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ZeroCouponInflationSwap

ZeroCouponInflationSwap instrument object

Description

Create and price a ZeroCouponInflationSwap instrument object for one or more Zero-Coupon Inflation Swap instruments using this workflow:

  1. Use fininstrument to create a ZeroCouponInflationSwap instrument object for one or more Zero-Coupon Inflation Swap instruments.

  2. Use ratecurve to specify an interest-rate model for the ZeroCouponInflationSwap instrument object.

  3. Use inflationcurve to specify an inflation curve model.

  4. Use finpricer to specify an Inflation pricing method for one or more ZeroCouponInflationSwap instruments.

  5. Use inflationCashflows to compute cash flows for each one of the ZeroCouponInflationSwap instruments.

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 models and pricing methods for a ZeroCouponInflationSwap instrument, see Choose Instruments, Models, and Pricers.

Creation

Description

example

ZCInflationSwap = fininstrument(InstrumentType,'Maturity',maturity_date,'Notional',notional_value,'FixedInflationRate',inflation_rate) creates a ZeroCouponInflationSwap object for one or more Zero-Coupon Inflation Swap instruments by specifying InstrumentType and sets the properties for the required name-value pair arguments Maturity, Notional, and FixedInflationRate.

example

ZCInflationSwap = fininstrument(___,Name,Value) sets optional properties using name-value pairs in addition to the required arguments in the previous syntax. For example, ZCInflationSwap = fininstrument("ZeroCouponInflationSwap", 'Maturity',Maturity,'Notional',Notional,'FixedInflationRate',FixedInflationRate,'StartDate',StartDate) creates a ZeroCouponInflationSwap instrument. You can specify multiple name-value pair arguments.

Input Arguments

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Instrument type, specified as a string with the value of "ZeroCouponInflationSwap", a character vector with the value of 'ZeroCouponInflationSwap', an NINST-by-1 string array with values of "ZeroCouponInflationSwap", or an NINST-by-1 cell array of character vectors with values of 'ZeroCouponInflationSwap'.

Data Types: char | cell | string

ZeroCouponInflationSwap Name-Value Pair Arguments

Specify required and 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: ZCInflationSwap = fininstrument("ZeroCouponInflationSwap", 'Maturity',Maturity,'Notional',Notional,'FixedInflationRate',FixedInflationRate,'StartDate',StartDate)
Required ZeroCouponInflationSwap Name-Value Pair Arguments

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Swap maturity date, specified as the comma-separated pair consisting of 'Maturity' and a scalar datetime, serial date number, date character vector, date string or an NINST-by-1 vector of datetimes, serial date numbers, cell array of date character vectors, or date string array.

If you use date character vectors or date strings, the format must be recognizable by datetime because the Maturity property is stored as a datetime.

Data Types: char | cell | double | string | datetime

Notional amount, specified as the comma-separated pair consisting of 'Notional' and a scalar numeric or an NINST-by-1 numeric vector.

Data Types: double

Inflation rate, specified as the comma-separated pair consisting of 'FixedInflationRate' and a scalar decimal or an NINST-by-1 vector of decimals.

Data Types: double

Optional ZeroCouponInflationSwap Name-Value Pair Arguments

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Date swap starts, specified as the comma-separated pair consisting of 'StartDate' and a scalar datetime, serial date number, date character vector, date string or an NINST-by-1 vector of datetimes, serial date numbers, cell array of date character vectors, or date string array. If not specified, when pricing the ZeroCouponInflationSwap instrument, the Inflation pricer uses the Settle date of the DiscountCurve as the StartDate.

If you use date character vectors or date strings, the format must be recognizable by datetime because the StartDate property is stored as a datetime.

Data Types: char | cell | double | string | datetime

Day count basis for the fixed leg, specified as the comma-separated pair consisting of 'Basis' and a scalar integer or an NINST-by-1 vector of integers for the following:

  • 0 — actual/actual

  • 1 — 30/360 (SIA)

  • 2 — actual/360

  • 3 — actual/365

  • 4 — 30/360 (PSA)

  • 5 — 30/360 (ISDA)

  • 6 — 30/360 (European)

  • 7 — actual/365 (Japanese)

  • 8 — actual/actual (ICMA)

  • 9 — actual/360 (ICMA)

  • 10 — actual/365 (ICMA)

  • 11 — 30/360E (ICMA)

  • 12 — actual/365 (ISDA)

  • 13 — BUS/252

For more information, see Basis.

