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# bondbyhw

Price bond from Hull-White interest-rate tree

## Syntax

``````[Price,PriceTree] = bondbyhw(HWTree,CouponRate,Settle,Maturity)``````
``````[Price,PriceTree] = bondbyhw(___,Name,Value)``````

## Description

example

``````[Price,PriceTree] = bondbyhw(HWTree,CouponRate,Settle,Maturity)``` prices bond from a Hull-White interest-rate tree. `bondbyhw` computes prices of vanilla bonds, stepped coupon bonds and amortizing bonds.```

example

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

## Examples

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Price a 4% bond using a Hull-White interest-rate tree.

Load `deriv.mat`, which provides `HWTree`. The `HWTree` structure contains the time and interest-rate information needed to price the bond.

`load deriv.mat;`

Define the bond using the required arguments. Other arguments use defaults.

```CouponRate = 0.04; Settle = '01-Jan-2004'; Maturity = '31-Dec-2008';```

Use `bondbyhw` to compute the price of the bond.

`Price = bondbyhw(HWTree, CouponRate, Settle, Maturity)`
```Warning: Not all cash flows are aligned with the tree. Result will be approximated. ```
```Price = 98.0483 ```

Price single stepped coupon bonds using market data.

Define the interest-rate term structure.

```Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = 'Jan-1-2010'; StartDates = ValuationDate; EndDates = {'Jan-1-2011'; 'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'}; Compounding = 1;```

Create the `RateSpec`.

```RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)```
```RS = struct with fields: FinObj: 'RateSpec' Compounding: 1 Disc: [4x1 double] Rates: [4x1 double] EndTimes: [4x1 double] StartTimes: [4x1 double] EndDates: [4x1 double] StartDates: 734139 ValuationDate: 734139 Basis: 0 EndMonthRule: 1 ```

Create the stepped bond instrument.

```Settle = '01-Jan-2010'; Maturity = {'01-Jan-2011';'01-Jan-2012';'01-Jan-2013';'01-Jan-2014'}; CouponRate = {{'01-Jan-2012' .0425;'01-Jan-2014' .0750}}; Period = 1;```

Build the HW tree using the following market data:

```VolDates = ['1-Jan-2011'; '1-Jan-2012'; '1-Jan-2013'; '1-Jan-2014']; VolCurve = 0.01; AlphaDates = '01-01-2014'; AlphaCurve = 0.1; HWVolSpec = hwvolspec(RS.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); HWTimeSpec = hwtimespec(RS.ValuationDate, VolDates, Compounding); HWT = hwtree(HWVolSpec, RS, HWTimeSpec);```

Compute the price of the stepped coupon bonds.

`PHW= bondbyhw(HWT, CouponRate, Settle,Maturity , Period)`
```PHW = 100.7246 100.0945 101.5900 102.0820 ```

Price two bonds with amortization schedules using the `Face` input argument to define the schedules.

Define the interest rate term structure.

```Rates = 0.035; ValuationDate = '1-Nov-2011'; StartDates = ValuationDate; EndDates = '1-Nov-2017'; Compounding = 1;```

Create the `RateSpec`.

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

Create the bond instrument. The bonds have a coupon rate of 4% and 3.85%, a period of one year, and mature on 1-Nov-2017.

```CouponRate = [0.04; 0.0385]; Settle ='1-Nov-2011'; Maturity = '1-Nov-2017'; Period = 1;```

Define the amortizing schedule.

```Face = {{'1-Nov-2015' 100;'1-Nov-2016' 85;'1-Nov-2017' 70}; {'1-Nov-2015' 100;'1-Nov-2016' 90;'1-Nov-2017' 80}};```

Build the HW tree and assume the volatility to be 10%.

```VolDates = ['1-Nov-2012'; '1-Nov-2013';'1-Nov-2014';'1-Nov-2015';'1-Nov-2016';'1-Nov-2017']; VolCurve = 0.1; AlphaDates = '01-01-2017'; AlphaCurve = 0.1; HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); HWTimeSpec = hwtimespec(RateSpec.ValuationDate, VolDates, Compounding); HWT = hwtree(HWVolSpec, RateSpec, HWTimeSpec);```

Compute the price of the amortizing bonds.

```Price = bondbyhw(HWT, CouponRate, Settle, Maturity, 'Period',Period,... 'Face', Face)```
```Price = 102.4791 101.7786 ```

## Input Arguments

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Interest-rate tree structure, created by `hwtree`

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 of serial date numbers or date character vectors.

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

Data Types: `char` | `double`

Maturity date, specified as a `NINST`-by-`1` vector of serial date numbers or date character vectors representing the maturity date for each bond.

Data Types: `char` | `double`

### Name-Value Pair 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 single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `[Price,PriceTree] = bondbyhw(HWTree,CouponRate,Settle,Maturity,'Period',4,'Face',10000)`

collapse all

Coupons per year, specified as 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 an `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

For more information, see basis.

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 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 an `NINST`-by-`1` vector using a serial nonnegative date number or date character vector.

Data Types: `double` | `char`

Irregular first coupon date, specified as an `NINST`-by-`1` vector using a serial nonnegative date number or date character vector.

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`

Irregular last coupon date, specified as an `NINST`-by-`1` vector using a serial nonnegative date number or date character vector.

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: `double` | `char`

Forward starting date of payments (the date from which a bond cash flow is considered), specified as a `NINST`-by-`1` vector using serial date numbers or date character vectors.

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

Data Types: `char` | `double`

Face or par value, specified as an `NINST`-by-`1` vector of nonnegative face values or an `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`

Derivatives pricing options, specified as structure that is created with `derivset`.

Data Types: `struct`

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

Data Types: `logical`

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

• `actual` — Non-business days are effectively ignored. Cash flows that fall on non-business 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 MATLAB date numbers using a `NHolidays`-by-`1` vector.

Data Types: `double`

## 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.PTree` contains the clean prices.

• `PriceTree.AITree` contains the accrued interest.

• `PriceTree.tObs` contains the observation times.

• `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.Probs` contains the probability arrays. Each element of the cell array contains the up, middle, and down transition probabilities for each node of the level.

## More About

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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.

#### Introduced before R2006a

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