JarrowYildirim

Create JarrowYildirim model object for YearYearInflationCap, YearYearInflationFloor, ZeroCouponInflationCap, ZeroCouponInflationFloor, YearYearInflationSwap, or ZeroCouponInflationSwap instrument

Since R2023b

Description

Create and price a YearYearInflationCap, YearYearInflationFloorZeroCouponInflationCap, ZeroCouponInflationFloor, YearYearInflationSwap, or ZeroCouponInflationSwap instrument object with a JarrowYildirim model using this workflow:

1. Use fininstrument to create a YearYearInflationCap, YearYearInflationFloor, ZeroCouponInflationCap, ZeroCouponInflationFloor, YearYearInflationSwap, or ZeroCouponInflationSwap instrument object.

2. Use finmodel to specify a JarrowYildirim model object for the YearYearInflationCap, YearYearInflationFloor, ZeroCouponInflationCap, ZeroCouponInflationFloor, YearYearInflationSwap, or ZeroCouponInflationSwap instrument object.

3. Use finpricer to specify an JarrowYildirim pricing method for a YearYearInflationCap, YearYearInflationFloor, ZeroCouponInflationCap, ZeroCouponInflationFloor, YearYearInflationSwap, or ZeroCouponInflationSwap 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 YearYearInflationCap, YearYearInflationFloor, ZeroCouponInflationCapZeroCouponInflationFloor, YearYearInflationSwap, or ZeroCouponInflationSwap instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

JarrowYildirimModelObj = finmodel(ModelType,NominalVolatility=nominal_volatility_value,RealVolatility=real_volatility_value,IndexVolatility=index_volatility_value,NominalConstant=nominal_constant_value,RealConstant=real_constant_value,NominalRealCorrelation=nominal_real_correlation_value,RealIndexCorrelation=real_index_correlation_value,NominalIndexCorrelation=nominal_index_correlation_value,CurrentIndex=current_index_value)

Description

JarrowYildirimModelObj = finmodel(ModelType,NominalVolatility=nominal_volatility_value,RealVolatility=real_volatility_value,IndexVolatility=index_volatility_value,NominalConstant=nominal_constant_value,RealConstant=real_constant_value,NominalRealCorrelation=nominal_real_correlation_value,RealIndexCorrelation=real_index_correlation_value,NominalIndexCorrelation=nominal_index_correlation_value,CurrentIndex=current_index_value), creates a JarrowYildirim model object by specifying ModelType and the required name-value arguments for NominalVolatility, RealVolatility, IndexVolatility, NominalConstant, RealConstant, NominalRealCorrelation, RealIndexCorrelation, NominalIndexCorrelation, and CurrentIndex to set properties using name-value arguments. For example, JarrowYildirimModelObj = finmodel("JarrowYildirim",NominalVolatility=0.008,RealVolatility=0.005,IndexVolatility=0.01,NominalConstant=0.04,RealConstant=0.05,NominalRealCorrelation=0.015,RealIndexCorrelation=-0.32,NominalIndexCorrelation=0.08,CurrentIndex=101)

creates a JarrowYildirim model object.

Input Arguments

ModelType — Model type

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

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

Data Types: char | string

Name-Value Arguments

Specify required 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: JarrowYildirimModelObj = finmodel("JarrowYildirim",NominalVolatility=0.008,RealVolatility=0.005,IndexVolatility=0.01,NominalConstant=0.04,RealConstant=0.05,NominalRealCorrelation=0.015,RealIndexCorrelation=-0.32,NominalIndexCorrelation=0.08,CurrentIndex=101)

NominalVolatility — Nominal rate volatility

positive decimal

Nominal rate volatility, specified as a scalar positive decimal.

Data Types: double

RealVolatility — Real rate volatility

positive decimal

Real rate volatility, specified as a scalar positive decimal.

Data Types: double

IndexVolatility — Inflation index volatility

positive decimal

Inflation index volatility, specified as a scalar positive decimal.

Data Types: double

NominalConstant — Nominal rate constant

positive decimal

Nominal rate constant (α_n), specified as a scalar positive decimal.

Data Types: double

RealConstant — Real rate constant

positive decimal

Real rate constant (α_r), specified as a scalar positive decimal.

Data Types: double

NominalRealCorrelation — Nominal rate and real rate correlation

decimal between -1 and 1

Nominal rate and real rate correlation, specified as a scalar decimal between -1 and 1.

Data Types: double

RealIndexCorrelation — Real rate and inflation index correlation

decimal between -1 and 1

Real rate and inflation index correlation, specified as a scalar decimal between -1 and 1.

