oasbyhjm
Determine option adjusted spread using Heath-Jarrow-Morton model
Syntax
Description
[
calculates option adjusted spread using a Heath-Jarrow-Morton model.OAS
,OAD
,OAC
]
= oasbyhjm(HJMTree
,Price
,CouponRate
,Settle
,Maturity
,OptSpec
,Strike
,ExerciseDates
)
oasbyhjm
computes prices of vanilla bonds with embedded
options, stepped coupon bonds with embedded options, amortizing bonds with
embedded options, and sinking fund bonds with call embedded option. For more
information, see More About.
[
adds optional name-value pair arguments.OAS
,OAD
,OAC
]
= oasbyhjm(___,Name,Value
)
Examples
Compute OAS Using the Heath-Jarrow-Morton (HJM) Model
This example shows how to compute OAS using the Heath-Jarrow-Morton (HJM) model using the following data.
ValuationDate = datetime(2010,11,1); Rates = [0.0356; 0.0427; 0.0478; 0.0529]; StartDates = ValuationDate; EndDates = datemnth(ValuationDate, 12:12:48)'; Compounding = 1; % define RateSpec RateSpec = intenvset('ValuationDate', ValuationDate,... 'StartDates', StartDates,'EndDates', EndDates, ... 'Rates', Rates,'Compounding', Compounding); % specify VolSpec and TimeSpec Sigma = 0.02; VS = hjmvolspec('Constant', Sigma); TS = hjmtimespec(ValuationDate, EndDates, Compounding); % build the HJM tree HJMTree = hjmtree(VS, RateSpec, TS); HJMTreenew = cvtree(HJMTree); % instrument information CouponRate = 0.05; Settle = ValuationDate; Maturity = datetime(2014,11,1); OptSpec = 'call'; Strike = 100; ExerciseDates = datetime(2010,11,1); Period = 1; Price = 97.5; % compute the OAS OAS = oasbyhjm(HJMTree, Price, CouponRate, Settle, Maturity, OptSpec, Strike,... ExerciseDates, 'Period', Period)
OAS = 49.3289
Compute OAS for an Amortizing Callable Bond Using an HJM Interest-Rate Tree Model
This example shows how to compute the OAS for amortizing callable bond using an HJM lattice model.
Create a RateSpec
.
Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = datetime(2012,1,1); StartDates = ValuationDate; EndDates = [datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1)]; Compounding = 1; RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding);
Build a HJM tree.
VolSpec = hjmvolspec('Constant', 0.01)
VolSpec = struct with fields:
FinObj: 'HJMVolSpec'
FactorModels: {'Constant'}
FactorArgs: {{1x1 cell}}
SigmaShift: 0
NumFactors: 1
NumBranch: 2
PBranch: [0.5000 0.5000]
Fact2Branch: [-1 1]
TimeSpec = hjmtimespec(ValuationDate, EndDates, Compounding)
TimeSpec = struct with fields:
FinObj: 'HJMTimeSpec'
ValuationDate: 734869
Maturity: [4x1 double]
Compounding: 1
Basis: 0
EndMonthRule: 1
HJMTree = hjmtree(VolSpec, RS, TimeSpec)
HJMTree = struct with fields:
FinObj: 'HJMFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2 3]
dObs: [734869 735235 735600 735965]
TFwd: {[4x1 double] [3x1 double] [2x1 double] [3]}
CFlowT: {[4x1 double] [3x1 double] [2x1 double] [4]}
FwdTree: {[4x1 double] [3x1x2 double] [2x2x2 double] [1x4x2 double]}
Define the callable bond.
CouponRate = 0.05;
Settle = datetime(2012,1,1);
Maturity = datetime(2016,1,1);
Period = 1;
Face = {
{datetime(2014,1,1) 100;
datetime(2015,1,1) 70;
datetime(2016,1,1) 50};
};
OptSpec = 'call';
Strike = [97 95 93];
ExerciseDates = [datetime(2014,1,1) datetime(2015,1,1) datetime(2016,1,1)];
Compute the OAS for a callable amortizing bond using the HJM tree.
