swaptionbycir
Price swaption from Cox-Ingersoll-Ross interest-rate tree
Syntax
Description
[
prices swaption with a Cox-Ingersoll-Ross (CIR) tree using a CIR++ model with the
Nawalka-Beliaeva (NB) approach.Price
,PriceTree
]
= swaptionbycir(CIRTree
,OptSpec
,Strike
,ExerciseDates
,Spread
,Settle
,Maturity
)
Note
Alternatively, you can use the Swaption
object to price a
swaption instrument. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.
[
adds optional name-value pair arguments.Price
,PriceTree
]
= swaptionbycir(___,Name,Value
)
Examples
Price a Swaption Using a CIR Interest-Rate Tree
Define a 3-year put swaption.
Rates =0.075 * ones (10,1); Compounding = 2; StartDates = [datetime(2017,1,1);datetime(2017,7,1);datetime(2018,1,1);datetime(2018,7,1);datetime(2019,1,1);datetime(2019,7,1);datetime(2020,1,1);datetime(2020,7,1);datetime(2021,1,1);datetime(2021,7,1)]; EndDates = [datetime(2017,7,1);datetime(2018,1,1);datetime(2018,7,1);datetime(2019,1,1);datetime(2019,7,1);datetime(2020,1,1);datetime(2020,7,1);datetime(2021,1,1);datetime(2021,7,1);datetime(2022,1,1)]; ValuationDate = datetime(2017,1,1);
Create a RateSpec
using the intenvset
function.
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; Maturity = datetime(2023,1,1); 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 0.6000 1.2000 1.8000 2.4000 3 3.6000 4.2000 4.8000 5.4000]
dObs: [736696 736915 737134 737353 737572 737791 738010 738229 738448 738667]
FwdTree: {1x10 cell}
Connect: {[3x1 double] [3x3 double] [3x5 double] [3x7 double] [3x9 double] [3x11 double] [3x13 double] [3x14 double] [3x15 double]}
Probs: {[3x1 double] [3x3 double] [3x5 double] [3x7 double] [3x9 double] [3x11 double] [3x13 double] [3x14 double] [3x15 double]}
Use the following arguments for a 1-year swap and a 3-year swaption.
ExerciseDates = datetime(2020,1,1);
SwapSettlement = ExerciseDates;
SwapMaturity = datetime(2022,1,1);
Spread = 0;
SwapReset = 2 ;
Principal = 100;
OptSpec = 'put';
Strike= 0.04;
Basis=1;
Price the swaption.
[Price,PriceTree] = swaptionbycir(CIRT,OptSpec,Strike,ExerciseDates,Spread,SwapSettlement,SwapMaturity,'SwapReset',SwapReset, ... 'Basis',Basis,'Principal',Principal)
Price = 3.1537
PriceTree = struct with fields:
FinObj: 'CIRPriceTree'
PTree: {1x11 cell}
tObs: [0 0.6000 1.2000 1.8000 2.4000 3 3.6000 4.2000 4.8000 5.4000 6]
Connect: {[3x1 double] [3x3 double] [3x5 double] [3x7 double] [3x9 double] [3x11 double] [3x13 double] [3x14 double] [3x15 double]}
Probs: {[3x1 double] [3x3 double] [3x5 double] [3x7 double] [3x9 double] [3x11 double] [3x13 double] [3x14 double] [3x15 double]}
Input Arguments
CIRTree
— Interest-rate tree structure
structure
Interest-rate tree structure, specified by using cirtree
.
Data Types: struct
OptSpec
— Definition of option
character vector with values 'call'
or
'put'
| cell array of character vector with values 'call'
or
'put'
| string array with values "call"
or
"put"
Definition of the option as 'call'
or 'put'
,
specified as a NINST
-by-1
cell array of
character vectors or string arrays. For more information, see More About.
Data Types: char
| cell
| string
Strike
— Strike swap rate values
decimal
Strike swap rate values, specified as a
NINST
-by-1
vector.
