BraceGatarekMusiela

Create BraceGatarekMusiela 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 BraceGatarekMusiela 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 BraceGatarekMusiela 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

BraceGatarekMusielaModelObj = finmodel(ModelType,Volatility=volatility_value,Correlation=correlation_value)

BraceGatarekMusielaModelObj = finmodel(___,Name=Value)

Description

BraceGatarekMusielaModelObj = finmodel(ModelType,Volatility=volatility_value,Correlation=correlation_value) creates a BraceGatarekMusiela model object by specifying ModelType and the required name-value arguments Volatility and Correlation to set the properties.

BraceGatarekMusielaModelObj = finmodel(___,Name=Value) sets optional properties using additional name-value arguments in addition to the required arguments in the previous syntax. For example, BraceGatarekMusielaModelObj = finmodel("BraceGatarekMusiela",Volatility=VolFunc,Correlation=Correlation,Period=1) creates a BraceGatarekMusiela model object. You can specify multiple name-value arguments.

Input Arguments

ModelType — Model type

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

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

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: BraceGatarekMusielaModelObj = finmodel("BraceGatarekMusiela",Volatility=VolFunc,Correlation=Correlation,Period=1)

Volatility — Volatility

function handle

Volatility, specified as Volatility 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.

Data Types: double | cell

Correlation — Correlation matrix

matrix

Correlation matrix, specified as Correlation and a (NumRates-1)-by-(NumRates-1) correlation matrix.

Data Types: double

Optional BraceGatarekMusiela Name-Value Arguments

NumFactors — Number of Brownian factors

NaN (default) | numeric

Number of Brownian factors, specified as NumFactors and a scalar numeric. The default is NaN which means that NumFactors is equal to the number of rates.

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

BraceGatarekMusielaModelObj — Brace-Gatarek-Musiela model

BraceGatarekMusiela object

Brace-Gatarek-Musiela model, returned as a BraceGatarekMusiela object.

Properties

Volatility — Volatility

function handle

Volatility, returned as a (NumRates-1)-by-1 cell array of function handles.

Data Types: double | cell

Correlation — Correlation matrix

matrix

Correlation matrix, returned as a (NumRates-1)-by-(NumRates-1) correlation matrix.

Data Types: double

NumFactors — Number of Brownian factors

NaN (default) | numeric

Number of Brownian factors, returned as a scalar numeric.

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 BraceGatarekMusiela Model and IRMonteCarlo Pricer to Price Floor Instrument

This example shows the workflow to price a Floor instrument when using a BraceGatarekMusiela model and an IRMonteCarlo pricing method.

Create Floor Instrument Object

Use fininstrument to create a Floor instrument object.

FloorOpt = fininstrument("Floor",Maturity=datetime(2022,9,15),Strike=0.05,Reset=1,Name="floor_option")

FloorOpt = 
  Floor with properties:

                      Strike: 0.0500
                    Maturity: 15-Sep-2022
                 ResetOffset: 0
                       Reset: 1
                       Basis: 0
                   Principal: 100
             ProjectionCurve: [0×0 ratecurve]
    DaycountAdjustedCashFlow: 0
       BusinessDayConvention: "actual"
                    Holidays: NaT
                        Name: "floor_option"

Create BraceGatarekMusiela Model Object

Use finmodel to create a LinearGaussian2F model object.

BGMVolFunc = @(a,t) (a(1)*t + a(2)).*exp(-a(3)*t) + a(4);
BGMVolParams = [.3 -.02 .7 .14];
numRates = 20;
VolFunc(1:numRates-1) = {@(t) BGMVolFunc(BGMVolParams,t)};
Beta = .08;
CorrFunc = @(i,j,Beta) exp(-Beta*abs(i-j));
Correlation = CorrFunc(meshgrid(1:numRates-1)',meshgrid(1:numRates-1),Beta);
BGM = finmodel("BraceGatarekMusiela",Volatility=VolFunc,Correlation=Correlation,Period=1);

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])]';
ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293]';
ZeroDates = Settle + ZeroTimes;
 
myRC = ratecurve('zero',Settle,ZeroDates,ZeroRates)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: [9×1 datetime]
                Rates: [9×1 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.

outPricer = finpricer("IRMonteCarlo",Model=BGM,DiscountCurve=myRC,SimulationDates=ZeroDates)

outPricer = 
  BGMMonteCarlo with properties:

          NumTrials: 1000
      RandomNumbers: []
      DiscountCurve: [1×1 ratecurve]
    SimulationDates: [01-Jul-2019    01-Jan-2020    01-Jan-2021    01-Jan-2022    01-Jan-2023    01-Jan-2024    01-Jan-2026    01-Jan-2029    01-Jan-2039]
              Model: [1×1 finmodel.BraceGatarekMusiela]

Price Floor Instrument

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

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

Price = 
14.7975

outPR = 
  priceresult with properties:

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

outPR.Results

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

    14.797    -398.43    1399.5

More About

Brace-Gatarek-Musiela Model

The Brace-Gatarek-Musiela (BGM) model, also known as the LIBOR market model, is a multifactor model that takes into account the term structure of interest rates and the volatility of interest rate movements.

The BGM model is based on the Heath-Jarrow-Morton (HJM) forward rate approach, it builds a process for LIBOR interest rates, assuming a conditional lognormal process for LIBOR. The BGM model is an interest-rate model that differs from short rate models in that it evolves a set of discrete forward rates.

Specifically, the lognormal BGM model specifies the following diffusion equation for each forward rate

$$\frac{d{F}_i(t)}{F_i}=-{\mu }_{i}dt+{\sigma }_i(t)d{W}_i$$

where:

W is an N-dimensional geometric Brownian motion with

$$d{W}_i(t)d{W}_j(t)={\rho }_{ij}$$

The BGM model relates drifts of the forward rates based on no-arbitrage arguments. Specifically, under the Spot LIBOR measure, drifts are expressed as

$${\mu }_i(t)=-{\sigma }_i(t){\displaystyle \sum _{j=q(t)}^i\frac{{\tau }_j{\rho }_{i,j}{\sigma }_j(t)F_j(t)}{1+{\tau }_j{F}_j(t)}}$$

where:

$${\rho }_{i,j}$$ represents the input argument Correlation.

$${\sigma }_j(t)$$ represents the input argument Volatility.

$$F_j(t)$$ represents the computation of the ZeroCurve.

$${\tau }_i$$ is the time fraction associated with the i th forward rate.

q(t) is an index defined by the relation

$$T_{q(t)-1}<t<T_{q(t)}$$

and the Spot LIBOR numeraire is defined as

$$B(t)=P(t,T_{q(t)}){\displaystyle \prod _{n=0}^{q(t)-1}(1+{\tau }_n{F}_n(T_n))}$$

Version History

Introduced in R2021b