Price = 99; BondType = 'amortizing'; OAS = oasbyhjm(HJMTree, Price, CouponRate, Settle, Maturity,... OptSpec, Strike, ExerciseDates, 'Period', Period, 'Face', Face,'BondType', BondType)
OAS = -19.1325
Input Arguments
HJMTree
— Interest-rate tree structure
structure
Interest-rate tree structure, specified by using hjmtree
.
Data Types: struct
Price
— Market prices of bonds with embedded options
numeric
Market prices of bonds with embedded options, specified as an NINST
-by-1
vector.
Data Types: double
CouponRate
— Bond coupon rate
positive decimal value
Bond coupon rate, specified as an NINST
-by-1
decimal
annual rate.
Data Types: double
Settle
— Settlement date
datetime array | string array | date character vector
Settlement date for the bond option, specified as a
NINST
-by-1
vector using a datetime
array, string array, or date character vectors.
Note
The Settle
date for every bond with an embedded
option is set to the ValuationDate
of the HJM
tree. The bond argument Settle
is ignored.
To support existing code, oasbyhjm
also
accepts serial date numbers as inputs, but they are not recommended.
Maturity
— Maturity date
datetime array | string array | date character vector
Maturity date, specified as an NINST
-by-1
vector using a
datetime array, string array, or date character vectors.
To support existing code, oasbyhjm
also
accepts serial date numbers as inputs, but they are not recommended.
OptSpec
— Definition of option
character vector with value 'call'
or 'put'
| cell array of character vectors with values 'call'
or 'put'
Definition of option, specified as a NINST
-by-1
cell
array of character vectors.
Data Types: char
| cell
Strike
— Option strike price values
nonnegative integer
Option strike price value, specified as a NINST
-by-1
or NINST
-by-NSTRIKES
depending
on the type of option:
European option —
NINST
-by-1
vector of strike price values.Bermuda option —
NINST
by number of strikes (NSTRIKES
) matrix of strike price values. Each row is the schedule for one option. If an option has fewer thanNSTRIKES
exercise opportunities, the end of the row is padded withNaN
s.American option —
NINST
-by-1
vector of strike price values for each option.
Data Types: double
ExerciseDates
— Option exercise dates
datetime array | string array | date character vector
Option exercise dates, specified as a NINST
-by-1
,
NINST
-by-2
, or
NINST
-by-NSTRIKES
vector using a
datetime array, string array, or date character vectors, depending on the
type of option:
For a European option, use a
NINST
-by-1
vector of dates. For a European option, there is only oneExerciseDates
on the option expiry date.For a Bermuda option, use a
NINST
-by-NSTRIKES
vector of dates. Each row is the schedule for one option.For an American option, use a
NINST
-by-2
vector of exercise date boundaries. The option can be exercised on any date between or including the pair of dates on that row. If only one non-NaN
date is listed, or ifExerciseDates
is aNINST
-by-1
vector, the option is exercised between the underlying bondSettle
date and the single listed exercise date.
To support existing code, oasbyhjm
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: OAS = oasbybk(BDTTree,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',4)
AmericanOpt
— Option type
0
European/Bermuda (default) | integer with values 0
or 1
Option type, specified as the comma-separated pair consisting of
'AmericanOpt'
and
NINST
-by-1
positive integer
flags with values:
0
— European/Bermuda1
— American
Data Types: double
Period
— Coupons per year
2
per year (default) | vector
Coupons per year, specified as the comma-separated pair consisting of
'Period'
and a
NINST
-by-1
vector.
Data Types: double
Basis
— Day-count basis
0
(actual/actual) (default) | integer from 0
to 13
Day-count basis, specified as the comma-separated pair consisting of
'Basis'
and a
NINST
-by-1
vector of integers.
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
1
(in effect) (default) | nonnegative integer with values 0
or 1
End-of-month rule flag, specified as the comma-separated pair consisting of
'EndMonthRule'
and a nonnegative integer using a
NINST
-by-1
vector. This rule
applies only when Maturity
is an end-of-month date
for a month having 30 or fewer days.
