Main Content


Price options on floating-rate notes for Cox-Ingersoll-Ross interest-rate tree



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


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


collapse all

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-2021'; 
CIRTimeSpec = cirtimespec(Settle, Maturity, NumPeriods); 
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 3]
        dObs: [736696 737061 737426 737791]
     FwdTree: {1x4 cell}
     Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
       Probs: {[3x1 double]  [3x3 double]  [3x5 double]}

The floater instrument has a spread of 10, a period of one year, and matures on Jan-1-2018.

Spread = 10;
Settle = 'Jan-1-2017';
Maturity = 'Jan-1-2019';
Period = 1;

Define the option for the floating-rate note.

OptSpec = {'call'};
Strike = 95;
ExerciseDates = 'Jan-1-2018';
AmericanOpt = [0;1];

Compute the price of the call options.

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


PriceTree = struct with fields:
     FinObj: 'CIRPriceTree'
      PTree: {1x5 cell}
     AITree: {1x5 cell}
       tObs: [0 1 2 3 4]
    Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
      Probs: {[3x1 double]  [3x3 double]  [3x5 double]}

Input Arguments

collapse all

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

Data Types: struct

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

Data Types: cell | char | string

Option strike price values, specified as nonnegative integers using an 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 a NINST-by-NSTRIKES or 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 ExerciseDate 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 ExerciseDate.

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

Option type specified as NINST-by-1 positive integer scalar flags with values:

  • 0 — European/Bermuda

  • 1 — American

Data Types: single | double

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 is set to the ValuationDate of the CIR tree. The floating-rate note argument Settle is ignored.

Data Types: double | cell | char | string

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

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] = optfloatbybk(CIRTree,OptSpec,Strike,ExerciseDates,AmericanOpt,Spread,Settle,Maturity,'FloatReset',4,'Basis',7)

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

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

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

Output Arguments

collapse all

Expected prices of the floating-rate note option at time 0 is 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 the clean prices.

  • PriceTree.AITree contains the accrued interest.

  • PriceTree.tObs contains the observation times.

  • PriceTree.Connect contains the connectivity vectors. Each element in the cell array describes how nodes in that level connect to the next. For a given tree level, there are NumNodes elements in the vector, and they contain the index of the node at the next level that the middle branch connects to. Subtracting 1 from that value indicates where the up-branch connects to, and adding 1 indicated where the down branch connects to.

  • PriceTree.Probs contains the probability arrays. Each element of the cell array contains the up, middle, and down transition probabilities for each node of the level.

More About

collapse all

Floating-Rate Note Options

A floating-rate note option is a put or call option on a floating-rate note.

Financial Instruments Toolbox™ supports three types of put and call options on floating-rate notes:

  • American option — An option that you exercise any time until its expiration date.

  • European option — An option that you exercise only on its expiration date.

  • Bermuda option — A Bermuda option resembles a hybrid of American and European options; you can only exercise it on predetermined dates, usually monthly.

For more information, see Floating-Rate Note 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