swaptionbybdt

Price swaption from Black-Derman-Toy interest-rate tree

Syntax

[Price,PriceTree] = swaptionbybdt(BDTTree,OptSpec,Strike,ExerciseDates,Spread,Settle,Maturity)

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

Description

[Price,PriceTree] = swaptionbybdt(BDTTree,OptSpec,Strike,ExerciseDates,Spread,Settle,Maturity) prices swaption using a Black-Derman-Toy tree.

Note

Alternatively, you can use the Swaption object to price swaption instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

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

Examples

Price a 5-Year Call Swaption Using a BDT Interest-Rate Tree

This example shows how to price a 5-year call swaption using a BDT interest-rate tree. Assume that interest rate and volatility are fixed at 6% and 20% annually between the valuation date of the tree until its maturity. Build a tree with the following data.

Rates = 0.06 * ones (10,1);      
StartDates = [datetime(2007,1,1) ; datetime(2008,1,1) ; datetime(2009,1,1) ; datetime(2010,1,1) ; datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1)];
 
EndDates = [datetime(2008,1,1) ; datetime(2009,1,1) ; datetime(2010,1,1) ; datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1) ; datetime(2017,1,1)];

ValuationDate = datetime(2007,1,1); 
Compounding = 1; 

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

% use VolSpec to compute interest-rate volatility
Volatility = 0.20 * ones (10,1);  VolSpec = bdtvolspec(ValuationDate,...
EndDates, Volatility);

% use TimeSpec to specify the structure of the time layout for a BDT tree
TimeSpec = bdttimespec(ValuationDate, EndDates, Compounding);

% build the BDT tree
BDTTree = bdttree(VolSpec, RateSpec, TimeSpec); 

% use the following swaption arguments
ExerciseDates = datetime(2012,1,1);
SwapSettlement = ExerciseDates;
SwapMaturity   = datetime(2015,1,1); 
Spread = 0;
SwapReset = 1; 
Principal = 100;
OptSpec = 'call';
Strike =.062;
Basis = 1;

% price the swaption
[Price, PriceTree] = swaptionbybdt(BDTTree, OptSpec, Strike, ExerciseDates, ...
Spread, SwapSettlement, SwapMaturity, 'SwapReset', SwapReset, ...
'Basis', Basis, 'Principal', Principal)

Price = 
2.0592

PriceTree = struct with fields:
    FinObj: 'BDTPriceTree'
      tObs: [0 1 2 3 4 5 6 7 8 9 10]
     PTree: {[2.0592]  [0.9218 3.4436]  [0.2189 1.7137 5.6694]  [0 0.4549 3.1715 9.1499]  [0 0 0.9524 5.8347 14.3819]  [0 0 0 2.0127 10.6474 21.7783]  [0 0 0 0 0 0 0]  [0 0 0 0 0 0 0 0]  [0 0 0 0 0 0 0 0 0]  [0 0 0 0 0 0 0 0 0 0]  [0 0 0 0 0 0 0 0 0 0]}

Price a 5-Year Call Swaption with Receiving and Paying Legs Using a BDT Interest-Rate Tree

This example shows how to price a 5-year call swaption with receiving and paying legs using a BDT interest-rate tree. Assume that interest rate and volatility are fixed at 6% and 20% annually between the valuation date of the tree until its maturity. Build a tree with the following data.

Rates = 0.06 * ones (10,1);      
StartDates = [datetime(2007,1,1) ; datetime(2008,1,1) ; datetime(2009,1,1) ; datetime(2010,1,1) ; datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1)];
EndDates = [datetime(2008,1,1) ; datetime(2009,1,1) ; datetime(2010,1,1) ; datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1) ; datetime(2016,1,1) ; datetime(2017,1,1)];
ValuationDate = datetime(2007,1,1); 
Compounding = 1;

Define the RateSpec.

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

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

Use VolSpec to compute interest-rate volatility.

Volatility = 0.20 * ones (10,1);  
VolSpec = bdtvolspec(ValuationDate,EndDates, Volatility);

Use TimeSpec to specify the structure of the time layout for a BDT tree.

TimeSpec = bdttimespec(ValuationDate, EndDates, Compounding);

Build the BDT tree.

BDTTree = bdttree(VolSpec, RateSpec, TimeSpec)

BDTTree = struct with fields:
      FinObj: 'BDTFwdTree'
     VolSpec: [1×1 struct]
    TimeSpec: [1×1 struct]
    RateSpec: [1×1 struct]
        tObs: [0 1 2 3 4 5 6 7 8 9]
        dObs: [733043 733408 733774 734139 734504 734869 735235 735600 735965 736330]
        TFwd: {[10×1 double]  [9×1 double]  [8×1 double]  [7×1 double]  [6×1 double]  [5×1 double]  [4×1 double]  [3×1 double]  [2×1 double]  [9]}
      CFlowT: {[10×1 double]  [9×1 double]  [8×1 double]  [7×1 double]  [6×1 double]  [5×1 double]  [4×1 double]  [3×1 double]  [2×1 double]  [10]}
     FwdTree: {1×10 cell}

Define the swaption arguments.

ExerciseDates = datetime(2012,1,1);
SwapSettlement = ExerciseDates;
SwapMaturity   = datetime(2015,1,1); 
Spread = 0;
SwapReset = [1 1]; % 1st column represents receiving leg, 2nd column represents paying leg
Principal = 100;
OptSpec = 'call';
Strike=.062;
Basis= [2 4]; % 1st column represents receiving leg, 2nd column represents paying leg

Price the swaption.

[Price, PriceTree] = swaptionbybdt(BDTTree, OptSpec, Strike, ExerciseDates, ...
Spread, SwapSettlement, SwapMaturity, 'SwapReset', SwapReset, ...
'Basis', Basis, 'Principal', Principal)

Price = 
2.0592

PriceTree = struct with fields:
    FinObj: 'BDTPriceTree'
      tObs: [0 1 2 3 4 5 6 7 8 9 10]
     PTree: {[2.0592]  [0.9218 3.4436]  [0.2189 1.7137 5.6694]  [0 0.4549 3.1715 9.1499]  [0 0 0.9524 5.8347 14.3819]  [0 0 0 2.0127 10.6474 21.7783]  [0 0 0 0 0 0 0]  [0 0 0 0 0 0 0 0]  [0 0 0 0 0 0 0 0 0]  [0 0 0 0 0 0 0 0 0 0]  [0 0 0 0 0 0 0 0 0 0]}

Input Arguments

BDTTree — Interest-rate tree structure

structure

Interest-rate tree structure, specified by using bdttree.

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

datetime array | string array | date character vector

Exercise dates for the swaption, specified as a NINST-by-1 vector or a NINST-by-2 vector using a datetime array, string array, 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.

To support existing code, swaptionbybdt also accepts serial date numbers as inputs, but they are not recommended.

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

datetime array | string array | date character vector

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

To support existing code, swaptionbybdt also accepts serial date numbers as inputs, but they are not recommended.

Maturity — Maturity date for swap

datetime array | string array | date character vector

Maturity date for each swap, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, swaptionbybdt also accepts serial date numbers as inputs, but they are not recommended.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: [Price,PriceTree] = swaptionbybdt(BDTTree,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

Swaption

A swaption (swap option) is a financial derivative that gives the holder the right, but not the obligation, to enter into an interest-rate swap agreement at a specified future date and under predetermined terms.

Swaptions are used by investors and institutions to hedge against interest rate fluctuations or to speculate on future changes in interest rates.

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.

Version History

Introduced before R2006a

R2022b: Serial date numbers not recommended

Although swaptionbybdt 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.