SABRBraceGatarekMusiela

Create SABRBraceGatarekMusiela model object for Cap, Floor, FixedBond, FloatBond, FloatBondOption, FixedBondOption, OptionEmbeddedFixedBond, or OptionEmbeddedFloatBond instrument

Since R2021b

Description

Create and price a Cap, Floor, FloatBond, FloatBondOption, FixedBond, FixedBondOption, OptionEmbeddedFixedBond, or OptionEmbeddedFloatBond instrument object with a SABRBraceGatarekMusiela model using this workflow:

1. Use fininstrument to create a Cap, Floor, FixedBond, FloatBond, FloatBondOption FixedBondOption, OptionEmbeddedFixedBond, or OptionEmbeddedFloatBond instrument object.

2. Use finmodel to specify a SABRBraceGatarekMusiela model object for the Cap, Floor, FixedBond, FloatBond, FloatBondOption, FixedBondOption, OptionEmbeddedFixedBond, or OptionEmbeddedFloatBond instrument object.

3. Use finpricer to specify an IRMonteCarlo pricing method for a Cap, Floor, FixedBond, FloatBond, FloatBondOption, FixedBondOption, OptionEmbeddedFixedBond, or OptionEmbeddedFloatBond instrument object.

For more information on this workflow, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

For more information on the available pricing methods for a Cap, Floor, FixedBond, FloatBond, FloatBondOption, FixedBondOption, OptionEmbeddedFixedBond, or OptionEmbeddedFloatBond instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

SABRBraceGatarekMusielaModelObj = finmodel(ModelType,Alpha=alpha_value,Beta=beta_value,VolatilityofVolatility=volatilityofvolatility_value,FwdFwdCorrelation=fwdfwdcorrelation_value,VolVolCorrelation=volvolcorrelation_value)

Description

SABRBraceGatarekMusielaModelObj = finmodel(ModelType,Alpha=alpha_value,Beta=beta_value,VolatilityofVolatility=volatilityofvolatility_value,FwdFwdCorrelation=fwdfwdcorrelation_value,VolVolCorrelation=volvolcorrelation_value) creates a classic SABRBraceGatarekMusiela model object with null forward to volatility correlation by specifying ModelType and the required name-value arguments Alpha, Beta, VolatilityofVolatility, FwdFwdCorrelation, and VolVolCorrelation to set properties using name-value arguments. For example, SABRBraceGatarekMusielaModelObj = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,FwdFwdCorrelation=FwdFwdCorrelation, VolVolCorrelation=VolVolCorrelation) creates a classic SABRBraceGatarekMusiela model object with null forward to volatility correlation.

SABRBraceGatarekMusielaModelObj = finmodel(___,Name=Value) sets optional properties using additional name-value arguments in addition to the required arguments in the previous syntax. You can specify multiple name-value arguments. For example, you can use name-value arguments to create the following variations of the SABRBraceGatarekMusiela model:

• To create a classic SABRBraceGatarekMusiela model object, use the FwdVolCorrelation name-value pair argument: SABRBraceGatarekMusielaModelObj = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,FwdFwdCorrelation=FwdFwdCorrelation,VolVolCorrelation=VolVolCorrelation,FwdVolCorrelation=FwdVolCorrelation)

• To create a classic SABRBraceGatarekMusiela model object in Rebonato parametric form with null forward-to-volatility correlation, use the Volatility name-value pair argument: SABRBraceGatarekMusielaModelObj = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,Volatility=VolFunc,FwdFwdCorrelation=FwdFwdCorrelation,VolVolCorrelation=VolVolCorrelation)

• To create a classic SABRBraceGatarekMusiela model object in Rebonato parametric form with FwdVolCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.02), use the Volatility and FwdVolCorrelation name-value arguments: SABRBraceGatarekMusielaModelObj = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,Volatility=VolFunc,FwdFwdCorrelation=FwdFwdCorrelation,VolVolCorrelation=VolVolCorrelation)

Input Arguments

ModelType — Model type

string with value "SABRBraceGatarekMusiela" | character vector with value 'SABRBraceGatarekMusiela'

Model type, specified as a string with the value of "SABRBraceGatarekMusiela" or a character vector with the value of 'SABRBraceGatarekMusiela'.

