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capbycir

Price cap instrument from Cox-Ingersoll-Ross interest-rate tree

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Syntax

[Price,PriceTree] = capbycir(CIRTree,Strike,Settle,Maturity)
[Price,PriceTree] = capbycir(___,Name,Value)

Description

[Price,PriceTree] = capbycir(CIRTree,Strike,Settle,Maturity) computes the price of a cap instrument from a Cox-Ingersoll-Ross (CIR) interest-rate tree. capbycir computes prices of vanilla caps and amortizing caps using a CIR++ model with the Nawalka-Beliaeva (NB) approach.

Note

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

example

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

example

Examples

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Open Live Script

Define the Strike for a cap.

Strike = 0.03;

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 = 'Jan-1-2017'; 
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(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 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]}

Price the 3% cap.

[Price,PriceTree] = capbycir(CIRT,Strike,Settle,Maturity) 
Price = 
7.9081

PriceTree = struct with fields:
     FinObj: 'CIRPriceTree'
       tObs: [0 1 2 3 4]
      PTree: {[7.9081]  [14.6338 7.8719 2.8464]  [17.0242 11.2109 6.4070 2.8162 0.8890]  [13.4818 9.7005 6.4840 3.9033 2.0165 0.8667 0.4804]  [-100 -100 -100 -100 -100 -100 -100]}
    Connect: {[3x1 double]  [3x3 double]  [3x5 double]}
      Probs: {[3x1 double]  [3x3 double]  [3x5 double]}

Input Arguments

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Interest-rate tree structure, specified by using cirtree.

Data Types: struct

Rate at which cap is exercised, specified as a NINST-by-1 vector of decimal values.

Data Types: double

Settlement date for the cap, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors. The Settle date for every cap is set to the ValuationDate of the CIR tree. The cap argument Settle is ignored.

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

Maturity date for the cap, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, capbycir 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] = capbycir(CIRTree,CouponRate,Settle,Maturity,'Basis',3)

Reset frequency payment per year, specified as the comma-separated pair consisting of 'CapReset' and a NINST-by-1 vector.

Data Types: double

Day-count basis representing the basis used when annualizing the input forward rate, specified as the comma-separated pair consisting of 'Basis' and 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 the comma-separated pair consisting of 'Principal' and a NINST-by-1 of notional principal amounts or a NINST-by-1 cell array.

For the 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.

Use Principal to pass a schedule to compute the price for an amortizing cap.

Data Types: double | cell

Output Arguments

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Expected price of the cap at time 0, returned as a NINST-by-1 vector.

Tree structure with values of the cap at each node, returned as a MATLAB® structure of trees containing vectors of instrument prices and a vector of observation times for each node:

  • PriceTree.PTree contains cap 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 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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Cap

A cap is a contract that includes a guarantee that sets the maximum interest rate to be paid by the holder, based on an otherwise floating interest rate.

The payoff for a cap is:

max(CurrentRate−CapRate,0)

For more information, see Cap.

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

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See Also

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