# shortrate man page

shortrate

## Synopsis

### Classes

class **GeneralizedHullWhite**

Generalized Hull-White model class.

class **AffineModel**

Affine model class.

class **TermStructureConsistentModel**

Term-structure consistent model class.

class **ShortRateModel**

Abstract short-rate model class.

class **OneFactorModel**

Single-factor short-rate model abstract class.

class **OneFactorAffineModel**

Single-factor affine base class.

class **BlackKarasinski**

Standard Black-Karasinski model class.

class **CoxIngersollRoss**

Cox-Ingersoll-Ross model class.

class **ExtendedCoxIngersollRoss**

Extended Cox-Ingersoll-Ross model class.

class **HullWhite**

Single-factor Hull-White (extended Vasicek) model class.

class **Vasicek**

Vasicek model class

class **TwoFactorModel**

Abstract base-class for two-factor models.

class **G2**

Two-additive-factor gaussian model class.

## 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 = ( heta(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**

[ dln{r_t} = ( heta(t) - alpha ln{r_t})dt + sigma dW_t ] No analytical tractability here.

**The extended Cox-Ingersoll-Ross model**

[ dr_t = ( heta(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} ac{(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:

[ ac{partial P}{partial t} + mu ac{partial P}{partial x} + ac{1}{2} sigma^2 ac{partial^2 P}{partial x^2} = r(t,x)P ].PP 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.

## Author

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