Data Types: char | string

Name-Value Arguments

Specify required and 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.

Example: SABRBraceGatarekMusielaModelObj = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,FwdFwdCorrelation=FwdFwdCorrelation, VolVolCorrelation=VolVolCorrelation)

Alpha — Initial SABR volatilities for each forward rate maturity

positive numeric

Initial SABR volatilities for each forward rate maturity, specified as Alpha and an (NumRates-1)-by-1 vector of positive numeric values.

Data Types: double

Beta — SABR exponent parameters for each forward rate maturity

numeric between 0 and 1

SABR exponent parameters for each forward rate maturity, specified as Beta and an (NumRates-1)-by-1 vector of numeric values between 0 and 1.

Data Types: double

VolatilityofVolatility — Variation in volatility

function handle

Variation in volatility, specified as VolatilityofVolatility and an (NumRates-1)-by-1 cell array of function handles. Each function handle must take time as an input and return a scalar volatility of volatility that must be positive numeric.

Data Types: double | cell

FwdFwdCorrelation — Correlation matrix for forward rates

numeric between -1 and 1

Correlation matrix for forward rates, specified as FwdFwdCorrelation and a (NumRates-1)-by-(NumRates-1) correlation matrix.

Data Types: double

VolVolCorrelation — Correlation matrix for volatilities

numeric between -1 and 1

Correlation matrix for volatilities, specified as VolVolFwdCorrelation and an (NumRates-1)-by-(NumRates-1) correlation matrix.

Data Types: double

Optional SABRBraceGatarekMusiela Name-Value Arguments

Volatility — Volatility

simulate stochastic volatilities without volatility functions (default) | function handle

Volatility, specified as Volatility and an (NumRates-1)-by-1 cell array of function handles. Specify these optional volatility function handles to use the Rebonato (2009) parametric form, which simulates stochastic volatilities with deterministic volatility functions and stochastic correction terms. Each function handle must take time as an input and return a scalar volatility that must be positive numeric.

Data Types: double

FwdVolCorrelation — Correlation matrix between forward rates and volatilities

(NumRates-1)-by-(NumRates-1) matrix of zeros (default) | numeric between -1 and 1

Correlation matrix between forward rates and volatilities, specified as FwdVolCorrelation and an (NumRates-1)-by-(NumRates-1) correlation matrix. The diagonal elements of the matrix are the SABR Rho parameters.

Data Types: double

Period — Period of forward rates

2 (default) | numeric

Period of forward rates, specified as Period and a scalar numeric. The default is 2, meaning forward rates are spaced at 0, .5, 1, 1.5, and so on.

Data Types: double

Output Arguments

SABRBraceGatarekMusielaModelObj — SABR-BGM model

SABRBraceGatarekMusiela object

SABR-BGM model, returned as a SABRBraceGatarekMusiela object.

Properties

Alpha — Initial SABR volatilities for each forward rate maturity

positive numeric

Initial SABR volatilities for each forward rate maturity, returned as an NumRates-1-by-1 vector of positive numeric values.

Data Types: double

Beta — SABR exponent parameters for each forward rate maturity

numeric between 0 and 1

SABR exponent parameters for each forward rate maturity, returned as an NumRates-1-by-1 vector of numeric values between 0 and 1.

Data Types: double

VolatilityofVolatility — Variation in volatility

function handle

Variation in volatility, returned as an NumRates-1-by-1 cell array of function handles.

Data Types: double | cell

FwdFwdCorrelation — Correlation matrix for forward rates

numeric between -1 and 1

Correlation matrix for forward rates, returned as an (NumRates-1)-by-(NumRates-1) correlation matrix.

Data Types: double

VolVolCorrelation — Correlation matrix for volatilities

numeric between -1 and 1

Correlation matrix for volatilities, returned as an (NumRates-1)-by-(NumRates-1) correlation matrix.

Data Types: double

Volatility — Volatility

simulate stochastic volatilities without volatility functions (default) | function handle

Volatility, returned as an NumRates-1-by-1 cell array of function handles.

