Contents

optembndbyhw

Price bonds with embedded options by Hull-White interest-rate tree

Syntax

[Price, PriceTree] = optembndbyhw(HWTree, CouponRate,
Settle, Maturity, OptSpec, Strike, ExerciseDates,
'Name1', Value1, 'Name2', Value2, ...)

Arguments

HWTree

Interest-rate tree structure created by hwtree.

CouponRate

Decimal annual rate. CouponRate is a NINST-by-1 vector or NINST-by-1 cell array of decimal annual rates, or decimal annual rate schedules. For the latter case of a variable coupon schedule, 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 rate. The date indicates the last day that the coupon rate is valid.

Settle

NINST-by-1 matrix for the settlement date. A vector of serial date numbers or date strings. Settle must be earlier than Maturity.

Maturity

NINST-by-1 matrix for the maturity date. A vector of serial date numbers or date strings.

OptSpec

NINST-by-1 cell array of strings 'call' or 'put'.

Strike

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 NaNs.

For an American option:

NINST-by-1 vector of strike price values for each option.

ExerciseDates

NINST-by-1 (European option) or NINST-by-NSTRIKES (Bermuda option) matrix of exercise dates. Each row is the schedule for one option. For a European option, there is only one exercise date, the option expiry date.

For an American option:

NINST-by-2 vector of exercise date boundaries. For each instrument, the option can be exercised on any coupon date between or including the pair of dates on that row. If only one non-NaN date is listed, or if ExerciseDates is NINST-by-1, the option can be exercised between the underlying bond Settle and the single listed exercise date.

'Name1', Value1 'Name2', Value2 ...

(Optional) The name/value pairs are a variable length list of parameters. All optional inputs are specified as matching parameter name/value pairs. The parameter name is specified as a character string, followed by the corresponding parameter value. Parameter name/value pairs may be specified in any order; names are case insensitive and partial string matches are allowed provided no ambiguities exist. Valid parameter names are as follows:

  • AmericanOpt is a NINST-by-1 matrix for flags options: 0 (European/Bermuda) or 1 (American). Default is 0.

  • Period is a NINST-by-1 matrix for coupons per year. Default is 2.

  • Basis is a day-count basis of the instrument. Basis is a vector of integers with the following supported values:

    • 0 = actual/actual (default)

    • 1 = 30/360 (SIA)

    • 2 = actual/360

    • 3 = actual/365

    • 4 = 30/360 (BMA)

    • 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/actual (ISDA)

    • 13 = BUS/252

    For more information, see basis.

  • EndMonthRule is a NINST-by-1 matrix for the end-of-month rule. This rule applies only when Maturity is an end-of-month date for a month having 30 or fewer days. When the value is 0, the end-of-month rule is ignored, meaning that a bond's coupon payment date is always the same numerical day of the month. When the value is 1, the end-of-month rule is set rule on (default), meaning that a bond's coupon payment date is always the last actual day of the month.

  • IssueDate is a NINST-by-1 matrix for the bond issue date.

  • FirstCouponDate is a NINST-by-1 matrix for a date when a bond makes its first coupon payment; used when bond has an irregular first coupon period. 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 is a NINST-by-1 matrix for a last coupon date of a bond before the maturity date; used when bond has an irregular last coupon period. 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 is a NINST-by-1 matrix for date when a bond actually starts (the date from which a bond cash flow is considered). To make an instrument forward-starting, specify this date as a future date. If you do not specify StartDate, the effective start date is the Settle date.

  • Face value. Face is a NINST-by-1 vector or 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. Default is 100.

  • Options is a derivatives pricing options structure created with derivset.

    Note:   The Settle date for every bond with embedded option is set to the ValuationDate of the HW tree; the bond's argument for Settle date is ignored.

Description

[Price, PriceTree] = optembndbyhw(HWTree, CouponRate,
Settle, Maturity, OptSpec, Strike, ExerciseDates,
'Name1', Value1, 'Name2', Value2, ...)
prices bonds with embedded options by a HW interest-rate tree.

Price is a number of instruments (NINST)-by-1 matrix of expected prices at time 0.

PriceTree is a structure of trees containing vectors of instrument prices and observation times for each node. Within PriceTree:

  • PriceTree.PTree contains the clean prices.

  • PriceTree.tObs contains the observation times.

optembndbyhw computes prices of vanilla bonds with embedded options, stepped coupon bonds with embedded options, and bonds with sinking fund option provisions.

Examples

expand all

Price a Callable Bond Using an HW Interest-Rate Tree Model

Create a HWTree with the following data:

ZeroRates = [ 0.035;0.04;0.045];
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', ZeroRates, 'StartDates', ValuationDate, 'EndDates', ...
EndDates, 'Compounding', Compounding, 'ValuationDate', ValuationDate)
RateSpec = 

           FinObj: 'RateSpec'
      Compounding: 1
             Disc: [3x1 double]
            Rates: [3x1 double]
         EndTimes: [3x1 double]
       StartTimes: [3x1 double]
         EndDates: [3x1 double]
       StartDates: 733043
    ValuationDate: 733043
            Basis: 0
     EndMonthRule: 1

Create a VolSpec.

