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Price embedded option on floating-rate note for Cox-Ingersoll-Ross interest-rate tree



[Price,PriceTree] = optemfloatbycir(CIRTree,Spread,Settle,Maturity,OptSpec,Strike,ExerciseDates) prices embedded options on floating-rate notes from a Cox-Ingersoll-Ross (CIR) interest rate tree. optemfloatbycir computes prices of vanilla floating-rate notes with embedded options using a CIR++ model with the Nawalka-Beliaeva (NB) approach.


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


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Create a RateSpec using the intenvset function.

Rates = [0.035; 0.042147; 0.047345; 0.052707]; 
Dates = {'Jan-1-2017'; 'Jan-1-2018'; 'Jan-1-2019'; 'Jan-1-2020'; 'Jan-1-2021'}; 
ValuationDate = 'Jan-1-2017'; 
EndDates = Dates(2:end)'; 
Compounding = 1; 
RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, 'EndDates',EndDates,'Rates', Rates, 'Compounding', Compounding); 

Create a CIR tree.

NumPeriods = length(EndDates); 
Alpha = 0.03; 
Theta = 0.02;  
Sigma = 0.1;   
Settle = '01-Jan-2017'; 
Maturity = '01-Jan-2020'; 
CIRTimeSpec = cirtimespec(Settle, Maturity, 3); 
CIRVolSpec = cirvolspec(Sigma, Alpha, Theta); 

CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec)
CIRT = struct with fields:
      FinObj: 'CIRFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 1 2]
        dObs: [736696 737061 737426]
     FwdTree: {[1.0350]  [1.0790 1.0500 1.0298]  [1.1275 1.0887 1.0594 ... ]}
     Connect: {[3x1 double]  [3x3 double]}
       Probs: {[3x1 double]  [3x3 double]}

Define the floater instruments with the embedded call option.

Spread = 10;
Settle = 'Jan-1-2017';
Maturity =  {'Jan-1-2019';'Jan-1-2020'};
Period = 1;
OptSpec = {'call'};
Strike = 101;
ExerciseDates = 'Jan-1-2019';

Compute the price of the floaters with the embedded call.

[Price,PriceTree] = optemfloatbycir(CIRT,Spread,Settle,Maturity,OptSpec,Strike,ExerciseDates)
Price = 2×1


PriceTree = struct with fields:
    FinObj: 'CIRPriceTree'
      tObs: [0 1 2 3]
     PTree: {[2x1 double]  [2x3 double]  [2x5 double]  [2x5 double]}

Input Arguments

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

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, date character vectors, string arrays, or datetime arrays 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 CIR tree. The floating-rate note argument Settle is ignored.

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

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

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

Definition of option, specified as a NINST-by-1 cell array of character vectors or string arrays with a value of 'call' or 'put'.

Data Types: cell | char | string

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, date character vectors, string arrays, or datetime arrays using an NINST-by-NSTRIKES or an NINST-by-2 vector of for the option exercise dates.

  • For 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 ExerciseDates on the option expiry date.

  • For an American option, the 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 ExerciseDates.

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

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] = optemfloatbycir(CIRTree,Spread,Settle,Maturity,OptSpec,Strike,ExerciseDates,'AmericanOpt',1,'FloatReset',6,'Basis',8)

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and a NINST-by-1 vector of 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 the output from 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.tObs contains the observation times.

  • PriceTree.PTree contains option prices.

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.


[1] Cox, J., Ingersoll, J., and S. Ross. "A Theory of the Term Structure of Interest Rates." Econometrica. Vol. 53, 1985.

[2] Brigo, D. and F. Mercurio. Interest Rate Models - Theory and Practice. Springer Finance, 2006.

[3] Hirsa, A. Computational Methods in Finance. CRC Press, 2012.

[4] Nawalka, S., Soto, G., and N. Beliaeva. Dynamic Term Structure Modeling. Wiley, 2007.

[5] Nelson, D. and K. Ramaswamy. "Simple Binomial Processes as Diffusion Approximations in Financial Models." The Review of Financial Studies. Vol 3. 1990, pp. 393–430.

Version History

Introduced in R2018a