Data Types: double

FwdVolCorrelation — Correlation matrix between forward rates and volatilities

(NumRates-1)-by-(NumRates-1) matrix of zeros (default) | numeric between -1 and 1

Correlation matrix between forward rates and volatilities, returned as an (NumRates-1)-by-(NumRates-1) correlation matrix.

Data Types: double

Period — Period of forward rates

2 (default) | numeric

Period of forward rates, returned as a scalar numeric.

Data Types: double

Examples

Use SABRBraceGatarekMusiela Model with Null Forward-Volatility Correlation and IRMonteCarlo Pricer to Price Cap Instrument

This example shows the workflow to price a Cap instrument when you use a SABRBraceGatarekMusiela model with null Forward-Volatility correlation and an IRMonteCarlo pricing method.

Create Cap Instrument Object

Use fininstrument to create a Cap instrument object.

CapOpt = fininstrument("Cap",Maturity=datetime(2021,7,1),Strike=0.035,Name="cap_option")

CapOpt = 
  Cap with properties:

                      Strike: 0.0350
                    Maturity: 01-Jul-2021
                 ResetOffset: 0
                       Reset: 1
                       Basis: 0
                   Principal: 100
             ProjectionCurve: [0×0 ratecurve]
    DaycountAdjustedCashFlow: 0
       BusinessDayConvention: "actual"
                    Holidays: NaT
                        Name: "cap_option"

Create Classic SABRBraceGatarekMusiela Model Object with Null Forward-Volatility Correlation

Use finmodel to create a SABRBraceGatarekMusiela model object that is a classic SABR-BGM model with null Forward-Volatility correlation.

Alpha = [0.4;0.34;0.31;0.28];
Beta = [0.5;0.5;0.5;0.5];

SABRBGMVolVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
SABRBGMVolVolParams = [.3 -.02 .7 .14];

numRates = 5;
VolVolFunc(1:numRates-1,1) = {@(t) SABRBGMVolVolFunc(SABRBGMVolVolParams,t)};

CorrFunc = @(i,j,B) exp(-B*abs(i-j));
FwdFwdCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.08);
VolVolCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.04);

SABRBGM_NF = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,FwdFwdCorrelation=FwdFwdCorrelation,VolVolCorrelation=VolVolCorrelation)

SABRBGM_NF = 
  SABRBraceGatarekMusiela with properties:

                    Period: 2
                     Alpha: [4×1 double]
                      Beta: [4×1 double]
                Volatility: {4×1 cell}
    VolatilityofVolatility: {4×1 cell}
         FwdFwdCorrelation: [4×4 double]
         VolVolCorrelation: [4×4 double]
         FwdVolCorrelation: [4×4 double]

Create ratecurve Object

Create a ratecurve object using ratecurve.

Settle = datetime(2019,1,1);
Type = 'zero';
ZeroTimes = [calmonths(6) calyears([1 2 3 4 5 7 10 20 30])]';
ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]';
ZeroDates = Settle + ZeroTimes;
 
myRC = ratecurve('zero',Settle,ZeroDates,ZeroRates)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: [10×1 datetime]
                Rates: [10×1 double]
               Settle: 01-Jan-2019
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create IRMonteCarlo Pricer Object

Use finpricer to create an IRMonteCarlo pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

simDates = datetime(2019,7,1)+calmonths(0:6:24);
outPricer = finpricer("IRMonteCarlo",Model=SABRBGM_NF,DiscountCurve=myRC,SimulationDates=simDates)

outPricer = 
  SABRBGMMonteCarlo with properties:

          NumTrials: 1000
      RandomNumbers: []
      DiscountCurve: [1×1 ratecurve]
    SimulationDates: [01-Jul-2019    01-Jan-2020    01-Jul-2020    01-Jan-2021    01-Jul-2021]
              Model: [1×1 finmodel.SABRBraceGatarekMusiela]

Price Cap Instrument

Use price to compute the price and sensitivities for the Cap instrument.

