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Price range floating note using Cox-Ingersoll-Ross tree



[Price,PriceTree] = rangefloatbycir(CIRTree,Spread,Settle,Maturity,RateSched) prices range floating note with a Cox-Ingersoll-Ross (CIR) interest-rate tree using a CIR++ model with the Nawalka-Beliaeva (NB) approach.


[Price,PriceTree] = rangefloatbycir(___,Name,Value) adds additional 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 1.0390 1.0270]}
     Connect: {[3x1 double]  [3x3 double]}
       Probs: {[3x1 double]  [3x3 double]}

Define the range note instrument that matures in Jan-1-2014 and has the following RateSchedule:

Spread = 100;
Settle = 'Jan-1-2017';
Maturity = 'Jan-1-2020';
RateSched(1).Dates = {'Jan-1-2018'; 'Jan-1-2019'  ; 'Jan-1-2020'};
RateSched(1).Rates = [0.045 0.055 ; 0.0525  0.0675; 0.06 0.08];

Compute the price of the range floating note.

[Price,PriceTree] = rangefloatbycir(CIRT,Spread,Settle,Maturity,RateSched)
Price = 91.6849
PriceTree = struct with fields:
     FinObj: 'CIRPriceTree'
      PTree: {[91.6849]  [88.9878 92.6039 95.1352]  [88.6954 91.8547 94.3896 96.2429 97.3723]  [100 100 100 100 100]}
     AITree: {[0]  [0 0 0]  [0 0 0 0 0]  [0 0 0 0 0]}
       tObs: [0 1 2 3]
    Connect: {[3x1 double]  [3x3 double]}
      Probs: {[3x1 double]  [3x3 double]}

Input Arguments

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Interest-rate tree structure, specified by using cirtree.

Data Types: struct

Number of basis points over the reference rate, specified as a NINST-by-1 vector.

Data Types: double

Settlement date for the floating range note, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors. The Settle date for every range floating instrument is set to the ValuationDate of the CIR tree. The floating range note argument Settle is ignored.

To support existing code, rangefloatbycir also accepts serial date numbers as inputs, but they are not recommended.

Maturity date for the floating-rate note, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, rangefloatbycir also accepts serial date numbers as inputs, but they are not recommended.

Range of rates within which cash flows are nonzero, specified as a NINST-by-1 vector of structures. Each element of the structure array contains two fields:

  • RateSched.DatesNDates-by-1 cell array of dates corresponding to the range schedule.

  • RateSched.RatesNDates-by-2 array with the first column containing the lower bound of the range and the second column containing the upper bound of the range. Cash flow for date RateSched.Dates(n) is nonzero for rates in the range RateSched.Rates(n,1) < Rate < RateSched.Rate (n,2).

Data Types: struct

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] = rangefloatbycir(CIRTree,Spread,Settle,Maturity,RateSched,'Reset',4,'Basis',5,'Principal',10000)

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


Payments on range floating notes are determined by the effective interest-rate between reset dates. If the reset period for a range 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 is more than one possible path for connecting the two payment dates.

Data Types: double

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

  • 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

Notional principal amount, specified as the comma-separated pair consisting of 'Principal' and a NINST-by-1 vector.

Data Types: double

End-of-month rule flag, specified as the comma-separated pair consisting of 'EndMonthRule' and a nonnegative integer with a value of 0 or 1 using a NINST-by-1 vector.

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

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

Data Types: logical

Output Arguments

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Expected prices of the range floating notes at time 0, returned as a NINST-by-1 vector.

Tree structure of instrument prices, returned as a structure containing trees of vectors of instrument prices and accrued interest, and a vector of observation times for each node. Values are:

  • 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

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Range Note

A range note is a structured (market-linked) security whose coupon rate is equal to the reference rate as long as the reference rate is within a certain range.

If the reference rate is outside of the range, the coupon rate is 0 for that period. This type of instrument entitles the holder to cash flows that depend on the level of some reference interest rate and are floored to be positive. The note holder gets direct exposure to the reference rate. In return for the drawback that no interest is paid for the time the range is left, they offer higher coupon rates than comparable standard products, vanilla floating notes. For more information, see Range Note.


[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

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