bondbybdt
Price bond from Black-Derman-Toy interest-rate tree
Syntax
Description
[
adds
additional name-value pair arguments.Price
,PriceTree
]
= bondbybdt(___,Name,Value
)
Examples
Price a Bond Using a BDT Tree
Price a 10% bond using a BDT interest-rate tree.
Load deriv.mat
, which provides BDTTree
. The BDTTree
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 = 0.10; Settle = datetime(2000,1,1); Maturity = datetime(2003,1,1); Period = 1;
Use bondbybdt
to compute the price of the bond.
Price = bondbybdt(BDTTree, CouponRate, Settle, Maturity, Period)
Price = 95.5030
Price a Stepped Coupon Bond
Price single stepped coupon bonds using market data.
Define the interest-rate term structure.
Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = datetime(2010,1,1); StartDates = ValuationDate; EndDates = [datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1)]; 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 = datetime(2010,1,1);
Maturity = [datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1)];
%CouponRate = {{'01-Jan-2012' .0425;'01-Jan-2014' .0750}};
CouponRate = {{datetime(2012,1,1) .0425; datetime(2014,1,1) .0750}};
Period = 1;
Build the BDT tree and assume the volatility to be 10% using the following market data:
Sigma = 0.1; BDTTimeSpec = bdttimespec(ValuationDate, EndDates); BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Sigma*ones(1, length(EndDates))'); BDTT = bdttree(BDTVolSpec, RS, BDTTimeSpec)
BDTT = struct with fields:
FinObj: 'BDTFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2 3]
dObs: [734139 734504 734869 735235]
TFwd: {[4x1 double] [3x1 double] [2x1 double] [3]}
CFlowT: {[4x1 double] [3x1 double] [2x1 double] [4]}
FwdTree: {[1.0350] [1.0444 1.0543] [1.0469 1.0573 1.0700] [1.0505 1.0617 1.0754 1.0921]}
Compute the price of the stepped coupon bonds.
PBDT= bondbybdt(BDTT, CouponRate, Settle,Maturity , Period)
PBDT = 4×1
100.7246
100.0945
101.5900
102.0820
Price Two Bonds with Amortization Schedules
Price two bonds with amortization schedules using the Face
input argument to define the schedule.
Define the interest-rate term structure.
Rates = 0.035; ValuationDate = datetime(2011,11,1); StartDates = ValuationDate; EndDates = datetime(2017,11,1); Compounding = 1;
Create the RateSpec
.
RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding);
Create the bond instrument. The bonds have a coupon rate of 4% and 3.85%, a period of one year, and mature on 1-Nov-2017.
CouponRate = [0.04; 0.0385]; Settle = datetime(2011,11,1); Maturity = datetime(2017,11,1); Period = 1;
Define the amortizing schedule.
Face = {{datetime(2015,11,1) 100; datetime(2016,11,1) 85; datetime(2017,11,1) 70}; {datetime(2015,11,1) 100; datetime(2016,11,1) 90; datetime(2017,11,1) 80}};
Build the BDT tree and assume the volatility to be 10%.
MatDates = [datetime(2012,11,1) ; datetime(2013,11,1) ; datetime(2014,11,1) ; datetime(2015,11,1) ; datetime(2016,11,1) ; datetime(2017,11,1)]; BDTTimeSpec = bdttimespec(ValuationDate, MatDates); Volatility = 0.1; BDTVolSpec = bdtvolspec(ValuationDate, MatDates, Volatility*ones(1,length(MatDates))'); BDTT = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec);
Compute the price of the amortizing bonds.
Price = bondbybdt(BDTT, CouponRate, Settle, Maturity, 'Period',Period,... 'Face', Face)
Price = 2×1
102.4791
101.7786
Input Arguments
BDTTree
— Interest-rate structure
structure
Interest-rate tree structure, created by bdttree
Data Types: struct
CouponRate
— Bond coupon rate
positive decimal value
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 date
datetime array | string array | date character vector
Settlement date, specified either as a scalar or
NINST
-by-1
vector using a datetime
array, string array, or date character vectors.
To support existing code, bondbybdt
also
accepts serial date numbers as inputs, but they are not recommended.
The Settle
date for every bond is set to
the ValuationDate
of the BDT tree. The bond 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 bond.
To support existing code, bondbybdt
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] = bondbybdt(BDTTree,CouponRate,Settle,Maturity,'Period',4,'Face',10000)
Period
— Coupons per year
2
per year (default) | vector
Coupons per year, specified as the comma-separated pair consisting of
'Period'
and an
NINST
-by-1
vector. Values for
Period
are 1
,
2
, 3
, 4
,
6
, and 12
.
Data Types: double
Basis
— Day-count basis
0
(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 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
IssueDate
— Bond issue date
datetime array | string array | date character vector
Bond issue date, specified as the comma-separated pair consisting of
'IssueDate'
and a
NINST
-by-1
vector using a
datetime array, string array, or date character vectors.
To support existing code, bondbybdt
also
accepts serial date numbers as inputs, but they are not recommended.
FirstCouponDate
— Irregular first coupon date
datetime array | string array | date character vector
Irregular first coupon date, specified as the comma-separated pair consisting of
'FirstCouponDate'
and a
NINST
-by-1
vector using a
datetime array, string array, or date character vectors.
To support existing code, bondbybdt
also
accepts serial date numbers as inputs, but they are not recommended.
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.
LastCouponDate
— Irregular last coupon date
datetime array | string array | date character vector
Irregular last coupon date, specified as the comma-separated pair consisting of
'LastCouponDate'
and a
NINST
-by-1
vector using a
datetime array, string array, or date character vectors.
To support existing code, bondbybdt
also
accepts serial date numbers as inputs, but they are not recommended.
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.
StartDate
— Forward starting date of payments
Settle
date (default) | datetime array | string array | date character vector
Forward 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 a
datetime array, string array, or date character vectors.
To support existing code, bondbybdt
also
accepts serial date numbers as inputs, but they are not recommended.
If you do not specify StartDate
, the effective
start date is the Settle
date.
Face
— Face value
100
(default) | nonnegative value | cell array of nonnegative values
Face 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 options
structure
Derivatives 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 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
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
(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 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
Output Arguments
Price
— Expected bond prices at time 0
vector
Expected bond 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
Vanilla Bond
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.
Stepped Coupon Bond
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.
Bond with an Amortization Schedule
An amortized bond is treated as an asset, with the discount amount being amortized to interest expense over the life of the bond.
Version History
Introduced before R2006aR2022b: Serial date numbers not recommended
Although bondbybdt
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.
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