# optbndbycir

Price bond option from Cox-Ingersoll-Ross interest-rate tree

## Description

example

[Price,PriceTree] = optbndbycir(CIRTree,OptSpec,Strike,ExerciseDates,AmericanOpt,CouponRate,Settle,Maturity) calculates the price for a bond option from a Cox-Ingersoll-Ross (CIR) interest-rate tree using a CIR++ model with the Nawalka-Beliaeva (NB) approach.

example

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

## Examples

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Compute the price for a European call option on a 4% bond with a strike of 96. The exercise date for the option is Jan. 01, 2018. The settle date for the bond is Jan. 01, 2017, and the maturity date is Jan. 01, 2020.

Create a RateSpec using the intenvset function.

Rates = [0.035; 0.042147; 0.047345; 0.052707];
Dates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)];
ValuationDate = 'Jan-1-2017';
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;
Settle = datetime(2017,1,1);
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 0.5000 1 1.5000]
dObs: [736696 736878 737061 737243]
FwdTree: {1x4 cell}
Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
Probs: {[3x1 double]  [3x3 double]  [3x5 double]}

Price the 'Call' option.

[Price,PriceTree] = optbndbycir(CIRT,'Call',96,datetime(2018,1,1),...
0,0.04,datetime(2017,1,1),datetime(2020,1,1))
Price = 2.6827
PriceTree = struct with fields:
FinObj: 'CIRPriceTree'
tObs: [0 0.5000 1 1.5000 2]
PTree: {1x5 cell}
Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
ExTree: {[0]  [0 0 0]  [0 0 1 1 1]  [0 0 0 0 0 0 0]  [0 0 0 0 0 0 0]}

Price the 'Put' option.

[Price,PriceTree] = optbndbycir(CIRT,'Put',96,datetime(2018,1,1),...
0,0.04,datetime(2017,1,1),datetime(2020,1,1))
Price = 0.6835
PriceTree = struct with fields:
FinObj: 'CIRPriceTree'
tObs: [0 0.5000 1 1.5000 2]
PTree: {1x5 cell}
Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
ExTree: {[0]  [0 0 0]  [1 1 0 0 0]  [0 0 0 0 0 0 0]  [0 0 0 0 0 0 0]}

The PriceTree.ExTree output for the 'Call' and 'Put' option 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.

## Input Arguments

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Interest-rate tree structure, specified by using cirtree.

Data Types: struct

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

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, optbndbycir also accepts serial date numbers as inputs, but they are not recommended.

Option type, specified as NINST-by-1 positive integer flags with values:

• 0 — European/Bermuda

• 1 — American

Data Types: double

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, optbndbycir 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, optbndbycir 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] = optbndbycir(CIRTree,OptSpec, Strike,ExerciseDates,AmericanOpt,CouponRate,Settle,Maturity,'Period'6,'Basis',7,'Face',1000)

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, optbndbycir 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, optbndbycir 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, optbndbycir 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, optbndbycir 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.

Data Types: double

## Output Arguments

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

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

• PriceTree.tObs contains the observation times.

• PriceTree.PTree contains the clean prices.

• 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.

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### Bond Option

A bond option gives the holder the right to sell a bond back to the issuer (put) or to redeem a bond from its current owner (call) at a specific price and on a specific date.

Financial Instruments Toolbox™ supports three types of put and call options on bonds:

• American option: An option that you exercise any time until its expiration date.

• European option: An option that you exercise only on its expiration date.

• Bermuda option: A Bermuda option resembles a hybrid of American and European options. You can exercise it on predetermined dates only, usually monthly.

## 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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