blackvolbyrebonato

Compute Black volatility for LIBOR Market Model using Rebonato formula

Syntax

outVol = blackvolbyrebonato(ZeroCurve,VolFunc,CorrMat,ExerciseDate,Maturity)

outVol = blackvolbyrebonato(___,Name,Value)

Description

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.

Examples

Price Swaption for LIBOR Market Model Using the Rebonato Formula

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

    0.2210
    0.2079

Input Arguments

ZeroCurve — Zero-curve for LiborMarketModel model

structure

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

Data Types: struct

VolFunc — Function handle for volatility

cell array of function handles

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

CorrMat — Correlation matrix

vector

Correlation matrix, specified by NumRates-by-NumRates.

Data Types: single | double

ExerciseDate — Swaption exercise date

serial date number | vector of serial date numbers | date character vector

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

Data Types: single | double | char | cell

Maturity — Swap maturity date

serial date number | vector of serial date numbers | date character vector

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 Pair 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)

'Period' — Compounding frequency of curve and reset of swaptions

2 (default) | positive integer from the set [1,2,3,4,6,12] | vector of positive integers from the set [1,2,3,4,6,12]

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

outVol — Black volatility for specified swaption

scalar | vector

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

Algorithms

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

$${({\upsilon }_{{}_{\alpha ,\beta }}^{LFM})}^2={\displaystyle \sum _{i,j=\alpha +1}^{\beta }\frac{w_i(0)w_j(0)F_i(0)F_j(0){\rho }_{i,j}}{S_{\alpha ,\beta }{(0)}^2}}{\displaystyle \underset{0}{\overset{T_{\alpha }}{\int }}{\sigma }_i(t){\sigma }_j(t)dt}$$

where:

$$w_i(t)=\frac{{\tau }_{i}P(t,T_i)}{{\displaystyle \sum _{k=\alpha +1}^{\beta }{\tau }_{\kappa }P(t,t_{\kappa })}}$$

References

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