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Price floating-rate note from Black-Karasinski interest-rate tree

`[`

prices a floating-rate note from a Black-Karasinski interest-rate tree. `Price`

,`PriceTree`

]
= floatbybk(`BKTree`

,`Spread`

,`Settle`

,`Maturity`

)

`floatbybk`

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`

]
= floatbybk(___,`Name,Value`

)

Price a 20-basis point floating-rate note using a Black-Karasinski interest-rate tree.

Load the file `deriv.mat`

, which provides `BKTree`

. The `BKTree`

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 `floatbybk`

to compute the price of the note.

Price = floatbybk(BKTree, 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 BK 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; BKVolSpec = bkvolspec(RateSpec.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); BKTimeSpec = bktimespec(RateSpec.ValuationDate, VolDates, Compounding); BKT = bktree(BKVolSpec, RateSpec, BKTimeSpec);

Compute the price of the amortizing floating-rate note.

`Price = floatbybk(BKT, 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 BK 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; BKVolSpec = bkvolspec(RateSpec.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); BKTimeSpec = bktimespec(RateSpec.ValuationDate, VolDates, Compounding); BKT = bktree(BKVolSpec, RateSpec, BKTimeSpec);

Create the floating-rate note instrument.

Settle ='1-April-2012'; Maturity = '1-April-2016'; Spread = [15;10]; Principal = 100;

Compute the price of the two vanilla floaters.

Price = floatbybk(BKT, Spread, Settle, Maturity)

`Price = `*2×1*
100.5519
100.3680

Compute the price of the collared floating-rate notes.

CapStrike = {{'1-April-2013' 0.045; '1-April-2014' 0.05;... '1-April-2015' 0.06}; 0.06}; FloorStrike = {{'1-April-2013' 0.035; '1-April-2014' 0.04;... '1-April-2015' 0.05}; 0.03}; PriceCollared = floatbybk(BKT, Spread, Settle, Maturity,... 'CapRate', CapStrike,'FloorRate', FloorStrike)

`PriceCollared = `*2×1*
102.8537
100.4918

When using `floatbybk`

to
price floating-rate notes, there are cases where the dates specified
in the BK tree Time Specs are not aligned with the cash flow dates.

Price floating-rate notes using the following data:

ValuationDate = '13-Sep-2013'; ForwardRatesVector = [ 0.0001; 0.0001; 0.0010; 0.0015]; EndDatesVector = ['13-Dec-2013'; '14-Mar-2014'; '13-Jun-2014'; '13-Sep-2014'];

Create the `RateSpec`

.

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

Build the BK tree.

Volcurve = 0.1; Alpha = 0.01; BKVolatilitySpec = bkvolspec(RateSpec.ValuationDate, ... EndDatesVector, Volcurve,... EndDatesVector, Alpha); BKTimeSpec = bktimespec(RateSpec.ValuationDate, EndDatesVector, 1); BKT = bktree(BKVolatilitySpec, RateSpec, BKTimeSpec);

Create the floating-rate note instrument using the following data;

```
Spread = 0;
Maturity = '13-Jun-2014';
reset = 4;
```

Compute the price of the floating-rate note.

Price = floatbybk(BKT, Spread, RateSpec.ValuationDate,... Maturity, 'FloatReset', reset)

Warning: Not all cash flows are aligned with the tree. Result will be approximated. > In floatengbytrintree at 214 In floatbybk at 136 Error using floatengbytrintree (line 319) Instrument '1 ' has cash flow dates that span across tree nodes. Error in floatbybk (line 136) [Price, PriceTree, CFTree] = floatengbytrintree(BKTree, 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 `BKTimeSpec`

. 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); BKTimeSpec = bktimespec(RateSpec.ValuationDate,resetDates,reset); BKT = bktree(BKVolatilitySpec, RateSpec, BKTimeSpec); Price = floatbybk(BKT, Spread, RateSpec.ValuationDate, ... Maturity, 'FloatReset', reset)

Price = 100.0004

`BKTree`

— Interest-rate structurestructure

Interest-rate tree structure, created by `bktree`

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

.

```
[Price,PriceTree] =
floatbybk(BKTree,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.

**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`

`'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.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.

A *floating-rate note* is a security like a bond,
but the interest rate of the note is reset periodically, relative to a reference index rate,
to reflect fluctuations in market interest rates.

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