-factor short rate models. The models
implemented in this library are widely used by practitioners. For the
moment, the ShortRateModel class defines the short-rate dynamics
with stochastic equations of the type
\f[
dx_i = \mu(t,x_i) dt + \sigma(t,x_i) dW_t
\f]
where \f$ r = f(t,x) \f$. If the model is affine (i.e. derived from the
QuantLib::AffineModel class), analytical formulas
for discount bonds and discount bond options are given (useful for
calibration).
\section singlefactormodels Single-factor models
\par The Hull & White model
\f[
dr_t = (\theta(t) - \alpha(t) r_t)dt + \sigma(t) dW_t
\f]
When \f$ \alpha \f$ and \f$ \sigma \f$ are constants, this model
has analytical formulas for discount bonds and discount bond options.
\par The Black-Karasinski model
\f[
d\ln{r_t} = (\theta(t) - \alpha \ln{r_t})dt + \sigma dW_t
\f]
No analytical tractability here.
\par The extended Cox-Ingersoll-Ross model
\f[
dr_t = (\theta(t) - k r_t)dt + \sigma \sqrt{r_t} dW_t
\f]
There are analytical formulas for discount bonds (and soon for discount
bond options).
\section calibration Calibration
The class CalibrationHelper is a base class that facilitates the
instantiation of market instruments used for calibration. It has
a method marketValue() that gives the market price using a Black
formula, and a modelValue() method that gives the price according to
a model
Derived classed are
QuantLib::CapHelper and
QuantLib::SwaptionHelper.
For the calibration itself, you must choose an optimization method that
will find constant parameters such that the value:
\f[
V = \sqrt{\sum_{i=1}^{n} \frac{(T_i - M_i)^2}{M_i}},
\f]
where \f$ T_i \f$ is the price given by the model and \f$ M_i \f$ is the
market price, is minimized. A few optimization methods are available in
the ql/Optimization directory.
\section twofactormodels Two-factor models
\section pricers Pricers
\par Analytical pricers
If the model is affine, i.e. discount bond options formulas exist, caps are
easily priced since they are a portfolio of discount bond options. Such a
pricer is implemented in QuantLib::AnalyticalCapFloor.
In the case of single-factor affine models, swaptions can be priced using
the Jamshidian decomposition, implemented in
QuantLib::JamshidianSwaption.
\par Using Trees
Each model derived from the single-factor model class has the ability to
return a trinomial tree. For yield-curve consistent models, the fitting
parameter can be determined either analytically (when possible) or
numerically. When a tree is built, it is then pretty straightforward to
implement a pricer for any path-independent derivative. Just implement
a class derived from NumericalDerivative (see
QuantLib::NumericalSwaption for example) and roll it back until
the present time...
Just look at QuantLib::TreeCapFloor and
QuantLib::TreeSwaption for working pricers.
*/
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