# Documentation

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# floatbyhw

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

## Syntax

Settle, Maturity)
Settle, Maturity, Reset, Basis, Principal, Options,
EndMonthRule)
[Price, PriceTree] = floatbyhw(HWTree, Spread, Settle,
Maturity,Name,Value)

## Input Arguments

HWTree

Interest-rate tree structure created by hwtree.

Number of instruments (NINST-by-1) vector of number of basis points over the reference rate.

Settle

Settlement dates. NINST-by-1 vector of dates representing the settlement dates of the floating-rate note.

 Note:   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.

Maturity

NINST-by-1 vector of dates representing the maturity dates of the floating-rate note.

### Ordered Input or Name-Value Pair Arguments

Enter the following optional inputs using an ordered syntax or as name-value pair arguments. You cannot mix ordered syntax with name-value pair arguments.

Reset

NINST-by-1 vector representing the frequency of payments per year.

 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 will be more than one possible path for connecting the two payment dates.

Default: 1

Basis

Day-count basis of the instrument. A vector of integers.

• 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

Default: 0 (actual/actual)

Principal

NINST-by-1 vector of notional principal amounts or NINST-by-1 cell array. For the latter case, each element of the cell array is a NumDates-by-2 matrix where the first column is dates and the second column is associated principal amount. The date indicates the last day that the principal value is valid.

Default: 100

Options

Derivatives pricing options structure created with derivset.

EndMonthRule

End-of-month rule. NINST-by-1 vector. This rule applies only when Maturity is an end-of-month date for a month having 30 or fewer days.

• 0 = Ignore rule, meaning that a bond coupon payment date is always the same numerical day of the month.

• 1 = Set rule on, meaning that a bond coupon payment date is always the last actual day of the month.

Default: 1

### 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.

 AdjustCashFlowsBasis Adjust the cash flows based on the actual period day count. NINST-by-1 of logicals. Default: false BusinessDayConvention Require payment dates to be business dates. NINST-by-1 cell array with possible choices of business day convention:actual followmodifiedfollowpreviousmodifiedprevious Default: actual Holidays Holidays used for business day convention. NHOLIDAYS-by-1 of MATLAB® date numbers. Default: If no dates are specified, holidays.m is used. CapRate 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. FloorRate 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.

## Description

Settle, Maturity)
computes the price of a floating-rate note from a Hull-White tree.

Settle, Maturity, Reset, Basis, Principal, Options,
EndMonthRule)
computes the price of a floating-rate note with optional inputs from a Hull-White tree.

[Price, PriceTree] = floatbyhw(HWTree, Spread, Settle,
Maturity,Name,Value)
computes the price of a floating-rate note from a Hull-White tree with additional options specified by one or more Name,Value pair arguments.

Price is an NINST-by-1 vector of expected prices of the floating-rate note at time 0.

PriceTree is a structure of trees containing vectors of instrument prices and accrued interest, and a vector of observation times for each node.

PriceTree.PTree contains the clean prices.

PriceTree.AITree contains the accrued interest.

PriceTree.tObs contains the observation times.

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.

## Examples

collapse all

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.

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

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: [5×1 double]
Rates: [5×1 double]
EndTimes: [5×1 double]
StartTimes: [5×1 double]
EndDates: [5×1 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';

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;
AlphaCurve = 0.1;

HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,...
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;
AlphaCurve = 0.1;

HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,...
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';
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.

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 will be 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