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

Syntax

LMM = LiborMarketModel(ZeroCurve,VolFunc,Correlation)

LMM = LiborMarketModel(___,Name,Value)

Description

LMM = LiborMarketModel(ZeroCurve,VolFunc,Correlation) creates a 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.

LMM = LiborMarketModel(___,Name,Value) sets Properties using name-value pairs. For example, LMM = LiborMarketModel(irdc,VolFunc,Correlation,'Period',1). You can specify multiple name-value pairs. Enclose each property name in single quotes.

Input Arguments

ZeroCurve — Zero curve used to evolve path of future interest rates

IRDataCurve object | RateSpec

Zero curve used to evolve the path of future interest rates, specified as an output from IRDataCurve or a RateSpec that is obtained from intenvset. The ZeroCurve input sets the ZeroCurve property.

Data Types: object | struct

VolFunc — Volatility function

cell array of function handles

Volatility function, specified using a NumRates-by-1 cell array of function handles and sets the VolFunc property. Each function handle must take time as an input and, return a scalar volatility.

Note

The number of rates to simulate using the simTermStructs function is determined by the size of the VolFunc and Correlation inputs which must be consistent. These can be any value and, together with the Period property, determines the kinds and number of rates being simulated. For example, if the Period is set to 4 (quarterly) and VolFunc has length of 120 and Correlation has size 120-by-120, then 120 quarterly rates are simulated. In other words, 30 years of the yield curve are simulated (0-3mos, 3mos-6mos, 6mos-9mos, and so on, all the way up to 30 years). Therefore, if VolFunc and Correlation have size 120, the output of a call to simTermStructs is (nPeriods+1) -by-121-by-nTrials.

Data Types: cell

Correlation — Correlation matrix

matrix

Correlation matrix, specified using a NumRates-by-NumRates correlation matrix and sets the Correlation property.

Note

The number of rates to simulate using the simTermStructs function is determined by the size of the VolFunc and Correlation inputs which must be consistent. These can be any value and, together with the Period property, determines the kinds and number of rates being simulated. For example, if the Period is set to 4 (quarterly) and VolFunc has length of 120 and Correlation has size 120-by-120, then 120 quarterly rates are simulated. In other words, 30 years of the yield curve are simulated (0-3mos, 3mos-6mos, 6mos-9mos, and so on, all the way up to 30 years). Therefore, if VolFunc and Correlation have size 120, the output of a call to simTermStructs is (nPeriods+1) -by-121-by-nTrials.

Data Types: double

Properties

ZeroCurve — Zero curve

IRDataCurve object | RateSpec

Zero curve, specified as an output from IRDataCurve or a RateSpec that is obtained from intenvset.

Data Types: object | struct

VolFunc — Volatility function

cell array of function handles

Volatility function, specified using a NumRates-by-1 cell array of function handles. Each function handle must take time as an input and, return a scalar volatility.

Data Types: cell

Correlation — Correlation matrix

matrix

Correlation matrix, specified using a NumRates-by-NumRates correlation matrix.

Data Types: double

NumFactors — Number of Brownian factors

NaN (default) | numeric

Number of Brownian factors, specified as a numeric value. The default is NaN, where the number of factors is equal to the number of rates.

Data Types: double

Period — Period of forward rates, specifically number of rates per year

2 (default) | numeric with value 1, 2, 4, or 12

Period of the forward rates, specifically the number of rates per year, specified as a numeric value of 1, 2, 4, or 12. The default value is 2, meaning forward rates are spaced at 0, .5, 1, 1.5, and so on.

Data Types: double

Object Functions

simTermStructs

Simulate term structures for LIBOR Market Model

Examples

Create a LIBOR Market Model Using an IRDataCurve

Create a LMM object using an IRDataCurve.

Settle = datetime(2007,12,15);
CurveTimes = [1:5 7 10 20]';
ZeroRates = [.01 .018 .024 .029 .033 .034 .035 .034]';
CurveDates = daysadd(Settle,360*CurveTimes,1);

irdc = IRDataCurve('Zero',Settle,CurveDates,ZeroRates);

LMMVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
LMMVolParams = [.3 -.02 .7 .14];
  
numRates = 20;
VolFunc(1:numRates,1) = {@(t) LMMVolFunc(LMMVolParams,t)};
  
Beta = .08;
CorrFunc = @(i,j,Beta) exp(-Beta*abs(i-j));
Correlation = CorrFunc(meshgrid(1:numRates)',meshgrid(1:numRates),Beta);
  
LMM = LiborMarketModel(irdc,VolFunc,Correlation,'Period',1)

LMM = 
  LiborMarketModel with properties:

       ZeroCurve: [1x1 IRDataCurve]
    VolFunctions: {20x1 cell}
     Correlation: [20x20 double]
      NumFactors: NaN
          Period: 1

Simulate the term structures for the specified LMM object.

[ZeroRates, ForwardRates] = simTermStructs(LMM, 10,'nTrials',100);

Create a LIBOR Market Model Using a RateSpec

Create a LMM object using a RateSpec.

Settle = datetime(2007,12,15);
CurveTimes = [1:5 7 10 20]';
ZeroRates = [.01 .018 .024 .029 .033 .034 .035 .034]';
CurveDates = daysadd(Settle,360*CurveTimes,1);

RateSpec = intenvset('Rates',ZeroRates,'EndDates',CurveDates,'StartDate',Settle);

LMMVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
LMMVolParams = [.3 -.02 .7 .14];
  
numRates = 20;
VolFunc(1:numRates,1) = {@(t) LMMVolFunc(LMMVolParams,t)};
  
Beta = .08;
CorrFunc = @(i,j,Beta) exp(-Beta*abs(i-j));
Correlation = CorrFunc(meshgrid(1:numRates)',meshgrid(1:numRates),Beta);
  
LMM = LiborMarketModel(RateSpec,VolFunc,Correlation,'Period',1)

LMM = 
  LiborMarketModel with properties:

       ZeroCurve: [1x1 IRDataCurve]
    VolFunctions: {20x1 cell}
     Correlation: [20x20 double]
      NumFactors: NaN
          Period: 1

Simulate the term structures for the specified LMM object.

[ZeroRates, ForwardRates] = simTermStructs(LMM, 10,'nTrials',100);

More About

LIBOR Market Model

The LIBOR Market Model, also called the BGM Model (Brace, Gatarek, Musiela Model) is a financial model of interest rates.

The quantities that are modeled are a set of forward rates (also called forward LIBORs) which have the advantage of being directly observable in the market, and whose volatilities are naturally linked to traded contracts.

References

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

Version History

Introduced in R2013a