[Price,outPR] = price(outPricer,CapOpt,["all"])

Price = 
0.8867

outPR = 
  priceresult with properties:

       Results: [1×3 table]
    PricerData: [1×1 struct]

outPR.Results

ans=1×3 table
    Price     Delta      Gamma 
    ______    ______    _______

    0.8867    118.82    -9486.5

You can access the simulated interest-rate Paths in the PricerData output.

outPR.PricerData

ans = struct with fields:
    SimulationTimes: [6×1 timetable]
              Paths: [6×8×1000 double]
      RandomNumbers: [1×1 struct]

Use Classic SABRBraceGatarekMusiela Model and IRMonteCarlo Pricer to Price Floor Instrument

This example shows the workflow to price a Floor instrument when you use a classic SABRBraceGatarekMusiela model and an IRMonteCarlo pricing method.

Create Floor Instrument Object

Use fininstrument to create a Floor instrument object.

FloorOpt = fininstrument("Floor",Maturity=datetime(2021,7,1),Strike=0.05,Reset=1,Name="floor_option")

FloorOpt = 
  Floor with properties:

                      Strike: 0.0500
                    Maturity: 01-Jul-2021
                 ResetOffset: 0
                       Reset: 1
                       Basis: 0
                   Principal: 100
             ProjectionCurve: [0×0 ratecurve]
    DaycountAdjustedCashFlow: 0
       BusinessDayConvention: "actual"
                    Holidays: NaT
                        Name: "floor_option"

Create Classic SABRBraceGatarekMusiela Model Object

Use finmodel to create a classic SABRBraceGatarekMusiela model object.

Alpha = [0.4;0.34;0.31;0.28];
Beta = [0.5;0.5;0.5;0.5];

SABRBGMVolVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
SABRBGMVolVolParams = [.3 -.02 .7 .14];

numRates = 5;
VolVolFunc(1:numRates-1,1) = {@(t) SABRBGMVolVolFunc(SABRBGMVolVolParams,t)};

CorrFunc = @(i,j,B) exp(-B*abs(i-j));
FwdFwdCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.08);
VolVolCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.04);
SABRRho = [0.0005;0.0006;0.0060;0.0055];
FwdVolCorrelation = diag(SABRRho);

SABRBGM_Classic = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,FwdFwdCorrelation=FwdFwdCorrelation,VolVolCorrelation=VolVolCorrelation,FwdVolCorrelation=FwdVolCorrelation)

SABRBGM_Classic = 
  SABRBraceGatarekMusiela with properties:

                    Period: 2
                     Alpha: [4×1 double]
                      Beta: [4×1 double]
                Volatility: {4×1 cell}
    VolatilityofVolatility: {4×1 cell}
         FwdFwdCorrelation: [4×4 double]
         VolVolCorrelation: [4×4 double]
         FwdVolCorrelation: [4×4 double]

Create ratecurve Object

Create a ratecurve object using ratecurve.

Settle = datetime(2019,1,1);
Type = 'zero';
ZeroTimes = [calmonths(6) calyears([1 2 3 4 5 7 10 20 30])]';
ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]';
ZeroDates = Settle + ZeroTimes;
 
myRC = ratecurve('zero',Settle,ZeroDates,ZeroRates)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: [10×1 datetime]
                Rates: [10×1 double]
               Settle: 01-Jan-2019
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create IRMonteCarlo Pricer Object

Use finpricer to create an IRMonteCarlo pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

simDates = datetime(2019,7,1)+calmonths(0:6:24);
outPricer = finpricer("IRMonteCarlo",Model=SABRBGM_Classic,DiscountCurve=myRC,SimulationDates=simDates)

outPricer = 
  SABRBGMMonteCarlo with properties:

          NumTrials: 1000
      RandomNumbers: []
      DiscountCurve: [1×1 ratecurve]
    SimulationDates: [01-Jul-2019    01-Jan-2020    01-Jul-2020    01-Jan-2021    01-Jul-2021]
              Model: [1×1 finmodel.SABRBraceGatarekMusiela]

Price Floor Instrument

Use price to compute the price and sensitivities for the Floor instrument.

