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<h1>Short-rate modelling framework</h1><hr><a name="_details"></a><h2>Detailed Description</h2>
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 <p class="formulaDsp">
<img class="formulaDsp" alt="\[ dx_i = \mu(t,x_i) dt + \sigma(t,x_i) dW_t \]" src="form_0.png">
<p>
where <img class="formulaInl" alt="$ r = f(t,x) $" src="form_1.png">. If the model is affine (i.e. derived from the <a class="el" href="class_quant_lib_1_1_affine_model.html">QuantLib::AffineModel</a> class), analytical formulas for discount bonds and discount bond options are given (useful for calibration).<h2><a class="anchor" name="singlefactormodels">
Single-factor models</a></h2>
<dl class="user" compact><dt><b>The Hull & White model</b></dt><dd><p class="formulaDsp">
<img class="formulaDsp" alt="\[ dr_t = (\theta(t) - \alpha(t) r_t)dt + \sigma(t) dW_t \]" src="form_2.png">
<p>
When <img class="formulaInl" alt="$ \alpha $" src="form_3.png"> and <img class="formulaInl" alt="$ \sigma $" src="form_4.png"> are constants, this model has analytical formulas for discount bonds and discount bond options.</dd></dl>
<dl class="user" compact><dt><b>The Black-Karasinski model</b></dt><dd><p class="formulaDsp">
<img class="formulaDsp" alt="\[ d\ln{r_t} = (\theta(t) - \alpha \ln{r_t})dt + \sigma dW_t \]" src="form_5.png">
<p>
No analytical tractability here.</dd></dl>
<dl class="user" compact><dt><b>The extended Cox-Ingersoll-Ross model</b></dt><dd><p class="formulaDsp">
<img class="formulaDsp" alt="\[ dr_t = (\theta(t) - k r_t)dt + \sigma \sqrt{r_t} dW_t \]" src="form_6.png">
<p>
There are analytical formulas for discount bonds (and soon for discount bond options).</dd></dl>
<h2><a class="anchor" name="calibration">
Calibration</a></h2>
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<p>
Derived classed are <a class="el" href="class_quant_lib_1_1_cap_helper.html">QuantLib::CapHelper</a> and <a class="el" href="class_quant_lib_1_1_swaption_helper.html">QuantLib::SwaptionHelper</a>.<p>
For the calibration itself, you must choose an optimization method that will find constant parameters such that the value: <p class="formulaDsp">
<img class="formulaDsp" alt="\[ V = \sqrt{\sum_{i=1}^{n} \frac{(T_i - M_i)^2}{M_i}}, \]" src="form_7.png">
<p>
where <img class="formulaInl" alt="$ T_i $" src="form_8.png"> is the price given by the model and <img class="formulaInl" alt="$ M_i $" src="form_9.png"> is the market price, is minimized. A few optimization methods are available in the ql/Optimization directory.<h2><a class="anchor" name="twofactormodels">
Two-factor models</a></h2>
<h2><a class="anchor" name="pricers">
Pricers</a></h2>
<dl class="user" compact><dt><b>Analytical pricers</b></dt><dd></dd></dl>
If the model is affine, i.e. discount bond options formulas exist, caps are easily priced since they are a portfolio of discount bond options. Such a pricer is implemented in QuantLib::AnalyticalCapFloor. In the case of single-factor affine models, swaptions can be priced using the Jamshidian decomposition, implemented in QuantLib::JamshidianSwaption.<p>
<dl class="user" compact><dt><b>Using Finite Differences</b></dt><dd></dd></dl>
(Doesn't work for the moment) For the moment, this is only available for single-factor affine models. If <img class="formulaInl" alt="$ x = x(t, r) $" src="form_10.png"> is the state variable and follows this stochastic process: <p class="formulaDsp">
<img class="formulaDsp" alt="\[ dx_t = \mu(t,x)dt + \sigma(t,x)dW_t \]" src="form_11.png">
<p>
any european-style instrument will follow the following PDE:<p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \frac{\partial P}{\partial t} + \mu \frac{\partial P}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 P}{\partial x^2} = r(t,x)P \]" src="form_12.png">
<p>
<p>
The adequate operator to feed a Finite Difference Model instance is defined in the QuantLib::OneFactorOperator class.<p>
<dl class="user" compact><dt><b>Using Trees</b></dt><dd></dd></dl>
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.
<p>
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<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_affine_model.html">AffineModel</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Affine model class. <a href="class_quant_lib_1_1_affine_model.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_term_structure_consistent_model.html">TermStructureConsistentModel</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Term-structure consistent model class. <a href="class_quant_lib_1_1_term_structure_consistent_model.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_short_rate_model.html">ShortRateModel</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Abstract short-rate model class. <a href="class_quant_lib_1_1_short_rate_model.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_model.html">OneFactorModel</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Single-factor short-rate model abstract class. <a href="class_quant_lib_1_1_one_factor_model.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_affine_model.html">OneFactorAffineModel</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Single-factor affine base class. <a href="class_quant_lib_1_1_one_factor_affine_model.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_black_karasinski.html">BlackKarasinski</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Standard Black-Karasinski model class. <a href="class_quant_lib_1_1_black_karasinski.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_cox_ingersoll_ross.html">CoxIngersollRoss</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Cox-Ingersoll-Ross model class. <a href="class_quant_lib_1_1_cox_ingersoll_ross.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_extended_cox_ingersoll_ross.html">ExtendedCoxIngersollRoss</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Extended Cox-Ingersoll-Ross model class. <a href="class_quant_lib_1_1_extended_cox_ingersoll_ross.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_hull_white.html">HullWhite</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Single-factor Hull-White (extended Vasicek) model class. <a href="class_quant_lib_1_1_hull_white.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_vasicek.html">Vasicek</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Vasicek model class <a href="class_quant_lib_1_1_vasicek.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_two_factor_model.html">TwoFactorModel</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Abstract base-class for two-factor models. <a href="class_quant_lib_1_1_two_factor_model.html#_details">More...</a><br></td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top">class </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_g2.html">G2</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Two-additive-factor gaussian model class. <a href="class_quant_lib_1_1_g2.html#_details">More...</a><br></td></tr>
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