floatbyhjm

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

Syntax

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

Description

example

``````[Price,PriceTree] = floatbyhjm(HJMTree,Spread,Settle,Maturity)``` prices a floating-rate note from a Heath-Jarrow-Morton interest-rate tree. `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.```

example

``````[Price,PriceTree] = floatbyhjm(___,Name,Value)``` adds additional name-value pair arguments.```

Examples

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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 = datetime(2000,1,1); Maturity = datetime(2003,1,1);```

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 = datetime(2011,11,15); StartDates = ValuationDate; EndDates = [datetime(2012,11,15) ; datetime(2013,11,15) ; datetime(2014,11,15) ; datetime(2015,11,15) ; datetime(2016,11,15)]; 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 = datetime(2011,11,15); Maturity = datetime(2015,11,15); Spread = 15;```

Define the floating-rate note amortizing schedule.

`Principal ={{datetime(2012,11,15) 100;datetime(2013,11,15) 70;datetime(2014,11,15) 40;datetime(2015,11,15) 10}};`

Build the HJM tree using the following data:

```MatDates = [datetime(2012,11,15) ; datetime(2013,11,15) ; datetime(2014,11,15) ; datetime(2015,11,15) ; datetime(2016,11,15) ; datetime(2017,11,15)]; 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 = datetime(2012,4,1); StartDates = ValuationDate; EndDates = [datetime(2013,4,1) ; datetime(2014,4,1) ; datetime(2015,4,1) ; datetime(2016,4,1) ; datetime(2017,4,1)]; 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 = [datetime(2013,4,1) ; datetime(2014,4,1) ; datetime(2015,4,1) ; datetime(2016,4,1) ; datetime(2017,4,1) ; datetime(2018,4,1)]; 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 = datetime(2012,4,1); Maturity = datetime(2016,4,1); 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 ```

Input Arguments

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Interest-rate tree structure, created by `hjmtree`

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, `floatbyhjm` 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 HJM 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, `floatbyhjm` 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] = floatbyhjm(HJMTree,Spread,Settle,Maturity,'Basis',3)```

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`

Derivatives pricing options structure, specified as the comma-separated pair consisting of `'Options'` and a structure using `derivset`.

Data Types: `struct`

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 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`

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, 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`

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.PBush` contains the clean prices.

• `PriceTree.AIBush` contains the accrued interest.

• `PriceTree.tObs` contains the observation times.