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optembndbybk

Price bonds with embedded options by Black-Karasinski interest-rate tree

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Syntax

[Price,PriceTree] = optembndbybk(BKTree,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates)
[Price,PriceTree] = optembndbybk(___,Name,Value)

Description

[Price,PriceTree] = optembndbybk(BKTree,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates) calculates price for bonds with embedded options from a Black-Karasinski interest-rate tree and returns exercise probabilities in PriceTree.

optembndbybk 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.

Note

Alternatively, you can use the OptionEmbeddedFixedBond object to price embedded fixed-rate bond option instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

example

[Price,PriceTree] = optembndbybk(___,Name,Value) adds optional name-value pair arguments.

example

Examples

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Open Live Script

Create a BKTree with the following data: 

ZeroRates = [ 0.035;0.04;0.045];
Compounding = 1;
StartDates = [datetime(2007,1,1) ; datetime(2008,1,1) ; datetime(2009,1,1)];
EndDates   = [datetime(2008,1,1) ; datetime(2009,1,1) ; datetime(2010,1,1)];
ValuationDate = datetime(2007,1,1);

Create a RateSpec. 

RateSpec = intenvset('Rates', ZeroRates, 'StartDates', ValuationDate, 'EndDates', ...
EndDates, 'Compounding', Compounding, 'ValuationDate', ValuationDate)
RateSpec = struct with fields:
           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 = [datetime(2008,1,1) ; datetime(2009,1,1) ; datetime(2010,1,1)];
VolCurve = 0.01;
AlphaDates = datetime(2010,1,1);
AlphaCurve = 0.1;
BKVolSpec = bkvolspec(ValuationDate, VolDates, VolCurve, AlphaDates, AlphaCurve)
BKVolSpec = struct with fields:
             FinObj: 'BKVolSpec'
      ValuationDate: 733043
           VolDates: [3x1 double]
           VolCurve: [3x1 double]
         AlphaCurve: 0.1000
         AlphaDates: 734139
    VolInterpMethod: 'linear'

Create a TimeSpec. 

BKTimeSpec = bktimespec(ValuationDate, EndDates, Compounding);

Build the BKTree. 

BKTree = bktree(BKVolSpec, RateSpec, BKTimeSpec)
BKTree = struct with fields:
      FinObj: 'BKFwdTree'
     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]  [1.0458 1.0450 1.0442]  [1.0571 1.0561 1.0551 1.0541 1.0531]}

To compute the price of an American puttable bond that pays an annual coupon of 5.25% , matures on January 1, 2010, and is callable on January 1, 2008 and January 1, 2010. 

BondSettlement = datetime(2007,1,1);
BondMaturity   = datetime(2010,1,1); 
CouponRate = 0.0525;
Period = 1;
OptSpec = 'put'; 
Strike = [100];  
ExerciseDates = [datetime(2008,1,1) datetime(2010,1,1)]; 
AmericanOpt = 1;

PricePutBondBK = optembndbybk(BKTree, CouponRate, BondSettlement, BondMaturity,...
OptSpec, Strike, ExerciseDates,'Period', 1, 'AmericanOpt', 1)
PricePutBondBK = 
102.3820
Open Live Script

Create a BKTree with the following data: 

ZeroRates = [ 0.025;0.03;0.045];
Compounding = 1;
StartDates = [datetime(2007,1,1); datetime(2008,1,1) ; datetime(2009,1,1)];
EndDates   = [datetime(2008,1,1); datetime(2009,1,1) ; datetime(2010,1,1)];
ValuationDate = datetime(2007,1,1);

Create a RateSpec. 

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

Build the BK tree with the following data. 

VolDates = [datetime(2008,1,1); datetime(2009,1,1) ; datetime(2010,1,1)];
VolCurve = 0.01;
AlphaDates = datetime(2010,1,1);
AlphaCurve = 0.1;
BKVolSpec = bkvolspec(ValuationDate, VolDates, VolCurve, AlphaDates, AlphaCurve);

BKTimeSpec = bktimespec(ValuationDate, EndDates, Compounding);
BKTree = bktree(BKVolSpec, RateSpec, BKTimeSpec);

Define the callable bond instruments.

