optfloatbycir
Price options on floating-rate notes for Cox-Ingersoll-Ross interest-rate tree
Syntax
Description
[
prices options on floating-rate notes from a Cox-Ingersoll-Ross (CIR) interest-rate tree.
Price
,PriceTree
]
= optfloatbycir(CIRTree
,OptSpec
,Strike
,ExerciseDates
,AmericanOpt
,Spread
,Settle
,Maturity
)optfloatbycir
computes prices of options on vanilla floating-rate
notes using a CIR++ model with the Nawalka-Beliaeva (NB) approach.
Note
Alternatively, you can use the FloatBondOption
object to price options on floating-rate notes. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.
[
adds optional name-value pair arguments. Price
,PriceTree
]
= optfloatbycir(___,Name,Value
)
Examples
Compute the Price of American and European Call Options on a Floating-Rate Note Using a CIR Interest-Rate Tree
Create a RateSpec
using the intenvset
function.
Rates = [0.035; 0.042147; 0.047345; 0.052707]; Dates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)]; ValuationDate = datetime(2017,1,1); EndDates = Dates(2:end)'; Compounding = 1; 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; Settle = datetime(2017,1,1); Maturity = datetime(2021,1,1); CIRTimeSpec = cirtimespec(Settle, 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 1 2 3]
dObs: [736696 737061 737426 737791]
FwdTree: {[1.0350] [1.0790 1.0500 1.0298] [1.1275 1.0887 1.0594 1.0390 1.0270] [1.1905 1.1406 1.1014 1.0718 1.0512 1.0390 1.0350]}
Connect: {[3x1 double] [3x3 double] [3x5 double]}
Probs: {[3x1 double] [3x3 double] [3x5 double]}
The floater instrument has a spread of 10, a period of one year, and matures on Jan-1-2018.
Spread = 10; Settle = datetime(2017,1,1); Maturity = datetime(2019,1,1); Period = 1;
Define the option for the floating-rate note.
OptSpec = {'call'};
Strike = 95;
ExerciseDates = datetime(2018,1,1);
AmericanOpt = [0;1];
Compute the price of the call options.
[Price,PriceTree] = optfloatbycir(CIRT, OptSpec,Strike,ExerciseDates,AmericanOpt,...
Spread, Settle, Maturity)
Price = 2×1
4.9230
5.1887
PriceTree = struct with fields:
FinObj: 'CIRPriceTree'
PTree: {[2x1 double] [2x3 double] [2x5 double] [2x7 double] [2x7 double]}
AITree: {[2x1 double] [2x3 double] [2x5 double] [2x7 double] [2x7 double]}
tObs: [0 1 2 3 4]
Connect: {[3x1 double] [3x3 double] [3x5 double]}
Probs: {[3x1 double] [3x3 double] [3x5 double]}
Input Arguments
CIRTree
— Interest-rate tree structure
structure
Interest-rate tree specified as a structure by using cirtree
.
Data Types: struct
OptSpec
— Definition of option
character vector with values 'call'
or
'put'
| cell array of character vectors with values of 'call'
or
'put'
| string array with values 'call'
or
'put'
Definition of option, specified as a
NINST
-by-1
cell array of character vectors or
string arrays with values of 'call'
or
'put'
.
Data Types: cell
| char
| string
Strike
— Option strike price values
nonnegative integer | vector of nonnegative integers
Option strike price values, specified as nonnegative integers using an
NINST
-by-NSTRIKES
vector of strike price values.
Data Types: double
ExerciseDates
— Exercise date for option (European, Bermuda, or American)
datetime array | string array | date character vector
Exercise date for option (European, Bermuda, or American) specified as a
NINST
-by-NSTRIKES
or
NINST
-by-2
vector using a datetime array, string
array, or date character vectors.
To support existing code, optfloatbycir
also
accepts serial date numbers as inputs, but they are not recommended.
For a European or Bermuda option, the
ExerciseDates
is a1
-by-1
(European) or1
-by-NSTRIKES
(Bermuda) vector of exercise dates. For a European option, there is only oneExerciseDate
on the option expiry date.For an American option, the
ExerciseDates
is a1
-by-2
vector of exercise date boundaries. The option exercises on any date between or including the pair of dates on that row. If there is only one non-NaN
date, or ifExerciseDates
is1
-by-1
, the option exercises between theSettle
date and the single listedExerciseDate
.
