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floatbyhw

Price floating-rate note from Hull-White interest-rate tree

Syntax

``````[Price,PriceTree] = floatbyhw(HWTree,CouponRate,Settle,Maturity)``````
``````[Price,PriceTree] = floatbyhw(___,Name,Value)``````

Description

example

``````[Price,PriceTree] = floatbyhw(HWTree,CouponRate,Settle,Maturity)``` prices a floating-rate note from a Hull-White interest-rate tree. `floatbyhw` 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] = floatbyhw(___,Name,Value)``` adds additional name-value pair arguments.```

Examples

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Price a 20-basis point floating-rate note using a Hull-White interest-rate tree.

Load the file `deriv.mat`, which provides `HWTree`. The `HWTree` 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-2005'; Maturity = '01-Jan-2006';```

Use `floatbyhw` to compute the price of the note.

`Price = floatbyhw(HWTree, Spread, Settle, Maturity)`
```Warning: Floating range notes are valued at Tree ValuationDate rather than Settle. ```
```Price = 100.3825 ```

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 HW tree and assume the volatility is 10%.

```VolDates = ['15-Nov-2012'; '15-Nov-2013';'15-Nov-2014';'15-Nov-2015';'15-Nov-2016';'15-Nov-2017']; VolCurve = 0.1; AlphaDates = '15-Nov-2017'; AlphaCurve = 0.1; HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); HWTimeSpec = hwtimespec(RateSpec.ValuationDate, VolDates, Compounding); HWT = hwtree(HWVolSpec, RateSpec, HWTimeSpec);```

Compute the price of the amortizing floating-rate note.

`Price = floatbyhw(HWT, 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 two 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 HW tree and assume the volatility to be 5%.

```VolDates = ['1-April-2013';'1-April-2014';'1-April-2015';... '1-April-2016';'1-April-2017';'1-April-2018']; VolCurve = 0.05; AlphaDates = '15-Nov-2018'; AlphaCurve = 0.1; HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); HWTimeSpec = hwtimespec(RateSpec.ValuationDate, VolDates, Compounding); HWT = hwtree(HWVolSpec, RateSpec, HWTimeSpec);```

Create the floating-rate note instrument.

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

Compute the price of a vanilla floater.

`Price = floatbyhw(HWT, Spread, Settle, Maturity)`
```Price = 100.3680 ```

Compute the price of the collared floating-rate notes.

```CapStrike = {{'1-April-2014' 0.045; '1-April-2015' 0.05;... '1-April-2016' 0.06}; 0.06}; FloorStrike = {{'1-April-2014' 0.035; '1-April-2015' 0.04;... '1-April-2016' 0.05}; 0.03}; PriceCollared = floatbyhw(HWT, Spread, Settle, Maturity,.... 'CapRate', CapStrike,'FloorRate', FloorStrike)```
```PriceCollared = 102.0458 100.9299 ```

When using `floatbyhw` to price floating-rate notes, there are cases where the dates specified in the HW tree `TimeSpec` are not aligned with the cash flow dates.

Price floating-rate notes using the following data:

```ValuationDate = '01-Sep-2013'; Rates = [0.0001; 0.0001; 0.0010; 0.0015]; EndDates = ['01-Dec-2013'; '01-Mar-2014'; '01-Jun-2014'; '01-Sep-2014'];```

Create the `RateSpec`.

```RateSpec = intenvset('ValuationDate',ValuationDate,'StartDates',... ValuationDate,'EndDates',EndDates,'Rates',Rates,'Compounding', 1);```

Build the HW tree.

```Volcurve = 0.1; Alpha = 0.01; HWVolatilitySpec = hwvolspec(RateSpec.ValuationDate, ... EndDates, Volcurve,... EndDates, Alpha); HWTimeSpec = hwtimespec(RateSpec.ValuationDate, EndDates, 1); HWT = hwtree(HWVolatilitySpec, RateSpec, HWTimeSpec); ```

Compute the price of the floating-rate note using the following data.

```Spread = 10; Settle = '01-Sep-2013'; Maturity = '01-Jun-2014'; Reset = 2; Price = floatbyhw(HWT, Spread, Settle, Maturity, 'Reset', Reset)```
```Error using floatengbytrintree (line 318) Instrument '1 ' has cash flow dates that span across tree nodes. Error in floatbyhw (line 136) [Price, PriceTree, CFTree] = floatengbytrintree(HWTree, Spread, Settle, Maturity, OArgs{:});```

This error indicates that it is not possible to determine the applicable rate used to calculate the payoff at the reset dates, given that the applicable rate needed cannot be calculated (the information was lost due to the recombination of the tree nodes). Note, 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. The simplest solution is to place the tree levels at the cash flow dates of the instrument, which is done by specifying `HWTimeSpec`. It is also acceptable to have reset dates between tree levels, as long as there are reset dates on the tree levels.

To recover from this error, build a tree that lines up with the instrument.

```Basis = intenvget(RateSpec, 'Basis'); EOM = intenvget(RateSpec, 'EndMonthRule'); resetDates = cfdates(ValuationDate, Maturity, Reset, Basis, EOM); HWTimeSpec = hwtimespec(RateSpec.ValuationDate,resetDates, Reset); HWT = hwtree(HWVolatilitySpec, RateSpec, HWTimeSpec); Price = floatbyhw(HWT, Spread, RateSpec.ValuationDate, ... Maturity, 'Reset', Reset)```
```Price = 100.0748```

Input Arguments

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Interest-rate tree structure, created by `hwtree`

Data Types: `struct`

Coupon annual rate, specified as a `NINST`-by-`1` vector.

Data Types: `double`

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 HW Tree. The floating-rate note argument `Settle` is ignored.

Data Types: `char` | `double`

Maturity date, specified as a `NINST`-by-`1` vector of serial date numbers or date character vectors representing the maturity date for each swap.

Data Types: `char` | `double`

Name-Value Pair Arguments

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`.

Example: `[Price,PriceTree] = floatbyhw(HWTree,CouponRate,Settle,Maturity,'Basis',3)`

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Frequency of payments per year, specified as `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 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 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 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 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 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 MATLAB date numbers using a `NHolidays`-by-`1` vector.

Data Types: `double`

Business day conventions, specified by 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 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 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.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.