Data Types: double
ExerciseDates
— Exercise dates for swaption
datetime array | string array | date character vector
Exercise dates for the swaption, specified as a
NINST
-by-1
vector or a
NINST
-by-2
vector using a datetime array, string
array, or date character vectors, depending on the option type.
For a European option,
ExerciseDates
are aNINST
-by-1
vector of exercise dates. Each row is the schedule for one option. When using a European option, there is only oneExerciseDate
on the option expiry date.For an American option,
ExerciseDates
are aNINST
-by-2
vector of exercise date boundaries. For each instrument, the option can be exercised on any coupon date between or including the pair of dates on that row. If only one non-NaN
date is listed, or ifExerciseDates
isNINST
-by-1
, the option can be exercised between theValuationDate
of the tree and the single listedExerciseDate
.
To support existing code, swaptionbycir
also
accepts serial date numbers as inputs, but they are not recommended.
Spread
— Number of basis points over reference rate
numeric
Number of basis points over the reference rate, specified as a
NINST
-by-1
vector.
Data Types: double
Settle
— Settlement date
datetime array | string array | date character vector
Settlement date (representing the settle date for each swap), specified as a
NINST
-by-1
vector using a datetime array, string
array, or date character vectors. The Settle
date for every swaption
is set to the ValuationDate
of the CIR tree. The swap argument
Settle
is ignored. The underlying swap starts at the maturity of
the swaption.
To support existing code, swaptionbycir
also
accepts serial date numbers as inputs, but they are not recommended.
Maturity
— Maturity date for swap
datetime array | string array | date character vector
Maturity date for each swap, specified as a
NINST
-by-1
vector using a datetime array, string
array, or date character vectors.
To support existing code, swaptionbycir
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] = swaptionbycir(CIRTree,OptSpec,
ExerciseDates,Spread,Settle,Maturity,'SwapReset',4,'Basis',5,'Principal',10000)
AmericanOpt
— Option type
0
(European) (default) | integer with values 0
or 1
Option type, specified as the comma-separated pair consisting of
'AmericanOpt'
and NINST
-by-1
positive integer flags with values:
0
— European1
— American
Data Types: double
SwapReset
— Reset frequency per year for underlying swap
1
(default) | numeric
Reset frequency per year for the underlying swap, specified as the
comma-separated pair consisting of 'SwapReset'
and a
NINST
-by-1
vector or
NINST
-by-2
matrix representing the reset
frequency per year for each leg. If SwapReset
is
NINST
-by-2
, the first column represents the
receiving leg, and the second column represents the paying leg.
Data Types: double
Basis
— Day-count basis of instrument
0
(actual/actual) (default) | integer from 0
to 13
Day-count basis representing the basis used when annualizing the input
forward-rate tree for each instrument, specified as the comma-separated pair
consisting of 'Basis'
and a
NINST
-by-1
vector or
NINST
-by-2
matrix representing the basis for
each leg. If Basis
is
NINST
-by-2
, the first column represents the
receiving leg, while the second column represents the paying 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 amount
100
(default) | numeric
Notional principal amount, specified as the comma-separated pair consisting of
'Principal'
and a
NINST
-by-1
vector.
Data Types: double
Output Arguments
Price
— Expected prices of swaptions at time 0
vector
Expected prices of the swaptions 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.
More About
Call Swaption
A call swaption or payer swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option pays the fixed rate and receives the floating rate.
Put Swaption
A put swaption or receiver swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option receives the fixed rate and pays the floating rate.
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 R2018aR2022b: Serial date numbers not recommended
Although swaptionbycir
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
bondbycir
| capbycir
| cfbycir
| fixedbycir
| floatbycir
| floorbycir
| oasbycir
| optbndbycir
| optfloatbycir
| optembndbycir
| optemfloatbycir
| rangefloatbycir
| swapbycir
| instswaption
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