capbybdt

Price cap instrument from Black-Derman-Toy interest-rate tree

Syntax

[Price,PriceTree] = capbybdt(BDTTree,Strike,Settle,Maturity)

[Price,PriceTree] = capbybdt(___,CapReset,Basis,Principal,Options)

Description

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

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

Examples

Price a 3% Cap Instrument Using a BDT Interest-Rate Tree

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

load deriv.mat;

Set the required values. Other arguments will use defaults.

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

Use capbybdt to compute the price of the cap instrument.

Price = capbybdt(BDTTree, Strike, Settle, Maturity)

Price = 28.4001

Price a 10% Cap Instrument Using a BDT Interest-Rate Tree

Set the required arguments for the three specifications required to create a BDT tree.

Compounding = 1; 
ValuationDate = '01-01-2000'; 
StartDate = ValuationDate; 
EndDates = ['01-01-2001'; '01-01-2002'; '01-01-2003'; 
'01-01-2004'; '01-01-2005']; 
Rates = [.1; .11; .12; .125; .13]; 
Volatility = [.2; .19; .18; .17; .16];

Create the specifications.

RateSpec = intenvset('Compounding', Compounding,... 
'ValuationDate', ValuationDate,... 
'StartDates', StartDate,... 
'EndDates', EndDates,... 
'Rates', Rates); 
BDTTimeSpec = bdttimespec(ValuationDate, EndDates, Compounding); 
BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Volatility);

Create the BDT tree from the specifications.

BDTTree = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec)

BDTTree = struct with fields:
      FinObj: 'BDTFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 1 2 3 4]
        dObs: [730486 730852 731217 731582 731947]
        TFwd: {[5x1 double]  [4x1 double]  [3x1 double]  [2x1 double]  [4]}
      CFlowT: {[5x1 double]  [4x1 double]  [3x1 double]  [2x1 double]  [5]}
     FwdTree: {1x5 cell}

Set the cap arguments. Remaining arguments will use defaults.

CapStrike = 0.10; 
Settlement = ValuationDate; 
Maturity = '01-01-2002'; 
CapReset = 1;

Use capbybdt to find the price of the cap instrument.

Price= capbybdt(BDTTree, CapStrike, Settlement, Maturity,...
CapReset)

Price = 1.7169

Compute the Price of an Amortizing Cap Using the BDT Model

Define the RateSpec.

Rates = [0.03583; 0.042147; 0.047345; 0.052707; 0.054302];
ValuationDate = '15-Nov-2011';
StartDates = ValuationDate;
EndDates = {'15-Nov-2012';'15-Nov-2013';'15-Nov-2014' ;'15-Nov-2015';'15-Nov-2016'};
Compounding = 1;
RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,...
'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)

RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 1
             Disc: [5x1 double]
            Rates: [5x1 double]
         EndTimes: [5x1 double]
       StartTimes: [5x1 double]
         EndDates: [5x1 double]
       StartDates: 734822
    ValuationDate: 734822
            Basis: 0
     EndMonthRule: 1

Define the cap instrument.

Settle ='15-Nov-2011';
Maturity = '15-Nov-2015';
Strike = 0.04;
CapReset = 1;
Principal ={{'15-Nov-2012' 100;'15-Nov-2013' 70;'15-Nov-2014' 40;'15-Nov-2015' 10}};

Build the BDT Tree.

BDTTimeSpec = bdttimespec(ValuationDate, EndDates);
Volatility = 0.10;  
BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Volatility*ones(1,length(EndDates))');
BDTTree = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec)

BDTTree = struct with fields:
      FinObj: 'BDTFwdTree'
     VolSpec: [1x1 struct]
    TimeSpec: [1x1 struct]
    RateSpec: [1x1 struct]
        tObs: [0 1 2 3 4]
        dObs: [734822 735188 735553 735918 736283]
        TFwd: {[5x1 double]  [4x1 double]  [3x1 double]  [2x1 double]  [4]}
      CFlowT: {[5x1 double]  [4x1 double]  [3x1 double]  [2x1 double]  [5]}
     FwdTree: {1x5 cell}

Price the amortizing cap.

Basis = 0;
Price = capbybdt(BDTTree, Strike, Settle, Maturity, CapReset, Basis, Principal)

Price = 1.4042

Input Arguments

BDTTree — Interest-rate tree structure

structure

Interest-rate tree structure, specified by using bdttree.

Data Types: struct

Strike — Rate at which cap is exercised

decimal

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

Data Types: double

Settle — Settlement date for cap

serial date number | date character vector | cell array of date character vectors

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 BDT tree. The cap argument Settle is ignored.

Data Types: double | char | cell

Maturity — Maturity date for cap

serial date number | date character vector | cell array of date character vectors

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

CapReset — Reset frequency payment per year

1 (default) | numeric

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

Data Types: double

Basis — Day-count basis of instrument

0 (actual/actual) (default) | integer from 0 to 13

(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

For more information, see Basis.

Data Types: double

Principal — Notional principal amount

100 (default) | numeric

(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

Options — Derivatives pricing options structure

structure

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

Data Types: struct

Output Arguments

Price — Expected price of cap at time 0

vector

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

PriceTree — Tree structure with values of cap at each node

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.

More About

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.