Short-rate modelling framework
Detailed Description
This framework (corresponding to the ql/ShortRateModels directory) implements some single-factor and two-factor short rate models. The models implemented in this library are widely used by practitionners. For the moment, the ShortRateModels::Model class defines the short-rate dynamics with stochastic equations of the type
where . 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).
- The Hull & White model
When and are constants, this model has analytical formulas for discount bonds and discount bond options.
- The Black-Karasinski model
No analytical tractability here.
- The extended Cox-Ingersoll-Ross model
There are analytical formulas for discount bonds (and soon for discount bond options).
The class CalibrationHelper is a base class that facilitates the instanciation 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:
where is the price given by the model and is the market price, is minimized. A few optimization methods are available in the ql/Optimization directory.
- 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.
- Using Finite Differences
(Doesn't work for the moment) For the moment, this is only available for single-factor affine models. If is the state variable and follows this stochastic process:
any european-style instrument will follow the following PDE:
The adequate operator to feed a Finite Difference Model instance is defined in the QuantLib::OneFactorOperator class.
- 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-independant 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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