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Price range floating note using Black-Karasinski tree


[Price,PriceTree] = rangefloatbybk(BKTree,Spread,Settle,Maturity,RateSched)
[Price,PriceTree] = rangefloatbybk(___Name,Value)



[Price,PriceTree] = rangefloatbybk(BKTree,Spread,Settle,Maturity,RateSched) prices range floating note using a Black-Karasinski tree.

Payments on range floating notes are determined by the effective interest-rate between reset dates. If the reset period for a range 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 is more than one possible path for connecting the two payment dates.


[Price,PriceTree] = rangefloatbybk(___Name,Value) adds optional name-value pair arguments.


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This example shows how to compute the price of a range note using a Black-Karasinski tree with the following interest-rate term structure data.

Rates = [0.035; 0.042147; 0.047345; 0.052707];
ValuationDate = 'Jan-1-2011';
StartDates = ValuationDate;
EndDates = {'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'};
Compounding = 1;

% define RateSpec
RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,...
'EndDates', EndDates, 'Rates', Rates, 'Compounding', Compounding);
% range note instrument matures in Jan-1-2014 and has the following RateSchedule:
Spread = 100;
Settle = 'Jan-1-2011';
Maturity = 'Jan-1-2014';
RateSched(1).Dates = {'Jan-1-2012'; 'Jan-1-2013'  ; 'Jan-1-2014'};
RateSched(1).Rates = [0.045 0.055 ; 0.0525  0.0675; 0.06 0.08];

% data to build the tree is as follows:
VolDates = ['1-Jan-2012'; '1-Jan-2013'; '1-Jan-2014';'1-Jan-2015'];
VolCurve = 0.01;
AlphaDates = '01-01-2015';
AlphaCurve = 0.1;

BKVS = bkvolspec(RS.ValuationDate, VolDates, VolCurve,... 
AlphaDates, AlphaCurve);
BKTS = bktimespec(RS.ValuationDate, VolDates, Compounding);
BKT = bktree(BKVS, RS, BKTS);

% price the instrument 
Price = rangefloatbybk(BKT, Spread, Settle, Maturity, RateSched)
Price = 102.7574

Input Arguments

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Interest-rate tree structure, specified by using bKtree.

Data Types: struct

Number of basis points over the reference rate, specified as a NINST-by-1 vector.

Data Types: double

Settlement date for the floating range note, specified as a NINST-by-1 vector of serial date numbers or date character vectors. The Settle date for every range floating instrument is set to the ValuationDate of the BK tree. The floating range note argument Settle is ignored.

Data Types: double | char | cell

Maturity date for the floating-rate note, specified as a NINST-by-1 vector of serial date numbers or date character vectors.

Data Types: double | char | cell

Range of rates within which cash flows are nonzero, specified as a NINST-by-1 vector of structures. Each element of the structure array contains two fields:

  • RateSched.DatesNDates-by-1 cell array of dates corresponding to the range schedule.

  • RateSched.RatesNDates-by-2 array with the first column containing the lower bound of the range and the second column containing the upper bound of the range. Cash flow for date RateSched.Dates(n) is nonzero for rates in the range RateSched.Rates(n,1) < Rate < RateSched.Rate (n,2).

Data Types: struct

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [Price,PriceTree] = rangefloatbybk(BKTree,Spread,Settle,Maturity,RateSched,'Reset',4,'Basis',5,'Principal',10000)

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Frequency of payments per year, specified as a NINST-by-1 vector.

Data Types: double

Day-count basis representing the basis used when annualizing the input forward rate tree, specified as a NINST-by-1 vector of integers.

  • 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

Notional principal amount, specified as a NINST-by-1 vector.

Data Types: double

Derivatives pricing options structure, specified using derivset.

Data Types: struct

End-of-month rule flag, specified as nonnegative integer with a value of 0 or 1 using a NINST-by-1 vector.

  • 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

Output Arguments

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Expected prices of the range floating notes at time 0, returned as a NINST-by-1 vector.

Tree structure of instrument prices, returned as a structure containing trees of vectors of instrument prices and accrued interest, and a vector of observation times for each node. Values are:

  • PriceTree.PTree contains the clean prices.

  • PriceTree.AITree contains the accrued interest.

  • PriceTree.tObs contains the observation times.

More About

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Range Note Instrument

A range note is a structured (market-linked) security whose coupon rate is equal to the reference rate as long as the reference rate is within a certain range.

If the reference rate is outside of the range, the coupon rate is 0 for that period. This type of instrument entitles the holder to cash flows that depend on the level of some reference interest rate and are floored to be positive. The note holder gets direct exposure to the reference rate. In return for the drawback that no interest is paid for the time the range is left, they offer higher coupon rates than comparable standard products, like vanilla floating notes.


Jarrow, Robert. "Modelling Fixed Income Securities and Interest Rate Options." Stanford Economics and Finance. 2nd Edition. 2002.

Introduced in R2012a

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