Settle = datetime(2007,1,1);
Maturity = [datetime(2008,1,1) ; datetime(2010,1,1)];
CouponRate = {{datetime(2008,1,1) .0315;datetime(2010,1,1) .0350}};  
OptSpec='call';
Strike= [86;90];
ExerciseDates= [datetime(2008,1,1) ; datetime(2010,1,1)];

Price the instruments. 

[Price, PriceTree]= optembndbybk(BKTree, CouponRate,  Settle, Maturity, OptSpec, Strike,...
ExerciseDates, 'Period', 1,'AmericanOp', 1)
Price = 2×1

   86.0000
   88.3060


PriceTree = struct with fields:
                FinObj: 'BKPriceTree'
                  tObs: [0 1 2 3]
                 PTree: {[2x1 double]  [2x3 double]  [2x5 double]  [2x5 double]}
              ProbTree: {[1]  [0.1667 0.6667 0.1667]  [0.0203 0.2206 0.5183 0.2206 0.0203]  [0.0203 0.2206 0.5183 0.2206 0.0203]}
                ExTree: {[2x1 double]  [2x3 double]  [2x5 double]  [2x5 double]}
            ExProbTree: {[2x1 double]  [2x3 double]  [2x5 double]  [2x5 double]}
    ExProbsByTreeLevel: [2x4 double]
               Connect: {[2]  [2 3 4]}

Examine the output PriceTree.ExTree. PriceTree.ExTree contains the exercise indicator arrays. Each element of the cell array is an array containing 1's where an option is exercised and 0's where it is not.

PriceTree.ExTree{4} 
ans = 2×5

     0     0     0     0     0
     1     1     1     1     1

There is no exercise for instrument 1 and instrument 2 is exercised at all nodes.

PriceTree.ExTree{3} 
ans = 2×5

     0     0     0     0     0
     0     0     0     0     0

There is no exercise for instrument 1 or instrument 2.

PriceTree.ExTree{2} 
ans = 2×3

     1     1     1
     0     0     0

There is exercise for instrument 1 at all nodes and instrument 2 is not exercised.

Next view the probability of reaching each node from the root node using PriceTree.ProbTree.

PriceTree.ProbTree{2}
ans = 1×3

    0.1667    0.6667    0.1667


PriceTree.ProbTree{3}
ans = 1×5

    0.0203    0.2206    0.5183    0.2206    0.0203


PriceTree.ProbTree{4}
ans = 1×5

    0.0203    0.2206    0.5183    0.2206    0.0203

Then view the exercise probabilities using PriceTree.ExProbTree. PriceTree.ExProbTree contains the exercise probabilities. Each element in the cell array is an array containing 0's where there is no exercise, or the probability of reaching that node where exercise happens. 

PriceTree.ExProbTree{4}
ans = 2×5

         0         0         0         0         0
    0.0203    0.2206    0.5183    0.2206    0.0203


PriceTree.ExProbTree{3}
ans = 2×5

     0     0     0     0     0
     0     0     0     0     0


PriceTree.ExProbTree{2}
ans = 2×3

    0.1667    0.6667    0.1667
         0         0         0

View the exercise probabilities at each tree level using PriceTree.ExProbsByTreeLevel. PriceTree.ExProbsByTreeLevel is an array with each row holding the exercise probability for a given option at each tree observation time. 

