QuantLib: Short-rate modelling framework

A free/open-source library for quantitative finance Reference manual - version 1.6

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Classes
Short-rate modelling framework

## Classes
class | GeneralizedHullWhite
| Generalized Hull-White model class. More...
class | AffineModel
| Affine model class. More...
class | TermStructureConsistentModel
| Term-structure consistent model class. More...
class | ShortRateModel
| Abstract short-rate model class. More...
class | OneFactorModel
| Single-factor short-rate model abstract class. More...
class | OneFactorAffineModel
| Single-factor affine base class. More...
class | BlackKarasinski
| Standard Black-Karasinski model class. More...
class | CoxIngersollRoss
| Cox-Ingersoll-Ross model class. More...
class | ExtendedCoxIngersollRoss
| Extended Cox-Ingersoll-Ross model class. More...
class | HullWhite
| Single-factor Hull-White (extended Vasicek) model class. More...
class | Vasicek
| Vasicek model class More...
class | TwoFactorModel
| Abstract base-class for two-factor models. More...
class | G2
| Two-additive-factor gaussian model class. More...

## 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

\[ dx_i = \mu(t,x_i) dt + \sigma(t,x_i) dW_t \]

where \( r = f(t,x) \). 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).

# Single-factor models

The Hull & White model

  \[ dr_t = (\theta(t) - \alpha(t) r_t)dt + \sigma(t) dW_t \]

  When \( \alpha \) and \( \sigma \) are constants, this model has analytical formulas for discount bonds and discount bond options.

The Black-Karasinski model

  \[ d\ln{r_t} = (\theta(t) - \alpha \ln{r_t})dt + \sigma dW_t \]

  No analytical tractability here.

The extended Cox-Ingersoll-Ross model

  \[ dr_t = (\theta(t) - k r_t)dt + \sigma \sqrt{r_t} dW_t \]

  There are analytical formulas for discount bonds (and soon for discount bond options).

# Calibration

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:

\[ V = \sqrt{\sum_{i=1}^{n} \frac{(T_i - M_i)^2}{M_i}}, \]

where \( T_i \) is the price given by the model and \( M_i \) is the market price, is minimized. A few optimization methods are available in the ql/Optimization directory.

# Two-factor models

# Pricers

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 \( x = x(t, r) \) is the state variable and follows this stochastic process:

\[ dx_t = \mu(t,x)dt + \sigma(t,x)dW_t \]

any european-style instrument will follow the following PDE:

\[ \frac{\partial P}{\partial t} + \mu \frac{\partial P}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 P}{\partial x^2} = r(t,x)P \]

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.