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<h1>Lattice methods</h1><hr><a name="_details"></a><h2>Detailed Description</h2>
The framework (corresponding to the ql/Lattices directory) contains basic building blocks for pricing instruments using lattice methods (trees). A lattice, i.e. an instance of the abstract class <a class="el" href="class_quant_lib_1_1_lattice.html" title="Lattice (tree, finite-differences) base class">QuantLib::Lattice</a>, relies on one or several trees (each one approximating a diffusion process) to price an instance of the DiscretizedAsset class. Trees are instances of classes derived from <a class="el" href="class_quant_lib_1_1_tree.html" title="Tree approximating a single-factor diffusion">QuantLib::Tree</a>, classes which define the branching between nodes and transition probabilities.<h2><a class="anchor" name="binomial">
Binomial trees</a></h2>
The binomial method is the simplest numerical method that can be used to price path-independent derivatives. It is usually the preferred lattice method under the Black-Scholes-Merton model. As an example, let's see the framework implemented in the <a class="el" href="bsmlattice_8hpp.html" title="Binomial trees under the BSM model.">bsmlattice.hpp</a> file. It is a method based on a binomial tree, with constant short-rate (discounting). There are several approaches to build the underlying binomial tree, like Jarrow-Rudd or Cox-Ross-Rubinstein.<h2><a class="anchor" name="trinomial">
Trinomial trees</a></h2>
When the underlying stochastic process has a mean-reverting pattern, it is usually better to use a trinomial tree instead of a binomial tree. An example is implemented in the <a class="el" href="class_quant_lib_1_1_trinomial_tree.html" title="Recombining trinomial tree class.">QuantLib::TrinomialTree</a> class, which is constructed using a diffusion process and a time-grid. The goal is to build a recombining trinomial tree that will discretize, at a finite set of times, the possible evolutions of a random variable <img class="formulaInl" alt="$ y $" src="form_13.png"> satisfying <p class="formulaDsp">
<img class="formulaDsp" alt="\[ dy_t = \mu(t, y_t) dt + \sigma(t, y_t) dW_t. \]" src="form_14.png">
<p>
At each node, there is a probability <img class="formulaInl" alt="$ p_u, p_m $" src="form_15.png"> and <img class="formulaInl" alt="$ p_d $" src="form_16.png"> to go through respectively the upper, the middle and the lower branch. These probabilities must satisfy <p class="formulaDsp">
<img class="formulaDsp" alt="\[ p_{u}y_{i+1,k+1}+p_{m}y_{i+1,k}+p_{d}y_{i+1,k-1}=E_{i,j} \]" src="form_17.png">
<p>
and <p class="formulaDsp">
<img class="formulaDsp" alt="\[ p_u y_{i+1,k+1}^2 + p_m y_{i+1,k}^2 + p_d y_{i+1,k-1}^2 = V^2_{i,j}+E_{i,j}^2, \]" src="form_18.png">
<p>
where k (the index of the node at the end of the middle branch) is the index of the node which is the nearest to the expected future value, <img class="formulaInl" alt="$ E_{i,j}=\mathbf{E}\left( y(t_{i+1})|y(t_{i})=y_{i,j}\right) $" src="form_19.png"> and <img class="formulaInl" alt="$ V_{i,j}^{2}=\mathbf{Var}\{y(t_{i+1})|y(t_{i})=y_{i,j}\} $" src="form_20.png">. If we suppose that the variance is only dependant on time <img class="formulaInl" alt="$ V_{i,j}=V_{i} $" src="form_21.png"> and set <img class="formulaInl" alt="$ y_{i+1} $" src="form_22.png"> to <img class="formulaInl" alt="$ V_{i}\sqrt{3} $" src="form_23.png">, we find that <p class="formulaDsp">
<img class="formulaDsp" alt="\[ p_{u} = \frac{1}{6}+\frac{(E_{i,j}-y_{i+1,k})^{2}}{6V_{i}^{2}} + \frac{E_{i,j}-y_{i+1,k}}{2\sqrt{3}V_{i}}, \]" src="form_24.png">
<p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ p_{m} = \frac{2}{3}-\frac{(E_{i,j}-y_{i+1,k})^{2}}{3V_{i}^{2}}, \]" src="form_25.png">
<p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ p_{d} = \frac{1}{6}+\frac{(E_{i,j}-y_{i+1,k})^{2}}{6V_{i}^{2}} - \frac{E_{i,j}-y_{i+1,k}}{2\sqrt{3}V_{i}}. \]" src="form_26.png">
<p>
<h2><a class="anchor" name="bidimensional">
Bidimensional lattices</a></h2>
To come...<h2><a class="anchor" name="discretizedasset">
The QuantLib::DiscretizedAsset class</a></h2>
This class is a representation of the price of a derivative at a specific time. It is roughly an array of values, each value being associated to a state of the underlying stochastic variables. For the moment, it is only used when working with trees, but it should be quite easy to make a use of it in finite-differences methods. The two main points, when deriving classes from <a class="el" href="class_quant_lib_1_1_discretized_asset.html" title="Discretized asset class used by numerical methods.">QuantLib::DiscretizedAsset</a>, are:<ol type=1>
