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Price options on floating-rate notes for Hull-White interest-rate tree


[Price,PriceTree] = optfloatbyhw(HWTree,OptSpec,Strike,ExerciseDates,AmericanOpt,Spread,Settle,Maturity) prices options on floating-rate notes from a Hull-White interest rate tree. optfloatbyhw computes prices of options on vanilla floating-rate notes.


Alternatively, you can use the FloatBondOption object to price floating-rate bond option instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

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


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Define the interest-rate term structure.

Rates = [0.03;0.034;0.038;0.04];
ValuationDate = datetime(2012,1,1);
StartDates = ValuationDate;
EndDates = [datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1)];
Compounding = 1;

Create the RateSpec.

RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,...
'EndDates', EndDates, 'Rates', Rates, 'Compounding', Compounding)
RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 1
             Disc: [4x1 double]
            Rates: [4x1 double]
         EndTimes: [4x1 double]
       StartTimes: [4x1 double]
         EndDates: [4x1 double]
       StartDates: 734869
    ValuationDate: 734869
            Basis: 0
     EndMonthRule: 1

Build the HW tree using the following:

VolDates = [datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1)];
VolCurve = 0.01;
AlphaDates = datetime(2016,1,1);
AlphaCurve = 0.1;

HWVolSpec = hwvolspec(RateSpec.ValuationDate, VolDates, VolCurve,... 
            AlphaDates, AlphaCurve);
HWTimeSpec = hwtimespec(RateSpec.ValuationDate, VolDates, Compounding);
HWT = hwtree(HWVolSpec, RateSpec, HWTimeSpec)
HWT = struct with fields:
      FinObj: 'HWFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 1 2 3]
        dObs: [734869 735235 735600 735965]
      CFlowT: {[4x1 double]  [3x1 double]  [2x1 double]  [4]}
       Probs: {[3x1 double]  [3x3 double]  [3x5 double]}
     Connect: {[2]  [2 3 4]  [2 3 4 5 6]}
     FwdTree: {[1.0300]  [1.0562 1.0381 1.0202]  [1.0831 1.0645 1.0462 1.0283 1.0106]  [1.1022 1.0833 1.0647 1.0464 1.0284 1.0108 0.9934]}

The floater instrument has a spread of 10, a period of one year, and matures on Jan-1-2016.

Spread = 10;
Settle = datetime(2012,1,1);
Maturity =  datetime(2016,1,1);
Period = 1;

Define the option for the floating-rate note.

OptSpec = {'call'};
Strike = 95;
ExerciseDates = datetime(2016,1,1);
AmericanOpt = [0;1];

Compute the price of the call options.

Price= optfloatbyhw(HWT,  OptSpec, Strike, ExerciseDates,AmericanOpt,...
Spread, Settle, Maturity)
Price = 2×1


Input Arguments

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Interest-rate tree specified as a structure by using hwtree.

Data Types: struct

Definition of option as 'call' or 'put' specified as a NINST-by-1 cell array of character vectors for 'call' or 'put'.

Data Types: cell | char

Option strike price values specified nonnegative integers using as NINST-by-NSTRIKES vector of strike price values.

Data Types: double

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, optfloatbyhw also accepts serial date numbers as inputs, but they are not recommended.

  • If 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.

  • If an American option, then 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.

Option type specified as NINST-by-1 positive integer scalar flags with values:

  • 0 — European/Bermuda

  • 1 — American

Data Types: double

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

Settlement dates of floating-rate note specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors.


The Settle date for every floating-rate note is set to the ValuationDate of the HW tree. The floating-rate note argument Settle is ignored.

To support existing code, optfloatbyhw also accepts serial date numbers as inputs, but they are not recommended.

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, optfloatbyhw 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]=optfloatbyhw(HWTree,OptSpec,Strike,ExerciseDates,AmericanOpt,Spread,Settle,Maturity,'FloatReset',4,'Basis',7)

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.


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

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 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

Structure containing derivatives pricing options, specified as the comma-separated pair consisting of 'Options' and a structure obtained from using derivset.

Data Types: struct

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

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Expected prices of the floating-rate note option at time 0 is returned as a scalar or an NINST-by-1 vector.

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

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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.

Version History

Introduced in R2013a

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