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# optbndbycir

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

## Syntax

``````[Price,PriceTree] = optbndbycir(CIRTree,OptSpec,Strike,ExerciseDates,AmericanOpt,CouponRate,Settle,Maturity)``````
``````[Price,PriceTree] = optbndbycir(___,Name,Value)``````

## 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 = {'Jan-1-2017'; 'Jan-1-2018'; 'Jan-1-2019'; 'Jan-1-2020'; 'Jan-1-2021'}; 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 = '01-Jan-2017'; Maturity = '01-Jan-2019'; 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,'01-Jan-2018',... 0,0.04,'01-Jan-2017','01-Jan-2020')```
```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,'01-Jan-2018',... 0,0.04,'01-Jan-2017','01-Jan-2020')```
```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 `NaN`s.

• 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` using serial date numbers, data character vectors, string arrays, or datetime arrays 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` | `string` | `datetime`

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 of serial date numbers, date character vectors, string arrays, or datetime arrays.

Note

The `Settle` date for every bond is set to the `ValuationDate` of the CIR tree. The bond argument `Settle` is ignored.

Data Types: `double` | `char` | `string`

Maturity date, specified as an `NINST`-by-`1` vector of serial date numbers, date character vectors, string arrays, or datetime arrays.

Data Types: `double` | `char` | `string` | `datetime`

### Name-Value Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

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

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 serial date numbers, date character vectors, string arrays, or datetime arrays.

Data Types: `double` | `char` | `string` | `datetime`

Irregular first coupon date, specified as the comma-separated pair consisting of `'FirstCouponDate'` and a `NINST`-by-`1` vector using serial date numbers date, date character vectors, string arrays, or datetime arrays.

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` | `string` | `datetime`

Irregular last coupon date, specified as the comma-separated pair consisting of `'LastCouponDate'` and a `NINST`-by-`1` vector using serial date numbers, date character vectors, string arrays, or datetime arrays.

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` | `string` | `datetime`

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 serial date numbers, date character vectors, string arrays, or datetime arrays.

If you do not specify `StartDate`, the effective start date is the `Settle` date.

Data Types: `char` | `double` | `string` | `datetime`

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.

## More About

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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.

For more information, see Bond Options.

## 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.

Introduced in R2018a

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