# floatbycir

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

## Syntax

## Description

`[`

prices a floating-rate note from a Cox-Ingersoll-Ross (CIR) interest-rate tree. `Price`

,`PriceTree`

]
= floatbycir(`CIRTree`

,`Spread`

,`Settle`

,`Maturity`

)

`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.

`[`

adds additional name-value pair arguments.`Price`

,`PriceTree`

]
= floatbycir(___,`Name,Value`

)

## Examples

### Price a Floating-Rate Note Using a CIR Interest-Rate Tree

Define a `Spread`

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

Spread = 20;

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: {1x4 cell}
Connect: {[3x1 double] [3x3 double] [3x5 double]}
Probs: {[3x1 double] [3x3 double] [3x5 double]}

Price the 20-basis point floating-rate note.

[Price,PriceTree] = floatbycir(CIRT,Spread,Settle,Maturity)

Price = 100.7143

`PriceTree = `*struct with fields:*
FinObj: 'CIRPriceTree'
PTree: {1x5 cell}
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

`CIRTree`

— Interest-rate tree structure

structure

Interest-rate tree structure, created by `cirtree`

**Data Types: **`struct`

`Spread`

— Number of basis points over the reference rate

vector

Number of basis points over the reference rate, specified as a
`NINST`

-by-`1`

vector.

**Data Types: **`double`

`Settle`

— Settlement date

datetime array | string array | date character vector

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`

— Maturity date

datetime array | string array | date character vector

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.*

**Example: **```
[Price,PriceTree] =
floatbycir(CIRTree,Spread,Settle,Maturity,'Basis',3)
```

`FloatReset`

— Frequency of payments per year

`1`

(default) | vector

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`

`Basis`

— Day count basis

`0`

(actual/actual) (default) | integer from `0`

to `13`

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

For more information, see Basis.

**Data Types: **`double`

`Principal`

— Notional principal amounts or principal value schedules

`100`

(default) | vector or cell array

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`

`EndMonthRule`

— End-of-month rule flag for generating dates when `Maturity`

is end-of-month date for month having 30 or fewer days

`1`

(in effect) (default) | nonnegative integer `[0,1]`

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`

`AdjustCashFlowsBasis`

— Flag to adjust cash flows based on actual period day count

`false`

(default) | value of `0`

(false) or `1`

(true)

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`

— Holidays used in computing business days

if not specified, the default is to use
`holidays.m`

(default) | MATLAB^{®} dates

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`

`BusinessDayConvention`

— Business day conventions

`actual`

(default) | character vector | cell array of character vectors

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`

`CapRate`

— Annual cap rate

decimal

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`

`FloorRate`

— Annual floor rate

decimal

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

`Price`

— Expected floating-rate note prices at time 0

vector

Expected floating-rate note prices at time 0, returned as a
`NINST`

-by-`1`

vector.

`PriceTree`

— Tree structure of instrument prices

structure

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.

## More About

### 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**

### R2022b: Serial date numbers not recommended

Although `floatbycir`

supports serial date numbers,
`datetime`

values are recommended instead. The
`datetime`

data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.

To convert serial date numbers or text to `datetime`

values, use the `datetime`

function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)

y = 2021

There are no plans to remove support for serial date number inputs.

## See Also

`bondbycir`

| `capbycir`

| `cfbycir`

| `fixedbycir`

| `floorbycir`

| `oasbycir`

| `optbndbycir`

| `optfloatbycir`

| `optembndbycir`

| `optemfloatbycir`

| `rangefloatbycir`

| `swapbycir`

| `swaptionbycir`

| `instfloat`

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