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.

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.

[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 = 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

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 = 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

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 = 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

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

datetime array | string array | date character vector

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 — Maturity date for cap

datetime array | string array | date character vector

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.

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.

Version History

Introduced before R2006a

R2022b: Serial date numbers not recommended

Although capbybdt supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.