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capbybdt

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

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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.

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

example

Examples

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Open Live Script

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 = datetime(2000,1,1);
Maturity = datetime(2004,1,1);

Use capbybdt to compute the price of the cap instrument. 

Price = capbybdt(BDTTree, Strike, Settle, Maturity)
Price = 
28.4001
Open Live Script

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

Compounding = 1; 
ValuationDate = datetime(2000,1,1); 
StartDate = ValuationDate; 
EndDates = [datetime(2001,1,1) ; datetime(2002,1,1) ; datetime(2003,1,1) ; 
datetime(2004,1,1) ; datetime(2005,1,1)]; 
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: {[1.1000]  [1.0979 1.1432]  [1.0976 1.1377 1.1942]  [1.0872 1.1183 1.1606 1.2179]  [1.0865 1.1134 1.1486 1.1948 1.2552]}

Set the cap arguments. Remaining arguments will use defaults. 

CapStrike = 0.10; 
Settlement = ValuationDate; 
Maturity = datetime(2002,1,1); 
CapReset = 1;

Use capbybdt to find the price of the cap instrument. 

Price= capbybdt(BDTTree, CapStrike, Settlement, Maturity,...
CapReset)
Price = 
1.7169
Open Live Script

Define the RateSpec. 

Rates = [0.03583; 0.042147; 0.047345; 0.052707; 0.054302];
ValuationDate = datetime(2011,11,15);
StartDates = ValuationDate;
EndDates = [datetime(2012,11,15) ; datetime(2013,11,15) ; datetime(2014,11,15) ; datetime(2015,11,15) ; datetime(2016,11,15)];
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 = datetime(2011,11,15);
Maturity = datetime(2015,11,15);
Strike = 0.04;
CapReset = 1;
Principal ={{datetime(2012,11,15) 100;datetime(2013,11,15) 70;datetime(2014,11,15) 40;datetime(2015,11,15) 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: {[1.0358]  [1.0437 1.0534]  [1.0469 1.0573 1.0700]  [1.0505 1.0617 1.0754 1.0921]  [1.0401 1.0490 1.0598 1.0731 1.0894]}

Price the amortizing cap. 

Basis = 0;
Price = capbybdt(BDTTree, Strike, Settle, Maturity, CapReset, Basis, Principal)
Price = 
1.4042

Input Arguments

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Interest-rate tree structure, specified by using bdttree.

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 BDT tree. The cap argument Settle is ignored.

To support existing code, capbybdt 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, capbybdt 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

For more information, see Basis.

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.

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.

Version History

Introduced before R2006a

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See Also

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