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Price bonds with embedded options by Heath-Jarrow-Morton interest-rate tree
[Price,PriceTree]
= optembndbyhjm(HJMTree,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates)
[Price,PriceTree]
= optembndbyhjm(___,Name,Value)
[
calculates
price for bonds with embedded options from a Heath-Jarrow-Morton interest-rate
tree.Price
,PriceTree
]
= optembndbyhjm(HJMTree
,CouponRate
,Settle
,Maturity
,OptSpec
,Strike
,ExerciseDates
)
optembndbyhjm
computes prices of vanilla
bonds with embedded options, stepped coupon bonds with embedded options,
and bonds with sinking fund option provisions. For more information,
see Definitions.
[
adds
optional name-value pair arguments.Price
,PriceTree
]
= optembndbyhjm(___,Name,Value
)
Create a HJMTree
with the following data:
Rates = [0.05;0.06;0.07]; Compounding = 1; StartDates = ['jan-1-2007';'jan-1-2008';'jan-1-2009']; EndDates = ['jan-1-2008';'jan-1-2009';'jan-1-2010']; ValuationDate = 'jan-1-2007';
Create a RateSpec
.
RateSpec = intenvset('Rates', Rates, 'StartDates', ValuationDate, 'EndDates', ... EndDates, 'Compounding', Compounding, 'ValuationDate', ValuationDate)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: 1
Disc: [3×1 double]
Rates: [3×1 double]
EndTimes: [3×1 double]
StartTimes: [3×1 double]
EndDates: [3×1 double]
StartDates: 733043
ValuationDate: 733043
Basis: 0
EndMonthRule: 1
Create a VolSpec
.
VolSpec = hjmvolspec('Constant', 0.01)
VolSpec = struct with fields:
FinObj: 'HJMVolSpec'
FactorModels: {'Constant'}
FactorArgs: {{1×1 cell}}
SigmaShift: 0
NumFactors: 1
NumBranch: 2
PBranch: [0.5000 0.5000]
Fact2Branch: [-1 1]
Create a TimeSpec
.
TimeSpec = hjmtimespec(ValuationDate, EndDates, Compounding)
TimeSpec = struct with fields:
FinObj: 'HJMTimeSpec'
ValuationDate: 733043
Maturity: [3×1 double]
Compounding: 1
Basis: 0
EndMonthRule: 1
Build the HJMTree
.
HJMTree = hjmtree(VolSpec, RateSpec, TimeSpec)
HJMTree = struct with fields:
FinObj: 'HJMFwdTree'
VolSpec: [1×1 struct]
TimeSpec: [1×1 struct]
RateSpec: [1×1 struct]
tObs: [0 1 2]
dObs: [733043 733408 733774]
TFwd: {[3×1 double] [2×1 double] [2]}
CFlowT: {[3×1 double] [2×1 double] [3]}
FwdTree: {[3×1 double] [2×1×2 double] [1×2×2 double]}
To compute the price of an American callable bond that pays a 6% annual coupon and matures and is callable on January 1, 2010.
BondSettlement = 'jan-1-2007'; BondMaturity = 'jan-1-2010'; CouponRate = 0.06; Period = 1; OptSpec = 'call'; Strike = [98]; ExerciseDates = '01-Jan-2010'; AmericanOpt = 1; [PriceCallBond,PT] = optembndbyhjm(HJMTree, CouponRate, BondSettlement, BondMaturity,... OptSpec, Strike, ExerciseDates, 'Period', 1,'AmericanOp',1)
PriceCallBond = 95.9492
PT = struct with fields:
FinObj: 'HJMPriceTree'
tObs: [0 1 2 3]
PBush: {[95.9492] [1×1×2 double] [1×2×2 double] [98 98 98 98]}
Price the following single stepped callable bonds using the following data: The data for the interest rate term structure is as follows:
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 RateSpec RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding); % Instrument Settle = '01-Jan-2010'; Maturity = {'01-Jan-2013';'01-Jan-2014'}; CouponRate = {{'01-Jan-2012' .0425;'01-Jan-2014' .0750}}; OptSpec='call'; Strike=100; ExerciseDates='01-Jan-2012'; %Callable in two years % Build the tree with the following data Volatility = [.2; .19; .18; .17]; CurveTerm = [ 1; 2; 3; 4]; HJMTimeSpec = hjmtimespec(ValuationDate, EndDates); HJMVolSpec = hjmvolspec('Proportional', Volatility, CurveTerm, 1e6); HJMT = hjmtree(HJMVolSpec, RS, HJMTimeSpec); % The first row corresponds to the price of the callable bond with maturity % of three years. The second row corresponds to the price of the callable % bond with maturity of four years. PHJM= optembndbyhjm(HJMT, CouponRate, Settle, Maturity ,OptSpec, Strike,... ExerciseDates, 'Period', 1)
PHJM =
100.0484
99.8009
A corporation issues a three year bond with a sinking fund obligation requiring the company to sink 1/3 of face value after the first year and 1/3 after the second year. The company has the option to buy the bonds in the market or call them at $99. The following data describes the details needed for pricing the sinking fund bond:
Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = 'Jan-1-2011'; StartDates = ValuationDate; EndDates = {'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'}; 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: [4×1 double]
Rates: [4×1 double]
EndTimes: [4×1 double]
StartTimes: [4×1 double]
EndDates: [4×1 double]
StartDates: 734504
ValuationDate: 734504
Basis: 0
EndMonthRule: 1
Build the HJM
tree.
