Lattice model (finance)
Encyclopedia
In finance
, a lattice model can be used to find the fair value
of a stock option; variants also exist for interest rate derivatives.
The model divides time between now and the option's expiration into N discrete periods. At the specific time n, the model has a finite number of outcomes at time n + 1 such that every possible change in the state of the world between n and n + 1 is captured in a branch. This process is iterated until every possible path between n = 0 and n = N is mapped. Probabilities are then estimated for every n to n + 1 path. The outcomes and probabilities flow backwards through the tree until a fair value of the option today is calculated.
The simplest lattice model for options is the binomial options pricing model
, while a more sophisticated variant is the Trinomial tree
. For multiple underlyers
multinomial lattices http://www.espenhaug.com/3d_lattice.html can be built, although the number of nodes increases exponentially with the number of underlyings. For Interest rate derivatives the lattice is built by discretizing a short rate model
, such as Hull-White or Black Derman Toy, or a forward rate
-based model such as the LIBOR market model
or HJM.
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
, a lattice model can be used to find the fair value
Fair value
Fair value, also called fair price , is a concept used in accounting and economics, defined as a rational and unbiased estimate of the potential market price of a good, service, or asset, taking into account such objective factors as:* acquisition/production/distribution costs, replacement costs,...
of a stock option; variants also exist for interest rate derivatives.
The model divides time between now and the option's expiration into N discrete periods. At the specific time n, the model has a finite number of outcomes at time n + 1 such that every possible change in the state of the world between n and n + 1 is captured in a branch. This process is iterated until every possible path between n = 0 and n = N is mapped. Probabilities are then estimated for every n to n + 1 path. The outcomes and probabilities flow backwards through the tree until a fair value of the option today is calculated.
The simplest lattice model for options is the binomial options pricing model
Binomial options pricing model
In finance, the binomial options pricing model provides a generalizable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein in 1979. Essentially, the model uses a “discrete-time” model of the varying price over time of the underlying...
, while a more sophisticated variant is the Trinomial tree
Trinomial Tree
The Trinomial tree is a lattice based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the Binomial options pricing model, and is conceptually similar...
. For multiple underlyers
Underlying
In finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this underlying...
multinomial lattices http://www.espenhaug.com/3d_lattice.html can be built, although the number of nodes increases exponentially with the number of underlyings. For Interest rate derivatives the lattice is built by discretizing a short rate model
Short rate model
In the context of interest rate derivatives, a short-rate model is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.-The short rate:...
, such as Hull-White or Black Derman Toy, or a forward rate
Forward rate
The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a forward rate.-Forward rate calculation:...
-based model such as the LIBOR market model
LIBOR Market Model
The LIBOR market model, also known as the BGM Model , is a financial model of interest rates...
or HJM.