swaptionbybk

Price swaption from Black-Karasinski interest-rate tree

Syntax

[Price,PriceTree] = swaptionbybk(BKTree,OptSpec,Strike,ExerciseDates,Spread,Settle,Maturity)

[Price,PriceTree] = swaptionbybk(___,Name,Value)

Description

[Price,PriceTree] = swaptionbybk(BKTree,OptSpec,Strike,ExerciseDates,Spread,Settle,Maturity) prices swaption using a Black-Karasinski tree.

[Price,PriceTree] = swaptionbybk(___,Name,Value) adds optional name-value pair arguments.

Examples

Price a 4-Year Call and Put Swaption Using a BK Interest-Rate Tree

This example shows how to price a 4-year call and put swaption using a BK interest-rate tree, assuming the interest rate is fixed at 7% annually.

Rates =0.07 * ones (10,1);
Compounding = 2; 
StartDates = ['jan-1-2007';'jul-1-2007';'jan-1-2008';'jul-1-2008';'jan-1-2009'; ...
'jul-1-2009'; 'jan-1-2010'; 'jul-1-2010';'jan-1-2011';'jul-1-2011'];  
EndDates =['jul-1-2007';'jan-1-2008';'jul-1-2008';'jan-1-2009';'jul-1-2009'; ...
'jan-1-2010'; 'jul-1-2010';'jan-1-2011';'jul-1-2011';'jan-1-2012'];
ValuationDate = 'jan-1-2007'; 

% define the RateSpec

RateSpec = intenvset('Rates', Rates, 'StartDates', StartDates, 'EndDates', EndDates,...
'Compounding', Compounding); 

% use BKVolSpec to compute the interest-rate volatility
Volatility = 0.10*ones(10,1);  
AlphaCurve = 0.05*ones(10,1);  
AlphaDates = EndDates;  
BKVolSpec = bkvolspec(ValuationDate, EndDates, Volatility, AlphaDates, AlphaCurve); 

% use BKTimeSpec to specify the structure of the time layout for the BK interest-rate tree
BKTimeSpec = bktimespec(ValuationDate, EndDates, Compounding);

% build the BK tree
BKTree = bktree(BKVolSpec, RateSpec, BKTimeSpec); 

% use the following arguments for a 1-year swap and 4-year swaption
ExerciseDates = 'jan-1-2011';
SwapSettlement = ExerciseDates;  
SwapMaturity   = 'jan-1-2012';  
Spread = 0;  
SwapReset = 2 ;   
Principal = 100;  
OptSpec = {'call' ;'put'};    
Strike= [ 0.07 ; 0.0725];    
Basis=1; 

% price the swaption
PriceSwaption = swaptionbybk(BKTree, OptSpec, Strike, ExerciseDates, ...
Spread, SwapSettlement, SwapMaturity, 'SwapReset', SwapReset, 'Basis', Basis, ...
'Principal', Principal)

PriceSwaption = 2×1

    0.3634
    0.4798

Price a 4-Year Call and Put Swaption with Receiving and Paying Legs Using a BK Interest-Rate Tree

This example shows how to price a 4-year call and put swaption with receiving and paying legs using a BK interest-rate tree, assuming the interest rate is fixed at 7% annually. Build a tree with the following data.

Rates =0.07 * ones (10,1);
Compounding = 2; 
StartDates = ['jan-1-2007';'jul-1-2007';'jan-1-2008';'jul-1-2008';'jan-1-2009'; ...
'jul-1-2009'; 'jan-1-2010'; 'jul-1-2010';'jan-1-2011';'jul-1-2011'];  
EndDates =['jul-1-2007';'jan-1-2008';'jul-1-2008';'jan-1-2009';'jul-1-2009'; ...
'jan-1-2010'; 'jul-1-2010';'jan-1-2011';'jul-1-2011';'jan-1-2012'];
ValuationDate = 'jan-1-2007';

Define the RateSpec.

RateSpec = intenvset('Rates', Rates, 'StartDates', StartDates, 'EndDates', EndDates,...
'Compounding', Compounding)

RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 2
             Disc: [10x1 double]
            Rates: [10x1 double]
         EndTimes: [10x1 double]
       StartTimes: [10x1 double]
         EndDates: [10x1 double]
       StartDates: [10x1 double]
    ValuationDate: 733043
            Basis: 0
     EndMonthRule: 1

Use BKVolSpec to compute interest-rate volatility.

Volatility = 0.10*ones(10,1);  
AlphaCurve = 0.05*ones(10,1);  
AlphaDates = EndDates;  
BKVolSpec = bkvolspec(ValuationDate, EndDates, Volatility, AlphaDates, AlphaCurve);

Use BKTimeSpec to specify the structure of the time layout for a BK tree.

BKTimeSpec = bktimespec(ValuationDate, EndDates, Compounding);

Build the BK tree.

