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Price embedded option on floating-rate note for Heath-Jarrow-Morton interest-rate tree


[Price,PriceTree] = optemfloatbyhjm(HJMTree,Spread,Settle,Maturity,OptSpec,Strike,ExerciseDates)
[Price,PriceTree] = optemfloatbyhjm(___,Name,Value)



[Price,PriceTree] = optemfloatbyhjm(HJMTree,Spread,Settle,Maturity,OptSpec,Strike,ExerciseDates) prices embedded options on floating-rate notes from a Heath-Jarrow-Morton interest rate tree. optemfloatbybk computes prices of vanilla floating-rate notes with embedded options.


[Price,PriceTree] = optemfloatbyhjm(___,Name,Value) adds optional name-value pair arguments.


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Define the interest-rate term structure.

Rates = [0.03;0.035;0.040;0.045];
ValuationDate = 'Jan-1-2012';
StartDates = {'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'};
EndDates = {'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'; 'Jan-1-2016'};
Compounding = 1;

Create the RateSpec.

RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates',...
StartDates, 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)
RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 1
             Disc: [4x1 double]
            Rates: [4x1 double]
         EndTimes: [4x1 double]
       StartTimes: [4x1 double]
         EndDates: [4x1 double]
       StartDates: [4x1 double]
    ValuationDate: 734869
            Basis: 0
     EndMonthRule: 1

Build the HJM tree.

VolSpec = hjmvolspec('Constant', 0.01);
TimeSpec = hjmtimespec(RateSpec.ValuationDate, EndDates, Compounding);
HJMTree = hjmtree(VolSpec, RateSpec, TimeSpec)
HJMTree = struct with fields:
      FinObj: 'HJMFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 1 2 3]
        dObs: [734869 735235 735600 735965]
        TFwd: {[4x1 double]  [3x1 double]  [2x1 double]  [3]}
      CFlowT: {[4x1 double]  [3x1 double]  [2x1 double]  [4]}
     FwdTree: {[4x1 double]  [3x1x2 double]  [2x2x2 double]  [1x4x2 double]}

The floater instrument has a spread of 15, a period of one year, and matures and is callable on Jan-1-2016.

Spread = 15;
Settle = 'Jan-1-2012';
Maturity =  'Jan-1-2016';
Period = 1;
OptSpec = {'call'};
Strike = 95;
ExerciseDates = 'Jan-1-2016';

Compute the price of the floater with the embedded call.

Price = optemfloatbyhjm(HJMTree, Spread, Settle, Maturity,...
OptSpec, Strike, ExerciseDates)
Price = 96.2355

Input Arguments

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Interest-rate tree specified as a structure by using hjmtree.

Data Types: struct

Number of basis points over the reference rate specified as a vector of nonnegative integers for the number of instruments (NINST)-by-1).

Data Types: single | double

Settlement dates of floating-rate note specified as serial date numbers or date character vectors using a NINST-by-1 vector of dates.


The Settle date for every floating-rate note with an embedded option is set to the ValuationDate of the HJM tree. The floating-rate note argument Settle is ignored.

Data Types: double | cell | char

Floating-rate note maturity date specified as serial date numbers or date character vectors using a NINST-by-1 vector of dates.

Data Types: double | cell | char

Definition of option as 'call' or 'put' specified as a NINST-by-1 cell array of character vectors for 'call' or 'put'.

Data Types: cell | char

Option strike price values specified nonnegative integers using as NINST-by-NSTRIKES vector of strike price values.

Data Types: single | double

Exercise date for option (European, Bermuda, or American) specified as serial date numbers or date character vectors using a NINST-by-NSTRIKES or NINST-by-2 vector of for the option exercise dates.

  • If a European or Bermuda option, the ExerciseDates is a 1-by-1 (European) or 1-by-NSTRIKES (Bermuda) vector of exercise dates. For a European option, there is only one ExerciseDate on the option expiry date.

  • If an American option, then ExerciseDates is a 1-by-2 vector of exercise date boundaries. The option exercises on any date between or including the pair of dates on that row. If there is only one non-NaN date, or if ExerciseDates is 1-by-1, the option exercises between the Settle date and the single listed ExerciseDate.

Data Types: double | cell | char

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [Price,PriceTree] = optemfloatbyhjm(HJMTree,Spread,Settle,Maturity,OptSpec,Strike,ExerciseDates,'AmericanOpt',1,'FloatReset',6,'Basis',8)

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Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and NINST-by-1 positive integer scalar flags with values:

  • 0 — European/Bermuda

  • 1 — American

Data Types: double

Frequency of payments per year, specified as the comma-separated pair consisting of 'FloatReset' and positive integers for the values [1,2,3,4,6,12] in a NINST-by-1 vector.


Payments on floating-rate notes (FRNs) are determined by the effective interest-rate between reset dates. If the reset period for an FRN spans more than one tree level, calculating the payment becomes impossible due to the recombining nature of the tree. That is, the tree path connecting the two consecutive reset dates cannot be uniquely determined because there will be more than one possible path for connecting the two payment dates.

Data Types: double

Day-count basis of the instrument, specified as the comma-separated pair consisting of 'Basis' and a positive integer using a NINST-by-1 vector. The Basis value represents the basis used when annualizing the input forward-rate tree.

  • 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

End-of-month rule flag, specified as the comma-separated pair consisting of 'EndMonthRule' and a nonnegative integer [0, 1] using a NINST-by-1 vector. This rule applies only when Maturity is an end-of-month date for a month having 30 or fewer days.

  • 0 = Ignore rule, meaning that a bond coupon payment date is always the same numerical day of the month.

  • 1 = Set rule on, meaning that a bond coupon payment date is always the last actual day of the month.

Data Types: double

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

Data Types: double | cell

Structure containing derivatives pricing options, specified as the comma-separated pair consisting of 'Options' and a structure obtained from using derivset.

Data Types: struct

Output Arguments

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Expected prices of the floating-rate note option at time 0 are returned as a scalar or an NINST-by-1 vector.

Structure of trees containing vectors of instrument prices and accrued interest and a vector of observation times for each node returned as:

  • PriceTree.PBush contains the clean prices.

  • PriceTree.tObs contains the observation times.

Introduced in R2013a

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