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Price swap instrument from Heath-Jarrow-Morton interest-rate tree

This example shows how to price an interest-rate swap with a fixed receiving leg and a floating paying leg. Payments are made once a year, and the notional principal amount is $100. The values for the remaining arguments are:

Coupon rate for fixed leg: 0.06 (6%)

Spread for floating leg: 20 basis points

Swap settlement date: Jan. 01, 2000

Swap maturity date: Jan. 01, 2003

Based on the information above, set the required arguments
and build the `LegRate`

, `LegType`

,
and `LegReset`

matrices:

Settle = '01-Jan-2000'; Maturity = '01-Jan-2003'; Basis = 0; Principal = 100; LegRate = [0.06 20]; % [CouponRate Spread] LegType = [1 0]; % [Fixed Float] LegReset = [1 1]; % Payments once per year

Price the swap using the `HJMTree`

included
in the MAT-file `deriv.mat`

. The `HJMTree`

structure
contains the time and forward-rate information needed to price the
instrument.

```
load deriv.mat;
```

Use `swapbyhjm`

to compute the price
of the swap.

[Price, PriceTree, CFTree] = swapbyhjm(HJMTree, LegRate,... Settle, Maturity, LegReset, Basis, Principal, LegType)

Price = 3.6923 PriceTree = FinObj: 'HJMPriceTree' tObs: [0 1 2 3 4] PBush: {1x5 cell} CFTree = FinObj: 'HJMCFTree' tObs: [0 1 2 3 4] CFBush: {[0] [1x1x2 double] [1x2x2 double] ... [1x8 double]}

Use `treeviewer`

to examine `CFTree`

graphically
and see the cash flows from the swap along both the up and the down
branches. A positive cash flow indicates an inflow (income - payments
> 0), while a negative cash flow indicates an outflow (income -
payments < 0).

treeviewer(CFTree)

In this example, you have sold a swap (receive fixed rate and
pay floating rate). At time `t = 3`

,
if interest rates go down, your cash flow is positive ($2.63), meaning
that you receive this amount. But if interest rates go up, your cash
flow is negative (-$1.58), meaning that you owe this amount.

`treeviewer`

price tree diagrams follow the
convention that increasing prices appear on the upper branch of a
tree and, so, decreasing prices appear on the lower branch. Conversely,
for interest-rate displays, *decreasing* interest
rates appear on the upper branch (prices are rising) and *increasing* interest
rates on the lower branch (prices are falling).

Using the previous data, calculate the swap rate, which is the coupon rate for the fixed leg, such that the swap price at time = 0 is zero.

LegRate = [NaN 20]; [Price, PriceTree, CFTree, SwapRate] = swapbyhjm(HJMTree,... LegRate, Settle, Maturity, LegReset, Basis, Principal, LegType)

Price = 0 PriceTree = FinObj: 'HJMPriceTree' tObs: [0 1 2 3 4] PBush:{[0] [1x1x2 double] [1x2x2 double] ... [1x8 double]} CFTree = FinObj: 'HJMCFTree' tObs: [0 1 2 3 4] CFBush:{[0] [1x1x2 double] [1x2x2 double] ... [1x8 double]} SwapRate = 0.0466

Price an amortizing swap using the `Principal`

input argument to define the amortization schedule.

Create the `RateSpec`

.

Rates = 0.035; ValuationDate = '1-Jan-2011'; StartDates = ValuationDate; EndDates = '1-Jan-2017'; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: 1
Disc: 0.8135
Rates: 0.0350
EndTimes: 6
StartTimes: 0
EndDates: 736696
StartDates: 734504
ValuationDate: 734504
Basis: 0
EndMonthRule: 1

Create the swap instrument using the following data:

Settle ='1-Jan-2011'; Maturity = '1-Jan-2017'; Period = 1; LegRate = [0.04 10];

Define the swap amortizing schedule.

Principal ={{'1-Jan-2013' 100;'1-Jan-2014' 80;'1-Jan-2015' 60;'1-Jan-2016' 40; '1-Jan-2017' 20}};

Build the HJM tree using the following data:

MatDates = {'1-Jan-2012'; '1-Jan-2013';'1-Jan-2014';'1-Jan-2015';'1-Jan-2016';'1-Jan-2017'}; HJMTimeSpec = hjmtimespec(RateSpec.ValuationDate, MatDates); Volatility = [.10; .08; .06; .04]; CurveTerm = [ 1; 2; 3; 4]; HJMVolSpec = hjmvolspec('Proportional', Volatility, CurveTerm, 1e6); HJMT = hjmtree(HJMVolSpec,RateSpec,HJMTimeSpec);

Compute the price of the amortizing swap.