0
= Ignore rule, meaning that a bond coupon payment date is always the same numerical day of the month.1
= Set rule on, meaning that a bond coupon payment date is always the last actual day of the month.
Data Types: double
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, oasbyhjm
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, oasbyhjm
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, oasbyhjm
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
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, oasbyhjm
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) | NINST
-by-1
vector | NINST
-by-1
cell array
Face or par value, specified as the comma-separated pair consisting of
'Face'
and a
NINST
-by-1
vector or a
NINST
-by-1
cell array where
each element is a NumDates
-by-2
cell array where the first column is dates using a datetime, string, or
date character vector, and the second column is associated face value.
The date indicates the last day that the face value is valid.
Data Types: double
| char
| string
| datetime
BondType
— Type of underlying bond
'vanilla'
for scalar
Face
values, 'callablesinking'
for scheduled Face
values (default) | cell array of character vectors with values
'vanilla'
,'amortizing'
, or
'callablesinking'
| string array with values "vanilla"
,
"amortizing"
, or
"callablesinking"
Type of underlying bond, specified as the comma-separated pair
consisting of 'BondType'
and a
NINST
-by-1
cell array of
character vectors or string array specifying if the underlying is a
vanilla bond, an amortizing bond, or a callable sinking fund bond. The
supported types are:
'vanilla
' is a standard callable or puttable bond with a scalarFace
value and a single coupon or stepped coupons.'callablesinking'
is a bond with a schedule ofFace
values and a sinking fund call provision with a single or stepped coupons.'amortizing'
is an amortizing callable or puttable bond with a schedule ofFace
values with single or stepped coupons.
Data Types: char
| string
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
Output Arguments
OAS
— Option adjusted spread in basis points
vector
Option adjusted spread in basis points, returned as a
NINST
-by-1
vector.
OAD
— Option adjusted duration
vector
Option adjusted duration, returned as a NINST
-by-1
vector.
OAC
— Option adjusted convexity
vector
Option adjusted convexity, returned as a NINST
-by-1
vector.
More About
Vanilla Bond with Embedded Option
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 vanilla bond with an embedded option is where an option contract has an underlying asset of a vanilla bond.
Stepped Coupon Bond with Callable and Puttable Features
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. Stepped coupon bonds can have options features (call and puts).
Sinking Fund Bond with Call Embedded Option
A sinking fund bond is a coupon bond with a sinking fund provision.
This provision obligates the issuer to amortize portions of the principal prior to maturity, affecting bond prices since the time of the principal repayment changes. This means that investors receive the coupon and a portion of the principal paid back over time. These types of bonds reduce credit risk, since it lowers the probability of investors not receiving their principal payment at maturity.
The bond may have a sinking fund call option provision allowing the issuer to retire the sinking fund obligation either by purchasing the bonds to be redeemed from the market or by calling the bond via a sinking fund call, whichever is cheaper. If interest rates are high, then the issuer buys back the requirement amount of bonds from the market since bonds are cheap, but if interest rates are low (bond prices are high), then most likely the issuer is buying the bonds at the call price. Unlike a call feature, however, if a bond has a sinking fund call option provision, it is an obligation, not an option, for the issuer to buy back the increments of the issue as stated. Because of this, a sinking fund bond trades at a lower price than a non-sinking fund bond.
Amortizing Callable or Puttable Bond
Amortizing callable or puttable bonds work under a scheduled
Face
.
An amortizing callable bond gives the issuer the right to call back the bond, but
instead of paying the Face
amount at maturity, it repays part
of the principal along with the coupon payments. An amortizing puttable bond, repays
part of the principal along with the coupon payments and gives the bondholder the
right to sell the bond back to the issuer.
References
[1] Fabozzi, F. Handbook of Fixed Income Securities. 7th Edition. McGraw-Hill, 2005.
[2] Windas, T. Introduction to Option-Adjusted Spread Analysis. 3rd Edition. Bloomberg Press, 2007.
Version History
Introduced in R2011aR2022b: Serial date numbers not recommended
Although oasbyhjm
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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