VolDates = ['jan-1-2008';'jan-1-2009';'jan-1-2010'];
VolCurve = 0.01;
AlphaDates = 'jan-1-2010';
AlphaCurve = 0.1;
HWVolSpec = hwvolspec(ValuationDate, VolDates, VolCurve, AlphaDates, AlphaCurve)
HWVolSpec = 

             FinObj: 'HWVolSpec'
      ValuationDate: 733043
           VolDates: [3x1 double]
           VolCurve: [3x1 double]
         AlphaCurve: 0.1000
         AlphaDates: 734139
    VolInterpMethod: 'linear'

Create a TimeSpec.

HWTimeSpec = hwtimespec(ValuationDate, EndDates, Compounding)
HWTimeSpec = 

           FinObj: 'HWTimeSpec'
    ValuationDate: 733043
         Maturity: [3x1 double]
      Compounding: 1
            Basis: 0
     EndMonthRule: 1

Build the HWTree.

HWTree = hwtree(HWVolSpec, RateSpec, HWTimeSpec)
HWTree = 

      FinObj: 'HWFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 1 2]
        dObs: [733043 733408 733774]
      CFlowT: {[3x1 double]  [2x1 double]  [3]}
       Probs: {[3x1 double]  [3x3 double]}
     Connect: {[2]  [2 3 4]}
     FwdTree: {[1.0350]  [1.0624 1.0451 1.0280]  [1x5 double]}

Compute the price of an American puttable bond that pays an annual coupon of 5.25%, matures on January 1, 2010, and is puttable from January 1, 2008 to January 1, 2010.

BondSettlement = 'jan-1-2007';
BondMaturity   = 'jan-1-2010';
CouponRate = 0.0525;
Period = 1;
OptSpec = 'put';
Strike = [100];
ExerciseDates = {'jan-1-2008' '01-Jan-2010'};
AmericanOpt = 1;

PricePutBondHW = optembndbyhw(HWTree, CouponRate, BondSettlement, BondMaturity,...
OptSpec, Strike, ExerciseDates,'Period', 1, 'AmericanOpt', 1)
PricePutBondHW =

  102.8801

Price Single Stepped Callable Bonds Using an HW Interest-Rate Tree Model

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
VolDates = ['1-Jan-2011'; '1-Jan-2012'; '1-Jan-2013'; '1-Jan-2014'];
VolCurve = 0.01;
AlphaDates = '01-01-2014';
AlphaCurve = 0.1;


HWVolSpec = hwvolspec(RS.ValuationDate, VolDates, VolCurve,...
AlphaDates, AlphaCurve);
HWTimeSpec = hwtimespec(RS.ValuationDate, VolDates, Compounding);
HWT = hwtree(HWVolSpec, RS, HWTimeSpec);

% 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.

PHW= optembndbyhw(HWT, CouponRate,  Settle, Maturity,OptSpec, Strike,...
ExerciseDates, 'Period', 1)
PHW =

  100.0521
   99.8322

Price a Sinking Fund Bond Using an HW Interest-Rate Tree Model

A corporation issues a two year bond with a sinking fund obligation requiring the company to sink 1/3 of face value after the first 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 = [0.1;0.1;0.1;0.1];
ValuationDate = 'Jan-1-2011';
StartDates = ValuationDate;
EndDates = {'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'};
Compounding = 1;

% Create RateSpec
RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates',...
StartDates, 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding);

% Build the HW tree
% The data to build the tree is as follows:
VolDates = ['1-Jan-2012'; '1-Jan-2013'; '1-Jan-2014';'1-Jan-2015'];
VolCurve = 0.01;
AlphaDates = '01-01-2015';
AlphaCurve = 0.1;

HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,...
AlphaDates, AlphaCurve);
HWTimeSpec = hwtimespec(RateSpec.ValuationDate, VolDates, Compounding);
HWT = hwtree(HWVolSpec, RateSpec, HWTimeSpec);


% Instrument
% The bond has a coupon rate of 9%, a period of one year and matures in
% 1-Jan-2013. Face decreases 1/3 after the first year.
CouponRate = 0.09;
Settle = 'Jan-1-2011';
Maturity =  'Jan-1-2013';
Period = 1;
Face = { ...
            {'Jan-1-2012'  100; ...
             'Jan-1-2013'   66.6666}; ...
        };

% Option provision
OptSpec = 'call';
Strike = 99;
ExerciseDates = 'Jan-1-2012';

% Price of non-sinking fund bond.
PNSF = bondbyhw(HWT, CouponRate, Settle, Maturity, Period)
PNSF =

   98.2645

Price of the bond with the option sinking provision.

PriceSF = optembndbyhw(HWT, CouponRate, Settle, Maturity,...
OptSpec, Strike, ExerciseDates, 'Period', Period, 'Face', Face)
PriceSF =

   98.1594

More About

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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 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 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.

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