# capbybk

Price cap instrument from Black-Karasinski interest-rate tree

## Description

example

[Price,PriceTree] = capbybk(BKTree,Strike,Settle,Maturity) computes the price of a cap instrument from a Black-Karasinski interest-rate tree. capbybk computes prices of vanilla caps and amortizing caps.

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] = capbybk(___,CapReset,Basis,Principal,Options) adds optional arguments.

## Examples

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Load the file deriv.mat, which provides BKTree. The BKTree structure contains the time and interest-rate information needed to price the cap instrument.

Set the required values. Other arguments will use defaults.

Strike = 0.03;
Settle = datetime(2004,1,1);
Maturity = datetime(2007,1,1);

Use capbybk to compute the price of the cap instrument.

Price = capbybk(BKTree, Strike, Settle, Maturity)
Price = 2.0965

Load deriv.mat to specify the BKTree and then define the cap instrument.

Settle = datetime(2004,1,1);
Maturity = datetime(2008,1,1);
Strike = 0.05;
CapReset = 1;
Principal ={{datetime(2005,1,1) 100;datetime(2006,1,1) 60;datetime(2007,1,1) 30;datetime(2008,1,1) 30};...
100};

Price the amortizing and vanilla caps.

Basis = 1;
Price = capbybk(BKTree, Strike, Settle, Maturity, CapReset, Basis, Principal)
Price = 2×1

0.2226
0.7422

## Input Arguments

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Interest-rate tree structure, specified by using bktree.

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 BK tree. The cap argument Settle is ignored.

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

(Optional) Reset frequency payment per year, specified as a NINST-by-1 vector.

Data Types: double

(Optional) Day-count basis representing the basis used when annualizing the input forward rate, 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

Data Types: double

(Optional) Notional principal amount, specified as a NINST-by-1 of notional principal amounts, or a NINST-by-1 cell array, where 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

(Optional) Derivatives pricing options structure, specified using derivset.

Data Types: struct

## 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:

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