[Price,outPR] = price(outPricer,FloorOpt,["all"])

Price = 
11.6086

outPR = 
  priceresult with properties:

       Results: [1×3 table]
    PricerData: [1×1 struct]

outPR.Results

ans=1×3 table
    Price      Delta     Gamma
    ______    _______    _____

    11.609    -171.15    13201

You can access the simulated interest-rate Paths in the PricerData output.

outPR.PricerData

ans = struct with fields:
    SimulationTimes: [6×1 timetable]
              Paths: [6×8×1000 double]
      RandomNumbers: [1×1 struct]

Use SABRBraceGatarekMusiela Model (Rebonato Parametric Form with Null Forward-Volatility Correlation) and IRMonteCarlo Pricer to Price FixedBond Instrument

This example shows the workflow to price a FixedBond instrument when you use a SABRBraceGatarekMusiela model in Rebonato parametric form with null Forward-Volatility correlation and an IRMonteCarlo pricing method.

Create FixedBond Instrument Object

Use fininstrument to create a FixedBond instrument object.

FixB = fininstrument("FixedBond",Maturity=datetime(2021,7,1),CouponRate=0.021,Period=2,Basis=1,Principal=100,Name="fixed_bond_instrument")

FixB = 
  FixedBond with properties:

                  CouponRate: 0.0210
                      Period: 2
                       Basis: 1
                EndMonthRule: 1
                   Principal: 100
    DaycountAdjustedCashFlow: 0
       BusinessDayConvention: "actual"
                    Holidays: NaT
                   IssueDate: NaT
             FirstCouponDate: NaT
              LastCouponDate: NaT
                   StartDate: NaT
                    Maturity: 01-Jul-2021
                        Name: "fixed_bond_instrument"

Create Rebonato Form with Null Forward-Volatility Correlation SABRBraceGatarekMusiela Model Object

Use finmodel to create a SABRBraceGatarekMusiela model object that is a SABR-BGM model in Rebonato parametric form with null Forward-Volatility correlation.

Alpha = [0.4;0.34;0.31;0.28];
Beta = [0.5;0.5;0.5;0.5];
numRates = 5;

SABRBGMVolVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
SABRBGMVolVolParams = [.3 -.02 .7 .14];

SABRBGMVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
SABRBGMVolParams = [.2 -.01 .8 .16];
VolFunc(1:numRates-1,1) = {@(t) SABRBGMVolFunc(SABRBGMVolParams,t)};

VolVolFunc(1:numRates-1,1) = {@(t) SABRBGMVolVolFunc(SABRBGMVolVolParams,t)};

CorrFunc = @(i,j,B) exp(-B*abs(i-j));
FwdFwdCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.08);
VolVolCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.04);

SABRBGM_Rebonato = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,Volatility=VolFunc,FwdFwdCorrelation=FwdFwdCorrelation, VolVolCorrelation=VolVolCorrelation)

SABRBGM_Rebonato = 
  SABRBraceGatarekMusiela with properties:

                    Period: 2
                     Alpha: [4×1 double]
                      Beta: [4×1 double]
                Volatility: {4×1 cell}
    VolatilityofVolatility: {4×1 cell}
         FwdFwdCorrelation: [4×4 double]
         VolVolCorrelation: [4×4 double]
         FwdVolCorrelation: [4×4 double]

Create ratecurve Object

Create a ratecurve object using ratecurve.

Settle = datetime(2019,1,1);
Type = 'zero';
ZeroTimes = [calmonths(6) calyears([1 2 3 4 5 7 10 20 30])]';
ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]';
ZeroDates = Settle + ZeroTimes;
 
myRC = ratecurve('zero',Settle,ZeroDates,ZeroRates)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: [10×1 datetime]
                Rates: [10×1 double]
               Settle: 01-Jan-2019
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create IRMonteCarlo Pricer Object

Use finpricer to create an IRMonteCarlo pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

simDates = datetime(2019,7,1)+calmonths(0:6:24);
outPricer = finpricer("IRMonteCarlo",Model=SABRBGM_Rebonato,DiscountCurve=myRC,SimulationDates=simDates)

outPricer = 
  SABRBGMMonteCarlo with properties:

          NumTrials: 1000
      RandomNumbers: []
      DiscountCurve: [1×1 ratecurve]
    SimulationDates: [01-Jul-2019    01-Jan-2020    01-Jul-2020    01-Jan-2021    01-Jul-2021]
              Model: [1×1 finmodel.SABRBraceGatarekMusiela]

Price FixedBond Instrument

Use price to compute the price and sensitivities for the FixedBond instrument.

