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Compute Black volatility for LIBOR Market Model using Rebonato formula



outVol = blackvolbyrebonato(ZeroCurve,VolFunc,CorrMat,ExerciseDate,Maturity) computes the Black volatility for a swaption using a LIBOR Market Model.


outVol = blackvolbyrebonato(___,Name,Value) adds optional name-value pair arguments.


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Define the input maturity and tenor for a LIBOR Market Model (LMM) specified by the cell array of volatility function handles, and a correlation matrix for the LMM.

Settle = datenum('11-Aug-2004');
% Zero Curve
CurveTimes = (1:10)';
CurveDates = daysadd(Settle,360*CurveTimes,1);
ZeroRates = [0.03 0.033 0.036 0.038 0.04 0.042 0.043 0.044 0.045 0.046]';
% Construct an IRCurve
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 = length(ZeroRates);
VolFunc(1:numRates-1) = {@(t) LMMVolFunc(LMMVolParams,t)};
Beta = .08;
CorrFunc = @(i,j,Beta) exp(-Beta*abs(i-j));
CorrMat = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),Beta);
ExerciseDate = datenum('11-Aug-2009');
Maturity = daysadd(ExerciseDate,360*[3;4],1);
Vol = blackvolbyrebonato(irdc,VolFunc,CorrMat,ExerciseDate,Maturity,'Period',1)
Vol = 2×1


Input Arguments

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Zero-curve for the LiborMarketModel, specified using IRDataCurve or RateSpec.

Data Types: struct

Function handle for volatility, specified by 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 | function_handle

Correlation matrix, specified by NumRates-by-NumRates.

Data Types: single | double

Swaption exercise dates, specified by a NumSwaptions-by-1 vector of serial date numbers or date character vectors.

Data Types: single | double | char | cell

Swap maturity dates, specified using a NumSwaptions-by-1 vector of serial date numbers or date character vectors.

Data Types: single | double | char | cell

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: Vol = blackvolbyrebonato(irdc,VolFunc,CorrMat,ExerciseDate,Maturity,'Period',1)

Compounding frequency of curve and reset of swaptions, specified as the comma-separated pair consisting of 'Period' and a positive integer for the values 1,2,4,6,12 in a NumSwaptions-by-1 vector.

Data Types: single | double

Output Arguments

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Black volatility, returned as a vector for the specified swaptions.


The Rebonato approximation formula relates the Black volatility for a European swaption, given a set of volatility functions and a correlation matrix





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

Introduced in R2013a