# LiborMarketModel

Create LIBOR Market Model

## Description

The LIBOR Market Model (LMM) is an interest-rate model that differs from short rate models in that it evolves a set of discrete forward rates.

Specifically, the lognormal LMM specifies the following diffusion equation for each forward rate

$$\frac{d{F}_{i}(t)}{{F}_{i}}=-{\mu}_{i}dt+{\sigma}_{i}(t)d{W}_{i}$$

where:

*W* is an N-dimensional geometric Brownian motion with

$$d{W}_{i}(t)d{W}_{j}(t)={\rho}_{ij}$$

The LMM relates drifts of the forward rates based on no-arbitrage arguments. Specifically, under the Spot LIBOR measure, drifts are expressed as

$${\mu}_{i}(t)=-{\sigma}_{i}(t){\displaystyle \sum _{j=q(t)}^{i}\frac{{\tau}_{j}{\rho}_{i,j}{\sigma}_{j}(t){F}_{j}(t)}{1+{\tau}_{j}{F}_{j}(t)}}$$

where:

$${\rho}_{i,j}$$ represents the input argument `Correlation`

.

$${\sigma}_{j}(t)$$ represents the input argument `VolFunc`

.

$${F}_{j}(t)$$ represents the computation of the input argument for
`ZeroCurve`

.

$${\tau}_{i}$$ is the time fraction associated with the *i* th forward
rate

*q(t)* is an index defined by the relation

$${T}_{q(t)-1}<t<{T}_{q(t)}$$

and the Spot LIBOR numeraire is defined as

$$B(t)=P(t,{T}_{q(t)}){\displaystyle \prod _{n=0}^{q(t)-1}(1+{\tau}_{n}{F}_{n}({T}_{n}))}$$

## Creation

### Description

creates a `LMM`

= LiborMarketModel(`ZeroCurve`

,`VolFunc`

,`Correlation`

)`LiborMarketModel`

(`LMM`

) object using the
required arguments for `ZeroCurve`

, `VolFunc`

,
`Correlation`

.

**Note**

Alternatively, you can use the `SABRBraceGatarekMusiela`

or `BraceGatarekMusiela`

model objects to create a LIBOR market model. For more
information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

sets Properties using name-value pairs. For example, `LMM`

= LiborMarketModel(___,`Name,Value`

)```
LMM =
LiborMarketModel(irdc,VolFunc,Correlation,'Period',1)
```

. You can specify
multiple name-value pairs. Enclose each property name in single quotes.

### Input Arguments

## Properties

## Object Functions

`simTermStructs` | Simulate term structures for LIBOR Market Model |

## Examples

## More About

## References

[1] Brigo, D. and F. Mercurio. *Interest Rate Models - Theory and
Practice.* Springer Finance, 2006.

## Version History

**Introduced in R2013a**

## See Also

`HullWhite1F`

| `intenvset`

| `IRDataCurve`

| `LinearGaussian2F`

| `simTermStructs`

| `SABRBraceGatarekMusiela`

| `BraceGatarekMusiela`