[Price,outPR] = price(outPricer,FixB,["all"])

Price = 
103.5433

outPR = 
  priceresult with properties:

       Results: [1×3 table]
    PricerData: [1×1 struct]

outPR.Results

ans=1×3 table
    Price     Delta     Gamma
    ______    ______    _____

    103.54    -253.5    628.2

You can access the simulated interest-rate Paths in the PricerData output.

outPR.PricerData

ans = struct with fields:
    SimulationTimes: [6×1 timetable]
              Paths: [6×8×1000 double]
      RandomNumbers: [1×1 struct]

Use SABRBraceGatarekMusiela Model (Rebonato Parameteric Form) and IRMonteCarlo Pricer to Price FloatBond Instrument

This example shows the workflow to price a FloatBond instrument when you use a SABRBraceGatarekMusiela model in Rebonato parametric form and an IRMonteCarlo pricing method.

Create FloatBond Instrument Object

Use fininstrument to create a FloatBond instrument object.

FloatB = fininstrument("FloatBond",Maturity=datetime(2021,7,1),Spread=0.025,Reset=2,Basis=1,Principal=100,EndMonthRule=false,Name="float_bond_instrument")

FloatB = 
  FloatBond with properties:

                      Spread: 0.0250
             ProjectionCurve: [0×0 ratecurve]
                 ResetOffset: 0
                       Reset: 2
                       Basis: 1
                EndMonthRule: 0
                   Principal: 100
    DaycountAdjustedCashFlow: 0
       BusinessDayConvention: "actual"
          LatestFloatingRate: NaN
                    Holidays: NaT
                   IssueDate: NaT
             FirstCouponDate: NaT
              LastCouponDate: NaT
                   StartDate: NaT
                    Maturity: 01-Jul-2021
                        Name: "float_bond_instrument"

Create Rebonato Form SABRBraceGatarekMusiela Model Object

Use finmodel to create a SABRBraceGatarekMusiela model object that is a SABR-BGM model in Rebonato parametric form.

Alpha = [0.4;0.34;0.31;0.28];
Beta = [0.5;0.5;0.5;0.5];
numRates = 5;

SABRBGMVolVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
SABRBGMVolVolParams = [.3 -.02 .7 .14];

SABRBGMVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
SABRBGMVolParams = [.2 -.01 .8 .16];
VolFunc(1:numRates-1,1) = {@(t) SABRBGMVolFunc(SABRBGMVolParams,t)};

VolVolFunc(1:numRates-1,1) = {@(t) SABRBGMVolVolFunc(SABRBGMVolVolParams,t)};

CorrFunc = @(i,j,B) exp(-B*abs(i-j));
FwdFwdCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.08);
VolVolCorrelation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),.04);
SABRRho = [0.0005;0.0006;0.0060;0.0055];
FwdVolCorrelation = diag(SABRRho);

SABRBGM_Rebonato_param = finmodel("SABRBraceGatarekMusiela",Alpha=Alpha,Beta=Beta,VolatilityofVolatility=VolVolFunc,Volatility=VolFunc,FwdFwdCorrelation=FwdFwdCorrelation,VolVolCorrelation=VolVolCorrelation,FwdVolCorrelation=FwdVolCorrelation)

SABRBGM_Rebonato_param = 
  SABRBraceGatarekMusiela with properties:

                    Period: 2
                     Alpha: [4×1 double]
                      Beta: [4×1 double]
                Volatility: {4×1 cell}
    VolatilityofVolatility: {4×1 cell}
         FwdFwdCorrelation: [4×4 double]
         VolVolCorrelation: [4×4 double]
         FwdVolCorrelation: [4×4 double]

Create ratecurve Object

Create a ratecurve object using ratecurve.

