swapbyhw
Price swap instrument from Hull-White interest-rate tree
Syntax
Description
[
prices a swap instrument from a Hull-White interest-rate tree. Price
,PriceTree
,CFTree
,SwapRate
]
= swapbyhw(HWTree
,LegRate
,Settle
,Maturity
)swapbyhw
computes prices of vanilla swaps, amortizing swaps and forward swaps.
Note
Alternatively, you can use the Swap
object to price swap
instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.
Examples
Price an Interest-Rate Swap
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, 2005
Swap maturity date: Jan. 01, 2008
Based on the information above, set the required arguments and build the LegRate
, LegType
, and LegReset
matrices:
Settle = datetime(2005,1,1); Maturity = datetime(2008,1,1); 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 HWTree
included in the MAT-file deriv.mat
. The HWTree
structure contains the time and forward-rate information needed to price the instrument.
load deriv.mat;
Use swapbyhw
to compute the price of the swap.
[Price, PriceTree, SwapRate] = swapbyhw(HWTree, LegRate, ... Settle, Maturity, LegReset, Basis, Principal, LegType)
Warning: Swaps are valued at Tree ValuationDate rather than Settle
Price = 5.9109
PriceTree = struct with fields:
FinObj: 'HWPriceTree'
PTree: {[5.9109] [-1.2692 3.0317 7.5253] [-5.1049 -2.1588 0.8799 4.0142 7.2471] [-3.3462 -1.7392 -0.1055 1.5553 3.2438] [0 0 0 0 0]}
tObs: [0 1 2 3 4]
Connect: {[2] [2 3 4] [2 2 3 4 4]}
Probs: {[3×1 double] [3×3 double] [3×5 double]}
SwapRate = NaN
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, SwapRate] = swapbyhw(HWTree, LegRate, ... Settle, Maturity, LegReset, Basis, Principal, LegType)
Warning: Swaps are valued at Tree ValuationDate rather than Settle
Price = 1.4211e-14
PriceTree = struct with fields:
FinObj: 'HWPriceTree'
PTree: {[1.4211e-14] [-5.5941 -1.4265 2.9289] [-7.9659 -5.0883 -2.1199 0.9423 4.1011] [-4.8231 -3.2407 -1.6319 0.0036 1.6662] [0 0 0 0 0]}
tObs: [0 1 2 3 4]
Connect: {[2] [2 3 4] [2 2 3 4 4]}
Probs: {[3×1 double] [3×3 double] [3×5 double]}
SwapRate = 0.0438
Price an Amortizing Swap
Price an amortizing swap using the Principal
input argument to define the amortization schedule.
Create the RateSpec
.
Rates = 0.035; ValuationDate = datetime(2011,1,1); StartDates = ValuationDate; EndDates = datetime(2017,1,1); 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 = datetime(2011,1,1); Maturity = datetime(2017,1,1); Period = 1; LegRate = [0.04 10];
Define the swap amortizing schedule.
Principal ={{datetime(2013,1,1) 100;datetime(2014,1,1) 80;datetime(2015,1,1) 60;datetime(2016,1,1) 40;datetime(2017,1,1) 20}};
Build the HW tree using the following data:
VolDates = [datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1) ; datetime(2017,1,1)]; VolCurve = 0.1; AlphaDates = datetime(2017,1,1); 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 swap.
Price = swapbyhw(HWT, LegRate, Settle, Maturity, 'Principal', Principal)
Price = 1.4574
Price a Forward Swap
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 = datetime(2012,1,1); StartDates = ValuationDate; EndDates = datetime(2018,1,1); 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 HW tree.
VolDates = [datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1) ; datetime(2017,1,1) ; datetime(2018,1,1)]; VolCurve = 0.1; AlphaDates = datetime(2018,1,1); 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 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 = datetime(2012,1,1);
Maturity = datetime(2017,1,1);
StartDate = datetime(2013,1,1);
LegRate = [0.0425 10];
Price = swapbyhw(HWT, 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] = swapbyhw(HWT, LegRate, Settle, Maturity, 'StartDate', StartDate)
Price = 4.2633e-14
SwapRate = 0.0384
Input Arguments
HWTree
— Interest-rate structure
structure
Interest-rate tree structure, created by hwtree
Data Types: struct
LegRate
— Leg rate
matrix
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 date
datetime array | string array | date character vector
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, swapbyhw
also
accepts serial date numbers as inputs, but they are not recommended.
The Settle
date for every swap is set to the
ValuationDate
of the HW tree. The swap argument
Settle
is ignored.
Maturity
— Maturity date
datetime array | string array | date character vector
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 swap.
To support existing code, swapbyhw
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,CFTree,SwapRate] = swapbyhw(HWTree,LegRate,Settle,Maturity,LegReset,Basis,Principal,LegType)
LegReset
— Reset frequency per year for each swap
[1 1]
(default) | vector
Reset 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 array
Notional 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 structure
structure
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 vectors
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
— Holidays used in computing business days
if not specified, the default is to use
holidays.m
(default) | MATLAB® dates
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
StartDate
— Date swap actually starts
Settle
date (default) | datetime array | string array | date character vector
Date swap actually starts, 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, swapbyhw
also
accepts serial date numbers as inputs, but they are not recommended.
Use this argument to price forward swaps, that is, swaps that start in a future date
Output Arguments
Price
— Expected swap prices at time 0
vector
Expected swap prices at time 0, returned as a NINST
-by-1
vector.
PriceTree
— Tree structure of instrument prices
structure
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.PTree
contains the clean prices.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 areNumNodes
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.
CFTree
— Swap cash flows
structure
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 leg
matrix
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
.
More About
Amortizing Swap
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.
Forward Swap
Agreement to enter into an interest-rate swap arrangement on a fixed date in future.
Version History
Introduced before R2006aR2022b: Serial date numbers not recommended
Although swapbyhw
supports serial date numbers,
datetime
values are recommended instead. The
datetime
data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.
To convert serial date numbers or text to datetime
values, use the datetime
function. For example:
t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)
y = 2021
There are no plans to remove support for serial date number inputs.
See Also
bondbyhw
| capbyhw
| cfbyhw
| floorbyhw
| fixedbyhw
| hwtree
| Swap
Topics
- Pricing Using Interest-Rate Tree Models
- Calibrating Hull-White Model Using Market Data
- Price a Swaption Using SABR Model and Analytic Pricer
- Swap
- Understanding Interest-Rate Tree Models
- Pricing Options Structure
- Supported Interest-Rate Instrument Functions
- Mapping Financial Instruments Toolbox Functions for Interest-Rate Instrument Objects
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