Data Types: double

Indexation lag in months, specified as the comma-separated pair consisting of 'Lag' and a scalar numeric or an NINST-by-1 numeric vector.

Data Types: double

User-defined name for the instrument, specified as the comma-separated pair consisting of 'Name' and a scalar string or character vector or an NINST-by-1 cell array of character vectors or string array.

Data Types: char | cell | string

Properties

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Swap maturity date, returned as a scalar datetime or an NINST-by-1 vector of datetimes.

Data Types: datetime

Notional amount, returned as a scalar numeric or an NINST-by-1 vector of decimals.

Data Types: double

Inflation rate, returned as a scalar decimal or an NINST-by-1 vector of decimals.

Data Types: double

Date swap starts, returned as a scalar datetime or an NINST-by-1 vector of datetimes.

Data Types: datetime

Day count basis for fixed leg, returned as a scalar integer or an NINST-by-1 vector of integers.

Data Types: double

Indexation lag in months, returned as a scalar numeric or an NINST-by-1 numeric vector.

Data Types: double

User-defined name for the instrument, returned as a string or an NINST-by-1 string array.

Data Types: string

Object Functions

inflationCashflowsCompute cash flows for ZeroCouponInflationSwap instrument

Examples

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This example shows the workflow to price a ZeroCouponInflationSwap instrument when you use an inflationcurve object and an Inflation pricing method.

Create ratecurve Object

Create a ratecurve object using ratecurve.

Settle = datetime(2021,1,15);
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;
ZeroCurve = ratecurve('zero',Settle,ZeroDates,ZeroRates)
ZeroCurve = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: [10x1 datetime]
                Rates: [10x1 double]
               Settle: 15-Jan-2021
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create inflationcurve Object

Create an inflationcurve object using inflationcurve.

BaseDate = datetime(2020, 10, 1);
InflationTimes = [0 calyears([1 2 3 4 5 7 10 20 30])]';
InflationIndexValues = [100 102 103.5 105 106.8 108.2 111.3 120.1 130.4 150.2]';
InflationDates = BaseDate + InflationTimes;
myInflationCurve = inflationcurve(InflationDates,InflationIndexValues)
myInflationCurve = 
  inflationcurve with properties:

                    Basis: 0
                    Dates: [10x1 datetime]
     InflationIndexValues: [10x1 double]
    ForwardInflationRates: [9x1 double]
              Seasonality: [12x1 double]

Create ZeroCouponInflationSwap Instrument Object

Use fininstrument to create a ZeroCouponInflationSwap instrument object.

StartDate = datetime(2021,1,1);
Maturity = datetime(2022,10,1);
FixedInflationRate = 0.015;
Notional = 2000;

ZCInflationSwap = fininstrument("ZeroCouponInflationSwap",'StartDate',StartDate,'Maturity',Maturity,'FixedInflationRate',FixedInflationRate,'Notional',Notional,'Name',"zero_coupon_inflation_swap_instrument")
ZCInflationSwap = 
  ZeroCouponInflationSwap with properties:

              Notional: 2000
    FixedInflationRate: 0.0150
                 Basis: 0
                   Lag: 3
             StartDate: 01-Jan-2021
              Maturity: 01-Oct-2022
                  Name: "zero_coupon_inflation_swap_instrument"

Create Inflation Pricer Object

Use finpricer to create an Inflation pricer object and use the ratecurve object with the 'DiscountCurve' name-value pair argument and the inflationcurve object with the 'InflationCurve' name-value pair argument.

outPricer = finpricer("Inflation",'DiscountCurve',ZeroCurve,'InflationCurve',myInflationCurve)
outPricer = 
  Inflation with properties:

     DiscountCurve: [1x1 ratecurve]
    InflationCurve: [1x1 inflationcurve]

Price ZeroCouponInflationSwap Instrument

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

[Price,outPR] = price(outPricer,ZCInflationSwap,"all")
Price = 9.5675
outPR = 
  priceresult with properties:

       Results: [1x1 table]
    PricerData: []

outPR.Results
ans=table
    Price 
    ______

    9.5675

This example shows the workflow to price multiple ZeroCouponInflationSwap instruments when you use an inflationcurve object and an Inflation pricing method.

