Lattice methods


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.


Classes

class  BinomialTree
 Binomial tree base class. More...
class  EqualProbabilitiesBinomialTree
 Base class for equal probabilities binomial tree. More...
class  EqualJumpsBinomialTree
 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
 Simple binomial lattice approximating the Black-Scholes model. More...
class  TreeLattice
 Tree-based lattice-method base class. More...
class  TreeLattice1D
 One-dimensional tree-based lattice. More...
class  TreeLattice2D
 Two-dimensional tree-based lattice. More...
class  TsiveriotisFernandesLattice
 Binomial lattice approximating the Tsiveriotis-Fernandes model. More...
class  Tree
 Tree approximating a single-factor diffusion More...
class  TrinomialTree
 Recombining trinomial tree class. More...