# optembndbycir

Price bonds with embedded options by Cox-Ingersoll-Ross interest-rate tree

## Description

example

[Price,PriceTree] = optembndbycir(CIRTree,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates) calculates price for bonds with embedded options from a Cox-Ingersoll-Ross (CIR) interest-rate tree and returns exercise probabilities in PriceTree.

optembndbycir 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 using a CIR++ model with the Nawalka-Beliaeva (NB) approach. For more information, see More About.

example

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

## Examples

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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(2017,1,1);
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 = '01-Jan-2018';
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 0.2500 0.5000 0.7500]
dObs: [736696 736787 736878 736969]
FwdTree: {1x4 cell}
Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
Probs: {[3x1 double]  [3x3 double]  [3x5 double]}

Define the bond with embedded option instrument.

BondSettlement = datetime(2017,1,1);
BondMaturity   = datetime(2020,1,1);
CouponRate = 0.035;
Period = 1;
OptSpec = 'put';
Strike = 100;
ExerciseDates = [datetime(2018,1,1) datetime(2019,1,1)];

Price the bond.

[Price,PriceTree] = optembndbycir(CIRT,CouponRate,BondSettlement,BondMaturity,OptSpec,...
Strike,ExerciseDates,'AmericanOpt',1,'Period',1)
Price = 103.3099
PriceTree = struct with fields:
FinObj: 'CIRPriceTree'
tObs: [0 0.2500 0.5000 0.7500 1]
PTree: {1x5 cell}
ProbTree: {1x5 cell}
ExTree: {[0]  [0 0 0]  [0 0 0 0 0]  [0 0 0 0 0 0 0]  [1 1 1 0 0 0 0]}
ExProbTree: {[0]  [0 0 0]  [0 0 0 0 0]  [0 0 0 0 0 0 0]  [0.0222 ... ]}
ExProbsByTreeLevel: [0 0 0 0 0.3089]
Connect: {[3x1 double]  [3x3 double]  [3x5 double]}

Create a CIRTree with the following data:

Rates = [0.025; 0.027; 0.028; 0.03];
Dates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)];
ValuationDate = datetime(2017,1,1);
EndDates = Dates(2:end)';
Compounding = 1;

Create a RateSpec.

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

Build the CIR tree with the following data.

NumPeriods = length(EndDates);
Alpha = 0.03;
Theta = 0.02;
Sigma = 0.1;
Maturity = datetime(2018,1,1);
CIRTimeSpec = cirtimespec(ValuationDate, Maturity, NumPeriods);
CIRVolSpec = cirvolspec(Sigma, Alpha, Theta);
CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec);

Define the callable bond instruments.

Settle = datetime(2017,1,1);
Maturity = [datetime(2018,1,1) ; datetime(2020,1,1)];
CouponRate = {{datetime(2019,1,1) .0325;datetime(2020,1,1) .0375}};
OptSpec='call';
Strike= [100;110];
ExerciseDates= [datetime(2018,1,1) ; datetime(2020,1,1)];

Price the instruments.

[Price, PriceTree]= optembndbycir(CIRT, CouponRate,  Settle, Maturity, OptSpec, Strike,...
ExerciseDates, 'Period', 1,'AmericanOpt',1)
Price = 2×1

98.1718
102.6458

PriceTree = struct with fields:
FinObj: 'CIRPriceTree'
tObs: [0 0.2500 0.5000 0.7500 1]
PTree: {1x5 cell}
ProbTree: {1x5 cell}
ExTree: {1x5 cell}
ExProbTree: {1x5 cell}
ExProbsByTreeLevel: [2x5 double]
Connect: {[3x1 double]  [3x3 double]  [3x5 double]}

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 = 2×7

0     0     0     0     0     0     0
0     0     0     0     0     0     0

There is no exercise for instrument 1 or instrument 2.

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

0     0     0
0     0     0

There is exercise for instrument 1 or instrument 2.

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

PriceTree.ProbTree{2}
ans = 1×3

0.2794    0.3750    0.3456

PriceTree.ProbTree{3}
ans = 1×5

0.0786    0.2095    0.3318    0.2592    0.1209

PriceTree.ProbTree{4}
ans = 1×7

0.0222    0.0885    0.1982    0.2678    0.2442    0.1360    0.0432

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 = 2×7

0     0     0     0     0     0     0
0     0     0     0     0     0     0

PriceTree.ExProbTree{3}
ans = 2×5

0     0     0     0     0
0     0     0     0     0

PriceTree.ExProbTree{2}
ans = 2×3

0     0     0
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 = 2×5

0     0     0     0     0
0     0     0     0     0

## Input Arguments

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Interest-rate tree structure, specified by using cirtree.

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 CIR tree. The bond argument Settle is ignored.

To support existing code, optembndbycir 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, optembndbycir 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 or string arrays with a value of 'call' or 'put'.

Data Types: char | cell | string

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, optembndbycir 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,PriceTree] = optembndbycir(BDTTree,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',1,'AmericanOpt',1)

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and an 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

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, optembndbycir 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, optembndbycir 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, optembndbycir 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, optembndbycir 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

## 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.tObs contains the observation times.

• PriceTree.PTree contains the clean prices.

• PriceTree.Connect contains the connectivity vectors. Each element in the cell array describes how nodes in that level connect to the next. For a given tree level, there are NumNodes elements in the vector, and they contain the index of the node at the next level that the middle branch connects to. Subtracting 1 from that value indicates where the up-branch connects to, and adding 1 indicated where the down branch connects to.

• 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.

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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 (calls 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 R2018a

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