# optfloatbycir

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

## Syntax

## Description

`[`

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

,`PriceTree`

]
= optfloatbycir(`CIRTree`

,`OptSpec`

,`Strike`

,`ExerciseDates`

,`AmericanOpt`

,`Spread`

,`Settle`

,`Maturity`

)`optfloatbycir`

computes prices of options on vanilla floating-rate
notes using a CIR++ model with the Nawalka-Beliaeva (NB) approach.

`[`

adds optional name-value pair arguments. `Price`

,`PriceTree`

]
= optfloatbycir(___,`Name,Value`

)

## Examples

### Compute the Price of American and European Call Options on a Floating-Rate Note Using a CIR Interest-Rate Tree

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(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: {[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]}

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

Spread = 10; Settle = datetime(2017,1,1); Maturity = datetime(2019,1,1); Period = 1;

Define the option for the floating-rate note.

```
OptSpec = {'call'};
Strike = 95;
ExerciseDates = datetime(2018,1,1);
AmericanOpt = [0;1];
```

Compute the price of the call options.

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

`Price = `*2×1*
4.9230
5.1887

`PriceTree = `*struct with fields:*
FinObj: 'CIRPriceTree'
PTree: {[2x1 double] [2x3 double] [2x5 double] [2x7 double] [2x7 double]}
AITree: {[2x1 double] [2x3 double] [2x5 double] [2x7 double] [2x7 double]}
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 specified as a structure by using `cirtree`

.

**Data Types: **`struct`

`OptSpec`

— Definition of option

character vector with values `'call'`

or
`'put'`

| cell array of character vectors with values of `'call'`

or
`'put'`

| string array with values `'call'`

or
`'put'`

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`

`Strike`

— Option strike price values

nonnegative integer | vector of nonnegative integers

Option strike price values, specified as nonnegative integers using an
`NINST`

-by-`NSTRIKES`

vector of strike price values.

**Data Types: **`double`

`ExerciseDates`

— Exercise date for option (European, Bermuda, or American)

datetime array | string array | date character vector

Exercise date for option (European, Bermuda, or American) specified as a
`NINST`

-by-`NSTRIKES`

or
`NINST`

-by-`2`

vector using a datetime array, string
array, or date character vectors.

To support existing code, `optfloatbycir`

also
accepts serial date numbers as inputs, but they are not recommended.

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`

.

`AmericanOpt`

— Option type

scalar | vector of positive integers`[0,1]`

Option type specified as `NINST`

-by-`1`

positive
integer scalar flags with values:

`0`

— European/Bermuda`1`

— American

**Data Types: **`double`

`Spread`

— Number of basis points over the reference rate

nonnegative integer | vector of nonnegative integers

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: **`double`

`Settle`

— Settlement dates of floating-rate note

`ValuationDate`

of CIR tree (default) | datetime array | string array | date character vector

Settlement dates of floating-rate note specified as a
`NINST`

-by-`1`

vector using a datetime array, string
array, or date character vectors.

**Note**

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.

To support existing code, `optfloatbycir`

also
accepts serial date numbers as inputs, but they are not recommended.

`Maturity`

— Floating-rate note maturity date

datetime array | string array | date character vector

Floating-rate note maturity date specified as a
`NINST`

-by-`1`

vector using a datetime array, string
array, or date character vectors.

To support existing code, `optfloatbycir`

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

`FloatReset`

— Frequency of payments per year

`1`

(default) | positive integer from the set`[1,2,3,4,6,12]`

| vector of positive integers from the set
`[1,2,3,4,6,12]`

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.

**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 will be more than one possible path for connecting the two payment dates.

**Data Types: **`double`

`Basis`

— Day-count basis of the instrument

`0`

(actual/actual) (default) | positive integers of the set `[1...13]`

| vector of positive integers of the set `[1...13]`

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`

— Principal values

`100`

(default) | vector of nonnegative values | cell array of nonnegative values

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`

`Options`

— Structure containing derivatives pricing options

structure

Structure containing derivatives pricing options, specified as the
comma-separated pair consisting of `'Options'`

and the output from
`derivset`

.

**Data Types: **`struct`

`EndMonthRule`

— End-of-month rule flag

`1`

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

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

`Price`

— Expected prices of the floating-rate note option at time 0

scalar | vector

Expected prices of the floating-rate note option at time 0 is returned as a scalar
or an `NINST`

-by-`1`

vector.

`PriceTree`

— Structure of trees containing vectors of option prices at each node

tree structure

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

### 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.

## 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 `optfloatbycir`

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`

| `floatbycir`

| `floorbycir`

| `oasbycir`

| `optbndbycir`

| `optembndbycir`

| `optemfloatbycir`

| `rangefloatbycir`

| `swapbycir`

| `swaptionbycir`

| `instoptfloat`

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