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

[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.

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

Examples

Price a Floating-Rate Note Using an HJM Tree

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

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

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

Input Arguments

HJMTree — Interest-rate structure

structure

Interest-rate tree structure, created by hjmtree

Data Types: struct

Spread — Number of basis points over the reference rate

vector

Number of basis points over the reference rate, specified as a NINST-by-1 vector.

Data Types: double

Settle — Settlement date

serial 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 date

serial 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

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

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

'FloatReset' — Frequency of payments per year

1 (default) | vector

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

'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 array

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

'Options' — Derivatives pricing options structure

structure

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 days

if not specified, the default is to use holidays.m (default) | MATLAB^® date numbers

Holidays 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 vectors

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

'CapRate' — Annual cap rate

decimal

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 rate

decimal

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

Price — Expected floating-rate note prices at time 0

vector

Expected floating-rate note prices at time 0, returned as a NINST-by-1 vector.

PriceTree — Tree structure of instrument prices

structure

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.

More About

Floating-Rate Note

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.