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Create LIBOR Market Model
The LIBOR Market Model (LMM) 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
where:
W is an Ndimensional geometric Brownian motion with
The LMM relates the drifts of the forward rates based on noarbitrage arguments. Specifically, under the Spot LIBOR measure, the drifts are expressed as
where:
is the time fraction associated with the i th forward rate
q(t) is an index defined by the relation
and the Spot LIBOR numeraire is defined as
OBJ = LiborMarketModel(ZeroCurve,VolFunc,Correlation) constructs a LIBOR Market Model object.
For example:
Settle = datenum('15Dec2007'); 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:numRates1) = {@(t) LMMVolFunc(LMMVolParams,t)}; Beta = .08; CorrFunc = @(i,j,Beta) exp(Beta*abs(ij)); Correlation = CorrFunc(meshgrid(1:numRates1)',meshgrid(1:numRates1),Beta); LMM = LiborMarketModel(irdc,VolFunc,Correlation,'Period',1);
The following properties are from the LiborMarketModel class.
ZeroCurve 
ZeroCurve is specified using the output from IRDataCurve or RateSpec. This is the zero curve used to evolve the path of future interest rates. Attributes:
 
VolFunc 
NumRatesby1 cell array of function handles. Each function handle must take time as an input and, return a scalar volatility. Attributes:
 
Correlation 
NumRatesbyNumRates correlation matrix. Attributes:
 
NumFactors 
Number of Brownian factors. The default is NaN, where the number of factors is equal to the number of rates. Attributes:
 
Period 
Period of the forward rates. The default is 2, meaning forward rates are spaced at 0, .5, 1, 1.5 and so on. Possible values for Period are: 1, 2, 4, and 12.
Attributes:

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.
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB^{®} documentation.
Brigo, D. and F. Mercurio, Interest Rate Models  Theory and Practice, Springer Finance, 2006.