AmericanOpt
— Option type
scalar | vector of positive integers[0,1]
Option type specified as NINST
-by-1
positive
integer scalar flags with values:
0
— European/Bermuda1
— American
Data Types: double
Spread
— Number of basis points over the reference rate
nonnegative integer | vector of nonnegative integers
Number of basis points over the reference rate specified as a vector of
nonnegative integers for the number of instruments
(NINST
)-by-1
).
Data Types: double
Settle
— Settlement dates of floating-rate note
ValuationDate
of CIR tree (default) | datetime array | string array | date character vector
Settlement dates of floating-rate note specified as a
NINST
-by-1
vector using a datetime array, string
array, or date character vectors.
Note
The Settle
date for every floating-rate note is set to the
ValuationDate
of the CIR tree. The floating-rate note argument
Settle
is ignored.
To support existing code, optfloatbycir
also
accepts serial date numbers as inputs, but they are not recommended.
Maturity
— Floating-rate note maturity date
datetime array | string array | date character vector
Floating-rate note maturity date specified as a
NINST
-by-1
vector using a datetime array, string
array, or date character vectors.
To support existing code, optfloatbycir
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] =
optfloatbybk(CIRTree,OptSpec,Strike,ExerciseDates,AmericanOpt,Spread,Settle,Maturity,'FloatReset',4,'Basis',7)
FloatReset
— Frequency of payments per year
1
(default) | positive integer from the set[1,2,3,4,6,12]
| vector of positive integers from the set
[1,2,3,4,6,12]
Frequency of payments per year, specified as the comma-separated pair consisting
of 'FloatReset'
and positive integers for the values
[1,2,3,4,6,12]
in a
NINST
-by-1
vector.
Note
Payments on floating-rate notes (FRNs) are determined by the effective interest-rate between reset dates. If the reset period for an FRN spans more than one tree level, calculating the payment becomes impossible due to the recombining nature of the tree. That is, the tree path connecting the two consecutive reset dates cannot be uniquely determined because there will be more than one possible path for connecting the two payment dates.
Data Types: double
Basis
— Day-count basis of the instrument
0
(actual/actual) (default) | positive integers of the set [1...13]
| vector of positive integers of the set [1...13]
Day-count basis of the instrument, specified as the comma-separated pair
consisting of 'Basis'
and a positive integer using a
NINST
-by-1
vector. The
Basis
value represents the basis used when annualizing the input
forward-rate tree.
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
— Principal values
100
(default) | vector of nonnegative values | cell array of nonnegative values
Principal values, specified as the comma-separated pair consisting of
'Principal'
and nonnegative values using a
NINST
-by-1
vector or
NINST
-by-1
cell array of notional principal
amounts.
When using a NINST
-by-1
cell array, each
element is a NumDates
-by-2
cell array where the
first column is dates, and the second column is associated principal amount. The date
indicates the last day that the principal value is valid.
Data Types: double
| cell
Options
— Structure containing derivatives pricing options
structure
Structure containing derivatives pricing options, specified as the
comma-separated pair consisting of 'Options'
and the output from
derivset
.
Data Types: struct
EndMonthRule
— End-of-month rule flag
1
(in effect) (default) | nonnegative integer [0,1]
End-of-month rule flag, specified as the comma-separated pair consisting of
'EndMonthRule'
and a nonnegative integer [0
,
1
] using a NINST
-by-1
vector. This rule applies only when Maturity
is an end-of-month
date for a month having 30 or fewer days.
0
= Ignore rule, meaning that a bond coupon payment date is always the same numerical day of the month.1
= Set rule on, meaning that a bond coupon payment date is always the last actual day of the month.
Data Types: double
Output Arguments
Price
— Expected prices of the floating-rate note option at time 0
scalar | vector
Expected prices of the floating-rate note option at time 0 is returned as a scalar
or an NINST
-by-1
vector.
PriceTree
— Structure of trees containing vectors of option prices at each node
tree structure
Structure of trees containing vectors of instrument prices and accrued interest and a vector of observation times for each node returned as:
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 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
Floating-Rate Note Options
A floating-rate note option is a put or call option on a floating-rate note.
Financial Instruments Toolbox™ supports three types of put and call options on floating-rate notes:
American option — An option that you exercise any time until its expiration date.
European option — An option that you exercise only on its expiration date.
Bermuda option — A Bermuda option resembles a hybrid of American and European options; you can only exercise it on predetermined dates, usually monthly.
For more information, see Floating-Rate Note Options.
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 optfloatbycir
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
| optembndbycir
| optemfloatbycir
| rangefloatbycir
| swapbycir
| swaptionbycir
| instoptfloat
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