floatbybdt

Price floating-rate note from Black-Derman-Toy interest-rate tree

Syntax

[Price,PriceTree] = floatbybdt(BDTTree,Spread,Settle,Maturity)

[Price,PriceTree] = floatbybdt(___,Name,Value)

Description

[Price,PriceTree] = floatbybdt(BDTTree,Spread,Settle,Maturity) prices a floating-rate note from a Black-Derman-Toy interest-rate tree.

floatbybdt computes prices of vanilla floating-rate notes, amortizing floating-rate notes, capped floating-rate notes, floored floating-rate notes and collared floating-rate notes.

Note

Alternatively, you can use the FloatBond object to price floating-rate bond instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

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

Examples

Price a Floating-Rate Note Using a BDT Tree

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

Load the file deriv.mat, which provides BDTTree. The BDTTree 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 floatbybdt to compute the price of the note.

Price = floatbybdt(BDTTree, Spread, Settle, Maturity)

Price = 
100.4865

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 = 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 BDT tree and assume volatility is 10%.

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)];
BDTTimeSpec = bdttimespec(ValuationDate, MatDates);
Volatility = 0.10;  
BDTVolSpec = bdtvolspec(ValuationDate, MatDates, Volatility*ones(1,length(MatDates))');
BDTT = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec);

Compute the price of the amortizing floating-rate note.

Price = floatbybdt(BDTT, 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.

Create the RateSpec.

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 BDT tree and assume volatility is 5%.

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)];
BDTTimeSpec = bdttimespec(ValuationDate, MatDates);
Volatility = 0.05;  
BDTVolSpec = bdtvolspec(ValuationDate, MatDates, Volatility*ones(1,length(MatDates))');
BDTT = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec);

Create the floating rate note instrument.

Settle = datetime(2012,4,1);
Maturity = datetime(2016,4,1);
Spread = 10;
Principal = 100;

Compute the price of a collared floating-rate note.

CapStrike = {{datetime(2013,4,1) 0.03;datetime(2015,4,1) 0.055}};
FloorStrike = {{datetime(2013,4,1) 0.025;datetime(2015,4,1) 0.04}};

Price = floatbybdt(BDTT, Spread, Settle, Maturity, 'CapRate',...
CapStrike, 'FloorRate', FloorStrike)

Price = 
101.2414

Pricing a Floating-Rate Note When the Reset Dates Are Not Tree Level Dates

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

Price floating-rate notes using the following data:

ValuationDate = datetime(2013,9,1); 
Rates = [0.0235; 0.0239; 0.0311; 0.0323]; 
EndDates = [datetime(2014,9,1);datetime(2015,9,1);datetime(2016,9,1);datetime(2017,9,1)];

Create the RateSpec.

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

Build the BDT tree.

VolCurve = [.10; .11; .11; .12];

BDTVolatilitySpec = bdtvolspec(RateSpec.ValuationDate, EndDates,... 
                               VolCurve); 

BDTTimeSpec = bdttimespec(RateSpec.ValuationDate, EndDates, 1); 

BDTT = bdttree(BDTVolatilitySpec, RateSpec, BDTTimeSpec); 

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

Spread = 5; 
Settle = datetime(2013,9,1);
Maturity = datetime(2013,12,1); 
Reset = 2; 

Price = floatbybdt(BDTT, Spread, Settle, Maturity, 'FloatReset', Reset)

Warning: Not all cash flows are aligned with the tree. Result will be approximated. 
> In floatengbybdt at 204
  In floatbybdt at 123 
Error using floatengbybdt (line 299)
Instrument '1 ' has cash flow dates that span across tree nodes.

Error in floatbybdt (line 123)
[Price, PriceTree, CFTree, TLPpal] = floatengbybdt(BDTTree, 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 BDTTimeSpec. 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);
BDTTimeSpec = bdttimespec(RateSpec.ValuationDate,resetDates,Reset);
BDTT = bdttree(BDTVolatilitySpec, RateSpec, BDTTimeSpec);

Price = floatbybdt(BDTT, Spread, RateSpec.ValuationDate, ...
                   Maturity, 'FloatReset', Reset)

Price =

  100.1087

Input Arguments

BDTTree — Interest-rate structure

structure

Interest-rate tree structure, created by bdttree

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

datetime array | string array | date character vector

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, floatbybdt 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 BDT tree. The floating-rate note argument Settle is ignored.

Maturity — Maturity date

datetime array | string array | date character vector

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, floatbybdt 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] = floatbybdt(BDTTree,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^® dates

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

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.PTree contains the clean prices.

• PriceTree.AITree 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.

Version History

Introduced before R2006a

R2022b: Serial date numbers not recommended

Although floatbybdt supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.