Lattice methods

## Classes

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

## 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

\[ dy_t = \mu(t, y_t) dt + \sigma(t, y_t) dW_t. \]

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, \( E_{i,j}=\mathbf{E}\left( y(t_{i+1})|y(t_{i})=y_{i,j}\right) \) and \( V_{i,j}^{2}=\mathbf{Var}\{y(t_{i+1})|y(t_{i})=y_{i,j}\} \). If we suppose that the variance is only dependant on time \( V_{i,j}=V_{i} \) and set \( y_{i+1} \) to \( V_{i}\sqrt{3} \), we find that

\[ p_{u} = \frac{1}{6}+\frac{(E_{i,j}-y_{i+1,k})^{2}}{6V_{i}^{2}} + \frac{E_{i,j}-y_{i+1,k}}{2\sqrt{3}V_{i}}, \]

\[ p_{m} = \frac{2}{3}-\frac{(E_{i,j}-y_{i+1,k})^{2}}{3V_{i}^{2}}, \]

\[ p_{d} = \frac{1}{6}+\frac{(E_{i,j}-y_{i+1,k})^{2}}{6V_{i}^{2}} - \frac{E_{i,j}-y_{i+1,k}}{2\sqrt{3}V_{i}}. \]

# Bidimensional lattices

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.