capbycir

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

Description

example

[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.

example

Examples

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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 = {'Jan-1-2017'; 'Jan-1-2018'; 'Jan-1-2019'; 'Jan-1-2020'; 'Jan-1-2021'};
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 = '01-Jan-2017';
Maturity = '01-Jan-2021';
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]  [1x5 double]  [1x7 double]}
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: {1x5 cell}
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 of serial date numbers, date character vectors, string arrays, or datetime arrays. The Settle date for every cap is set to the ValuationDate of the CIR tree. The cap argument Settle is ignored.

Data Types: double | char | cell | string | datetime

Maturity date for the cap, specified as a NINST-by-1 vector of serial date numbers, date character vectors, string arrays, or datetime arrays.

Data Types: double | char | cell | string | datetime

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 quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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

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.

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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:

$\mathrm{max}\left(CurrentRate-CapRate,0\right)$

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.

Introduced in R2018a