oasbycir
Determine option adjusted spread using Cox-Ingersoll-Ross model
Syntax
Description
[
calculates the option adjusted spread from a Cox-Ingersoll-Ross (CIR) interest-rate tree
using a CIR++ model with the Nawalka-Beliaeva (NB) approach.OAS
,OAD
,OAC
]
= oasbycir(CIRTree
,Price
,CouponRate
,Settle
,Maturity
,OptSpec
,Strike
,ExerciseDates
)
oasbycir
computes prices of vanilla bonds with embedded options,
stepped coupon bonds with embedded options, amortizing bonds with embedded options, and
sinking fund bonds with embedded option. For more information, see More About.
[
adds optional name-value pair arguments.OAS
,OAD
,OAC
]
= oasbycir(___,Name,Value
)
Examples
Compute OAS Using a CIR Interest-Rate Tree
Create a RateSpec
using the intenvset
function.
ValuationDate = datetime(2018,10,25); Rates = [0.0355; 0.0382; 0.0427; 0.0489]; StartDates = ValuationDate; EndDates = datemnth(ValuationDate, 12:12:48)'; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, 'EndDates',EndDates,'Rates', Rates, 'Compounding', Compounding);
Create a CIR
tree.
NumPeriods = length(EndDates); Alpha = 0.03; Theta = 0.02; Sigma = 0.1; Maturity = datetime(2023,10,25); CIRTimeSpec = cirtimespec(ValuationDate, Maturity, NumPeriods); CIRVolSpec = cirvolspec(Sigma, Alpha, Theta); CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec)
CIRT = struct with fields:
FinObj: 'CIRFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1.2500 2.5000 3.7500]
dObs: [737358 737814 738271 738727]
FwdTree: {[1.0454] [1.0952 1.0574 1.0312] [1.1706 1.1188 1.0802 1.0534 1.0376] [1.2285 1.1624 1.1110 1.0726 1.0460 1.0304 1.0252]}
Connect: {[3x1 double] [3x3 double] [3x5 double]}
Probs: {[3x1 double] [3x3 double] [3x5 double]}
Define the OAS instrument.
CouponRate = 0.045; Settle = ValuationDate; Maturity = '25-October-2019'; OptSpec = 'call'; Strike = 100; ExerciseDates = {'25-October-2018','25-October-2019'}; Period = 1; AmericanOpt = 0; Price = 97;
Compute the OAS.
[OAS,OAD] = oasbycir(CIRT,Price,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',Period,'AmericanOpt',AmericanOpt)
OAS = 411.4425
OAD = 0.9282
Compute OAS for an Amortizing Callable Bond Using a CIR Interest-Rate Tree
his example shows how to compute the OAS for an amortizing callable bond using a CIR lattice model.
Create a RateSpec
using the intenvset
function.
Rates = [0.025; 0.032; 0.037; 0.042]; Dates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)]; ValuationDate = datetime(2016,1,1); EndDates = Dates(2:end)'; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, 'EndDates',EndDates,'Rates', Rates, 'Compounding', Compounding);
Create a CIR
tree.
NumPeriods = length(EndDates); Alpha = 0.03; Theta = 0.02; Sigma = 0.1; Maturity = datetime(2019,1,1); CIRTimeSpec = cirtimespec(ValuationDate, Maturity, NumPeriods); CIRVolSpec = cirvolspec(Sigma, Alpha, Theta); CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec)
CIRT = struct with fields:
FinObj: 'CIRFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 0.7500 1.5000 2.2500]
dObs: [736330 736604 736878 737152]
FwdTree: {[1.0187] [1.0338 1.0188 1.0083] [1.0577 1.0380 1.0230 1.0124 1.0061] [1.0964 1.0716 1.0517 1.0364 1.0257 1.0193 1.0172]}
Connect: {[3x1 double] [3x3 double] [3x5 double]}
Probs: {[3x1 double] [3x3 double] [3x5 double]}
Define the callable bond.
BondSettlement = datetime(2016,1,1);
BondMaturity = datetime(2020,1,1);
CouponRate = 0.035;
Period = 1;
OptSpec = 'call';
Strike = 100;
Face = {
{datetime(2018,1,1) 100;
datetime(2019,1,1) 70;
datetime(2020,1,1) 50};
};
ExerciseDates = [datetime(2018,1,1) ; datetime(2019,1,1)];
Compute OAS for a callable amortizing bond using the CIR tree.
Price = 99; BondType = 'amortizing'; OAS = oasbycir(CIRT, Price, CouponRate, BondSettlement, Maturity,... OptSpec, Strike, ExerciseDates, 'Period', Period, 'Face', Face,'BondType', BondType)
OAS = 2×1
80.4801
84.3684
Input Arguments
CIRTree
— Interest-rate tree structure
structure
Interest-rate tree structure, specified by using cirtree
.
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 CIR tree. The bond argument
Settle
is ignored.
To support existing code, oasbycir
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, oasbycir
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'
| string array with values "call"
or
"put"
Definition of option, specified as a
NINST
-by-1
cell array of character vectors or
string arrays with values 'call'
or
'put'
.
Data Types: char
| cell
| string
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.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 can be exercised betweenValuationDate
of the stock tree and the single listedExerciseDates
.
.
To support existing code, oasbycir
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 =
oasbycir(CIRTree,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 a
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, oasbycir
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, oasbycir
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, oasbycir
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, oasbycir
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
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] Cox, J., Ingersoll, J., and S. Ross. "A Theory of the Term Structure of Interest Rates." Econometrica. Vol. 53, 1985.
[2] Brigo, D. and F. Mercurio. Interest Rate Models - Theory and Practice. Springer Finance, 2006.
[3] Hirsa, A. Computational Methods in Finance. CRC Press, 2012.
[4] Nawalka, S., Soto, G., and N. Beliaeva. Dynamic Term Structure Modeling. Wiley, 2007.
[5] Nelson, D. and K. Ramaswamy. "Simple Binomial Processes as Diffusion Approximations in Financial Models." The Review of Financial Studies. Vol 3. 1990, pp. 393–430.
Version History
Introduced in R2018aR2022b: Serial date numbers not recommended
Although oasbycir
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.
See Also
bondbycir
| capbycir
| cfbycir
| fixedbycir
| floatbycir
| floorbycir
| optbndbycir
| optfloatbycir
| optembndbycir
| optemfloatbycir
| rangefloatbycir
| swapbycir
| swaptionbycir
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