Documentation

# capbyhjm

Price cap instrument from Heath-Jarrow-Morton interest-rate tree

## Description

example

[Price,PriceTree] = capbyhjm(HJMTree,Strike,Settle,Maturity) computes the price of a cap instrument from a Heath-Jarrow-Morton interest-rate tree. capbyhjm computes prices of vanilla caps and amortizing caps.

example

[Price,PriceTree] = capbyhjm(___,CapReset,Basis,Principal,Options) adds optional arguments.

## Examples

collapse all

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

Set the required values. Other arguments will use defaults.

Strike = 0.03;
Settle = '01-Jan-2000';
Maturity = '01-Jan-2004';

Use capbyhjm to compute the price of the cap instrument.

Price = capbyhjm(HJMTree, Strike, Settle, Maturity)
Price = 6.2831

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

Settle = '01-Jan-2000';
Maturity = '01-Jan-2004';
Strike = 0.045;
CapReset = 1;
Principal ={{'01-Jan-2001' 100;'01-Jan-2002' 80;'01-Jan-2003' 70;'01-Jan-2004' 30}};

Price the amortizing cap.

Basis = 1;
Price = capbyhjm(HJMTree, Strike, Settle, Maturity, CapReset, Basis, Principal)
Price = 1.4588

## Input Arguments

collapse all

Interest-rate tree structure, specified by using hjmtree.

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 or date character vectors. The Settle date for every cap is set to the ValuationDate of the HJM tree. The cap argument Settle is ignored.

Data Types: double | char | cell

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

Data Types: double | char | cell

(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

collapse all

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.tObs contains the observation times.

• PriceTree.PBush contains the clean prices.

collapse all

### 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:

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