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/*
Copyright (C) 2002, 2003 Sadruddin Rejeb
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \defgroup lattices Lattice methods
The framework (corresponding to the ql/methods/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.
\section binomial 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.
\section trinomial 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 \f$ y \f$ satisfying
\f[
dy_t = \mu(t, y_t) dt + \sigma(t, y_t) dW_t.
\f]
At each node, there is a probability \f$ p_u, p_m \f$ and \f$ p_d \f$ to go
through respectively the upper, the middle and the lower branch.
These probabilities must satisfy
\f[
p_{u}y_{i+1,k+1}+p_{m}y_{i+1,k}+p_{d}y_{i+1,k-1}=E_{i,j}
\f]
and
\f[
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,
\f]
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, \f$ E_{i,j}=\mathbf{E}\left( y(t_{i+1})|y(t_{i})=y_{i,j}\right) \f$
and \f$ V_{i,j}^{2}=\mathbf{Var}\{y(t_{i+1})|y(t_{i})=y_{i,j}\} \f$.
If we suppose that the variance is only dependent on time
\f$ V_{i,j}=V_{i} \f$ and set \f$ y_{i+1} \f$ to \f$ V_{i}\sqrt{3} \f$,
we find that
\f[
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}},
\f]
\f[
p_{m} = \frac{2}{3}-\frac{(E_{i,j}-y_{i+1,k})^{2}}{3V_{i}^{2}},
\f]
\f[
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}}.
\f]
\section bidimensional Bidimensional lattices
To come...
\section discretizedasset 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:
-# Define the initialisation procedure (e.g. terminal payoff for european
stock options).
-# 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.
*/
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