Data Types: double

NominalIndexCorrelation — Nominal rate and inflation index correlation

decimal between -1 and 1

Nominal rate and inflation index correlation, specified as a scalar decimal between -1 and 1.

Data Types: double

CurrentIndex — Current inflation index

positive numeric

Current inflation index, specified as a positive scalar numeric.

Data Types: double

Output Arguments

JarrowYildirimModelObj — Jarrow-Yildirim model

JarrowYildirim object

Jarrow-Yildirim model, returned as a JarrowYildirim object.

Properties

NominalVolatility — Nominal rate volatility

positive decimal

Nominal rate volatility, returned as a scalar positive decimal.

Data Types: double

RealVolatility — Real rate volatility

positive decimal

Real rate volatility, returned as a scalar positive decimal.

Data Types: double

IndexVolatility — Inflation index volatility

positive decimal

Inflation index volatility, returned as a scalar positive decimal.

Data Types: double

NominalConstant — Nominal rate constant

positive decimal

Nominal rate constant (α_n), returned as a scalar positive decimal.

Data Types: double

RealConstant — Real rate constant

positive decimal

Real rate constant (α_r), returned as a scalar positive decimal.

Data Types: double

NominalRealCorrelation — Nominal rate and real rate correlation

decimal between -1 and 1

Nominal rate and real rate correlation, returned as a scalar decimal.

Data Types: double

RealIndexCorrelation — Real rate and inflation index correlation

decimal between -1 and 1

Real rate and inflation index correlation, returned as a scalar decimal.

Data Types: double

NominalIndexCorrelation — Nominal rate and inflation index correlation

decimal between -1 and 1

Nominal rate and inflation index correlation, returned as a scalar decimal.

Data Types: double

CurrentIndex — Current inflation index

positive numeric

Current inflation index, returned as a positive scalar numeric.

Data Types: double

Examples

Use JarrowYildirim Model and JarrowYildirim Pricer to Price YearYearInflationCap Instrument

This example shows the workflow to price a YearYearInflationCap instrument when using a JarrowYildirim model and a JarrowYildirim pricing method.

Create YearYearInflationCap Instrument Object

Use fininstrument to create a YearYearInflationCap instrument object.

YYInflationCap = fininstrument("YearYearInflationCap",Maturity=datetime(2033, 10, 1), Strike=0.05, Notional=1000,Name="YYInflationCap")

YYInflationCap = 
  YearYearInflationCap with properties:

      Notional: 1000
        Strike: 0.0500
         Basis: 0
    IssueIndex: NaN
      Maturity: 01-Oct-2033
          Name: "YYInflationCap"

Create JarrowYildirim Model Object

Use finmodel to create a JarrowYildirim model object.

JarrowYildirimModel = finmodel("JarrowYildirim", ...
        NominalVolatility=0.008,RealVolatility=0.005,IndexVolatility=0.01, ...
        NominalConstant=0.04,RealConstant=0.05, ...
        NominalRealCorrelation=0.015,RealIndexCorrelation=-0.32, ...
        NominalIndexCorrelation=0.08,CurrentIndex=101)

JarrowYildirimModel = 
  JarrowYildirim with properties:

          NominalVolatility: 0.0080
             RealVolatility: 0.0050
            IndexVolatility: 0.0100
            NominalConstant: 0.0400
               RealConstant: 0.0500
     NominalRealCorrelation: 0.0150
       RealIndexCorrelation: -0.3200
    NominalIndexCorrelation: 0.0800
               CurrentIndex: 101

Create ratecurve Object

Create a ratecurve object using ratecurve for the NominalCurve and RealCurve.

Settle = datetime(2023,10,1);

ZeroTimes = [calmonths(6) calyears([1 2 3 5 7 10 20 30])];
NominalRates = [4.70 4.68 4.14 3.83 3.56 3.51 3.48 3.77 3.66]'./100;
RealRates = [1.47 1.55 1.31 1.30 1.33 1.28 1.25 1.33 1.42]'./100;
ZeroDates = Settle + ZeroTimes;
NominalCurve = ratecurve("zero",Settle,ZeroDates,NominalRates);
RealCurve = ratecurve("zero",Settle,ZeroDates,RealRates);

Create JarrowYildirim Pricer Object

Use finpricer to create a JarrowYildirim object and use the ratecurve object for the NominalCurve and RealCurve name-value arguments.

