# bondbyhw

Price bond from Hull-White interest-rate tree

## 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

## 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.

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

CouponRate = 0.04;
Settle = datetime(2004,1,1);
Maturity = datetime(2006,1,1);

Use bondbyhw to compute the price of the bond.

Period = 1;
Price = bondbyhw(HWTree, CouponRate, Settle, Maturity, Period)
Price = 101.6002

Price single stepped coupon bonds using market data.

Define the interest-rate term structure.

Rates = [0.035; 0.042147; 0.047345; 0.052707];
ValuationDate = datetime(2010,1,1);
StartDates = ValuationDate;
EndDates = [datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1)];
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 = datetime(2010,1,1);
Maturity = [datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1)];
CouponRate = {{datetime(2012,1,1) .0425;datetime(2014,1,1) .0750}};
Period = 1;

Build the HW tree using the following market data:

VolDates = [datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1)];
VolCurve = 0.01;
AlphaCurve = 0.1;

HWVolSpec = hwvolspec(RS.ValuationDate, VolDates, VolCurve,...
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 = 4×1

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 = datetime(2011,11,1);
StartDates = ValuationDate;
EndDates = datetime(2017,11,1);
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 = datetime(2011,11,1);
Maturity = datetime(2017,11,1);
Period = 1;

Define the amortizing schedule.

Face = {{datetime(2015,11,1) 100;datetime(2016,11,1) 85;datetime(2017,11,1) 70};
{datetime(2015,11,1) 100;datetime(2016,11,1) 90;datetime(2017,11,1) 80}};

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

VolDates = [datetime(2012,11,1) ; datetime(2013,11,1) ; datetime(2014,11,1) ; datetime(2015,11,1) ; datetime(2016,11,1) ; datetime(2017,11,1)];
VolCurve = 0.1;
AlphaCurve = 0.1;

HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,...
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 = 2×1

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 using a datetime array, string array, or date character vectors.

To support existing code, bondbyhw 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 HW 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, bondbyhw 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] = bondbyhw(HWTree,CouponRate,Settle,Maturity,'Period',4,'Face',10000)

Coupons per year, specified as the comma-separated pair consisting of 'Period' and a 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, bondbyhw 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, bondbyhw 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, bondbyhw 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, bondbyhw 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 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 the comma-separated pair consisting of 'Options' and a structure that is created with derivset.

Data Types: struct

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 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 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.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.

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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.

## Version History

Introduced before R2006a

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