# lattices (3) - Linux Man Pages

## NAME

Lattice methods -

### Classes

class ExtendedBinomialTree< T >
Binomial tree base class.
class ExtendedEqualProbabilitiesBinomialTree< T >
Base class for equal probabilities binomial tree.
class ExtendedEqualJumpsBinomialTree< T >
Base class for equal jumps binomial tree.
class ExtendedJarrowRudd
Jarrow-Rudd (multiplicative) equal probabilities binomial tree.
class ExtendedCoxRossRubinstein
Cox-Ross-Rubinstein (multiplicative) equal jumps binomial tree.
class ExtendedTrigeorgis
Trigeorgis (additive equal jumps) binomial tree
class ExtendedTian
Tian tree: third moment matching, multiplicative approach
class ExtendedLeisenReimer
Leisen & Reimer tree: multiplicative approach.
class BinomialTree< T >
Binomial tree base class.
class EqualProbabilitiesBinomialTree< T >
Base class for equal probabilities binomial tree.
class EqualJumpsBinomialTree< T >
Base class for equal jumps binomial tree.
class JarrowRudd
Jarrow-Rudd (multiplicative) equal probabilities binomial tree.
class CoxRossRubinstein
Cox-Ross-Rubinstein (multiplicative) equal jumps binomial tree.
class Trigeorgis
Trigeorgis (additive equal jumps) binomial tree
class Tian
Tian tree: third moment matching, multiplicative approach
class LeisenReimer
Leisen & Reimer tree: multiplicative approach.
class BlackScholesLattice< T >
Simple binomial lattice approximating the Black-Scholes model.
class TreeLattice< Impl >
Tree-based lattice-method base class.
class TreeLattice1D< Impl >
One-dimensional tree-based lattice.
class TreeLattice2D< Impl, T >
Two-dimensional tree-based lattice.
class TsiveriotisFernandesLattice< T >
Binomial lattice approximating the Tsiveriotis-Fernandes model.
class Tree< T >
Tree approximating a single-factor diffusion
class TrinomialTree
Recombining trinomial tree class.

## Detailed Description

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 QuantLib::Lattice, 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 QuantLib::Tree, classes which define the branching between nodes and transition probabilities.

## Binomial trees

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 bsmlattice.hpp 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.

## Trinomial trees

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 QuantLib::TrinomialTree 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 \$ y \$ satisfying [ dy_t = mu(t, y_t) dt + igma(t, y_t) dW_t. ] At each node, there is a probability \$ p_u, p_m \$ and \$ p_d \$ to go through respectively the upper, the middle and the lower branch. These probabilities must satisfy [ p_{u}y_{i+1,k+1}+p_{m}y_{i+1,k}+p_{d}y_{i+1,k-1}=E_{i,j} ] and [ 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, ] 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, \$ E_{i,j}=mathbf{E}

suppose that the variance is only dependant on time \$ V_{i,j}=V_{i} \$ and set \$ y_{i+1} \$ to \$ V_{i}qrt{3} \$, we find that [ p_{u} = ac{1}{6}+ac{(E_{i,j}-y_{i+1,k})^{2}}{6V_{i}^{2}} + ac{E_{i,j}-y_{i+1,k}}{2qrt{3}V_{i}}, ] [ p_{m} = ac{2}{3}-ac{(E_{i,j}-y_{i+1,k})^{2}}{3V_{i}^{2}}, ] [ p_{d} = ac{1}{6}+ac{(E_{i,j}-y_{i+1,k})^{2}}{6V_{i}^{2}} - ac{E_{i,j}-y_{i+1,k}}{2qrt{3}V_{i}}. ]

To come...

## The QuantLib::DiscretizedAsset class

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 QuantLib::DiscretizedAsset, are:

1.
Define the initialisation procedure (e.g. terminal payoff for european stock options).
2.
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.

## Author

Generated automatically by Doxygen for QuantLib from the source code.