Settle = datetime(2019,1,1);
Type = 'zero';
ZeroTimes = [calmonths(6) calyears([1 2 3 4 5 7 10 20 30])]';
ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]';
ZeroDates = Settle + ZeroTimes;
 
myRC = ratecurve('zero',Settle,ZeroDates,ZeroRates)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: [10×1 datetime]
                Rates: [10×1 double]
               Settle: 01-Jan-2019
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create IRMonteCarlo Pricer Object

Use finpricer to create an IRMonteCarlo pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

simDates = datetime(2019,7,1)+calmonths(0:6:24);
outPricer = finpricer("IRMonteCarlo",Model=SABRBGM_Rebonato_param,DiscountCurve=myRC,SimulationDates=simDates)

outPricer = 
  SABRBGMMonteCarlo with properties:

          NumTrials: 1000
      RandomNumbers: []
      DiscountCurve: [1×1 ratecurve]
    SimulationDates: [01-Jul-2019    01-Jan-2020    01-Jul-2020    01-Jan-2021    01-Jul-2021]
              Model: [1×1 finmodel.SABRBraceGatarekMusiela]

Price FloatBond Instrument

Use price to compute the price and sensitivities for the FloatBond instrument.

[Price,outPR] = price(outPricer,FloatB,["all"])

Price = 
106.1830

outPR = 
  priceresult with properties:

       Results: [1×3 table]
    PricerData: [1×1 struct]

outPR.Results

ans=1×3 table
    Price      Delta     Gamma 
    ______    _______    ______

    106.18    -9.2496    16.927

You can access the simulated interest-rate Paths in the PricerData output.

outPR.PricerData

ans = struct with fields:
    SimulationTimes: [6×1 timetable]
              Paths: [6×8×1000 double]
      RandomNumbers: [1×1 struct]

More About

SABR-BGM Model

The SABR-BGM model incorporates stochastic volatility, meaning that the volatility parameters in the BGM model are not constant but follow a stochastic process. This allows for the modeling of volatility smiles and skews observed in the options market.

The SABR-BGM model combines the BGM model and the SABR model by introducing the SABR β_k exponent and the SABR volatility α_k(t) to the BGM forward rate SDE.

SABR-BGM Model

• The volatility-of-volatility function ν_k(t) is a deterministic function of time.

• The correlation matrix is 2N-by-2N:

  SABR-BGM Correlation Matrix

  

A special case of null (zero) Forward-Volatility correlations has some advantages:

• Sets the Forward-Volatility correlations φ_{k, 1} and the SABR Rho ρ_k to zero, which means fewer parameters to calibrate

• Still fits market data

• Gives lower variance and faster convergence in Monte Carlo simulation

SABR-BGM Model in Rebonato Parametric Form

To facilitate the calibration of the SABR-BGM model using market data, Rebonato et. al. (2009) introduced the parametric form.

In the parametric form, the SABR volatility α_k(t) is decomposed into a product of two components: the deterministic volatility function g(T_k-1 - t) and the stochastic correction term s_k(t).

SABR-BGM Model in Rebonato Parametric Form

• The volatility function g(T_k-1 - t) is a deterministic function of time.

• The volatility-of-volatility function h(T_k-1 - t) is a deterministic function of time.

• The correlation matrix is 2N-by-2N:

  SABR-BGM Model in Rebonato Parametric Form Correlation Matrix

  

  A special case of null (zero) Forward-Volatility correlations has some advantages:

  • Sets the Forward-Volatility correlations φ_{k, 1} and the SABR Rho ρ_k to zero, which means fewer parameters to calibrate

  • Still fits market data

  • Gives lower variance and faster convergence in Monte Carlo simulation

References

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

[2] Crispoldi, C., Wigger, G., and P. Larkin. SABR and SABR LIBOR Market Models in Practice. Palgrave MacMillan, 2015.

[3] Hagan, P. and A. Lesniewski. LIBOR market Model with SABR Style Volatility. Working paper JPMorgan Chase, 2008.

[4] Rebonato, R., McKay, K., and R. White. The SABR/LIBOR Market Model: pricing, Calibration, and Hedging for Complex Interest-Rate Derivatives. Wiley, 2009.

Version History

Introduced in R2021b