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 = datetime(2004,8,11);
  
% 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 = datetime(2009,8,11);
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: double

ExerciseDate — Swaption exercise date

datetime array | string array | date character vector

Swaption exercise dates, specified by a NumSwaptions-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, blackvolbyrebonato also accepts serial date numbers as inputs, but they are not recommended.

Maturity — Swap maturity date

datetime array | string array | date character vector

Swap maturity dates, specified using a NumSwaptions-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, blackvolbyrebonato also accepts serial date numbers as inputs, but they are not recommended.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

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.

More About

Black Volatility

Black volatility is the volatility input used in the Black model to calculate the theoretical price of an option on a futures contract.

Black volatility represents the market's expectation of the future volatility of the underlying asset's price over the life of the option.

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.

Version History

Introduced in R2013a

R2022b: Serial date numbers not recommended

Although blackvolbyrebonato supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.