JarrowYildirimPricer = finpricer("analytic",Model=JarrowYildirimModel,NominalCurve=NominalCurve,RealCurve=RealCurve)

JarrowYildirimPricer = 
  JarrowYildirim with properties:

           Model: [1×1 finmodel.JarrowYildirim]
    NominalCurve: [1×1 ratecurve]
       RealCurve: [1×1 ratecurve]

Price YearYearInflationCap Instrument

Use price to compute the price for the YearYearInflationCap instrument.

YYInflationCapPrice = price(JarrowYildirimPricer,YYInflationCap)

YYInflationCapPrice = 
8.3100

More About

Jarrow-Yildirim Model

The Jarrow-Yildirim model is a mathematical model that describes the dynamics of interest rates and the associated nominal and real interest rates.

The Jarrow-Yildirim model is a modification of the Heath-Jarrow-Morton (HJM) model that uses a foreign currency analogy for the relationships among the real interest rates, nominal interest rates, and inflation rates:

• Real zero-coupon bond price ↔ Foreign currency

• Nominal zero-coupon bond price ↔ Domestic currency

• Inflation rate ↔ Spot exchange rate

Algorithms

The Jarrow-Yildirim model under real-world measure P is represented as

$$\begin{array}{l}d{f}_n(t,T)={\alpha }_n(t,T)dt+{\varsigma }_n(t,T)d{W}_n^P(t)\\ d{f}_r(t,T)={\alpha }_r(t,T)dt+{\varsigma }_r(t,T)d{W}_r^P(t)\\ dI(t)=I(t){\mu }_I(t)dt+{\sigma }_{I}I(t)d{W}_I^P(t)\\ {\varsigma }_n(t,T)={\sigma }_n{e}^{-a_n(T-t)}\\ {\varsigma }_r(t,T)={\sigma }_r{e}^{-a_r(T-t)}\\ f_x(t,T)=-\frac{\partial ln{P}_x(t,T)}{\partial T}\\ x\in \{n,r\}\end{array}$$

where:

• $$f_n(t,T)$$ Nominal instantaneous forward rate for maturity T.

• $$f_r(t,T)$$ Real instantaneous forward rate for maturity T.

• I(t) is the inflation index.

• $${\alpha }_n(t,T)\$,\${\alpha }_r(t,T)\$,\${\mu }_I(t)$$ is the adapted processes.

• $${\sigma }_n$$ is the nominal rate volatility (positive constant).

• $${\sigma }_r$$ is the real rate volatility (positive constant).

• $${\sigma }_I$$ is the inflation index volatility (positive constant).

• $$a_n$$, $$a_r$$ are the positive constants.

• $$P_x(t,T)$$ is the zero coupon bond price (where n is nominal and r is real).

The Jarrow-Yildirim model under nominal risk-neutral measure Q_n is represented as

$$\begin{array}{l}dn(t)=[{\vartheta }_n(t)-a_{n}n(t)]dt+{\sigma }_{n}d{W}_n(t)\\ dr(t)=[{\vartheta }_r(t)-{\rho }_{r,I}{\sigma }_I{\sigma }_r-a_{r}r(t)]dt+{\sigma }_{r}d{W}_r(t)\\ dI(t)=I(t)[n(t)-r(t)]dt+{\sigma }_{I}I(t)d{W}_I(t)\\ n(t)=f_n(t,t)\\ r(t)=f_r(t,t)\\ {\vartheta }_x(t)=\frac{\partial f_x(0,T)}{\partial T}+a_x{f}_x(0,t)+\frac{{\sigma }_x^2}{2a_x}(1-e^{-2a_{x}t})\\ x\in \{n,r\}\end{array}$$

where:

• n(t) is the instantaneous nominal rate.

• r(t) is the instantaneous real rate.

• I(t) is the inflation index.

• $$(W_n,W_r,W_I)$$ is the Brownian motion with correlations $${\rho }_{n,r}$$, $${\rho }_{n,I}$$, $${\rho }_{r,I}$$.

• $${\sigma }_n$$ is the nominal rate volatility (positive constant).

• $${\sigma }_r$$ is the real rate volatility (positive constant).

• $${\sigma }_I$$ is the inflation index volatility (positive constant).

• $$a_n$$, $$a_r$$ are the positive constants.

• $$f_x(t,T)$$ is the instantaneous forward rate (where n is nominal and r is real).

References

[1] Jarrow, R. and Yildirim, Y. "Pricing Treasury Inflation Protected Securities and Related Derivatives using an HJM Model." Journal of Financial and Quantitative Analysis. Vol. 38, 2003.

Version History

Introduced in R2023b