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floatbybk

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

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Syntax

[Price,PriceTree] = floatbybk(BKTree,Spread,Settle,Maturity)
[Price,PriceTree] = floatbybk(___,Name,Value)

Description

[Price,PriceTree] = floatbybk(BKTree,Spread,Settle,Maturity) prices a floating-rate note from a Black-Karasinski interest-rate tree.

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.

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.

example

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

example

Examples

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Open Live Script

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 = datetime(2005,1,1);
Maturity = datetime(2006,1,1);

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
Open Live Script

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 BK tree and assume the volatility is 10%. 

VolDates = [datetime(2012,11,15) ; datetime(2013,11,15) ; datetime(2014,11,15) ; datetime(2015,11,15) ; datetime(2016,11,15) ; datetime(2017,11,15)];
VolCurve = 0.1;
AlphaDates = datetime(2017,11,15);
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
Open Live Script

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 = 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 BK tree and assume the volatility to be 5%. 

VolDates = [datetime(2013,4,1) ; datetime(2014,4,1) ; datetime(2015,4,1) ; datetime(2016,4,1) ; datetime(2017,4,1) ; datetime(2018,4,1)];
VolCurve = 0.05;
AlphaDates = datetime(2018,11,15);
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 = datetime(2012,4,1);
Maturity = datetime(2016,4,1);
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 = {{datetime(2013,4,1) 0.045;datetime(2014,4,1) 0.05;...
datetime(2015,4,1) 0.06}; 0.06};
         
FloorStrike = {{datetime(2013,4,1) 0.035; datetime(2014,4,1) 0.04;...
datetime(2015,4,1) 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      = datetime(2013,9,13); 
ForwardRatesVector = [ 0.0001; 0.0001; 0.0010; 0.0015]; 
EndDatesVector     = [datetime(2013,12,13);datetime(2014,3,14);datetime(2014,6,13);datetime(2014,9,13)];

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    = datetime(2014,6,13); 
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

Input Arguments

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Interest-rate tree structure, created by bktree

Data Types: struct

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

Data Types: double

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, floatbybk 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 BK tree. The floating-rate note argument Settle is ignored.

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, floatbybk 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] = floatbybk(BKTree,Spread,Settle,Maturity,'Basis',3)

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

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

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

Derivatives pricing options structure, specified as the comma-separated pair consisting of 'Options' and a structure 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 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

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 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

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

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

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

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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.

More About

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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

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See Also

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