Price bond from Black-Karasinski interest-rate tree
[
adds
additional name-value pair arguments.Price
,PriceTree
]
= bondbybk(___,Name,Value
)
Price a 4% bond using a Black-Karasinski interest-rate tree.
Load deriv.mat
, which provides BKTree
. The BKTree
structure contains the time and interest-rate information needed to price the bond.
load deriv.mat;
Define the bond using the required arguments. Other arguments use defaults.
CouponRate = 0.04; Settle = '01-Jan-2004'; Maturity = '1-Jan-2008';
Use bondbybk
to compute the price of the bond.
Period = 1; Price = bondbybk(BKTree, CouponRate, Settle, Maturity, Period)
Price = 99.3296
Price single stepped coupon bonds using market data.
Define the interest-rate term structure.
Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = 'Jan-1-2010'; StartDates = ValuationDate; EndDates = {'Jan-1-2011'; 'Jan-1-2012';... 'Jan-1-2013'; 'Jan-1-2014'}; Compounding = 1;
Create the RateSpec
.
RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)
RS = struct with fields:
FinObj: 'RateSpec'
Compounding: 1
Disc: [4x1 double]
Rates: [4x1 double]
EndTimes: [4x1 double]
StartTimes: [4x1 double]
EndDates: [4x1 double]
StartDates: 734139
ValuationDate: 734139
Basis: 0
EndMonthRule: 1
Create the stepped bond instrument.
Settle = '01-Jan-2010'; Maturity = {'01-Jan-2011';'01-Jan-2012';'01-Jan-2013';'01-Jan-2014'}; CouponRate = {{'01-Jan-2012' .0425;'01-Jan-2014' .0750}}; Period = 1;
Build the BK tree using the following market data:
VolDates = ['1-Jan-2011'; '1-Jan-2012'; '1-Jan-2013'; '1-Jan-2014']; VolCurve = 0.01; AlphaDates = '01-01-2014'; AlphaCurve = 0.1; BKVolSpec = bkvolspec(RS.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); BKTimeSpec = bktimespec(RS.ValuationDate, VolDates, Compounding); BKT = bktree(BKVolSpec, RS, BKTimeSpec);
Compute the price of the stepped coupon bonds.
PBK= bondbybk(BKT, CouponRate, Settle,Maturity , Period)
PBK = 4×1
100.7246
100.0945
101.5900
102.0820
Price a bond with an amortization schedule using the Face
input argument to define the schedule.
Define the interest-rate term structure.
Rates = 0.065; ValuationDate = '1-Jan-2011'; StartDates = ValuationDate; EndDates= '1-Jan-2017'; Compounding = 1;
Create the RateSpec
.
RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: 1
Disc: 0.6853
Rates: 0.0650
EndTimes: 6
StartTimes: 0
EndDates: 736696
StartDates: 734504
ValuationDate: 734504
Basis: 0
EndMonthRule: 1
Create the bond instrument. The bond has a coupon rate of 7%, a period of one year, and matures on 1-Jan-2017.
CouponRate = 0.07; Settle ='1-Jan-2011'; Maturity = '1-Jan-2017'; Period = 1; Face = {{'1-Jan-2015' 100;'1-Jan-2016' 90;'1-Jan-2017' 80}};
Build the BK tree with the following market data:
VolDates = ['1-Jan-2012'; '1-Jan-2013';... '1-Jan-2014';'1-Jan-2015';'1-Jan-2016';'1-Jan-2017']; VolCurve = 0.01; AlphaDates = '01-01-2017'; 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 bond.
Price = bondbybk(BKT, CouponRate, Settle, Maturity, 'Period', Period,... 'Face', Face)
Price = 102.3155
Compare the results with price of a vanilla bond.
