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Price floating-rate note from Black-Karasinski interest-rate tree
[Price,PriceTree]
= floatbybk(BKTree,CouponRate,Settle,Maturity)
[Price,PriceTree]
= floatbybk(___,Name,Value)
[
prices
a floating-rate note from a Black-Karasinski interest-rate tree. Price
,PriceTree
]
= floatbybk(BKTree
,CouponRate
,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: [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'; 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 =
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 =
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, 'Reset', 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, 'Reset', reset)
Price = 100.0004
BKTree
— Interest-rate structureInterest-rate tree structure, created by bktree
Data Types: struct
CouponRate
— Coupon annual rateCoupon annual 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 swap.
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 single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.
[Price,PriceTree] = floatbybk(BKTree,CouponRate,Settle,Maturity,'Basis',3)
'Reset'
— Frequency of payments per year1
(default) | vectorFrequency of payments per year, specified as 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 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 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 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 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 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 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 by 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 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 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 one 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.
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