bondbyhw
Price bond from Hull-White interest-rate tree
Syntax
Description
[
adds
additional name-value pair arguments.Price
,PriceTree
]
= bondbyhw(___,Name,Value
)
Examples
Price a 4% bond using a Hull-White interest-rate tree.
Load deriv.mat
, which provides HWTree
. The HWTree
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 = datetime(2004,1,1); Maturity = datetime(2006,1,1);
Use bondbyhw
to compute the price of the bond.
Period = 1; Price = bondbyhw(HWTree, CouponRate, Settle, Maturity, Period)
Price = 101.6002
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: [4×1 double]
Rates: [4×1 double]
EndTimes: [4×1 double]
StartTimes: [4×1 double]
EndDates: [4×1 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 = {{datetime(2012,1,1) .0425;datetime(2014,1,1) .0750}}; Period = 1;
Build the HW tree using the following market data:
VolDates = [datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1)]; VolCurve = 0.01; AlphaDates = datetime(2014,1,1); AlphaCurve = 0.1; HWVolSpec = hwvolspec(RS.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); HWTimeSpec = hwtimespec(RS.ValuationDate, VolDates, Compounding); HWT = hwtree(HWVolSpec, RS, HWTimeSpec);
Compute the price of the stepped coupon bonds.
PHW= bondbyhw(HWT, CouponRate, Settle,Maturity , Period)
PHW = 4×1
100.7246
100.0945
101.5900
102.0820
Price two bonds with amortization schedules using the Face
input argument to define the schedules.
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 HW tree and assume the volatility to be 10%.
VolDates = [datetime(2012,11,1) ; datetime(2013,11,1) ; datetime(2014,11,1) ; datetime(2015,11,1) ; datetime(2016,11,1) ; datetime(2017,11,1)]; VolCurve = 0.1; AlphaDates = datetime(2017,1,1); AlphaCurve = 0.1; HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); HWTimeSpec = hwtimespec(RateSpec.ValuationDate, VolDates, Compounding); HWT = hwtree(HWVolSpec, RateSpec, HWTimeSpec);
Compute the price of the amortizing bonds.
Price = bondbyhw(HWT, CouponRate, Settle, Maturity, 'Period',Period,... 'Face', Face)
Price = 2×1
102.4791
101.7786
Input Arguments
Interest-rate tree structure, created by hwtree
.
Data Types: struct
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
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, bondbyhw
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 HW tree. The bond 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 bond.
To support existing code, bondbyhw
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] = bondbyhw(HWTree,CouponRate,Settle,Maturity,'Period',4,'Face',10000)
Coupons 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
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
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
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, bondbyhw
also
accepts serial date numbers as inputs, but they are not recommended.
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, bondbyhw
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.
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, bondbyhw
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.
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, bondbyhw
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 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
Derivatives pricing options, specified as the comma-separated pair consisting of
'Options'
and a structure that
is created with derivset
.
Data Types: struct
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
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 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
Expected bond 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 areNumNodes
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
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.
Version History
Introduced before R2006aAlthough bondbyhw
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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