PriceVanilla = bondbybk(BKT, CouponRate, Settle, Maturity, Period)
PriceVanilla = 102.4205
BKTree
— Interest-rate structureInterest-rate tree structure, created by bktree
Data Types: struct
CouponRate
— Bond coupon rate Bond coupon rate, specified as an NINST
-by-1
decimal
annual rate or NINST
-by-1
cell
array, where each element is a NumDates
-by-2
cell
array. The first column of the NumDates
-by-2
cell
array is dates and the second column is associated rates. The date
indicates the last day that the coupon rate is valid.
Data Types: double
| cell
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 bond is set to
the ValuationDate
of the BK tree. The bond 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 bond.
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] = bondbybk(BKTree,CouponRate,Settle,Maturity,'Period',4,'Face',10000)
'Period'
— Coupons per year2
per year (default) | vectorCoupons per year, specified as the comma-separated pair consisting of
'Period'
and a
NINST
-by-1
vector. Values for Period
are
1
, 2
,
3
, 4
,
6
, and
12
.
Data Types: double
'Basis'
— Day-count basis0
(actual/actual) (default) | integer from 0
to 13
Day-count basis of the instrument, 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
'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
'IssueDate'
— Bond issue dateBond issue date, specified as the comma-separated pair consisting of
'IssueDate'
and a
NINST
-by-1
vector using a serial nonnegative date number or
date character vector.
Data Types: double
| char
'FirstCouponDate'
— Irregular first coupon dateIrregular first coupon date, specified as the comma-separated pair consisting of
'FirstCouponDate'
and a
NINST
-by-1
vector using a serial nonnegative date number or
date character vector.
When FirstCouponDate
and LastCouponDate
are
both specified, FirstCouponDate
takes precedence
in determining the coupon payment structure. If you do not specify
a FirstCouponDate
, the cash flow payment dates
are determined from other inputs.
Data Types: double
| char
'LastCouponDate'
— Irregular last coupon dateIrregular last coupon date, specified as the comma-separated pair consisting of
'LastCouponDate'
and a
NINST
-by-1
vector using a serial nonnegative date number or
date character vector.
In the absence of a specified FirstCouponDate
,
a specified LastCouponDate
determines the coupon
structure of the bond. The coupon structure of a bond is truncated
at the LastCouponDate
, regardless of where it falls,
and is followed only by the bond's maturity cash flow date. If you
do not specify a LastCouponDate
, the cash flow
payment dates are determined from other inputs.
Data Types: double
| char
'StartDate'
— Forward starting date of paymentsSettle
date (default) | serial date number | date character vectorForward starting date of payments (the date from which a bond cash flow is considered),
specified as the comma-separated pair consisting
of 'StartDate'
and a
NINST
-by-1
vector using serial date numbers or date character
vectors.
If you do not specify StartDate
, the effective
start date is the Settle
date.
Data Types: char
| double
'Face'
— Face value100
(default) | nonnegative value | cell array of nonnegative valuesFace or par value, specified as the comma-separated pair consisting of
'Face'
and a
NINST
-by-1
vector of nonnegative face values or an
NINST
-by-1
cell array of face values or face value schedules.
For the latter case, each element of the cell
array is a
NumDates
-by-2
cell array, where the first column is dates and
the second column is its associated face value.
The date indicates the last day that the face
value is valid.
Data Types: cell
| double
'Options'
— Derivatives pricing optionsDerivatives pricing options, specified as the comma-separated pair consisting of
'Options'
and a structure that
is created with derivset
.
Data Types: struct
'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
'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
(or
NINST
-by-2
if BusinessDayConvention
is
different for each leg) 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
'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
Price
— Expected bond prices at time 0Expected bond 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 vanilla coupon bond is a security representing an obligation to repay a borrowed amount at a designated time and to make periodic interest payments until that time.
The issuer of a bond makes the periodic interest payments until the bond matures. At maturity, the issuer pays to the holder of the bond the principal amount owed (face value) and the last interest payment.
A step-up and step-down bond is a debt security with a predetermined coupon structure over time.
With these instruments, coupons increase (step up) or decrease (step down) at specific times during the life of the bond.
An amortized bond is treated as an asset, with the discount amount being amortized to interest expense over the life of the bond.
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