`Price = swapbyhjm(HJMT, LegRate, Settle, Maturity, 'Principal', Principal)`

Price = 1.4574

Price a forward swap using the `StartDate`

input argument to define the future starting date of the swap.

Create the `RateSpec`

.

Rates = 0.0374; ValuationDate = '1-Jan-2012'; StartDates = ValuationDate; EndDates = '1-Jan-2018'; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: 1
Disc: 0.8023
Rates: 0.0374
EndTimes: 6
StartTimes: 0
EndDates: 737061
StartDates: 734869
ValuationDate: 734869
Basis: 0
EndMonthRule: 1

Build an HJM tree.

MatDates = {'1-Jan-2013'; '1-Jan-2014';'1-Jan-2015';'1-Jan-2016';'1-Jan-2017';'1-Jan-2018'}; HJMTimeSpec = hjmtimespec(RateSpec.ValuationDate, MatDates); Volatility = [.10; .08; .06; .04]; CurveTerm = [ 1; 2; 3; 4]; HJMVolSpec = hjmvolspec('Proportional', Volatility, CurveTerm, 1e6); HJMT = hjmtree(HJMVolSpec,RateSpec,HJMTimeSpec);

Compute the price of a forward swap that starts in a year (Jan 1, 2013) and matures in four years with a forward swap rate of 4.25%.

Settle ='1-Jan-2012'; Maturity = '1-Jan-2017'; StartDate = '1-Jan-2013'; LegRate = [0.0425 10]; Price = swapbyhjm(HJMT, LegRate, Settle, Maturity, 'StartDate', StartDate)

Price = 1.4434

Using the previous data, compute the forward swap rate, the coupon rate for the fixed leg, such that the forward swap price at time = 0 is zero.

```
LegRate = [NaN 10];
[Price, ~,~, SwapRate] = swapbyhjm(HJMT, LegRate, Settle, Maturity, 'StartDate', StartDate)
```

Price = 0

SwapRate = 0.0384

`HJMTree`

— Interest-rate structurestructure

Interest-rate tree structure, created by `hjmtree`

**Data Types: **`struct`

`LegRate`

— Leg ratematrix

Leg rate, specified as a `NINST`

-by-`2`

matrix,
with each row defined as one of the following:

`[CouponRate Spread]`

(fixed-float)`[Spread CouponRate]`

(float-fixed)`[CouponRate CouponRate]`

(fixed-fixed)`[Spread Spread]`

(float-float)

`CouponRate`

is the decimal annual rate.
`Spread`

is the number of basis points over the reference rate. The
first column represents the receiving leg, while the second column represents the
paying leg.

**Data Types: **`double`

`Settle`

— Settlement dateserial date number | character vector

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 swap is set to the
`ValuationDate`

of the HJM tree. The swap argument
`Settle`

is ignored.

**Data Types: **`char`

| `double`

`Maturity`

— Maturity dateserial date number | character vector

Maturity date, specified as a `NINST`

-by-`1`

vector
of serial date numbers or date character vectors representing the
maturity date for each swap.

**Data Types: **`char`

| `double`

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`

.

`[Price,PriceTree,CFTree,SwapRate] = swapbyhjm(HJMTree,LegRate,Settle,Maturity,LegReset,Basis,Principal,LegType)`

`LegReset`

— Reset frequency per year for each swap`[1 1]`

(default) | vectorReset frequency per year for each swap, specified as the comma-separated pair consisting of
`'LegReset'`

and a `NINST`

-by-`2`

vector.

**Data Types: **`double`

`Basis`

— Day-count basis representing the basis for each leg`0`

(actual/actual) (default) | integer from `0`

to `13`

Day-count basis representing the basis for each leg, specified as the comma-separated pair
consisting of `'Basis'`

and a
`NINST`

-by-`1`

array (or
`NINST`

-by-`2`

if `Basis`

is
different for each leg).

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`

`Principal`

— Notional principal amounts or principal value schedules`100`

(default) | vector or cell arrayNotional principal amounts or principal value schedules, specified as the comma-separated pair
consisting of `'Principal'`

and a vector or cell array.