PriceTree.ExProbsByTreeLevel
ans = 2×4

    1.0000    1.0000         0         0
         0         0         0    1.0000
Open Live Script

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

% Instrument
Settle = datetime(2010,1,1);
Maturity = [datetime(2013,1,1) ; datetime(2014,1,1)];
CouponRate = {{datetime(2012,1,1) .0425;datetime(2014,1,1) .0750}};  
OptSpec='call';
Strike=100;
ExerciseDates= datetime(2012,1,1);  %Callable in two years

% Build the tree with the following 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;


BKVolSpec = bkvolspec(RS.ValuationDate, VolDates, VolCurve,... 
AlphaDates, AlphaCurve);
BKTimeSpec = bktimespec(RS.ValuationDate, VolDates, Compounding);
BKT = bktree(BKVolSpec, RS, BKTimeSpec);

% 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.
PBK= optembndbybk(BKT, CouponRate,  Settle, Maturity ,OptSpec, Strike,...
ExerciseDates, 'Period', 1)
PBK = 2×1

  100.0945
  100.0945
Open Live Script

This example shows how to price an amortizing callable bond and a vanilla callable bond using a BK 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 BK tree.

VolDates = [datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1)];
VolCurve = 0.01;
AlphaDates = datetime(2016,1,1);
AlphaCurve = 0.1;

BKVolSpec = bkvolspec(RS.ValuationDate, VolDates, VolCurve,... 
AlphaDates, AlphaCurve);
BKTimeSpec = bktimespec(RS.ValuationDate, VolDates, Compounding);
BKT = bktree(BKVolSpec, RS, BKTimeSpec);

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)];

Price a callable amortizing bond using the BDT tree.

BondType = 'amortizing';
Pcallbonds = optembndbybk(BKT, CouponRate,  Settle, Maturity ,OptSpec, Strike,...
ExerciseDates, 'Period', 1,'Face',Face,'BondType', BondType)
Pcallbonds = 
98.7475

Input Arguments

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Interest-rate tree structure, specified by using bktree.

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 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 is set to the ValuationDate of the BK tree. The bond argument Settle is ignored.

To support existing code, optembndbybk also accepts serial date numbers as inputs, but they are not recommended.

Maturity date, specified as an NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, optembndbybk also accepts serial date numbers as inputs, but they are not recommended.

Definition of option, specified as a NINST-by-1 cell array of character vectors.

Data Types: char

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 than NSTRIKES exercise opportunities, the end of the row is padded with NaNs.

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

Data Types: double

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 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.

To support existing code, optembndbybk 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 = optembndbybk(BKTree,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',1,'AmericanOp',1)

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and NINST-by-1 positive integer flags with values:

  • 0 — European/Bermuda

  • 1 — American

Data Types: double

Coupons per year, specified as the comma-separated pair consisting of 'Period' and a NINST-by-1 vector.

Data Types: double

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

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

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, optembndbybk 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, optembndbybk 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, optembndbybk 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, optembndbybk 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 or a NINST-by-1 cell array where each element is a NumDates-by-2 cell array where the first column is dates and the second column is associated face value. The date indicates the last day that the face value is valid.

Note

Instruments without a Face schedule are treated as either vanilla bonds or stepped coupon bonds with embedded options.

Data Types: double

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 scalar Face value and a single coupon or stepped coupons.

  • 'callablesinking' is a bond with a schedule of Face values and a sinking fund call provision with a single or stepped coupons.

  • 'amortizing' is an amortizing callable or puttable bond with a schedule of Face values with single or stepped coupons.

Data Types: char | string

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

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Expected price of the embedded option at time 0, returned as a NINST-by-1 matrix.

Structure containing trees of vectors of instrument prices, a vector of observation times for each node, and exercise probabilities. Values are:

  • PriceTree.PTree contains the clean prices.

  • PriceTree.tObs contains the observation times.

  • PriceTree.ExTree contains the exercise indicator arrays. Each element of the cell array is an array containing 1's where an option is exercised and 0's where it isn't.

  • PriceTree.ProbTree contains the probability of reaching each node from root node.

  • PriceTree.ExProbTree contains the exercise probabilities. Each element in the cell array is an array containing 0's where there is no exercise, or the probability of reaching that node where exercise happens.

  • PriceTree.ExProbsByTreeLevel is an array with each row holding the exercise probability for a given option at each tree observation time.

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 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.

Version History

Introduced in R2008a

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See Also

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