# bondbycir

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

## Syntax

``````[Price,PriceTree] = bondbycir(CIRTree,CouponRate,Settle,Maturity)``````
``````[Price,PriceTree] = bondbycir(___,Name,Value)``````

## Description

example

``````[Price,PriceTree] = bondbycir(CIRTree,CouponRate,Settle,Maturity)``` prices bond from a Cox-Ingersoll-Ross (CIR) interest-rate tree. `bondbycir` computes prices of vanilla bonds, stepped coupon bonds, and amortizing bonds using a CIR++ model with the Nawalka-Beliaeva (NB) approach. ```

example

``````[Price,PriceTree] = bondbycir(___,Name,Value)``` adds additional name-value pair arguments.```

## Examples

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Define the `CouponRate` for a bond.

`CouponRate = 0.035;`

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 = 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; Settle = datetime(2017,1,1); Maturity = datetime(2021,1,1); 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 1 2 3] dObs: [736696 737061 737426 737791] FwdTree: {[1.0350] [1.0790 1.0500 1.0298] [1.1275 1.0887 1.0594 1.0390 1.0270] [1.1905 1.1406 1.1014 1.0718 1.0512 1.0390 1.0350]} Connect: {[3x1 double] [3x3 double] [3x5 double]} Probs: {[3x1 double] [3x3 double] [3x5 double]} ```

Price the bond.

`[Price,PriceTree] = bondbycir(CIRT,CouponRate,Settle,Maturity) `
```Price = 94.0880 ```
```PriceTree = struct with fields: FinObj: 'CIRPriceTree' tObs: [0 1 2 3 4] dObs: [736696 737061 737426 737791 738157] PTree: {[94.0880] [86.8351 93.6116 98.7823] [83.9957 89.8068 94.6078 98.1958 100.4146] [87.0721 90.8422 94.0486 96.6208 98.5012 99.6471 100.0320] [100 100 100 100 100 100 100]} AITree: {[0] [0 0 0] [0 0 0 0 0] [0 0 0 0 0 0 0] [0 0 0 0 0 0 0]} Connect: {[3x1 double] [3x3 double] [3x5 double]} ```

## Input Arguments

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Interest-rate tree structure, created by `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, specified either as a scalar or `NINST`-by-`1` vector using a datetime array, string array, or date character vectors.

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

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

Maturity date, specified as a `NINST`-by-`1` vector using a datetime array, string array, or date character vectors representing the maturity date for each bond.

To support existing code, `bondbycir` 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] = bondbycir(CIRTree,CouponRate,Settle,Maturity,'Period',4,'Face',10000)```

Coupons per year, specified as the comma-separated pair consisting of `'Period'` and an `NINST`-by-`1` vector. Values for `Period` are `1`, `2`, `3`, `4`, `6`, and `12`.

Data Types: `double`

Day-count basis of the instrument, specified as the comma-separated pair consisting of `'Basis'` and a `NINST`-by-`1` vector.

• 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 for generating dates when `Maturity` is an end-of-month date for a month having 30 or fewer days, specified as the comma-separated pair consisting of `'EndMonthRule'` and a nonnegative integer [`0`, `1`] using a `NINST`-by-`1` vector.

• `0` = Ignore rule, meaning that a payment date is always the same numerical day of the month.

• `1` = Set rule on, meaning that a payment date is always the last actual day of the month.

Data Types: `logical`

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, `bondbycir` 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, `bondbycir` 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, `bondbycir` 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, `bondbycir` 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 of nonnegative face values or a `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.

Data Types: `cell` | `double`

Flag to adjust cash flows based on actual period day count, specified as the comma-separated pair consisting of `'AdjustCashFlowsBasis'` and a `NINST`-by-`1` vector of logicals with values of `0` (false) or `1` (true).

Data Types: `logical`

Business day conventions, specified as the comma-separated pair consisting of `'BusinessDayConvention'` and a character vector or a `N`-by-`1` (or `NINST`-by-`2` if `BusinessDayConvention` is different for each leg) cell array of character vectors of business day conventions. The selection for business day convention determines how nonbusiness days are treated. Nonbusiness days are defined as weekends plus any other date that businesses are not open (e.g. statutory holidays). Values are:

• `actual` — Nonbusiness days are effectively ignored. Cash flows that fall on nonbusiness days are assumed to be distributed on the actual date.

• `follow` — Cash flows that fall on a non-business day are assumed to be distributed on the following business day.

• `modifiedfollow` — Cash flows that fall on a non-business day are assumed to be distributed on the following business day. However if the following business day is in a different month, the previous business day is adopted instead.

• `previous` — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day.

• `modifiedprevious` — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day. However if the previous business day is in a different month, the following business day is adopted instead.

Data Types: `char` | `cell`

Holidays used in computing business days, specified as the comma-separated pair consisting of `'Holidays'` and MATLAB dates using a `NHolidays`-by-`1` vector.

Data Types: `datetime`

## Output Arguments

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Expected bond prices at time 0, returned as a `NINST`-by-`1` vector.

Tree structure of instrument prices, returned as a MATLAB structure of trees containing vectors of instrument prices and accrued interest, and a vector of observation times for each node. Within `PriceTree`:

• `PriceTree.tObs` contains the observation times.

• `PriceTree.dObs` contains the observation dates.

• `PriceTree.PTree` contains the clean prices.

• `PriceTree.AITree` contains the accrued interest.

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### Vanilla Bond

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.

### Stepped Coupon Bond

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.

### Bond with an Amortization Schedule

An amortized bond is treated as an asset, with the discount amount being amortized to interest expense over the life of the bond.

## 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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