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<div class="title">Short-rate modelling framework</div> </div>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<p>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><p class="formulaDsp">
\[ dx_i = \mu(t,x_i) dt + \sigma(t,x_i) dW_t \]
</p>
<p> where \( r = f(t,x) \). If the model is affine (i.e. derived from the <a class="el" href="class_quant_lib_1_1_affine_model.html" title="Affine model class.">QuantLib::AffineModel</a> class), analytical formulas for discount bonds and discount bond options are given (useful for calibration).</p>
<h1><a class="anchor" id="singlefactormodels"></a>
Single-factor models</h1>
<dl class="section user"><dt>The Hull & White model</dt><dd><p class="formulaDsp">
\[ dr_t = (\theta(t) - \alpha(t) r_t)dt + \sigma(t) dW_t \]
</p>
When \( \alpha \) and \( \sigma \) are constants, this model has analytical formulas for discount bonds and discount bond options.</dd></dl>
<dl class="section user"><dt>The Black-Karasinski model</dt><dd><p class="formulaDsp">
\[ d\ln{r_t} = (\theta(t) - \alpha \ln{r_t})dt + \sigma dW_t \]
</p>
No analytical tractability here.</dd></dl>
<dl class="section user"><dt>The extended Cox-Ingersoll-Ross model</dt><dd><p class="formulaDsp">
\[ dr_t = (\theta(t) - k r_t)dt + \sigma \sqrt{r_t} dW_t \]
</p>
There are analytical formulas for discount bonds (and soon for discount bond options).</dd></dl>
<h1><a class="anchor" id="calibration"></a>
Calibration</h1>
<p>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>
<p>Derived classed are <a class="el" href="class_quant_lib_1_1_cap_helper.html" title="calibration helper for ATM cap">QuantLib::CapHelper</a> and <a class="el" href="class_quant_lib_1_1_swaption_helper.html" title="calibration helper for ATM swaption">QuantLib::SwaptionHelper</a>.</p>
<p>For the calibration itself, you must choose an optimization method that will find constant parameters such that the value: </p><p class="formulaDsp">
\[ V = \sqrt{\sum_{i=1}^{n} \frac{(T_i - M_i)^2}{M_i}}, \]
</p>
<p> 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.</p>
<h1><a class="anchor" id="twofactormodels"></a>
Two-factor models</h1>
<h1><a class="anchor" id="pricers"></a>
Pricers</h1>
<dl class="section user"><dt>Analytical pricers</dt><dd></dd></dl>
<p>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="section user"><dt>Using Finite Differences</dt><dd></dd></dl>
<p>(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: </p><p class="formulaDsp">
\[ dx_t = \mu(t,x)dt + \sigma(t,x)dW_t \]
</p>
<p> any european-style instrument will follow the following PDE:</p>
<p class="formulaDsp">
\[ \frac{\partial P}{\partial t} + \mu \frac{\partial P}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 P}{\partial x^2} = r(t,x)P \]
</p>
<p>The adequate operator to feed a Finite Difference Model instance is defined in the <a class="el" href="group__findiff.html#gab2342f59174f2223a9889acf4a86496a" title="Interest-rate single factor model differential operator.">QuantLib::OneFactorOperator</a> class.</p>
<dl class="section user"><dt>Using Trees</dt><dd></dd></dl>
<p>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. </p>
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