Sigma = 0.1;
HJMTimeSpec = hjmtimespec(ValuationDate, EndDates, Compounding);
HJMVolSpec = hjmvolspec('Constant', Sigma);
HJMT = hjmtree(HJMVolSpec, RateSpec, HJMTimeSpec)
HJMT = struct with fields:
FinObj: 'HJMFwdTree'
VolSpec: [1×1 struct]
TimeSpec: [1×1 struct]
RateSpec: [1×1 struct]
tObs: [0 1 2 3]
dObs: [734504 734869 735235 735600]
TFwd: {[4×1 double] [3×1 double] [2×1 double] [3]}
CFlowT: {[4×1 double] [3×1 double] [2×1 double] [4]}
FwdTree: {[4×1 double] [3×1×2 double] [2×2×2 double] [1×4×2 double]}
Define the sinking fund instrument. The bond has a coupon rate of 4.5%, a period of one year and matures in 1-Jan-2013. Face decreases 1/3 after the first year.
CouponRate = 0.045; Settle = 'Jan-1-2011'; Maturity = 'Jan-1-2013'; Period = 1; Face = { {'Jan-1-2012' 100; ... 'Jan-1-2013' 66.6666}};
Define the option provision.
OptSpec = 'call'; Strike = 99; ExerciseDates = 'Jan-1-2012';
Price of non-sinking fund bond.
PNSF = bondbyhjm(HJMT, CouponRate, Settle, Maturity, Period)
PNSF = 100.5663
Price of the bond with the option sinking provision.
PriceSF = optembndbyhjm(HJMT, CouponRate, Settle, Maturity,... OptSpec, Strike, ExerciseDates, 'Period', Period, 'Face', Face)
PriceSF = 98.8736
HJMTree
— Interest-rate tree structureInterest-rate tree structure, specified by using hjmtree
.
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 for the bond option, specified as a NINST
-by-1
vector
of serial date numbers or date character vectors.
Note:
The |
Data Types: double
| char
Maturity
— Maturity dateMaturity date, specified as an NINST
-by-1
vector
of serial date numbers or date character vectors.
Data Types: double
| char
OptSpec
— Definition of option '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
Strike
— Option strike price valuesOption 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 than NSTRIKES
exercise opportunities, the
end of the row is padded with NaN
s.
American option — NINST
-by-1
vector
of strike price values for each option.
Data Types: double
ExerciseDates
— Option exercise datesOption exercise dates, specified as a NINST
-by-1
, NINST
-by-2
,
or NINST
-by-NSTRIKES
using serial
date numbers 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 one ExerciseDates
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 if ExerciseDates
is a NINST
-by-1
vector,
the option can be exercised between ValuationDate
of
the stock tree and the single listed ExerciseDates
.
Data Types: double
| char
Specify optional comma-separated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.
Price = optembndbyhjm(HJMTree,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',1,'AmericanOp',1)
'AmericanOpt'
— Option type0
European/Bermuda (default) | integer with values 0
or 1
Option type, specified as NINST
-by-1
positive
integer flags with values:
0
— European/Bermuda
1
— American
Data Types: double
'Period'
— Coupons per year2
per year (default) | vectorCoupons per year, specified as an NINST
-by-1
vector.
Data Types: double
'Basis'
— Day-count basis0
(actual/actual) (default) | integer from 0
to 13
Day-count basis, specified as 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 flag1
(in effect) (default) | nonnegative integer with values 0
or 1
End-of-month rule flag is specified as 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 dateBond issue date, specified as an NINST
-by-1
vector
using serial date numbers or date character vectors.
Data Types: double
| char
'FirstCouponDate'
— Irregular first coupon dateIrregular first coupon date, specified as an NINST
-by-1
vector
using serial date numbers or date character vectors.
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 a NINST
-by-1
vector
using serial date numbers or date character vectors.
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: char
| double
'StartDate'
— Forward starting date of paymentsForward starting date of payments (the date from which a bond
cash flow is considered), specified as 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 anNINST
-by-1
vector.
Data Types: double
'Options'
— Derivatives pricing optionsDerivatives pricing options, specified as structure that is
created with derivset
.
Data Types: struct
Price
— Expected prices of embedded option at time 0
Expected price of the embedded option at time 0
,
returned as a NINST
-by-1
matrix.
PriceTree
— Structure containing trees of vectors of instrument prices and accrued interest for each nodeStructure containing trees of vectors of instrument prices and accrued interest, and a vector of observation times for each node. Values are:
PriceTree.PBush
contains the clean
prices.
PriceTree.tObs
contains the observation
times.
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.
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).
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 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 will buy back the requirement amount of bonds from the market since bonds will be cheap, but if interest rates are low (bond prices are high), then most likely the issuer will be buying the bonds at the call price. Unlike a call feature, however, if a bond has a sinking fund 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.
cfamounts
| hjmprice
| hjmtree
| instoptembnd
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