`Principal`

accepts a `NINST`

-by-`1`

vector
or `NINST`

-by-`1`

cell array (or `NINST`

-by-`2`

if `Principal`

is
different for each leg) of the notional principal amounts or principal
value schedules. For schedules, each element of the cell array is
a `NumDates`

-by-`2`

array where
the first column is dates and the second column is its associated
notional principal value. The date indicates the last day that the
principal value is valid.

**Data Types: **`cell`

| `double`

`LegType`

— Leg type`[1 0]`

for each instrument (default) | matrix with values `[1 1]`

(fixed-fixed), ```
[1
0]
```

(fixed-float), `[0 1]`

(float-fixed),
or `[0 0]`

(float-float)Leg type, specified as the comma-separated pair consisting of `'LegType'`

and
a `NINST`

-by-`2`

matrix with values ```
[1
1]
```

(fixed-fixed), `[1 0]`

(fixed-float), ```
[0
1]
```

(float-fixed), or `[0 0]`

(float-float). Each row
represents an instrument. Each column indicates if the corresponding leg is fixed
(`1`

) or floating (`0`

). This matrix defines the
interpretation of the values entered in `LegRate`

.
`LegType`

allows `[1 1]`

(fixed-fixed),
`[1 0]`

(fixed-float), `[0 1]`

(float-fixed), or
`[0 0]`

(float-float) swaps

**Data Types: **`double`

`Options`

— Derivatives pricing options structurestructure

Derivatives pricing options structure, specified as the comma-separated pair consisting of
`'Options'`

and a structure obtained from using `derivset`

.

**Data Types: **`struct`

`EndMonthRule`

— End-of-month rule flag for generating dates when `Maturity`

is end-of-month date for month having 30 or fewer days`1`

(in effect) (default) | nonnegative integer `[0,1]`

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`

(or
`NINST`

-by-`2`

if `EndMonthRule`

is different for each leg).

`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`

`AdjustCashFlowsBasis`

— Flag to adjust cash flows based on actual period day count`false`

(default) | value of `0`

(false) or `1`

(true)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`

(or
`NINST`

-by-`2`

if
`AdjustCashFlowsBasis`

is different for each leg) of logicals with
values of `0`

(false) or `1`

(true).

**Data Types: **`logical`

`BusinessDayConvention`

— Business day conventions`actual`

(default) | character vector | cell array of character vectorsBusiness 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`

— Holidays used in computing business daysif not specified, the default is to use

`holidays.m`

(default) | MATLABHolidays used in computing business days, specified as the comma-separated pair consisting of
`'Holidays'`

and MATLAB date numbers using a
`NHolidays`

-by-`1`

vector.

**Data Types: **`double`

`StartDate`

— Date swap actually starts`Settle`

date (default) | serial date number | character vectorDate swap actually starts, specified as the comma-separated pair consisting of
`'StartDate'`

and a
`NINST`

-by-`1`

vector of dates using a serial date
number or a character vector.

Use this argument to price forward swaps, that is, swaps that start in a future date

**Data Types: **`char`

| `double`

`Price`

— Expected swap prices at time 0vector

Expected swap prices at time 0, returned as a `NINST`

-by-`1`

vector.

`PriceTree`

— Tree structure of instrument pricesstructure

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

:

`PriceTree.tObs`

contains the observation times.`PriceTree.PBush`

contains the clean prices.

`CFTree`

— Swap cash flowsstructure

Swap cash flows, returned as a tree structure with a vector
of the swap cash flows at each node. This structure contains only `NaN`

s
because with binomial recombining trees, cash flows cannot be computed
accurately at each node of a tree.

`SwapRate`

— Rates applicable to fixed legmatrix

Rates applicable to the fixed leg, returned as a `NINST`

-by-`1`

vector
of rates applicable to the fixed leg such that the swaps’ values
are zero at time 0. This rate is used in calculating the swaps’
prices when the rate specified for the fixed leg in `LegRate`

is `NaN`

.
The `SwapRate`

output is padded with `NaN`

for
those instruments in which `CouponRate`

is not set
to `NaN`

.

In an amortizing swap, the notional principal decreases periodically because it is tied to an underlying financial instrument with a declining (amortizing) principal balance, such as a mortgage.

Agreement to enter into an interest-rate swap arrangement on a fixed date in future.

`capbyhjm`

| `cfbyhjm`

| `floorbyhjm`

| `hjmtree`

| `treeviewer`

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