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 structureInterest-rate tree structure, created by bktree
Data Types: struct
Spread — Number of basis points over the reference rateNumber of basis points over the reference rate, specified as a
NINST-by-1 vector.
Data Types: double
Settle — Settlement dateSettlement 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 dateMaturity 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 year1
(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 13Day 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 schedules100 (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 structureDerivatives 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 days1 (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 countfalse (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 daysholidays.m (default) | MATLAB® date numbersHolidays 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 conventionsactual (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 rateAnnual 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 rateAnnual 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 0Expected floating-rate note prices at time 0, returned as a NINST-by-1 vector.
PriceTree — Tree structure of instrument pricesTree 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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