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Price embedded option on floating-rate note for Black-Karasinski interest-rate tree



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


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


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Define the interest-rate term structure.

Rates = [0.03;0.034;0.038;0.04];
ValuationDate = 'Jan-1-2012';
StartDates = ValuationDate;
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: 734869
    ValuationDate: 734869
            Basis: 0
     EndMonthRule: 1

Build the BK tree.

VolDates = ['1-Jan-2013'; '1-Jan-2014'; '1-Jan-2015';'1-Jan-2016'];
VolCurve = 0.01;
AlphaDates = '01-01-2016';
AlphaCurve = 0.1;

BKVolSpec = bkvolspec(RateSpec.ValuationDate, VolDates, VolCurve,... 
            AlphaDates, AlphaCurve);
BKTimeSpec = bktimespec(RateSpec.ValuationDate, VolDates, Compounding);
BKT = bktree(BKVolSpec, RateSpec, BKTimeSpec)
BKT = struct with fields:
      FinObj: 'BKFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 1 2 3]
        dObs: [734869 735235 735600 735965]
      CFlowT: {[4x1 double]  [3x1 double]  [2x1 double]  [4]}
       Probs: {[3x1 double]  [3x3 double]  [3x5 double]}
     Connect: {[2]  [2 3 4]  [2 3 4 5 6]}
     FwdTree: {1x4 cell}

The floater instrument has a spread of 15, a period of one year, and matures and is callable on Jan-1-2015.

Spread = 15;
Settle = 'Jan-1-2012';
Maturity =  'Jan-1-2015';
Period = 1;
OptSpec = {'call'};
Strike =101;
ExerciseDates = 'Jan-1-2015';

Compute the price of the floater with the embedded call.

Price= optemfloatbybk(BKT, Spread, Settle, Maturity,...
OptSpec, Strike, ExerciseDates)
Price = 100.4201

Input Arguments

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

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 BK 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 | char | cell

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

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.PTree contains option prices.

  • PriceTree.tObs contains the observation times.

More About

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Floating-Rate Note with Embedded Options

A floating-rate note with an embedded option enables floating-rate notes to have early redemption features.

A FRN with an embedded option gives investors or issuers the option to retire the outstanding principal prior to maturity. An embedded call option gives the right to retire the note prior to the maturity date (callable floater), and an embedded put option gives the right to sell the note back at a specific price (puttable floater).

For more information, see Floating-Rate Note with Embedded Options.

Version History

Introduced in R2013a