optfloatbycir

Price options on floating-rate notes for Cox-Ingersoll-Ross interest-rate tree

Syntax

[Price,PriceTree] = optfloatbycir(CIRTree,OptSpec,Strike,ExerciseDates,AmericanOpt,Spread,Settle,Maturity)

[Price,PriceTree] = optfloatbycir(___,Name,Value)

Description

[Price,PriceTree] = optfloatbycir(CIRTree,OptSpec,Strike,ExerciseDates,AmericanOpt,Spread,Settle,Maturity) prices options on floating-rate notes from a Cox-Ingersoll-Ross (CIR) interest-rate tree. 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.

[Price,PriceTree] = optfloatbycir(___,Name,Value) adds optional name-value pair arguments.

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 a 1-by-1 (European) or 1-by-NSTRIKES (Bermuda) vector of exercise dates. For a European option, there is only one ExerciseDate on the option expiry date.

• For an American option, the ExerciseDates is a 1-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 if ExerciseDates is 1-by-1, the option exercises between the Settle date and the single listed ExerciseDate.

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/Bermuda

• 1 — 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 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.

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 R2018a

R2022b: 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.