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Price floating-rate note from Heath-Jarrow-Morton interest-rate tree

```
[Price,PriceTree]
= floatbyhjm(HJMTree,Spread,Settle,Maturity)
```

```
[Price,PriceTree]
= floatbyhjm(___,Name,Value)
```

`[`

prices a floating-rate note from a
Heath-Jarrow-Morton interest-rate tree. `Price`

,`PriceTree`

]
= floatbyhjm(`HJMTree`

,`Spread`

,`Settle`

,`Maturity`

)

`floatbyhjm`

computes prices of vanilla floating-rate notes, amortizing
floating-rate notes, capped floating-rate notes,
floored floating-rate notes and collared
floating-rate notes.

`[`

adds
additional name-value pair arguments.`Price`

,`PriceTree`

]
= floatbyhjm(___,`Name,Value`

)

Price a 20-basis point floating-rate note using an HJM forward-rate tree.

Load the file `deriv.mat`

, which provides `HJMTree`

. The `HJMTree`

structure contains the time and interest-rate information needed to price the note.

`load deriv.mat;`

Define the floating-rate note using the required arguments. Other arguments use defaults.

Spread = 20; Settle = '01-Jan-2000'; Maturity = '01-Jan-2003';

Use `floatbyhjm`

to compute the price of the note.

Price = floatbyhjm(HJMTree, Spread, Settle, Maturity)

Price = 100.5529

Price an amortizing floating-rate note using the `Principal`

input argument to define the amortization schedule.

Create the `RateSpec`

.

Rates = [0.03583; 0.042147; 0.047345; 0.052707; 0.054302]; ValuationDate = '15-Nov-2011'; StartDates = ValuationDate; EndDates = {'15-Nov-2012';'15-Nov-2013';'15-Nov-2014' ;'15-Nov-2015';'15-Nov-2016'}; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: 1
Disc: [5x1 double]
Rates: [5x1 double]
EndTimes: [5x1 double]
StartTimes: [5x1 double]
EndDates: [5x1 double]
StartDates: 734822
ValuationDate: 734822
Basis: 0
EndMonthRule: 1

Create the floating-rate instrument using the following data:

Settle ='15-Nov-2011'; Maturity = '15-Nov-2015'; Spread = 15;

Define the floating-rate note amortizing schedule.

Principal ={{'15-Nov-2012' 100;'15-Nov-2013' 70;'15-Nov-2014' 40;'15-Nov-2015' 10}};

Build the HJM tree using the following data:

MatDates = {'15-Nov-2012'; '15-Nov-2013';'15-Nov-2014';'15-Nov-2015';'15-Nov-2016';'15-Nov-2017'}; HJMTimeSpec = hjmtimespec(RateSpec.ValuationDate, MatDates); Volatility = [.10; .08; .06; .04]; CurveTerm = [ 1; 2; 3; 4]; HJMVolSpec = hjmvolspec('Proportional', Volatility, CurveTerm, 1e6); HJMT = hjmtree(HJMVolSpec,RateSpec,HJMTimeSpec);

Compute the price of the amortizing floating-rate note.

`Price = floatbyhjm(HJMT, Spread, Settle, Maturity, 'Principal', Principal)`

Price = 100.3059

Price a collar with a floating-rate note using the `CapRate`

and `FloorRate`

input argument to define the collar pricing.

Price a portfolio of collared floating-rate notes using the following data:

Rates = [0.0287; 0.03024; 0.03345; 0.03861; 0.04033]; ValuationDate = '1-April-2012'; StartDates = ValuationDate; EndDates = {'1-April-2013';'1-April-2014';'1-April-2015' ;... '1-April-2016';'1-April-2017'}; Compounding = 1;

Create the `RateSpec`

.

RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding);

Build the HJM tree with the following data:

MatDates = {'1-April-2013'; '1-April-2014';'1-April-2015';... '1-April-2016';'1-April-2017';'1-April-2018'}; HJMTimeSpec = hjmtimespec(RateSpec.ValuationDate, MatDates); Volatility = [.10; .08; .06; .04]; CurveTerm = [ 1; 2; 3; 4]; HJMVolSpec = hjmvolspec('Proportional', Volatility, CurveTerm, 1e6); HJMT = hjmtree(HJMVolSpec,RateSpec,HJMTimeSpec);

Create the floating-rate note instrument.

Settle ='1-April-2012'; Maturity = '1-April-2016'; Spread = 10; Principal = 100;

Compute the price of two capped collared floating-rate notes.

CapStrike = [0.04;0.055]; PriceCapped = floatbyhjm(HJMT, Spread, Settle, Maturity,... 'CapRate', CapStrike)

`PriceCapped = `*2×1*
98.9986
100.2051

Compute the price of two collared floating-rate notes.

FloorStrike = [0.035;0.040]; PriceCollared = floatbyhjm(HJMT, Spread, Settle, Maturity,... 'CapRate', CapStrike, 'FloorRate', FloorStrike)

`PriceCollared = `*2×1*
99.9246
102.2321

`HJMTree`

— Interest-rate structurestructure

Interest-rate tree structure, created by `hjmtree`

**Data Types: **`struct`

`Spread`

— Number of basis points over the reference ratevector

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

-by-`1`

vector.

**Data Types: **`double`

`Settle`

— Settlement dateserial date number | character vector

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

-by-`1`

vector
of serial date numbers or date character vectors.

The `Settle`

date for every floating-rate note is set to the
`ValuationDate`

of the HJM tree. The floating-rate note argument
`Settle`

is ignored.

**Data Types: **`char`

| `double`

`Maturity`

— Maturity dateserial date number | character vector

Maturity date, specified as a `NINST`

-by-`1`

vector of
serial date numbers or date character vectors representing the maturity date for each
floating-rate note.

**Data Types: **`char`

| `double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside single quotes (`' '`

). You can
specify several name and value pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

```
[Price,PriceTree] =
floatbyhjm(HJMTree,Spread,Settle,Maturity,'Basis',3)
```

`'FloatReset'`

— Frequency of payments per year`1`

(default) | vectorFrequency of payments per year, specified as
the comma-separated pair consisting of
`'FloatReset'`

and a
`NINST`

-by-`1`

vector.

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 arrayNotional 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`

`'Options'`

— Derivatives pricing options structurestructure

Derivatives pricing options structure, specified as the comma-separated pair consisting of
`'Options'`

and a structure using
`derivset`

.

**Data Types: **`struct`

`'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 daysif not specified, the default is to use

`holidays.m`

(default) | MATLABHolidays used in computing business days, specified as the comma-separated pair consisting of
`'Holidays'`

and MATLAB date numbers using a
`NHolidays`

-by-`1`

vector.

**Data Types: **`double`

`'BusinessDayConvention'`

— Business day conventions`actual`

(default) | character vector | cell array of character vectorsBusiness 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 non-business days are
treated. Non-business days are defined as weekends
plus any other date that businesses are not open
(e.g. statutory holidays). Values are:

`actual`

— Non-business days are effectively ignored. Cash flows that fall on non-business 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 ratedecimal

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 ratedecimal

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, where each element is a
`NumDates`

-by-`2`

cell array, and 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`

`Price`

— Expected floating-rate note prices at time 0vector

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

-by-`1`

vector.

`PriceTree`

— Tree structure of instrument pricesstructure

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.PBush`

contains the clean prices.`PriceTree.AIBush`

contains the accrued interest.`PriceTree.tObs`

contains the observation times.

`bondbyhjm`

| `capbyhjm`

| `cfbyhjm`

| `fixedbyhjm`

| `floorbyhjm`

| `hjmtree`

| `swapbyhjm`

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