# floatbycir

Price floating-rate note from Cox-Ingersoll-Ross interest-rate tree

## Description

[Price,PriceTree] = floatbycir(CIRTree,Spread,Settle,Maturity) prices a floating-rate note from a Cox-Ingersoll-Ross (CIR) interest-rate tree.

floatbycir computes prices of vanilla floating-rate notes, amortizing floating-rate notes, capped floating-rate notes, floored floating-rate notes, and collared floating-rate notes using a CIR++ model with the Nawalka-Beliaeva (NB) approach.

Note

Alternatively, you can use the FloatBond object to price floating-rate note instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

example

example

## Examples

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Define a Spread of 20-basis points for a floating-rate note.

Create a RateSpec using the intenvset function.

Rates = [0.035; 0.042147; 0.047345; 0.052707];
Dates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)];
ValuationDate = datetime(2017,1,1);
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 = datetime(2017,1,1);
Maturity = datetime(2021,1,1);
CIRTimeSpec = cirtimespec(ValuationDate, 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: {[1.0350]  [1.0790 1.0500 1.0298]  [1.1275 1.0887 1.0594 1.0390 1.0270]  [1.1905 1.1406 1.1014 1.0718 1.0512 1.0390 1.0350]}
Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
Probs: {[3x1 double]  [3x3 double]  [3x5 double]}

Price the 20-basis point floating-rate note.

Price =
100.7143
PriceTree = struct with fields:
FinObj: 'CIRPriceTree'
PTree: {[100.7143]  [100.5113 100.5385 100.5589]  [100.3333 100.3508 100.3650 100.3756 100.3821]  [100.1680 100.1753 100.1816 100.1866 100.1903 100.1925 100.1932]  [100 100 100 100 100 100 100]}
AITree: {[0]  [0 0 0]  [0 0 0 0 0]  [0 0 0 0 0 0 0]  [0 0 0 0 0 0 0]}
tObs: [0 1 2 3 4]
Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
Probs: {[3x1 double]  [3x3 double]  [3x5 double]}

## Input Arguments

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Interest-rate tree structure, created by cirtree

Data Types: struct

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

Data Types: double

Settlement date, specified either as a scalar or a NINST-by-1 vector using a datetime array, string array, or date character vectors.

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

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.

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

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

### 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.

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

Note

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 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.

• 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

Data Types: double

Notional principal amounts, specified as the comma-separated pair consisting of 'Principal' and a vector or cell array.

Principal accepts a NINST-by-1 vector or NINST-by-1 cell array, where each element of the cell array is a NumDates-by-2 cell array, and the first column is dates and the second column is its associated notional principal value. The date indicates the last day that the principal value is valid.

Data Types: cell | double

End-of-month rule flag for generating dates when Maturity is an end-of-month date for a month having 30 or fewer days, specified as the comma-separated pair consisting of 'EndMonthRule' and a nonnegative integer [0, 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

Flag to adjust cash flows based on actual period day count, specified as the comma-separated pair consisting of 'AdjustCashFlowsBasis' and a NINST-by-1 vector of logicals with values of 0 (false) or 1 (true).

Data Types: logical

Holidays used in computing business days, specified as the comma-separated pair consisting of 'Holidays' and MATLAB dates using a NHolidays-by-1 vector.

Data Types: datetime

Business day conventions, specified as the comma-separated pair consisting of 'BusinessDayConvention' and a character vector or a N-by-1 cell array of character vectors of business day conventions. The selection for business day convention determines how nonbusiness days are treated. Nonbusiness days are defined as weekends plus any other date that businesses are not open (e.g. statutory holidays). Values are:

• actual — Nonbusiness days are effectively ignored. Cash flows that fall on nonbusiness days are assumed to be distributed on the actual date.

• follow — Cash flows that fall on a non-business day are assumed to be distributed on the following business day.

• modifiedfollow — Cash flows that fall on a non-business day are assumed to be distributed on the following business day. However if the following business day is in a different month, the previous business day is adopted instead.

• previous — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day.

• modifiedprevious — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day. However if the previous business day is in a different month, the following business day is adopted instead.

Data Types: char | cell

Annual cap rate, specified as the comma-separated pair consisting of 'CapRate' and a NINST-by-1 decimal annual rate or NINST-by-1 cell array, where each element is a NumDates-by-2 cell array, and the cell array first column is dates, and the second column is associated cap rates. The date indicates the last day that the cap rate is valid.

Data Types: double | cell

Annual floor rate, specified as the comma-separated pair consisting of 'FloorRate' and a NINST-by-1 decimal annual rate or NINST-by-1 cell array.

For the NINST-by-1 cell array, each element is a NumDates-by-2 cell array, where the cell array first column is dates, and the second column is associated floor rates. The date indicates the last day that the floor rate is valid.

Data Types: double | cell

## Output Arguments

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Expected floating-rate note prices at time 0, returned as a NINST-by-1 vector.

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

• 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.

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### Floating-Rate Note

A floating-rate note is a security like a bond, but the interest rate of the note is reset periodically, relative to a reference index rate, to reflect fluctuations in market interest rates.

## References

[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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