# swapbycir

Price swap instrument from Cox-Ingersoll-Ross interest-rate tree

## Syntax

## Description

`[`

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

,`PriceTree`

,`SwapRate`

]
= swapbycir(`CIRTree`

,`LegRate`

,`Settle`

,`Maturity`

)`swapbycir`

computes prices of vanilla swaps, amortizing swaps, and
forward swaps using a CIR++ model with the Nawalka-Beliaeva (NB) approach.

**Note**

Alternatively, you can use the `Swap`

object to price a swap
instrument. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

`[`

adds additional name-value pair arguments.`Price`

,`PriceTree`

,`SwapRate`

]
= swapbycir(___,`Name,Value`

)

## Examples

### Price an Interest-Rate Swap Using a CIR Interest-Rate Tree

Define an interest-rate swap with a fixed receiving leg and a floating paying leg. Payments are made once a year and the notional principal amount is $100.

Basis = 0; Principal = 100; LegRate = [0.06 20]; % [CouponRate Spread] LegType = [1 0]; % [Fixed Float] LegReset = [1 1]; % Payments once per year

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 = 5; Alpha = 0.03; Theta = 0.02; Sigma = 0.1; Settle = datetime(2017,1,1); Maturity = datetime(2022,1,1); 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 4]
dObs: [736696 737061 737426 737791 738156]
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] [1.2349 1.1740 1.1248 1.0861 1.0570 1.0366 1.0246 1.0206]}
Connect: {[3x1 double] [3x3 double] [3x5 double] [3x7 double]}
Probs: {[3x1 double] [3x3 double] [3x5 double] [3x7 double]}

Price the interest-rate swap.

[Price,PriceTree] = swapbycir(CIRT,LegRate,Settle,Maturity,'LegReset',LegReset,'Basis',3,'Principal',100,'LegType',LegType)

Price = 2.5522

`PriceTree = `*struct with fields:*
FinObj: 'CIRPriceTree'
tObs: [0 1 2 3 4 5]
PTree: {[2.5522] [-9.0229 -0.0249 6.9681] [-16.5229 -8.1674 -1.0761 4.3321 7.7223] [-19.1049 -12.2245 -6.1556 -1.1426 2.6034 4.9197 5.7042] [-14.3229 -9.8801 -5.9413 -2.5909 0.0972 2.0626 3.2603 3.6626] [0 0 0 0 0 0 0 0]}
Connect: {[3x1 double] [3x3 double] [3x5 double] [3x7 double]}

## Input Arguments

`CIRTree`

— Interest-rate structure

structure

Interest-rate tree structure, created by `cirtree`

**Data Types: **`struct`

`LegRate`

— Leg rate

matrix

Leg rate, specified as a `NINST`

-by-`2`

matrix,
with each row defined as one of the following:

`[CouponRate Spread]`

(fixed-float)`[Spread CouponRate]`

(float-fixed)`[CouponRate CouponRate]`

(fixed-fixed)`[Spread Spread]`

(float-float)

`CouponRate`

is the decimal annual rate.
`Spread`

is the number of basis points over the reference rate. The
first column represents the receiving leg, while the second column represents the
paying leg.

**Data Types: **`double`

`Settle`

— Settlement date

datetime array | string array | date character vector

Settlement date, specified either as a scalar or
`NINST`

-by-`1`

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

To support existing code, `swapbycir`

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

The `Settle`

date for every swap is set to the
`ValuationDate`

of the CIR tree. The swap 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 swap.

To support existing code, `swapbycir`

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,SwapRate] =
swapbycir(CIRTree,LegRate,Settle,Maturity,LegReset,Basis,Principal,LegType)
```

`LegReset`

— Reset frequency per year for each swap

`[1 1]`

(default) | vector

Reset frequency per year for each swap, specified as the comma-separated pair
consisting of `'LegReset'`

and a
`NINST`

-by-`2`

vector.

**Data Types: **`double`

`Basis`

— Day-count basis representing the basis for each leg

`0`

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

to `13`

Day-count basis representing the basis for each leg, specified as the
comma-separated pair consisting of `'Basis'`

and a
`NINST`

-by-`1`

array (or
`NINST`

-by-`2`

if `Basis`

is
different for each leg).

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 or principal value schedules, 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 (or
`NINST`

-by-`2`

if `Principal`

is different for each leg) of the notional principal amounts or principal value
schedules. For schedules, each element of the cell array is a
`NumDates`

-by-`2`

array where 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`

`LegType`

— Leg type

`[1 0]`

for each instrument (default) | matrix with values `[1 1]`

(fixed-fixed), ```
[1
0]
```

(fixed-float), `[0 1]`

(float-fixed), or ```
[0
0]
```

(float-float)

Leg type, specified as the comma-separated pair consisting of
`'LegType'`

and a `NINST`

-by-`2`

matrix with values:

`[1 1]`

(fixed-fixed) swap`[1 0]`

(fixed-float) swap`[0 1]`

(float-fixed) swap`[0 0]`

(float-float) swap

Each row represents an instrument. Each column indicates if the corresponding leg
is fixed (`1`

) or floating (`0`

). This matrix
defines the interpretation of the values entered in `LegRate`

.

**Data Types: **`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`

(or
`NINST`

-by-`2`

if `EndMonthRule`

is different for each leg).

`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`

(or
`NINST`

-by-`2`

if
`AdjustCashFlowsBasis`

is different for each leg) of logicals with
values of `0`

(false) or `1`

(true).

**Data Types: **`logical`

`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`

(or
`NINST`

-by-`2`

if
`BusinessDayConvention`

is different for each leg) 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`

`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`

`StartDate`

— Date swap actually starts

`Settle`

date (default) | datetime array | string array | date character vector

Date swap actually starts, specified as the comma-separated pair consisting of
`'StartDate'`

and a
`NINST`

-by-`1`

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

To support existing code, `swapbycir`

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

Use this argument to price forward swaps, that is, swaps that start in a future date

## Output Arguments

`Price`

— Expected swap prices at time 0

vector

Expected swap 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 swaption instrument prices and
a vector of observation times for each node. Within `PriceTree`

:

`PriceTree.tObs`

contains the observation times.`PriceTree.PTree`

contains the clean prices.

`SwapRate`

— Rates applicable to fixed leg

matrix

Rates applicable to the fixed leg, returned as a
`NINST`

-by-`1`

vector of rates applicable to the
fixed leg such that the swaps’ values are zero at time 0. This rate is used in
calculating the swaps’ prices when the rate specified for the fixed leg in
`LegRate`

is `NaN`

. The
`SwapRate`

output is padded with `NaN`

for those
instruments in which `CouponRate`

is not set to
`NaN`

.

## More About

### Amortizing Swap

In an amortizing swap, the notional principal decreases periodically because it is tied to an underlying financial instrument with a declining (amortizing) principal balance, such as a mortgage.

### Forward Swap

Agreement to enter into an interest-rate swap arrangement on a fixed date in future.

## 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 `swapbycir`

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`

| `optfloatbycir`

| `optembndbycir`

| `optemfloatbycir`

| `rangefloatbycir`

| `swaptionbycir`

| `instswap`

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