Create ratecurve Object

Create a ratecurve object using ratecurve.

Settle = datetime(2021,12,15);
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;
ZeroCurve = ratecurve('zero',Settle,ZeroDates,ZeroRates)
ZeroCurve = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: [10x1 datetime]
                Rates: [10x1 double]
               Settle: 15-Dec-2021
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create inflationcurve Object

Create an inflationcurve object using inflationcurve.

BaseDate = datetime(2020, 10, 1);
InflationTimes = [0 calyears([1 2 3 4 5 7 10 20 30])]';
InflationIndexValues = [100 102 103.5 105 106.8 108.2 111.3 120.1 130.4 150.2]';
InflationDates = BaseDate + InflationTimes;
myInflationCurve = inflationcurve(InflationDates,InflationIndexValues)
myInflationCurve = 
  inflationcurve with properties:

                    Basis: 0
                    Dates: [10x1 datetime]
     InflationIndexValues: [10x1 double]
    ForwardInflationRates: [9x1 double]
              Seasonality: [12x1 double]

Create ZeroCouponInflationSwap Instrument Object

Use fininstrument to create a ZeroCouponInflationSwap instrument object.

StartDate = datetime([2021,5,1 ; 2021,6,1 ; 2021,7,1]);
Maturity = datetime([2022,10,1 ; 2022,11,1 ;2022,12,1]);
FixedInflationRate = 0.015;
Notional = [20000 ; 30000 ; 40000] ;

ZCInflationSwap = fininstrument("ZeroCouponInflationSwap",'StartDate',StartDate,'Maturity',Maturity,'FixedInflationRate',FixedInflationRate,'Notional',Notional,'Name',"zero_coupon_inflation_swap_instrument")
ZCInflationSwap=3×1 object
  3x1 ZeroCouponInflationSwap array with properties:

    Notional
    FixedInflationRate
    Basis
    Lag
    StartDate
    Maturity
    Name

Create Inflation Pricer Object

Use finpricer to create an Inflation pricer object and use the ratecurve object with the 'DiscountCurve' name-value pair argument and the inflationcurve object with the 'InflationCurve' name-value pair argument.

outPricer = finpricer("Inflation",'DiscountCurve',ZeroCurve,'InflationCurve',myInflationCurve)
outPricer = 
  Inflation with properties:

     DiscountCurve: [1x1 ratecurve]
    InflationCurve: [1x1 inflationcurve]

Price ZeroCouponInflationSwap Instruments

Use price to compute the prices and sensitivities for the ZeroCouponInflationSwap instruments.

[Price,outPR] = price(outPricer,ZCInflationSwap,"all")
Price = 3×1

   59.4576
   80.6037
   89.4137

outPR=1×3 object
  1x3 priceresult array with properties:

    Results
    PricerData

outPR.Results
ans=table
    Price 
    ______

    59.458

ans=table
    Price 
    ______

    80.604

ans=table
    Price 
    ______

    89.414

More About

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Algorithms

To price a zero-coupon inflation-indexed swap (ZCIS), use an inflation curve and a nominal discount curve (model-free approach), where the cash flows are discounted using the nominal discount curve.

Cash flows at maturity t=TM:FixedLeg=N×[(1+k)M1]InflationLeg=N×[I(TM)I01]

where

  • N is the reference notional of the swap.

  • k is the fixed inflation rate.

  • M is the number of years for the life of the swap.

  • I(TM) is the inflation index at the maturity date with some lag (for example, three months).

  • I0 is the inflation index at the start date with some lag (for example, three months).

References

[1] Brody, D. C., Crosby, J., and Li, H. "Convexity Adjustments in Inflation-Linked Derivatives." Risk Magazine. November 2008, pp. 124–129.

[2] Kerkhof, J. "Inflation Derivatives Explained: Markets, Products, and Pricing." Fixed Income Quantitative Research, Lehman Brothers, July 2005.

[3] Zhang, J. X. "Limited Price Indexation (LPI) Swap Valuation Ideas." Wilmott Magazine. no. 57, January 2012, pp. 58–69.

Introduced in R2021a