BKTree = bktree(BKVolSpec, RateSpec, BKTimeSpec);

Define the arguments for a 1-year swap and 4-year swaption.

ExerciseDates = 'jan-1-2011';
SwapSettlement = ExerciseDates;  
SwapMaturity   = 'jan-1-2012';  
Spread = 0;  
SwapReset = [2 2]; % 1st column represents swaption receiving leg, 2nd column represents swaption paying leg
Principal = 100;  
OptSpec = {'call' ;'put'};    
Strike= [ 0.07 ; 0.0725];    
Basis= [1 3]; % 1st column represents swaption receiving leg, 2nd column represents swaption paying leg

Price the swaption.

PriceSwaption = swaptionbybk(BKTree, OptSpec, Strike, ExerciseDates, ...
Spread, SwapSettlement, SwapMaturity, 'SwapReset', SwapReset, 'Basis', Basis, ...
'Principal', Principal)

PriceSwaption = 2×1

    0.3634
    0.4798

Input Arguments

BKTree — Interest-rate tree structure

structure

Interest-rate tree structure, specified by using bktree.

Data Types: struct

OptSpec — Definition of option

character vector with values 'call' or 'put' | cell array of character vector with values 'call' or 'put'

Definition of the option as 'call' or 'put', specified as a NINST-by-1 cell array of character vectors. For more information, see More About.

Data Types: char | cell

Strike — Strike swap rate values

decimal

Strike swap rate values, specified as a NINST-by-1 vector.

Data Types: double

ExerciseDates — Exercise dates for swaption

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

Exercise dates for the swaption, specified as a NINST-by-1 vector or NINST-by-2 using serial date numbers or date character vectors, depending on the option type.

• For a European option, ExerciseDates are a NINST-by-1 vector of exercise dates. Each row is the schedule for one option. When using a European option, there is only one ExerciseDate on the option expiry date.

• For an American option, ExerciseDates are a NINST-by-2 vector of exercise date boundaries. For each instrument, the option can be exercised on any coupon date between or including the pair of dates on that row. If only one non-NaN date is listed, or if ExerciseDates is NINST-by-1, the option can be exercised between the ValuationDate of the tree and the single listed ExerciseDate.

Data Types: double | char | cell

Spread — Number of basis points over reference rate

numeric

Number of basis points over the reference rate, specified as a NINST-by-1 vector.

Data Types: double

Settle — Settlement date

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

Settlement date (representing the settle date for each swap), specified as a NINST-by-1 vector of serial date numbers or date character vectors. The Settle date for every swaption is set to the ValuationDate of the BK tree. The swap argument Settle is ignored. The underlying swap starts at the maturity of the swaption.

Data Types: double | char

Maturity — Maturity date for swap

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

Maturity date for each swap, specified as a NINST-by-1 vector of dates using serial date numbers or date character vectors.

Data Types: double | char | cell

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [Price,PriceTree] = swaptionbybk(BKTree,OptSpec, ExerciseDates,Spread,Settle,Maturity,'SwapReset',4,'Basis',5,'Principal',10000)

'AmericanOpt' — Option type

0 (European) (default) | integer with values 0 or 1

(Optional) Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and NINST-by-1 positive integer flags with values:

• 0 — European

• 1 — American

Data Types: double

'SwapReset' — Reset frequency per year for underlying swap

1 (default) | numeric

Reset frequency per year for the underlying swap, specified as the comma-separated pair consisting of 'SwapReset' and a NINST-by-1 vector or NINST-by-2 matrix representing the reset frequency per year for each leg. If SwapReset is NINST-by-2, the first column represents the receiving leg, while the second column represents the paying leg.

Data Types: double

'Basis' — Day-count basis of instrument

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

Day-count basis representing the basis used when annualizing the input forward rate tree for each instrument, specified as the comma-separated pair consisting of 'Basis' and a NINST-by-1 vector or NINST-by-2 matrix representing the basis for each leg. If Basis is NINST-by-2, the first column represents the receiving leg, while the second column represents the paying leg.

• 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

Notional principal amount, specified as the comma-separated pair consisting of 'Principal' and a NINST-by-1 vector.

Data Types: double

'Options' — Derivatives pricing options structure

structure

Derivatives pricing options structure, specified as the comma-separated pair consisting of 'Options' and a structure obtained from using derivset.

Data Types: struct

Output Arguments

Price — Expected prices of swaptions at time 0

vector

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

PriceTree — Tree structure of instrument prices

structure

Tree structure of instrument prices, returned as a MATLAB^® structure of trees containing vectors of swaption instrument prices and a vector of observation times for each node. Within PriceTree:

• PriceTree.PTree contains the clean prices.

• PriceTree.tObs contains the observation times.

More About

Call Swaption

A call swaption or payer swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option pays the fixed rate and receives the floating rate.

Put Swaption

A put swaption or receiver swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option receives the fixed rate and pays the floating rate.