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Price caps using Normal or Bachelier pricing model



[CapPrice,Caplets] = capbynormal(RateSpec,Strike,Settle,Maturity,Volatility) prices caps using the Normal (Bachelier) pricing model for negative rates. capbynormal computes prices of vanilla caps and amortizing caps.


[CapPrice,Caplets] = capbynormal(___,Name,Value) adds optional name-value pair arguments.


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Consider an investor who gets into a contract that caps the interest rate on a $100,000 loan at –.08% quarterly compounded for 3 months, starting on January 1, 2009. Assuming that on January 1, 2008 the zero rate is .069394% continuously compounded and the volatility is 20%, use this data to compute the cap price. First, calculate the RateSpec, and then use capbynormal to compute the CapPrice.

ValuationDate = 'Jan-01-2008';
EndDates ='April-01-2010';
Rates = 0.0069394;
Compounding = -1;
Basis = 1;

RateSpec = intenvset('ValuationDate', ValuationDate, ...
'StartDates', ValuationDate,'EndDates', EndDates, ...
'Rates', Rates,'Compounding', Compounding,'Basis', Basis);

Settle = 'Jan-01-2009'; % cap starts in a year
Maturity = 'April-01-2009';
Volatility = 0.20;
CapRate = -0.008;
CapReset = 4;

CapPrice = capbynormal(RateSpec,  CapRate, Settle, Maturity, Volatility,...
'Reset',CapReset,'ValuationDate',ValuationDate,'Principal', Principal,...
'Basis', Basis)
CapPrice = 2.1682e+03

Define the RateSpec.

Settle = datenum('20-Jan-2016');
ZeroTimes = [.5 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 = datemnth(Settle,12*ZeroTimes);
RateSpec = intenvset('StartDate',Settle,'EndDates',ZeroDates,'Rates',ZeroRates)
RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 2
             Disc: [10x1 double]
            Rates: [10x1 double]
         EndTimes: [10x1 double]
       StartTimes: [10x1 double]
         EndDates: [10x1 double]
       StartDates: 736349
    ValuationDate: 736349
            Basis: 0
     EndMonthRule: 1

Define the cap instrument and price with capbyblk.

ExerciseDate = datenum('20-Jan-2026');

[~,ParSwapRate] = swapbyzero(RateSpec,[NaN 0],Settle,ExerciseDate)
ParSwapRate = 0.0216
Strike = .01;
BlackVol = .3;
NormalVol = BlackVol*ParSwapRate;

Price = capbyblk(RateSpec,Strike,Settle,ExerciseDate,BlackVol)
Price = 11.8693

Price the cap instrument using capbynormal.

Price_Normal = capbynormal(RateSpec,Strike,Settle,ExerciseDate,NormalVol)
Price_Normal = 12.5495

Price the cap instrument using capbynormal for a negative strike.

 Price_Normal = capbynormal(RateSpec,-.005,Settle,ExerciseDate,NormalVol)
Price_Normal = 24.4816

Input Arguments

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Interest-rate term structure (annualized and continuously compounded), specified by the RateSpec obtained from intenvset. For information on the interest-rate specification, see intenvset.

Data Types: struct

Rate at which cap is exercised, specified as a NINST-by-1 vector of decimal values.

Data Types: double

Settlement date for the cap, specified as a NINST-by-1 vector of serial date numbers, date character vectors, datetime objects, or string objects.

Data Types: double | char | datetime | string

Maturity date for the cap, specified as a NINST-by-1 vector of serial date numbers, date character vectors, datetime objects, or string objects.

Data Types: double | char | datetime | string

Normal volatilities values, specified as a NINST-by-1 vector of numeric values.

For more information on the Normal model, see Work with Negative Interest Rates Using Functions.

Data Types: double

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: [CapPrice,Caplets] = capbynormal(RateSpec,Strike,Settle,Maturity,Volatility,'Reset',CapReset,'Principal',100000,'Basis',7)

Reset frequency payment per year, specified as the comma-separated pair consisting of 'Reset' and a NINST-by-1 vector.

Data Types: double

Notional principal amount, specified as the comma-separated pair consisting of 'Principal' and a NINST-by-1 of notional principal amounts, or a NINST-by-1 cell array. Each element in the NINST-by-1 cell array is a NumDates-by-2 cell array, where the first column is dates, and the second column is the associated principal amount. The date indicates the last day that the principal value is valid.

Use Principal to pass a schedule to compute the price for an amortizing cap.

Data Types: double | cell

Day-count basis of instrument representing the basis used when annualizing the input forward rate, specified as the comma-separated pair consisting of 'Basis'and a NINST-by-1 vector of integers. Values are:

  • 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

Observation date of the investment horizon, specified as the comma-separated pair consisting of 'ValuationDate' and a serial date number, date character vector, datetime object, or string array.

Data Types: double | char | datetime | string

The rate curve to be used in projecting the future cash flows, specified as the comma-separated pair consisting of 'ProjectionCurve' and rate curve structure. This structure must be created using intenvset. Use this optional input if the forward curve is different from the discount curve.

Data Types: struct

Output Arguments

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Expected price of the cap, returned as a NINST-by-1 vector.

Caplets, returned as a NINST-by-NCF array of caplets, padded with NaNs.

More About

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A cap is a contract that includes a guarantee that sets the maximum interest rate to be paid by the holder, based on an otherwise floating interest rate.

The payoff for a cap is:


For more information, see Cap.

Version History

Introduced in R2017a