<li>Define the initialisation procedure (e.g. terminal payoff for european stock options).</li><li>Define the method adjusting values, when necessary, at each time steps (e.g. apply the step condition for american or bermudan options). Some examples are found in QuantLib::DiscretizedSwap and QuantLib::DiscretizedSwaption. </li></ol>
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<tr><td colspan="2"><br><h2>Classes</h2></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_binomial_tree.html">BinomialTree</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Binomial tree base class. <a href="class_quant_lib_1_1_binomial_tree.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_equal_probabilities_binomial_tree.html">EqualProbabilitiesBinomialTree</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Base class for equal probabilities binomial tree. <a href="class_quant_lib_1_1_equal_probabilities_binomial_tree.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_equal_jumps_binomial_tree.html">EqualJumpsBinomialTree</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Base class for equal jumps binomial tree. <a href="class_quant_lib_1_1_equal_jumps_binomial_tree.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_jarrow_rudd.html">JarrowRudd</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Jarrow-Rudd (multiplicative) equal probabilities binomial tree. <a href="class_quant_lib_1_1_jarrow_rudd.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_ross_rubinstein.html">CoxRossRubinstein</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Cox-Ross-Rubinstein (multiplicative) equal jumps binomial tree. <a href="class_quant_lib_1_1_cox_ross_rubinstein.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_additive_e_q_p_binomial_tree.html">AdditiveEQPBinomialTree</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Additive equal probabilities binomial tree. <a href="class_quant_lib_1_1_additive_e_q_p_binomial_tree.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_trigeorgis.html">Trigeorgis</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Trigeorgis (additive equal jumps) binomial tree <a href="class_quant_lib_1_1_trigeorgis.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_tian.html">Tian</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Tian tree: third moment matching, multiplicative approach <a href="class_quant_lib_1_1_tian.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_leisen_reimer.html">LeisenReimer</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Leisen & Reimer tree: multiplicative approach. <a href="class_quant_lib_1_1_leisen_reimer.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_scholes_lattice.html">BlackScholesLattice</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Simple binomial lattice approximating the Black-Scholes model. <a href="class_quant_lib_1_1_black_scholes_lattice.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_tree_lattice.html">TreeLattice</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Tree-based lattice-method base class. <a href="class_quant_lib_1_1_tree_lattice.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_tree_lattice1_d.html">TreeLattice1D</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">One-dimensional tree-based lattice. <a href="class_quant_lib_1_1_tree_lattice1_d.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_tree_lattice2_d.html">TreeLattice2D</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Two-dimensional tree-based lattice. <a href="class_quant_lib_1_1_tree_lattice2_d.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_tsiveriotis_fernandes_lattice.html">TsiveriotisFernandesLattice</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Binomial lattice approximating the Tsiveriotis-Fernandes model. <a href="class_quant_lib_1_1_tsiveriotis_fernandes_lattice.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_tree.html">Tree</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Tree approximating a single-factor diffusion <a href="class_quant_lib_1_1_tree.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_trinomial_tree.html">TrinomialTree</a></td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Recombining trinomial tree class. <a href="class_quant_lib_1_1_trinomial_tree.html#_details">More...</a><br></td></tr>
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